What Drives Movements in the Unemployment Rate? A Decomposition of the Beveridge Curve

Finance and Economics Discussion Series Divisions of Research & Statistics and Monetary Affairs Federal Reserve Board, Washington, D.C. What Drives Movements in the Unemployment Rate? A Decomposition of the Beveridge Curve Regis Barnichon and Andrew Figura 2010-48 NOTE: Staff working papers in the Finance and Economics Discussion Series (FEDS) are preliminary materials circulated to stimulate discussion and critical comment. The analysis and conclusions set forth are those of the authors and do not indicate concurrence by other members of the research staff or the Board of Governors. References in publications to the Finance and Economics Discussion Series (other than acknowledgement) should be cleared with the author(s) to protect the tentative character of these papers.

What drives movements in the unemployment rate? A decomposition of the Beveridge curve. (cid:3) Regis Barnichon Andrew Figura Federal Reserve Board Federal Reserve Board 20 August 2010 Abstract This paper presents a framework to interpret movements in the Beveridge curve and analyzeunemployment(cid:135)uctuations. Wedecomposetheunemploymentrateintothreemain components: (1) a component driven by changes in labor demand (cid:150)movements along the Beveridge curve and shifts in the Beveridge curve due to layo⁄s(cid:150)(2) a component driven by changes in labor supply (cid:150)shifts in the Beveridge curve due to quits, movements inand-out of the labor force and demographics(cid:150)and (3) a component driven by changes in the e¢ ciency of matching unemployed workers to jobs. We (cid:133)nd that cyclical movements in unemployment are dominated by changes in labor demand, but that changes in labor supplyduetomovementsin-and-outofthelaborforcealsoplayanimportantrole. Further, cyclicalchangesinlabordemandleadcyclicalchangesinlaborsupply. Changesinmatching e¢ ciency generally play a small role but can decline substantially in recessions. At lowfrequencies, labor demand displays no trend, and changes in labor supply explain virtually all of the secular trend in unemployment since 1976. JEL classi(cid:133)cations: J6, E24, E32 (cid:3)We would like to thank Alessia Campolmi, Shigeru Fujita, Bart Hobijn, Chris Nekarda, Rob Valletta and seminarparticipantsattheChicagoFed,theNationalBankofHungary,theNewYorkFedandtheSanFrancisco Fed. The views expressed here do not necessarily re(cid:135)ect those of the Federal Reserve Board or of the Federal Reserve System. Any errors are our own. 1

Keywords: Gross Worker Flows, Job Finding Rate, Employment Exit Rate, Matching Function. 2

1 Introduction The unemployment rate is an important indicator of economic activity. Understanding its movements is useful in assessing the causes of economic (cid:135)uctuations and their impact on welfare, as well as assessing in(cid:135)ationary pressures in the economy. The Beveridge curve captures the downward sloping relationship between the unemployment rate and the job vacancy rate and is widely used as an indicator of the state of the labor market. Movements along the Beveridge curve, i.e., changes in unemployment due to changes in vacancies, are typically interpreted as cyclical movements in labor demand. However, shifts in the Beveridge curve are di¢ cult to interpret. While they are sometimes seen as indicating movements in the level of (cid:147)equilibrium(cid:148)or(cid:147)structural(cid:148)unemployment,theycaninfactbecausedbyanumberofdiverse factors; changes in the intensity of layo⁄s and quits, changes in labor force participation, or changes in the e¢ ciency of matching workers to jobs. In this paper, we present a framework to isolate the di⁄erent components of the Beveridge curve, and we use that framework to decompose unemployment rate movements into three categories: (1) (cid:133)rm-induced changes in unemployment, i.e. movements in labor demand, (2) worker-induced changes in unemployment, i.e. movements in labor supply, and (3) changes in the e¢ ciency of matching unemployed workers to jobs. The(cid:133)rstcontributionofthispaperistopresentaframeworktorigorouslystudymovements intheBeveridgecurve. WeaccomplishourBeveridgecurvedecompositionby(cid:133)rstisolatingthe in(cid:135)ows and out(cid:135)ows of unemployment, following Shimer (2007). Using an aggregate matching function tying vacancy posting and unemployment to transitions from unemployment into employment, we decompose the out(cid:135)ow component into a component driven by changes in vacancies, i.e. movements along a stable Beveridge curve, and a component driven by changes inthee¢ ciencyofmatchingworkerstojobs. WeinterpretmovementsalongastableBeveridge curve as changes in labor demand. To interpret the in(cid:135)ows of unemployment, we use CPS micro data to distinguish movements in layo⁄s, i.e. changes in labor demand, from changes in demographics, quits or movements in-and-out of the labor force, i.e. changes in labor supply. 3

The second contribution of this paper is to provide a comprehensive decomposition of the unemployment rate covering all frequencies over 1976-2009. We (cid:133)nd that labor demand and laborsupplycontributeapproximatelyequallytounemployment(cid:146)svariance, butthatthesetwo forces play very di⁄erent roles at di⁄erent frequencies. At business cycle frequencies, labor demand accounts for three quarters of unemployment(cid:146)s variance, a result in line with the approach taken by the search literature and the canonical Mortensen-Pissarides (1994) model to focus on vacancy posting and job separation when studying unemployment (cid:135)uctuations. However, movements in-and-out of the labor force explain close to a quarter of unemployment(cid:146)s variance, a result at odds with the conventional wisdom that movements in-and-out of the labor force played little role at business cycle frequencies (see e.g. Hall, 2005, Shimer, 2007, and Elsby, Michaels and Solon, 2009). Finally, changes in matching e¢ ciency play on average a small role but can decline substantially in recessions. For instance, in the 2008-2009 recession, lower matching e¢ ciency added about 11 2 percentage points to the unemployment rate. We also study the timing of the di⁄erent forces moving the unemployment rate. At the beginning of a recession, the Beveridge curve shifts out because of an increase in temporary layo⁄s. Aquarterlater, unemploymentmovesalongtheBeveridgecurveas(cid:133)rmsadjustvacancies. The Beveridge curve also shifts out further because of an increase in permanent layo⁄s. Then, another quarter later, labor supply responds to the economic situation; the Beveridge curve shifts inslightlybecause quits declinebutshifts outfurtherasworkersdisplayastronger attachmenttothelaborforce. Whileonlysuggestive, thischaineventcouldindicatethatlabor supply responds to labor demand at cyclical frequencies. Atlowfrequencies, we(cid:133)ndlittleevidenceofanytrendinlabordemand. Incontrast, unemployment(cid:146)s trend since 1976 can be entirely accounted for by secular changes in labor supply, in particular the aging of the baby boom, the increase in women(cid:146)s labor force participation and the increasing attachment of women to the labor force. The secular leftward shift in the Beveridge curve since 1976 correlates with a decline in the time-series volatility of business 4

growth rates since 1976 and a decline in the job destruction rate (Davis, Faberman, Haltiwanger, Jarmin and Miranda, 2010). Thus, our results suggest that an explanation of these phenomena lies with secular changes in labor supply rather than with secular changes in labor demand. Our paper is related to two strands in the literature. The (cid:133)rst strand investigates the relative responsibility of unemployment in(cid:135)ows and out(cid:135)ows in accounting for changes in unemployment.1 We take this literature one step further by decomposing the labor market (cid:135)ows into economically meaningful components that allow us to say something about the economic forces driving movements in unemployment. Our use of an aggregate matching function and the Beveridge curve to accomplish this decomposition harks back to an earlier strand in the literature (e.g. Lipsey, 1965, Abraham, 1987, Blanchard and Diamond, 1989) that relied on the Beveridge curve to distinguish between changes in labor demand (movements along the Beveridge curve) and shifts in sectoral reallocation (shifts in the Beveridge curve). We build on this literature to better identify causes of Beveridge curve shifts. The next section lays the theoretical groundwork for our decomposition. Section 3 estimates an aggregate matching function and decomposes changes in the unemployment rate into changes in labor demand, changes in labor supply, and changes in the matching function. Section 4 discusses the implications of our results. Section 5 concludes. 2 A Beveridge curve decomposition In this section, we present a method to quantitatively decompose movements in the Beveridge curve. We decompose unemployment (cid:135)uctuations into three categories; changes in labor demand (cid:150)movements along the Beveridge curve and shifts in the Beveridge curve due to layo⁄s(cid:150), changes in labor supply (cid:150)shifts in the Beveridge curve due to quits and movements in and out of the labor force(cid:150), and changes in matching e¢ ciency. 1See, e.g., Shimer (2007), Elsby, Michaels and Solon (2009), Fujita and Ramey (2009), Elsby, Hobijn and Sahin (2009). 5

