THE LABOR WEDGE AS A MATCHING FRICTION

Transcription

1 THE LABOR WEDGE AS A MATCHING FRICTION ANTON A. CHEREMUKHIN AND PAULINA RESTREPO-ECHAVARRIA RESEARCH DEPARTMENT WORKING PAPER 1004 Federal Reserve Bank of Dallas

2 The Labor Wedge as a Maching Fricion Anon A. Cheremukhin and Paulina Resrepo-Echavarria July 22, 2010 Absrac The labor wedge accouns for a large fracion of business cycle ucuaions. This paper uses a search and maching model o decompose he labor wedge ino hree classes of labor marke fricions and evaluae heir role. We nd ha fricions o job desrucion and bargaining commonly considered in he search lieraure are no helpful in explaining he labor wedge. We also idenify an asymmeric e ec of separaion, bargaining and maching fricions on unemploymen, as well as a poenial soluion o Shimer s puzzle. JEL: E20, E32, J22, J63, J64. Keywords:Labor Wedge, Business Cycles, Search and Maching For he las 25 years, macro and labor economiss have poined o large cyclical variaions in he relaionship beween he marginal rae of subsiuion beween leisure and consumpion and he marginal produc of labor Anon Cheremukhin: Federal Reserve Bank of Dallas, 2200 N Pearl S, Dallas TX 75201, cherosha@gmail.com, Paulina Resrepo-Echavarria: Deparmen of Economics, The Ohio Sae Universiy, 410 Arps Hall, 1945 N High Sree, Columbus OH 43210, paur@ucla.edu. Auhors hank Andrew Akeson, Gary Hansen, Richard Rogerson, Rober Shimer, Pierre-Olivier Weill, Mark Wrigh, hree anonymous referees, as well as paricipans of SED 2008, LAMES 2008 and Moneary Economics Proseminar a UCLA for helpful commens. The auhors are especially hankful o Roger Farmer, Chrisian Hellwig and Lee Ohanian for heir ime and suggesions. All errors are our own. The views expressed in his paper are hose of he auhors and do no necessarily re ec he views of he Federal Reserve Bank of Dallas or he Federal Reserve Sysem. Firs draf: April 9,

3 as an imporan feaure of business cycles. In heir business cycle accouning framework, Chari, Kehoe and McGraan (2007) (CKM) label his relaionship a "labor wedge" and argue ha i accouns for 60% of oupu ucuaions. In his paper we look a he labor wedge hrough he lens of a search and maching model. We decompose he labor wedge ino hree classes of labor marke fricions modeled as exogenous separaion, bargaining and maching shocks and use he business cycle accouning mehodology o evaluae heir quaniaive imporance. Our main nding is ha imperfecions in he job desrucion and bargaining processes commonly considered in he search lieraure are no helpful in explaining he labor wedge. Equilibrium search and maching models provide a naural framework for sudying labor marke fricions manifesed by he labor wedge. Maching fricions are also a common way of inroducing adjusmen along he exensive margin ino business cycle models. However, Shimer (2005a) poined ou ha he maching fricion iself works as a labor adjusmen cos and alone canno explain variaions in he wo key labor marke variables: unemploymen and vacancies. In order o help solve his puzzle various addiional fricions have been inroduced ino search and maching models. One class of fricions inroduced by Morensen and Pissarides (1994) is endogenous job desrucion, which we capure using an exogenous shock o he rae a which workers are separaed from heir jobs. The oher class of fricions proposed by Hall (2005a) and analyzed by Shimer (2010) replaces he Nash bargaining soluion wih a backward-looking wage-seing rule. In our model his form of wage sickiness is represened more generally by exogenous variaions in he relaive bargaining power of workers and rms. Finally, we inroduce a hird, residual source of fricions by allowing for exogenous variaions in he e ciency of he maching process. Our modeling approach augmens he represenaive agen RBC model wih a search and maching fricion as in Merz (1995) and Andolfao (1996). The sandard assumpion ha labor is raded in a spo marke is replaced by a search fricion which pus an addiional consrain on how much labor can be employed. The model endogenously deermines he level of unemploymen, he number of vacancies and he labor force paricipaion rae as in Farmer and Hollenhors (2006). To model he fricions menioned above we inroduce hree shocks which joinly deermine he labor wedge in he model: he separaion shock, he maching shock and he bargaining shock. 2