2.1 Steady-state unemployment Let U ; E ; and I denote the number of unemployed, employed and inactive (out of the labor t t t force) individuals, respectively, at instant t R+ . Letting (cid:21)A t B denote the hazard rate of 2 transiting from state A E;U;I to state B E;U;I , unemployment, employment and 2 f g 2 f g inactivity will satisfy the system of di⁄erential equations U_ = (cid:21)EUE +(cid:21)IUI ((cid:21)UE +(cid:21)UI)U t t t t t t t t (cid:0) 8 > > > E_ t = (cid:21)U t EU t +(cid:21)I t EI t (cid:0) ((cid:21)E t U +(cid:21)E t I)E t (1) > < I_ = (cid:21)EIE +(cid:21)UIU ((cid:21)IE +(cid:21)IU)I t t t t t t t t (cid:0) > > > > : As (cid:133)rst argued by Shimer (2007), the magnitudes of the hazard rates is such that the half-life of a deviation of unemployment from its steady state value is about a month. As a result, at a quarterly frequency, the unemployment rate u = Ut is very well approximated by its t LFt steady-state value uss so that t s u t uss (2) t ’ s +f (cid:17) t t t with s and f de(cid:133)ned by2 t t s = (cid:21)EU + (cid:21)E t I(cid:21)I t U t t 1 (cid:21)II 8 (cid:0) t > < f t = (cid:21)U t E + (cid:21) 1 U t I (cid:21) (cid:21) I I t I E : (cid:0) t > : 2.2 Modeling (cid:21)UE with a matching function The job (cid:133)nding rate is de(cid:133)ned as the ratio of new hires to the stock of unemployed, so that the job (cid:133)nding rate can be written as (cid:21)UE = mt with m the number of new matches at t ut t instant t: By modeling m with a constant returns to scale Cobb-Douglas matching function, t a speci(cid:133)cation widely used in the search and matching literature (see e.g. Pissarides, 2001), 2Expression (2) generalizes the simpler two-states case without movements in-and-out of the labor force in which uss = (cid:21)E t U : With movements in-and-out of the labor force, workers can transition between U and t (cid:21)E t U+(cid:21)U t E E either directly (U-E) or in two steps by (cid:133)rst leaving the labor force (U-I) and then by (cid:133)nding a job directly from inactivity (I-U). f , the (cid:147)U-E transition probability(cid:148)that matters for steady-state unemployment rate is t then a weighted average of (cid:21)UE and (cid:21)UI(cid:21)IE , with weights of 1 and 1 , the average time that an inactive worker spends in I. s has a s t imilar exp t ress t ion. 1 (cid:0) (cid:21)I t I t 6

we can express m as t m = m u(cid:27)v1 (cid:27) t 0 t t(cid:0) with m a positive constant, v the number of job openings and u the number of unemployed. 0 t t In this context, we can model the job (cid:133)nding rate (cid:21)UE as t v ln(cid:21)UE = (1 (cid:27))ln t +m +(cid:23) : (3) t (cid:0) u 0 t t where (cid:23) allows the e¢ ciency of matching workers to (cid:133)rms to vary over time. t 2.3 Decomposing movements in the Beveridge curve Writing the steady-state approximation for unemployment (2) and modeling the job (cid:133)nding rate with a matching function, we can write s s uss t t (4) t (cid:17) s t + (cid:21) 1 U t (cid:0) I (cid:21) (cid:21) I t I t I E +(cid:21)U t E ’ s t +(cid:21)U t IE +m 0 u v s t t s 1 (cid:0) (cid:27) (cid:16) (cid:17) with (cid:21)UIE = (cid:21)U t I(cid:21)I t E . Expression (4) is the theoretical underpinning of the Beveridge curve, t 1 (cid:21)II (cid:0) t the downward sloping relation between unemployment and vacancy posting. Steady-state unemployment moves along the Beveridge curve as (cid:133)rms adjust vacancies. In contrast, as illustrated in Figure 1, the Beveridge curve shifts because of layo⁄s, quits or movements in and out of the labor force, i.e. when s or (cid:21)UIE moves. t t However, while the matching function is remarkably successful at modeling the job (cid:133)nding 1 (cid:27) rate, the relation (cid:21)UE = m vt (cid:0) is not exact, and the labor market may temporarily t 0 ut (cid:16) (cid:17) deviate from its average matching e¢ ciency. To separate movements along the Beveridge ss;bc curve from shocks to the matching function, we de(cid:133)ne u as the steady-state unemployment t ss;bc rate implied by a stable Beveridge curve, i.e. by a stable matching function. Formally, u t 7

is de(cid:133)ned by s ss;bc t u = : (5) t 1 (cid:27) s +(cid:21)UIE +m vt (cid:0) t t 0 uss;bc (cid:18) t (cid:19) 1 (cid:27) Denoting (cid:21)^UE = m vt (cid:0) the job (cid:133)nding rate predicted by a stable matching funct 0 uss;bc (cid:18) t (cid:19) tion, we can rewrite (4) as s uss = t (6) t s t +(cid:21)U t IE +(cid:21)^U t E e"t where " = ln(cid:21)UE ln(cid:21)^UE captures deviations of the job (cid:133)nding rate from the value implied t t t (cid:0) by a stable Beveridge curve, i.e. a stable relationship between unemployment and vacancies.3 Log-linearizing (2) around the mean of the hazard rates gives us:4 dlnuss = (cid:11)EIdln(cid:21)EI +(cid:11)IUdln(cid:21)IU +(cid:11)EUdln(cid:21)EU (7) t t t t (cid:11)IEdln(cid:21)IE (cid:11)UIdln(cid:21)UI (cid:11)UEdln(cid:21)UE +(cid:17) t t t t (cid:0) (cid:0) (cid:0) with (cid:11)AB some positive constants depending on the mean of (cid:21)AB .5 In this context, we t can d(cid:8)ecomp(cid:9)ose unemployment movements in a Beveridge curve fr(cid:8)amew(cid:9)ork from dlnuss = dlnubc+dlnu shifts +dlnu eff +(cid:17) (8) t t t t t where dlnubc (cid:11)UEdln(cid:21)^UE = (cid:11)UE(1 (cid:27))dln v t UE represents movements along the Bevt (cid:17) (cid:0) t (cid:0) (cid:0) uss;bc t eridge curve, dlnu eff (cid:11)UEd" captures the shifts in the Beveridge curve caused by changes t t (cid:17) 3Note that " is di⁄erent from (cid:23) . While (3) is useful to highlight movements in matching e¢ ciency, this t t regressionconditionsonactualunemployment,nottheunemploymentthatwouldhaveprevailedhadtherebeen no changes in matching e¢ ciency. To properly identify changes in matching e¢ ciency, one needs to determine uss;bc; the unemployment rate implied by a stable matching function and the current levels of s and (cid:21)UIE. t t t Deviations of the actual job (cid:133)nding rate from the job (cid:133)nding rate implied by uss;bc can then be interpreted as t due to a change in matching e¢ ciency. 4A (cid:133)rst-order approximation is very good on average, but (cid:17) can become non-negligible during episodes of t high unemployment rate. Thus, for our quantitative exercises, we rely on a second-order approximation, which performs extremely well. The expressions for the second-order coe¢ cients are shown in the Appendix. 5Formally, (cid:11)EI = (1 u(cid:22)ss)(cid:21)EI(cid:21)IU , (cid:11)UE = (cid:21)IU(cid:21)UE+(cid:21)IE(cid:21)UE , (cid:11)IE = (cid:21)IE(cid:21)EU (1 u(cid:22)ss) (cid:21)UI(cid:21)IE+(cid:21)IE(cid:21)UE , (cid:0) s s+f s (cid:0) (cid:0) s+f (cid:11)UI = (cid:21)UI(cid:21)IE , (cid:11)EU =(1 u(cid:22)ss)(cid:21)IE(cid:21)EU+(cid:21)IU(cid:21)EU , (cid:11)IU =(1 u(cid:22)ss)(cid:21)EI(cid:21)IU+(cid:21)IU(cid:21)EU (cid:21)IU(cid:21)UE : s+f (cid:0) s (cid:0) s (cid:0) s+f 8

in matching e¢ ciency, and shifts in the Beveridge curve are given by dlnu shift (cid:11)EUdln(cid:21)EU +(cid:11)EIdln(cid:21)EI +(cid:11)IUdln(cid:21)IU (cid:11)IEdln(cid:21)IE (cid:11)UIdln(cid:21)UI t t t t t t (cid:17) (cid:0) (cid:0) Shifts in the Beveridge curve can occur through changes in workers(cid:146)attachment to the labor force or through changes in the probability that workers separate from their job and join the unemployment pool, either through a layo⁄ or through a quit. Finally, the residual term (cid:17) t corresponds to the approximation error. We can then assess the separate contributions of di⁄erent movements in the Beveridge curve by noting as Fujita and Ramey (2009) that Var(dlnuss) = Cov(dlnuss;dlnubc)+Cov(dlnuss;dlnu shifts )+Cov(dlnuss;dlnu eff )+Cov(dlnuss;(cid:17) ): t t t t t t t t t (9) Cov(dlnubc;dlnuss) so that, for example, t t measures the fraction of unemployment(cid:146)s variance due var(dlnuss) t to movements along the Beveridge curve: 2.4 Interpreting shifts in the Beveridge curve TheBeveridgecurvecanshiftiftheemployment-unemploymenttransitionprobabilitychanges. However, an employed worker can join the unemployment pool for two reasons: a layo⁄or a quit. While a layo⁄ is a (cid:133)rm-induced movement in unemployment, a quit is a decision of the worker. Thus, from a conceptual point of view, it is important to distinguish these two concepts empirically. In addition, shifts in the Beveridge curve can occur through changes in workers(cid:146)attachment to the labor force. Thus, to identify and interpret the di⁄erent forces that can shift the Beveridge curve, we separate job leavers, job losers and labor force entrants, and we classify jobless workers according to the event that led to their unemployment status: a permanent layo⁄p, a temporary layo⁄t, a quit q and a labor force entrance o. Further, anumberofresearchers(e.g. AbrahamandShimer, 2001)emphasizethatchanges in demographics have been an important force behind the secular trend in unemployment. In 9

particular, as the labor force gets older, the average turn-over rates declines, and the unemployment rate goes down. Thus, to better interpret the low-frequency shifts in the Beveridge curve, we extend our decomposition (8) and isolate the direct e⁄ect of demographics on unemployment. p Formally, for each demographic group i, there are four unemployment rates by reason: u ; i ut, u q and uo and the associated hazard rates (cid:21) jE ;(cid:21) Ej ;(cid:21) jI ; j p;t;q and (cid:21)oE;(cid:21)Io;(cid:21)oI . i i i f i i i g 2 f g f i i i g In this case, the system of di⁄erential equations (1) satis(cid:133)ed by the number of unemployed U , employed E and inactive I in demographic group i becomes it it it U_j = (cid:21) Ej E ((cid:21) jE +(cid:21) jI )U j , j p;t;q it it it (cid:0) it it it 2 f g 8 U_o = (cid:21)IoI ((cid:21)oE +(cid:21)oI)Uo > > > it it it (cid:0) it it it (10) > > > > < E_ it = (cid:21) p it E U i p t +(cid:21)t it EU i t t +(cid:21) q it E U i q t +(cid:21)o it EU i o t +(cid:21)I it EI it (cid:0) ((cid:21)E it l +(cid:21) E it q +(cid:21)E it I)E it > > > I_ it = (cid:21)E it IE it +(cid:21)o it IU i o t(cid:0) ((cid:21)I it E +(cid:21)I it o)I it > > > > : N With U = U p +Ut +U q +Uo , the aggregate steady-state unemployment rate uss t it it it it t i=1 satis(cid:133)es (2) with X th(cid:0)e average transition r(cid:1)ates given by N (cid:21)UB = U i j t(cid:21) jB , B E;I t Ut it 2 f g 8 i=1j p;t;q;o > > > X N 2fX g N > > > (cid:21)EU = Eit(cid:21) Ej and (cid:21)EI = Eit(cid:21)EI (11) > > > t Et it t Et it > < X i=1j 2Xf p;t;q g X i=1 N N > > > > (cid:21)I t U = I I i t t(cid:21)I it o and (cid:21)I t E = I I i t t(cid:21)I it E > > i=1 i=1 > X X > > > : 10

Using the steady-state approximations, we can approximate (11) with N (cid:21)UB ! uj it ;ss (cid:21) jB , B E;I 8 t ’ it us t s it 2 f g i=1j p;t;q;o > > > X N 2fX g N > > > (cid:21)EU ! es it s (cid:21) Ej and (cid:21)EI ! es it s (cid:21)EI (12) > > > t ’ ites t s it t ’ ites t s it > < X i=1j 2Xf p;t;q g X i=1 N N > > > > > > (cid:21)I t U ’ i=1 ! it i i s i s t t s s (cid:21)I it o and (cid:21)I t E ’ i=1 ! it i i s i s t t s s (cid:21)I it E > X X > > > : where ! = LFit is the share of group i in the labor force and uss; ess and iss denote respecit LFt it it it tively the steady-state unemployment rate, employment rate and inactivity rate of group i. The steady-state unemployment rate for category i satis(cid:133)es uss = sit since the system of it sit+fit di⁄erential equations (10) holds independently for each demographic group.6 To isolate the direct e⁄ect of demographics, we log-linearize (12) and get for (cid:21)EU t dln(cid:21)EU = N ! es i s(cid:21)E i U dln(cid:21)EU +dln ! es it s = dln(cid:21)~EU +dln(cid:21) EU;demog (13) t i ess(cid:21)EU it it ess t t i=1 (cid:18) (cid:18) t (cid:19)(cid:19) X and similarly for the other transition rates.7 The (cid:133)rst term corresponds to movements in N (cid:21)~EU ! es i s (cid:21)EU, thehazardratethatholdstheshareofeachdemographicgroupconstant. t (cid:17) iess it i=1 X N EU;demog ess(cid:21)EU ess The second term, dln(cid:21) ! i i dln! it , corresponds to movements in the t (cid:17) iess(cid:21)EU ites t s i=1 relative size of the labor force in eacXh group ! , as well as changes in the share of each group it ess in the employment pool ( i ). ess N Finally,toseparatequitsfromlayo⁄s,notethat(cid:21)EU = (cid:21) Ej and(cid:21) Ej = ! es it s (cid:21) Ej , t t t itess it t j p;t;q i=1 2Xf g X j p;t;q . 8 2 f g 6See the Appendix for analytical expressions of the steady-state values. 7Throughout the paper, we present the derivations to a (cid:133)rst-order for clarity of exposition. However, for the quantitative results, we used a second-order approximation. For instance, for (cid:21)EU, we took a second-order expansion of ln(cid:21)EU in (12), and we split the contributions of the cross-order terms in half between each two t components. 11

2.5 A labor demand/labor supply decomposition Using (13), we isolate the contribution of demographics to movements in unemployment and separate layo⁄s from quits and movements in-and-out of the labor force and rewrite (8) as dlnus t s = dlnub t c+dlnu s t hifts;layoffs +dlnu s t hifts;quits +dlnu s t hifts;LF (cid:0) NLF +dlnu d t emog +dlnu e t ff +(cid:17) t : Ld Ls (14) | {z } | {z } where8 dlnubc = (cid:11)UEdln(cid:21)^UE t t (cid:0) 8 dlnu shifts;layoffs = (cid:11)EU dln(cid:21)~Ep +dln(cid:21)~Et and dlnu shifts;quits = (cid:11)EUdln(cid:21)~Eq > > t t t t t > > > > > > > > dlnu s t hifts;LF (cid:0) NLF = (cid:11)E(cid:16)Idln(cid:21)~E t I +(cid:11)IUdln (cid:17) (cid:21)~I t U (cid:0) (cid:11)IEdln(cid:21)~I t E (cid:0) (cid:11)UIdln(cid:21)~U t I > > > > dlnu demog = (cid:11)EIdln(cid:21) EI;demog +(cid:11)IUdln(cid:21) IU;demog +(cid:11)EUdln(cid:21) Eq;demog < t t t t > > > (cid:0) (cid:11)IEdln(cid:21) I t E;demog (cid:0) (cid:11)UIdln(cid:21) U t I;demog > > > > dlnu eff = (cid:11)UEd" (cid:11)UEdln(cid:21) UE;demog : > > > t (cid:0) t (cid:0) t > > > > : We group the (cid:133)rms(cid:146)induced movements in unemployment (due to vacancies or layo⁄s) under the heading "labor demand" and the workers(cid:146)induced movements in unemployment (due to quits, movements in and out of the labor force and changes in demographics) under the heading "labor supply". Importantly, we do not presume that labor demand and labor supply are independent forces as changes in one factor could in(cid:135)uence the other. Rather, we think of the labor demand/labor supply classi(cid:133)cation as a useful framework to think about the mechanisms (changes in (cid:133)rms(cid:146)behavior or changes in workers(cid:146)behavior) at play behind unemployment (cid:135)uctuations. 8See the Appendix for the exact expressions for (cid:21)~AB , dln(cid:21)AB;demog or dln(cid:21)UA;reason: t t t 12