4 The separaion shock represens he proporion of employed workers ha ge separaed from heir jobs every period. The maching shock represens he e ciency of he maching echnology. The bargaining shock represens he proporions in which he lifeime surplus of a newly formed mach is spli beween he worker and he rm and hus pins down wages. To evaluae he relaive imporance of each shock we use he business cycle accouning mehodology employed in Cole and Ohanian (2002) and CKM. For ideni caion purposes in addiion o hree labor marke shocks, our model includes a TFP shock, an invesmen shock and a governmen consumpion shock. We use daa on real GDP, consumpion, invesmen, hours, unemploymen and vacancies o recover he six shocks: TFP, invesmen, governmen consumpion, separaion, maching and bargaining. We use he model as a diagnosic ool and measure he conribuions of each shock o each of he six variables by running a counerfacual exercise: we he shocks back ino he model one a a ime and all bu one a a ime. Compared o he neoclassical growh model in our model i is di cul o direcly apply he business cycle accouning mehodology. Firs, in order o recover he exac values of he underlying shocks we would have o ierae over he soluion of he model unil we nd a xed poin of a complicaed sysem of forward-looking equaions. Insead, we compue he rs order approximaion of he model around a sochasic rend and use he Kalman ler o recover he shocks. Second, here is subsanial conroversy on he values of many labor marke parameers, so we use a Bayesian esimaion sraegy o recover hem, and le he daa speak for iself. Our accouning exercise leads us o a sriking conclusion: variaions in job desrucion and impedimens o he bargaining process joinly accoun for less han 10 percen of variaions in he labor wedge. More han 90 percen of variaions in he labor wedge are aribued o he residual shock o maching e ciency. This implies ha commonly used fricions, such as endogenous variaions in job desrucion and wage sickiness, do no play a signi can role in deermining he labor wedge. Our resuls indicae ha a labor marke fricion responsible for variaions in he labor wedge mus be isomorphic o changes in maching e ciency. This broad class of fricions includes variaions in per capia coss associaed wih creaing jobs, variaions in ime and e or devoed o search by unemployed workers, variaions in he level of congesion and in he degree of compeiion beween peers characerizing he maching process. Apar from he main resul ha shocks o maching e ciency play a lead- 3

5 ing role in explaining he labor wedge, accouning for more han 90% of is variaion, our resuls have wo main implicaions. Firs, boh job creaion and job desrucion shocks play an imporan role in oupu and unemploymen ucuaions. This las resul is relevan for he debae beween Shimer (2005b) and Fujia and Ramey (2007) on wheher job desrucion or job creaion is more imporan for ucuaions in oupu and unemploymen. We nd ha boh job desrucion (separaion) and job creaion (bargaining and maching) shocks play a role, bu a di eren poins in ime. A he beginning of a recession a decline in TFP coincides wih an increase in he separaion rae. As unemploymen increases, he reservaion value (hrea-poin) of he workers falls while an increase in he bargaining power of workers keeps he wage fairly consan. The corresponding decrease in he bargaining power of he rms leads o a decline in vacancies. Laer he e ciency of he maching process falls, keeping unemploymen a a high level. Thus, undersanding he causes of excessive job desrucion a he beginning of recessions, as well as sources of wage rigidiy and insu cien maching during he recovery are equally imporan for undersanding he behavior of he labor marke. The second implicaion of our resuls is a poenial soluion o Shimer s puzzle. Shimer (2005a) poined ou ha sandard Morensen-Pissaridesype models are incapable of simulaneously generaing high volailiy of unemploymen and vacancies and low volailiy of wages. Our model while maching series for unemploymen and vacancies by consrucion, predics a pah for wages which maches remarkably well he behavior of wages in he daa. The fac ha we mach he volailiy of wage daa wihou using i in he esimaion procedure, boh serves as an over-idenifying resricion on our empirical exercise 1, and sresses he imporance of changes in he ouside opion of workers as he main driving force behind he variaions in bargaining power. Hall (2005b) suggesed ha changes in he bargaining power can be generaed by ucuaions in he marginal produc of labor when wages are sicky. Our resuls show ha in order o he daa, addiional signi can ucuaions in he ouside opion of workers are necessary. The paper is organized as follows. Secion 2 lays ou he heoreical framework and inroduces he six shocks, Secion 3 describes he mehodology we use o esimae he model and recover he shocks, Secion 4 explains 1 Noe ha he abiliy of a search model o mach volailiies of unemploymen and vacancies does no auomaically imply maching he behavior of wages, as shown by Lubik (2009) 4

6 he resuls and Secion 5 concludes. 1 Theoreical Framework This secion lays ou he seup of he model. We modify he sandard one secor real business cycle model by adding a search echnology for moving labor beween producive aciviies and leisure. We assume ha he economy is populaed by a coninuum of families. Each family operaes a backyard echnology and compleely insures is members agains variaions in heir labor incomes. Members of a family canno work in heir own backyard, bu can be employed in wo marke aciviies: head-huning which is compeiive and a producive aciviy where he wage is se according o a speci c wageseing rule. 1.1 Model The economy faces six exogenous shocks. A oal facor produciviy (TFP) shock A, an invesmen speci c echnology shock T, a governmen expendiure shock G, a shock o he separaion rae of employmen L, a shock o maching e ciency B and a shock o he bargaining power of workers. This las shock represens he fracion of he lifeime surplus of he mach ha goes o he worker, hence, as we will show laer on, deermines he wage w in he producive secor. A he beginning of period values of shocks A, T, L, G, B,, capial K, labor supply L s 1 and demand L d 1, he job nding and vacancy lling raes are given. The head of each family decides how many members V s o send o look for a head-huning job and how many members of oher families V d o hire in he head-huning marke o search for unemployed workers o ll in posiions in he backyard rm. Each head-huner represens one vacancy and is paid a wage q. The head of he family also decides how many members L s o send o work in producive aciviies and how many members of oher families L d o employ o operae her own backyard echnology. Finally, he head of he family assigns U members o apply for jobs in oher families backyards, allocaes resources o consumpion C of is members and invess ino capial K +1 nex period. We inroduce head-huners ino he model in order o measure coss of 5

7 searching for a worker and coss of searching for a job in he same unis of disuiliy of labor. We disinguish beween labor demand and supply in boh markes in order o derive shadow prices of boh ypes of employmen and compue he value of a mach. Each family head maximizes he expeced lifeime uiliy of is members (1), subjec o a budge consrain (2) and labor supply and demand accumulaion consrains (3) and (4): max E fc ;L s ;Ld ;V s;v d;u;;k +1g 1 X =0 U (C ; L s ; V s ; U ) ; (1) C + K +1 (1 K )K +G A F (K ; L d )+w (L s L d )+q (V s T V d ) (2) L s = (1 L )L s 1 + U M U (3) L d = (1 L )L d 1 + V d M V ; (4) where M is he oal number of maches formed in he economy in period. In equaion (3), labor supply in period depends on las period s labor supply minus he number of workers ha go separaed from heir job plus he new formed maches. The separaion rae L denoes he exogenously given rae a which workers are separaed from heir jobs and capures he various fricions leading o variaions in job desrucion over he cycle. The erm M U sands for he job nding rae and represens he increase in employmen when here is one more individual searching for a job (U increases by one uni). In equaion (4), labor demand accumulaes in he same way as labor supply wih he di erence ha he erm V d M V is he vacancy lling rae imes he number of head-huners demanded and means ha for every new individual ha works as a head-huner V d, he sock of employed workers M increases by V. The markes for labor and head-huning clear when L s = L d = L and V s = V d = V. The law of moion of aggregae employmen sais es: and in equilibrium U = U, V = V and L = (1 L )L 1 + M ; (5) M = M = B M(U ; V ): (6) 6

8 In equaion (6) B represens he e ciency of he maching echnology, deermining he number of maches formed for each combinaion of he numbers of workers and head-huners seeking for a mach. The resource consrain and producion funcion are given by C + 1 T (K +1 (1 K )K ) + G = Y ; (7) Y = A F (K ; L d ): (8) We derive he opimaliy condiions of he model: 1 U = E T 0 C +1 U 0 C A +1 F 0 K +1 (K ; L ) + 1 (1 K ) T +1 w + U! 0 0 L UC = U 0 E +1 C U 0 +1 (1 L+1 ) C A F 0 L (K ; L ) w = E U 0 C +1 U 0 C +1 (1 L+1 )! (9) (10) (11) U 0 V U 0 C + q = 0 (12) M V q = 0 (13) U 0 U U 0 C = M U (14) In he equaions above is he Lagrange muliplier associaed wih he labor supply accumulaion consrain and is he Lagrange muliplier associaed wih he labor demand accumulaion consrain, boh measured in unis of marginal uiliy of consumpion. Since T, A, L, G and B are exogenous, we have a sysem of en equaions and eleven variables, fk +1 ; L ; C ; M ; Y ; V ; U ; ; ; w ; q g. The model is missing an equilibrium condiion because equaions (10) and (11) deermine wo di eren ways of moving labor beween leisure and employmen in producive aciviies and here is only one price w. Therefore, we inroduce a bargaining shock o close he model. 7