3 Empirical results 3.1 Measuring individuals(cid:146)transition rates To identify the individuals(cid:146)transition rates, we use CPS gross (cid:135)ows measuring the number of workers moving from state A S to state B S each month. We classify jobless workers 2 2 accordingtotheeventthatledtotheirunemploymentstatus: apermanentlayo⁄, atemporary layo⁄, a quit and a labor force entrance.9 Further, we split workers into N = 8 categories; male vs. female in the three age categories 25-35, 35-45, 45-55, and male and female together for ages 16-25 and over 55. For each demographic group, there are 6 possible states with S = Up;Ut;Uq;Uo;E;I . To account for time aggregation bias, we consider a continuous environm(cid:8)ent in which data a(cid:9)re available at discrete dates t and proceed in a similar fashion to Shimer (2007). Denote NAB((cid:28)) t the number of workers who were in state A at t N and are in state B at t+(cid:28) with (cid:28) [0;1] 2 2 and de(cid:133)ne nAB((cid:28)) = N t AB((cid:28)) the share of workers who were in state A at t. t NAX((cid:28)) t X S Assuming that (cid:21)ABP2 , the hazard rate that moves a worker from state A at t to state B at t t+1, is constant from t to t+1, nAB((cid:28)) satis(cid:133)es the di⁄erential equation:10 t n_AB((cid:28)) = nAC((cid:28))(cid:21)CB nAB((cid:28)) (cid:21)BC, A = B: (15) t t t t t (cid:0) 8 6 C=B C=B X6 X6 We then solve this system of di⁄erential equations numerically to obtain the transition rates for each demographic group. We use data from the CPS from January 1976 through December 2009andcalculatethequarterlyseriesforthetransitionratesover1976Q1-2009Q4byaveraging the monthly series. 9To address Shimer(cid:146)s (2007) worry that the quit/layo⁄ distinction may be hard to interpret in the CPS because a sizeable fraction of households who report being a job leaver in month t subsequently report being a job loser at t+1, we discarded all the observations with "impossible" transitions (such as job leaver to job loser). 10Becauseanunemployedworkercannotchangereasonforunemploymentorbecauseajobloser/leavercannot bealaborforceentrant,sometransitionsareforbidden,andweimpose(cid:21)AB =0forsuchtransitions(forexample, t (cid:21)pq =0, (cid:21)Ip =0, etc..) 13

3.2 Estimating a matching function We estimate a matching function by regressing v ln(cid:21)UE = (1 (cid:27))ln t +c+(cid:23) (16) t (cid:0) u t t using our measure of the job (cid:133)nding rate (cid:21)UE as the dependent variable. We estimate (16) with monthly data using the composite help-wanted index presented in Barnichon (2010) as a measure of vacancy posting. We use non-detrended data over 1967:Q1- 2009:Q4 and allow for (cid:133)rst-order serial correlation in the residual. To take into account movements in the size of the labor force, we rescale the composite help-wanted index by the size of the labor force. Table 1 presents the result. The elasticity (cid:27) is precisely estimated at 0:62, a value inside the plausible range (cid:27) [0:5;0:7] identi(cid:133)ed by Petrongolo and Pissarides (2001). A 2 legitimate concern with this regression is that equation (16) may be subject to an endogeneity bias. We then estimate (16) using lagged values of v and u as instruments. As column (2) t t shows,theendogeneitybiasappearstobesmallastheelasticityislittlechangedat0:60. Figure 2 plots the residual of equation (16) estimated over 1967-2009. While the matching function appears relatively stable over time, a testimony of the success of the matching function, the residual can become large. In the third quarter of 2009, the residual reached an all time low of three standard-deviations. 3.3 A decomposition of unemployment (cid:135)uctuations 3.3.1 Aggregate decomposition In this section, we use (14) to decompose unemployment (cid:135)uctuations into: (i) movements due to changes in labor demand, (ii) movements due to changes in labor supply, and (iii) changes in matching e¢ ciency. To better visualize the contribution of each category in history, we log-linearize unemploy- 14

ment around the base date 2000q3.11 That base date is attractive because it corresponds to the highest reading for vacancy posting per capita as well as the lowest value for lnu shift .12 t Figure 3 plots (log) unemployment and its components relative to their 2000q3 values. To express the y-axis in units of unemployment rate, we use a logarithmic scale. Figure 3 suggests that both changes in labor demand and changes in labor supply contribute to unemployment(cid:146)s (cid:135)uctuations. However, the secular trend in unemployment appears to originate in changes in labor supply, while changes in labor demand appear to be mainly cyclical. A variance decomposition con(cid:133)rms this impression, and Table 2 shows that while labor demand and labor supply contribute to respectively 50 and 30 percent of unemployment(cid:146)s variance on average, movements in labor supply account for virtually all the trend in unemployment since 1976.13 In contrast, changes in labor demand account for 82 percent of unemployment(cid:146)s cyclical (cid:135)uctuations (excluding movements due to changes in matching e¢ ciency). Nonetheless, the contribution of changes in labor supply at cyclical frequencies is far from negligible at 18 percent. With a contribution of 13 percent, changes in matching e¢ ciency generally have a small impact on the equilibrium unemployment rate, a corollary of the success of the matching function in modeling the job (cid:133)nding rate. However, Figure 3 shows some marked decrease in matching e¢ ciencies in the aftermath of the 82 peaks in unemployment and during the 2008- 2009 recession. Without any loss in matching e¢ ciency, Figure 3 shows that unemployment would have been about 50 basis points lower over 1984-1988 and about 150 basis points lower in 2009.14 11For a Taylor-expansion around an extremum point such as 2000Q3, we use a second-order approximation (see the Appendix) to ensure that the approximation remains good. To classify the cross-order terms (in, say, labor demand versus labor supply), we split their contribution in half between each two components. The red line in Figure 3 plots the exact value of the steady-state unemployment rate, which is very close to our approximation. 12Thus, 2000q3 corresponds to the date with the most leftward Beveridge curve, and that base year can be used as a reference point from which we can quickly visualize the rise and fall in trend unemployment as well as the cyclical (cid:135)uctuations over the last 35 years. 13To separate trend and cyclical unemployment, we decompose changes in unemployment into a trend component (from an HP-(cid:133)lter, (cid:21)=105) and a cyclical component. 14In a companion paper (Barnichon and Figura, 2010), we investigate the forces behind changes in matching e¢ ciency. 15

3.3.2 Digging further To better interpret changes in labor demand and changes in labor supply, we now study the behavior of their subcomponents. Figure 4 and 5 plot the decomposition of labor demand and labor supply following (14). Wecanseethatthereisnocleartrendinanyofthecomponentsofunemploymentduetolabor demand. In contrast, labor supply seems responsible for the secular decline in unemployment since 1976. Table 3 presents the results of a variance decomposition using (14) and con(cid:133)rms this visual inspection. While movements along the Beveridge curve, layo⁄s and movements in-and-out of the labor force each account for about a third of unemployment(cid:146)s variance, the picture is very di⁄erent when one considers high and low-frequency movements separately. Demographics and movements in-and out of the labor force are the prime driving forces of secular shifts in unemployment but labor demand (movements along the Beveridge curve and layo⁄s) is the main driving force at business cycle frequencies. We thus discuss each frequency range separately. Business cycle (cid:135)uctuations: As Table 3 shows, movements along the Beveridge curve and shifts due to layo⁄s are the two main determinants of unemployment (cid:135)uctuations and account forrespectively37and46percentofthecyclical(cid:135)uctuationsinunemployment. However,Table 3 shows that the cyclical contribution of movements in-and-out of the labor force is far from negligible at around 23 percent. Quits have a small but negative contribution of -7 percent, a resultconsistentwithElsby, MichaelsandSolon(cid:146)s(2009)(cid:133)ndingusingunemploymentduration data that quits to unemployment move countercyclically. To better interpret these results, Table 4 presents the correlation matrix for the main determinants of unemployment (cid:135)uctuations at business cycle frequencies. Shifts in the Beveridge curve due to layo⁄s and movements along the Beveridge curve are strongly positively correlated, in line with the usual assumption that they both respond to (cid:133)rms(cid:146)labor demand. The correlation with shifts due to temporary layo⁄s is less strong, because, as we can see in Figure 16

4, (cid:133)rms(cid:146)increasing reliance on permanent layo⁄s at the expense of temporary layo⁄s muted the cyclicality of temporary layo⁄s in the second-half of the sample. Shifts in the Beveridge curve due to movements in-and-out of the labor force are strongly positively correlated with shifts due to layo⁄s and to movements along the Beveridge curve. As we can see in Figure 5, movements in-and-out of the labor force contribute to some of the rise in the unemployment rate in recessions. To visualize the role played by movements in-and-out of the labor force, Figures 6 to 9 plot the evolution of the four hazard rates related to movements in-and-out of inactivity for speci(cid:133)c demographic groups. A general observation is that attachment to the labor force is countercyclical, with workers more likely to join/stay in the labor force during recessions. This is particularly true for prime-age females as shown in Figure 6:15 Comparing prime-age women with prime-age men in Figures 6 and 7, the behavior of (cid:21)UI and (cid:21)IU shows that women(cid:146)s attachment to the labor force more countercyclical than for men. This phenomenon may be a sign of the added worker e⁄ect, according to which women are more likely to join/remain in the labor force when their husband has lost his job.16 Further, older workers can also experience strong cyclical movements in (cid:21)IU (Figure 8).17 Finally, Table 5 reports the timing of the peak correlation between any two series and shows that changes in unemployment follow a particular chain of events. Temporary layo⁄s leadpermanentlayo⁄sandchangesinjobposting, whichthemselvesleadquitsandmovements in-and-out of the labor force. Thus, at the beginning of a recession, the Beveridge curve shifts out because temporary layo⁄s increase. A quarter later, unemployment moves along the Beveridge curve as (cid:133)rms adjust vacancies and the Beveridge curve shifts out further because of more permanent layo⁄s. Then, another quarter later, labor supply responds to the economic situation; the Beveridge curve shifts in slightly because quits decline but also shifts out further 15This could be due to the extension of unemployment bene(cid:133)ts duration during recessions. In fact, during the mid-70s and early 80s recessions, there was comparatively little increase in unemployment coverage, and the large increases in unemployment were not caused by large movements in (cid:21)UI and (cid:21)IU. In contrast, a large t t increaseinunemploymentinsurancecoverageintheearly-90srecessioncoincidedwithunusuallylargeincreases in dlnuUI and dlnuIU given the magnitude of the recession. t t 16See Sahin, Song and Hobijn (2009) for a discussion of the added-worker e⁄ect in the 2008-2009 recession. 17This is particularly true in the 2008-2009 recession (especially women) and could be due to the nature of the recession as older workers had to come out of retirement because of large losses in stock market wealth. 17

as workers show a stronger attachment to the labor force. While only suggestive, this chain event could indicate that labor supply responds to labor demand at cyclical frequencies. Low-frequency movements: Shimer (1998, 2001) and Abraham and Shimer (2001) identi(cid:133)ed two forces that could be responsible for the low-frequency movements in unemployment since 1976: the aging of the baby boom and the increase in women(cid:146)s labor force participation rate. Consistent with this result, Figure 5 shows that the trend in labor supply originates in demographics and movements in-and-out of the labor force. Table 4 con(cid:133)rms this idea quantitatively and shows that the two forces can explain virtually all of the trend in unemployment. To explore this result in more details, we now look at the behavior of speci(cid:133)c demographics groups since 1976. demog The right panel of Figure 10 plots the trends in dlnu for six demographic groups and t showsthatthedeclineintheshareofyoungworkers(maleandfemale)contributedtothetrend in unemployment. Indeed, younger workers have higher turnover and a higher unemployment rate than prime age or old workers, and a decline in the youth share automatically reduces the aggregate unemployment rate. At the same time, another demographic change had an opposite e⁄ect on unemployment. The increase in the share of prime age female inside the labor force until the mid-90s dampened the baby boom(cid:146)s e⁄ect as women historically had a higher job (cid:133)nding rate and lower job separation rate than men. shifts;LF NLF The left panel of Figure 10 plots the trends in dlnu t (cid:0) for six demographic groups and highlights a downward trend in unemployment caused by a change in the behavior of women, consistent with the (cid:133)ndings of Abraham and Shimer (2001). Looking at Figure 6 and the behavior of prime age women(cid:146)s transition rates over 1976-2009, two changes are apparent.18 First, the secular increase in (cid:21)IU until the mid-90s and the secular increase in (cid:21)IE captures the fact that women were getting increasingly likely to join the labor force, either by directly (cid:133)nding a job (as is increasingly the case) or by going (cid:133)rst through the unemployment pool. Second, women display an increasing attachment to the labor force as (cid:21)UI and (cid:21)EI 18Abraham and Shimer (2001) also documented these two changes using annual transition probabilities. 18

follow downward trends since 1976, meaning that women are increasingly likely to remain in the unemployment pool after an employment spell rather than drop out of the labor force. As shown in Figure 5, quits to unemployment present little evidence of a trend, except perhaps in the last 10-15 years. This trend can be traced back to a secular decline in the rate of quits to unemployment amongst men and women aged 16 to 35.19 Looking forward, two more recent labor supply trends are worth mentioning. First, Figure 8 plots the transition rates for men and women aged over 55. A trend apparent since the late 90s is that older workers are increasingly likely to join the labor force as (cid:21)IU and (cid:21)IE are following upward trends.20 We can also notice an increase in labor force attachment as both (cid:21)UI and (cid:21)EI are following downward trends. Second, Figure 9 shows that young workers are less likely to join the labor force ((cid:21)IEand (cid:21)IU are both on downward trends since the mid-90s). This could be related to the increase in the number of years of education as young workers stay longer in school before joining the labor force. Using (14), we can infer the consequence of such trends in terms of steady-state unemployment. Because of the larger demographic weight of older people, the contribution older workers is larger and unemployment rate would increase slightly. Extrapolatingthetrendinlaborforceparticipationbehaviorsince2000foryoungand old workers implies a steady-state unemployment rate about a quarter of a percentage point higher in 2015.21 4 Theoretical implications Business cycle (cid:135)uctuations: At business cycle frequencies, our results can be summarized as follows: (i) movements along the Beveridge curve and job separation (layo⁄s and quits) accountforalargeshare(76percent)butnotallofunemployment(cid:146)svariance,(ii)movementsin- 19The other demographic groups present little evidence of a trend. See also Duca and Campbell (2007). While our evidence only pertains to quits to unemployment, it is likely that a similar secular decline occurred for all quits as Fallick and Fleischman (2004) and Rogerson and Shimer (2010) also report a secular decline in job-to-job transitions since 1994. 20This is especially true for women. 21Formally, we extrapolated the trend growth rates in labor force participation ((cid:21)IU;(cid:21)UI,(cid:21)EI and (cid:21)IE) for young and old workers over 2010-2016 using the 2000-2007 average growth rate of the HP-(cid:133)lter trends. 19

and-outofthelaborforceaccountforaquarterofunemployment(cid:146)svarianceandlagmovements in layo⁄s and vacancy posting by a quarter, (iii) quits are procyclical and lag layo⁄s by a quarter, (iv) changes in matching e¢ ciency are generally small but can at times account for signi(cid:133)cant changes in the unemployment rate. The Mortensen-Pissarides (1994) search and matching model has become the canonical model of equilibrium unemployment. In that model, and consistent with (i), unemployment (cid:135)uctuationsaredrivenbychangesinjobpostingandjobseparation. However, considering(ii), 25 percent of unemployment (cid:135)uctuations remains unaccounted for. This result is surprising given the conventional wisdom that movements in-and-out of the labor force played little role at business cycle frequencies (see e.g. Hall, 2005, Shimer, 2005, 2007 and Elsby, Michaels and Solon, 2009). Thus, introducing a labor force participation decision in the model is an important avenue for future research (see Garibaldi and Wasmer, 2005 and Haefke and Reiter, 2006 for e⁄orts in that direction). In addition, accounting for movements in-and-out of the labor force would help explain some of the unemployment volatility puzzle.22 Moreover,intheMPmodel,quitsandlayo⁄sareindistinguishablesinceamatchterminates when it is jointly optimal for both parties to separate. However, in the data, quits and layo⁄s display very di⁄erent time series properties: quits are negatively correlated with layo⁄s, and quits lag layo⁄s by one quarter. Finally, while shocks to matching e¢ ciency are rarely considered in search models, (iv) suggests that they may be a useful addition to the set of shocks considered to explain unemployment (cid:135)uctuations. Low-frequency movements: At low-frequencies, our main (cid:133)nding is the absence of any signi(cid:133)cant trend in labor demand and the fact that movements in labor supply account for all of the trend in unemployment. This result suggests that any explanation of the trend in unemployment since 1976 lies with demographics and changes in workers behavior rather than 22The unemployment volatility puzzle is the fact that the standard MP model cannot replicate the volatility of unemployment given productivity shocks of plausible magnitude (Shimer, 2005). 20

with any direct changes in (cid:133)rms(cid:146)labor demand. Davis, Faberman, Haltiwanger, Jarmin and Miranda (2010) link the secular decline in the job destruction rate to the secular decline in the unemployment in(cid:135)ow rate. Since we can attribute all of the latter to demographics and behavioral changes in labor supply (in particular, a stronger attachment of women to the labor force), our evidence suggests that the secular decline in job destruction is related to changes in labor supply rather than to changes in labor demand.23 Davis, Haltiwanger, Jarmin and Miranda (2007) also document a decline in cross-sectional dispersion of business growth rates and in the time-series volatility of business growth rates since 1976. Again, the absence of a trend in labor demand suggests that labor supply may have played an important role here. For example, since older workers have longer tenures and have a lower turn-over rate than young workers, some of the decline in business growth rate volatility may be due to the aging of the baby boom. In contrast, any labor demand based explanation (such as a decline in the variance of idiosyncratic shocks hitting (cid:133)rms) must also justify the absence of any signi(cid:133)cant trend in labor demand (such as why the layo⁄ rate did not decline). 5 Conclusion ThispaperpresentsaframeworktointerpretmovementsintheBeveridgecurveanddecompose the components of unemployment (cid:135)uctuations. We (cid:133)nd that movements in labor demand are themaindeterminantsofcyclical(cid:135)uctuationsinunemploymentbutthatmovementsin-and-out of the labor force play an important role and account for almost a quarter of unemployment(cid:146)s variance. Further, labor demand leads labor supply, possibly indicating a causal interpretation as workers are more likely to join/stay in the labor force during recessions. Possible explanations include wealth e⁄ects and the added-worker e⁄ect for spouses. At low-frequencies, labor demand appears to play no direct role. Unemployment(cid:146)s trend since 1976 can be entirely 23Of course, stronger attachment of workers to the labor force could in turn have been triggered by labor demand changes such as increased economic uncertainty. However, the fact that we (cid:133)nd no trend in labor demand suggests a less direct link. 21

accounted for by secular changes in labor supply, in particular the aging of the baby boom, the increase in women(cid:146)s labor force participation and the increasing attachment of women to the labor force. Finally, while changes in matching e¢ ciency generally play a small role, they can decline substantially in recessions. For instance, in the 2008-2009 recession, lower matching e¢ ciency added about 11 percentage points to the unemployment rate. In a companion paper 2 (Barnichon and Figura, 2010), we explore the possible mechanisms behind such large changes in matching e¢ ciency. Appendix Analytical expressions for three labor market states To (cid:133)nd the steady-state unemployment rate uss, employment rate ess and inactivity rate iss it it it j of each demographic group i, note that U , U , E and U satisfy the system of it it it it j p;t;q;o di⁄erential equations (1) so that U ss;jn o 2f , Us g s, Ess and Iss are the solutions of the it it it it j p;t;q;o n o 2f g system U ss;j (cid:21) jE +(cid:21)IEIss = ( (cid:21) Ej +(cid:21)EI)E it it it it it it it 8 j p;t;q;o j p;t;q 2fX g 2Xf g > > > (cid:21)EIEss+ U ss;j (cid:21) jI = ((cid:21)IE +(cid:21)Io)Iss > > it it it it it it it > > > > > j 2fX p;t;q;o g > > > U ss;j = (cid:21)Ej Ess, j p;t;q > > < it (cid:21)jE+(cid:21)jI it 8 2 f g U ss;o = (cid:21)Io Iss > it (cid:21)oE+(cid:21)oI it > > > > > > U i s t s = U i s t s;j > > > j p;t;q;o > > 2fX g > > The steady-stat > : e unemployment rate uss is then obtained from uss = U i s t s and satis(cid:133)es it it LFss it s uss it it (cid:17) s +f it it 22

with s and f de(cid:133)ned by it it s = (cid:21)EI(cid:21)IU +(cid:21)IE(cid:21)EU +(cid:21)IU(cid:21)EU t it it it it it it 8 > < f it = (cid:21)U it I(cid:21)I it E +(cid:21)I it U(cid:21)U it E +(cid:21)I it E(cid:21)U it E > : and where the transition rates are given by (cid:21)UE = us it s;j (cid:21) jE it uss it it 8 j p;t;q;o > > > > > > > (cid:21)U it I = 2fX g u u s it s i s t s ;j (cid:21) j it I > > > j 2fX p;t;q;o g > > < (cid:21)EU = (cid:21) Ej it it j p;t;q > > 2Xf g > > > (cid:21)IU = (cid:21)Io > > it it > > > > > : where uss = U i s t s , u ss;j = U i s t s;j . it LFss it LFss it it Correction for the 1994 CPS redesign As explained in Polivka and Miller (1998), the 1994 redesign of the CPS caused a discontinuity in the way workers were classi(cid:133)ed between permanent job losers (i.e. other job losers), temporary job losers (i.e. on layo⁄s), job leavers, reentrants to the labor force and new entrants to the labor force (although we do not distinguish between the last two categories). As a result, the transition probabilities display a discontinuity in the (cid:133)rst month of 1994. To "correct" the series for the redesign, we proceed as follows. We start from the monthly transition probabilities obtained from matched data for each demographic group. We remove the 94m1 value for each transition probability (since its value corresponds to the redesigned survey, not the pre-94 survey), and instead estimate a value consistent with the pre-94 survey. To do so, we use the transition probability average value over 1993m6-1993m12 (the monthly probabilities can be very noisy so we average them over 6 months to smooth them out)24 that 24Taking a the average over 3-months or 12-months does not change the the result. 23

we multiply by the average growth rate of the transition probability over 1994m1-2009. That way, wecapturethelong-runtrendinthetransitionprobability. Over1994m2-2009, wesimply adjust the transition probability by the di⁄erence between the average of the original values over 94m1-94m6 (to control for the in(cid:135)uence of noise or seasonality) and the inferred 94m1 value. By eliminating the jumps in the transition probabilities in 1994m1, we are assuming that these discontinuities were solely caused by the CPS redesign. Thus, the validity of our approach rests on the fact that 1994m1 was not a month with large "true" movements in transition probabilities. We think that this is unlikely because there is no such large movements in the aggregate job (cid:133)nding rate and aggregate job separation rate obtained from duration data (Shimer, 2007 and Elsby, Michaels and Solon, 2009) that do not su⁄er from these discontinuities. (these authors treat the 1994 discontinuity by using data from the (cid:133)rst and (cid:133)fth rotation group, for which the unemployment duration measure (and thus their transition probability measures) was una⁄ected by the redesign. Moreover, Abraham and Shimer (2001) used independent data from the Census Employment Survey to evaluate the e⁄ect of the CPS redesign on the average transition probabilities from matched data. They found that only (cid:21)UI and (cid:21)IU were signi(cid:133)cantly a⁄ected, and that, after correction of these discontinuities (using the CES employment-population ratio), none of the transition probabilities displayed large movements in 1994. Finally, we checked ex-post that our procedure had little e⁄ect on the stocks, i.e. on the measure of the aggregate unemployment rate and on the unemployment rate of each demographic group, consistent with Polivka and Miller(cid:146)s conclusion (1998) that the redesign did not a⁄ect the measure of unemployment. 24

Analytical expressions for the Beveridge curve decomposition N N N dln(cid:21)UB = ! uj i ;ss (cid:21)j i B dln(cid:21) jB + uj;ss (cid:21)jB dln uj t ;ss + ! us i s(cid:21)U i B dln! us it s 8 t i u (cid:21)UB it uss (cid:21)UB us t s iuss(cid:21)UB itus t s i=1j p;t;q i=1j p;t;q i=1 > > X 2Xf g X 2Xf g X > N > > < = dln(cid:21)~UB +dln(cid:21) UB;demog with dln(cid:21) UB;demog = ! us i s(cid:21)U i B dln! us it s , B E;I t t t iuss(cid:21)UB itus t s 2 f g > > X i=1 > > > : N N dln(cid:21)EU = ! es i s (cid:21)E i j dln(cid:21) Ej + ! es i s (cid:21)E i j dln! es it s 8 t iess(cid:21)EU it iess(cid:21)EU ites t s i=1j p;t;q i=1j p;t;q > > X 2Xf g X 2Xf g > < = dln(cid:21)~EU +dln(cid:21) EU;demog t t > > > : N N dln(cid:21)EI = ! es i s(cid:21)E i I dln(cid:21)EI + ! es i s(cid:21)E i I dln! es it s 8 t iess(cid:21)EI it iess(cid:21)EI ites t s i=1 i=1 > X X > < = dln(cid:21)~EI +dln(cid:21) EI;demog t t > > : N N dln(cid:21)IB = ! is i s(cid:21)I i B dln(cid:21)IB + ! is i s(cid:21)I i B dln! is it s 8 t iiss(cid:21)IB it iiss(cid:21)IB itis t s B E;U i=1 i=1 > X X 2 f g > < = dln(cid:21)~IB +dln(cid:21) IB;demog t t > > wher : e the aggregate hazard rates (cid:21)~AB that hold composition (by demographics and unemt ployment reason) constant are de(cid:133)ned by N (cid:21)~UB = ! uj i ;ss (cid:21) jB , B E;I t i uss it 2 f g 8 i=1j p;t;q > > > X N 2Xf g N > > > (cid:21)~EU = ! es i s (cid:21) Ej and (cid:21)~EI = ! es i s (cid:21)EI > > t iess it t iess it > > < X i=1j 2Xf p;t;q g X i=1 N N > > > (cid:21)~I t U = ! i i i s i s s s (cid:21)I it o and (cid:21)~I t E = ! i i i s i s s s (cid:21)E it I: > > > i=1 i=1 > X X > > > : A second-order decomposition A second-order Taylor expansion of s uss = t t s +f t t 25

with s and f de(cid:133)ned by t t s = (cid:21)EI(cid:21)IU +(cid:21)IE(cid:21)EU +(cid:21)IU(cid:21)EU t t t t t t t 8 > < f t = (cid:21)U t I(cid:21)I t E +(cid:21)I t U(cid:21)U t E +(cid:21)I t E(cid:21)U t E > : gives us dlnuss = (cid:11)UI d(cid:21)U t I + 1 (cid:21)IE2 (cid:21)UI (cid:21)UI 2 + (17) t (cid:0) (cid:21)UI 2(s+f)2 t (cid:0) (cid:11)UE d(cid:21)U t E + 1 (cid:21)IU +(cid:21) (cid:0)IE 2 (cid:21)UE (cid:1) (cid:21)UE 2 (cid:0) (cid:21)UE 2 (s+f)2 t (cid:0) (cid:0) (cid:1) (cid:11)IE d(cid:21)I t E + 1 (cid:21)EU2 + (cid:21)E(cid:0)I +(cid:21)UI +(cid:21) (cid:1)UE 2 (cid:21)IE (cid:21)IE 2 (cid:0) (cid:21)IE 2 " (cid:0) s2 (cid:0) (s+f)2 (cid:1) # t (cid:0) (cid:0) (cid:1) +(cid:11)EI d(cid:21)E t I + 1 (cid:21)IU2 + (cid:21)IU2 (cid:21)EI (cid:21)EI 2 (cid:21)EI 2 " (cid:0) s2 (s+f)2 # t (cid:0) (cid:0) (cid:1) +(cid:11)EU d(cid:21)E t U + 1 (cid:21)IE +(cid:21)IU 2 + (cid:21)IE +(cid:21)IU 2 (cid:21)EU (cid:21)EU 2 (cid:21)EU 2 " (cid:0) (cid:0) s2 (cid:1) (cid:0) (s+f)2 (cid:1) # t (cid:0) (cid:0) (cid:1) +(cid:11)IU d(cid:21)I t U + 1 (cid:21)EI +(cid:21)EU 2 (cid:21)EI +(cid:21)EU 2 + (cid:21)EI +(cid:21)EU +(cid:21)UE 2 (cid:21)IU (cid:21)IU 2 (cid:21)IU 2 " (cid:0) (cid:0) s2 (cid:1) (cid:0) (s+f)2 (cid:1) # t (cid:0) (cid:0) (cid:1) (cid:0) (cid:1) +cross-order terms +(cid:17) t with(cid:11)EI = (1 uss)(cid:21)EI(cid:21)IU ,(cid:11)UE = (cid:21)IU(cid:21)UE+(cid:21)IE(cid:21)UE ,(cid:11)IE = (cid:21)IE(cid:21)EU (1 uss) (cid:21)UI(cid:21)IE+(cid:21)IE(cid:21)UE , (cid:0) s s+f s (cid:0) (cid:0) s+f (cid:11)UI = (cid:21)UI(cid:21)IE , (cid:11)EU = (1 uss)(cid:21)IE(cid:21)EU+(cid:21)IU(cid:21)EU , (cid:11)IU = (1 uss)(cid:21)EI(cid:21)IU+(cid:21)IU(cid:21)EU (cid:21)IU(cid:21)UE : s+f (cid:0) s (cid:0) s (cid:0) s+f To classify the cross-order terms (in, say, labor demand versus labor supply), we split their contribution in half between each two components. Finally, to separate movements along the Beveridge curve from changes in matching e¢ - 1 (cid:27) ciency, note that " = ln(cid:21)UE ln(cid:21)^UE with (cid:21)^UE = m vt (cid:0) . To a second-order, we t t (cid:0) t t 0 uss;bc (cid:18) t (cid:19) can write d" = d(cid:21)U t E d(cid:21)^U t E d(cid:21)U t E2 d(cid:21)^U t E2 , so that by de(cid:133)ning d"1 = d(cid:21)U t E d(cid:21)^U t E and t (cid:21)UE (cid:0) (cid:21)UE (cid:0) (cid:21)UE2 (cid:0) (cid:21)UE2 t (cid:21)UE (cid:0) (cid:21)UE (cid:18) (cid:19) 26

d"2 = d(cid:21)U t E2 d(cid:21)^U t E2 , we can replace d(cid:21)UE and d (cid:21)UE 2 in (17) using t (cid:21)UE2 (cid:0) (cid:21)UE2 t t (cid:0) (cid:1) d(cid:21)UE d(cid:21)^UE t = t +d"1 (cid:21)UE (cid:21)UE t d (cid:21)UE 2 d (cid:21)^U t E 2 t = +d"2 (cid:21)UE (cid:16)(cid:21)UE(cid:17) t (cid:0) (cid:1) 27

References [1] Abraham, K. (cid:147)Help-Wanted Advertising, Job Vacancies, and Unemployment,(cid:148)Brookings Paper on Economic Activity, 1:207-248, 1987. [2] Abraham, K. and R. Shimer. (cid:147)Changes in Unemployment Duration and Labor-Force Attachment.(cid:148)In The Roaring Nineties: Can Full Employment Be Sustained?, ed. Alan B. Krueger and Robert M. Solow, 367-420. New York: Russell Sage Foundation and Century Foundation Press, 2001. [3] Barnichon, R. (cid:147)Building a composite Help-Wanted index,(cid:148)mimeo, 2010. [4] Barnichon, R. and A. Figura. (cid:147)What happened to matching e¢ ciency?,(cid:148)mimeo, 2010. [5] Blanchard O. and P. Diamond. (cid:147)The Beveridge Curve,(cid:148)Brookings Paper on Economic Activity, 1:1-60, 1989. [6] Davis, Steven J., Jason Faberman, and John Haltiwanger. (cid:147)The Flow Approach to Labor Markets: New Evidence and Micro-Macro Links.(cid:148)Journal of Economic Perspectives, 20(3), 3-26, 2006. [7] Duca J. and C. Campbell.(cid:147)The impact of evolving labor practices and demographics on U.S. in(cid:135)ation and unemployment,(cid:148)Dallas Fed Working Paper, 2007. [8] Elsby, M. B. Hobijn and A. Sahin. (cid:147)Unemployment Dynamics in the OECD,(cid:148)Working Paper, 2008. [9] Elsby, M. R. Michaels and G. Solon. (cid:147)The Ins and Outs of Cyclical Unemployment,(cid:148) American Economic Journal: Macroeconomics, 2009. [10] Fujita, S. and G. Ramey. (cid:147)The Cyclicality of Separation and Job Finding Rates,(cid:148)International Economic Review, 2009. 28

[11] Lipsey, R. (cid:147)Structural and De(cid:133)cient-Demand Unemployment Reconsidered,(cid:148)in Employment Policy and the Labor Market, ed. Arther M. Ross, 210-255, UC Berkeley Press, 1965. [12] Pissarides, C. Equilibrium Unemployment Theory, 2nd Edition, MIT Press, 2001 [13] Sahin, Aysegul, Joseph Song, and Bart Hobijn, (cid:147)The Unemployment Gender Gap During the Current Recession, (cid:148)mimeo, FRB-NY, 2009. [14] Shimer, R., (cid:147)The Impact of Young Workers on the Aggregate Labor Market,(cid:148)Quarterly Journal of Economics, 116, 969-1008, 2001. [15] Shimer, R. (cid:147)Reassessing the Ins and Outs of Unemployment,(cid:148)NBER Working Paper No. 13421, 2007. 29

U=s/(s+l UIE +m (V/U) 1 s e e ) t t t t 0 t t V s orl UIE: layoffs, quits, mvts LF NLF t t or matching efficiency e t Shift in the Beveridge curve U Figure 1: Shifts in the Beveridge curve. 0.6 Residual Job finding rate, US data 0.8 Job finding rate from matching function 1.0 1.2 .2 1.4 1.6 .1 1.8 .0 2.0 .1 .2 1970 1975 1980 1985 1990 1995 2000 2005 laudiser (log)jobfindingrate Figure 2: Empirical (log) job (cid:133)nding rate, model job (cid:133)nding rate and residual, 1967-2009. 30

0.11 s L 0.1 Ld shocks to matching efficiency tn 0.09 e m y o lp 0.08 m e n u fo 0.07 s tn io p e 0.06 g a tn e c r e P 0.05 0.04 1976 1981 1986 1991 1996 2001 2006 Figure 3: Decomposition of unemployment (cid:135)uctuations into labor demand movements, labor supply movements and shocks to matching e¢ ciency over 1976-2009. The y-axis uses a logarithmic scale. The decomposition uses 2000Q3 as the base year. The colored areas sum to the approximated steady-state unemployment. The dashed red line is the exact value of steady-state unemployment. 31

0.08 Permanent Layoffs Temporary Layoffs Mvts along BC 0.07 Ld tn e m yo lp m 0.06 e n u fo stn io p e g a 0.05 tn e cre P 0.04 1976 1981 1986 1991 1996 2001 2006 Figure 4: Decomposition of labor demand movements into movements along the Beveridge curve and Beveridge curve shifts from permanent layo⁄s or temporary layo⁄s, 1976-2009. The decomposition uses 2000Q3 as the base year. The y-axis uses a logarithmic scale. 0.07 Quits Mvts LF NLF Demographics Ls tn 0.06 e m yo lp m e n u fo stn 0.05 io p e g a tn e cre P 0.04 1976 1981 1986 1991 1996 2001 2006 Figure 5: Decomposition of labor supply movements into Beveridge curve shifts due to quits, movements in-and-out of the labor force and demographics, 1976-2009. The decomposition uses 2000Q3 as the base year. The y-axis uses a logarithmic scale. 32

l UI l IU 0.45 0.06 0.4 0.055 0.35 0.05 0.3 0.045 0.25 0.04 0.2 1976 1981 1986 1991 1996 2001 2006 1976 1981 1986 1991 1996 2001 2006 l EI l IE 0.045 0.08 0.04 0.07 0.035 0.06 0.03 0.05 0.025 0.02 0.04 1976 1981 1986 1991 1996 2001 2006 1976 1981 1986 1991 1996 2001 2006 Figure 6: Transition rates for in-and-out of the labor force movements for women aged 25-55, 1976-2009. The dashed line represents the corresponding HP-(cid:133)lter trend ((cid:21) = 105). l UI l IU 0.25 0.16 0.14 0.2 0.12 0.15 0.1 0.1 0.08 0.05 0.06 1976 1981 198619911996 20012006 1976 1981 19861991 19962001 2006 l EI l IE 0.014 0.12 0.11 0.012 0.1 0.01 0.09 0.008 0.08 0.006 1976 1981 198619911996 20012006 1976 1981 19861991 19962001 2006 Figure 7: Transition rates for in-and-out of the labor force movements for men aged 25-55, 1976-2009. The dashed line represents the corresponding HP-(cid:133)lter trend ((cid:21) = 105). 33

l UI x 10 3 l IU 0.7 6 0.6 5 e e tar draz0 0 . . 4 5 tar draz3 4 a a H H 0.3 2 0.2 1 1976198119861991199620012006 1976198119861991199620012006 l EI l IE 0.06 0.02 0.055 e tar draz0 0 .0 .0 4 5 5 e tar draz 0.018 a H a H0.016 0.04 0.014 1976198119861991199620012006 1976198119861991199620012006 Figure8: Transitionratesforin-and-outofthelaborforcemovementsformenandwomenaged over 55, 1976-2009. The dashed line represents the corresponding HP-(cid:133)lter trend ((cid:21) = 105). l UI l IU 0.55 0.16 0.14 0.5 e e tar draza 0.45 tar draza 0. 0 1 . 2 1 H H 0.4 0.08 0.35 0.06 1976198119861991199620012006 1976198119861991199620012006 l EI l IE 0.085 0.16 0.08 0.14 e e tar draza 0 0 .0 .0 7 7 5 tar draza 0. 0 1 . 2 1 H H 0.065 0.08 0.06 0.06 1976198119861991199620012006 1976198119861991199620012006 Figure 9: Transition rates for in-and-out of the labor force movements for men and women aged16-25,1976-2009. ThedashedlinerepresentsthecorrespondingHP-(cid:133)ltertrend((cid:21) = 105). 34

dlnui shifts, LF NLF dlnui demog 0.04 0.1 w1625 w1625 w2555 w2555 w5485 w5485 0.03 m1625 0.08 m1625 m2555 m2555 m5485 m5485 0.02 0.06 0.01 0.04 )X )X X E/t 0 X E/t 0.02 (nl (nl 0.01 0 0.02 0.02 0.03 0.04 0.04 0.06 1976 1981 1986 1991 1996 2001 2006 1976 1981 1986 1991 1996 2001 2006 Figure 10: HP-(cid:133)lter trends ((cid:21) = 105) in Beveridge curve shifts due to changes in labor supply or to changes in demographics, 1976-2009. All variables are expressed as log-deviations from their average values. 35

Table 1: Estimating a Cobb-Douglas matching function Dependent variable: UE UE Sample (quarterly frequency) 1967-2009 1967-2009 Regression (1) (2) Estimation OLS GMM σ 0.62*** 0.61*** (0.01) (0.01) R2 0.89 -- Note: Standard-errors are reported in parentheses. In equation (2), I use 3 lags of v and u as instruments. I allow for first-order serial correlation in the residual. Table 2: Variance decomposition of steady-state unemployment, 1976:Q1-2009:Q4 Shocks to the Changes in Ld Changes in Ls matching function Raw data 0.59 0.31 0.10 Trend component 0.16 0.84 -- Cyclical component 0.68 0.19 0.13 Note: Trend component denotes the trend from an HP-filter (105) and cyclical component the deviation of the raw data from that trend. Table 3: Variance decomposition of steady-state unemployment, 1976:Q1-2009:Q4 Trend Cyclical Raw data component component Mvts along BC 0.24 -0.13 0.37 Ld Layoffs 0.25 0.05 0.46 Quits -0.04 0.06 -0.07 Ls Mvts LF-NLF 0.28 0.61 0.23 Demographics 0.12 0.42 0.02 Matching efficiency 0.13 -- -- Note: Trend component denotes the trend from an HP-filter (105) and cyclical component the deviation of the raw data from that trend. Mvts along BC refers to movements along the Beveridge curve and Mvts LF-NLF refers to movements in-and-out of the labor force. Table 4: Correlation matrix of the determinants of cyclical unemployment, 1976-2009 Temporary Permanent Mvts along Quits Mvts LF-NLF layoffs layoffs BC Temporary layoffs 1 0.56 0.54 -0.52 0.42 Permanent layoffs - 1 0.88 -0.65 0.71 Mvts along BC - - 1 -0.68 0.71 Quits - - - 1 -0.62 Mvts LF-NLF - - - - 1 Note: All variables are detrended with an HP-filter (105). Table 5: Lead-lag structure of the determinants of cyclical unemployment, 1976-2009 Temporary Permanent Mvts along Quits Mvts LF-NLF layoffs layoffs BC Temporary layoffs 0 1 1 2 2 Permanent layoffs - 0 0 0 1 Mvts along BC - - 0 0 1 Quits - - - 0 0 Mvts LF-NLF - - - - 0 Note: The table reports the value of j for which corr(Xt,Yt+j) is highest (in absolute value). 35