9 1.2 Inroducing Bargaining Shocks Noe ha equaions (10) and (11) can be ieraed forward o solve for he corresponding mulipliers: = A F 0 L (K ; L ) w + E 1 X = U 0 L U 0 C + w + E s=+1 1X s=+1 s A s F 0 L s (K s ; L s ) w s sy s U 0 L s U 0 C s + w s! sy k=+1 k=+1 (1 Lk ) (15) (1 Lk ) (16) The Lagrange muliplier in he labor demand (supply) accumulaion equaion is he expeced sum of insananeous marginal values of he mach for he represenaive rm (worker), discouned and adjused for he probabiliy of he mach being dissolved in any given period. Similarly, he sum + of he wo Lagrange mulipliers can be inerpreed as he lifeime surplus of he mach an expeced sum of insananeous marginal values of he mach discouned and adjused for he probabiliy of he mach being dissolved in any given period. = A F 0 L (K ; L )+ U 0 L U 0 C +E 1X s=+1 s A s F 0 L s (K s ; L s ) + U 0 L s U 0 C s! sy k=+1 (1 Lk ) (17) A sandard way o close search and maching models is o assume ha he worker and he rm use Nash bargaining over he wage and spli he surplus in consan proporions. However, Shimer (2005a) and Hall (2005b) argue ha for a Morensen-Pissarides-ype model o he daa one needs variaions in he bargaining power. In order o incorporae such a fricion we close he model by assuming ha he surplus is spli beween he worker and he rm according o a ime-varying rule. 2 We de ne a variable o represen he fracion of he lifeime surplus of he mach going o he worker, and (1 ) he fracion of he lifeime surplus of he mach going o he rm. We assume ha = (1 ) and =. Hence, we refer o as he bargaining power of he worker as well as he bargaining shock. Noice ha allocaions are economically ine cien A. 2 For a more general descripion of how we consruc he bargaining shock see Appendix 8

10 whenever is di eren from he elasiciy of he maching funcion wih respec o he number of unemployed. 3 Subsiuing he bargaining shock and he lifeime surplus of he mach ino equaions (10), (11), (12), (13) and (14) yields: = A F 0 L (K ; L ) + U! 0 0 L UC + E +1 U 0 C U 0 +1 (1 L+1 ) (18) C M (1 ) = U 0 V (19) V U 0 C = U 0 U M U U 0 C (20) Equaion (18) governs he behavior of he oal value of he mach. Today s value is equal o he insananeous gain from a newly formed mach plus he fuure value, discouned and adjused for he possibiliy of being desroyed omorrow. Equaions (19) and (20) equae he marginal bene s and marginal coss of head-huning and searching for a job respecively and hus pin down vacancies and unemploymen. Equaions (21) and (22) below deermine he real wage and he discoun facor for labor: w + U 0 L U 0 C! = 0 U C E +1 0 U C 1 E +1 U 0 +1 C U 0 C A F 0 L (K ; L ) + U 0 L U 0 C! (21) +1 = +1 (1 L+1 ) (22) A compeiive equilibrium of he model economy is a soluion o equaions (5)-(9), (12), and (18)-(22) where fk +1 ; C ; L ; V ; U ; ; w ; q ; ; Y ; M g are endogenous variables and fa ; T ; L ; G ; B ; g are he exogenous shocks of he model. The exogenous variables behave according o sochasic processes o be de ned laer. 3 See Hosios (1990) for a deailed discussion. 9

11 1.3 Idenifying assumpions Mos of he funcional forms we use are sandard in he lieraure. We assume ha he producion funcion is Cobb-Douglas wih consan reurns o scale: F (K; L) = K L 1 (23) We assume ha he maching funcion is also Cobb-Douglas and consisen wih consan reurns o scale: M(U; V ) = U V 1 (24) We posulae a uiliy funcion consisen wih a balanced growh pah and where fracions of ime spen head-huning and searching for a job ener symmerically wih he ime spen on he producion aciviy: U(C; L; U; V ) = log C (L + V + U) (25) The funcional form for hours is he only non-sandard assumpion we make. I implies ha workers ge he same disuiliy from working in producive aciviies as when searching for a job or head-huning. This assumpion may seem somewha exreme, bu we assume ha alhough individuals spend only a few hours per week searching for a job, hey also spend ime in oher aciviies ha generae disuiliy: in expanding heir nework by making phone calls, geing echnical raining, coninuing heir educaion, helping heir relaives or working in home producion. Anoher implicaion of his assumpion is ha he coss of searching for a job from he poin of view of an unemployed worker and of a rm wih a vacan posiion are equalized. While micro daa sheds lile ligh on how o discipline hese coss, his assumpion delivers a clear-cu inerpreaion of he shocks. In he decomposiion we perform his assumpion disinguishes he variaions in labor marke variables aribued o he bargaining shock from hose aribued o he maching shock. I insures ha any variaions in he search coss on he worker and rm sides show up as variaions in he maching e ciency shock, while variaions in he wage-seing pracices show up as he bargaining shock. In he decomposiion we perform, his uiliy funcion also a ecs he behavior of he reservaion value of he workers and search coss. The parameer deermines how much he labor coss fall as families spend less ime in he labor marke. 10

12 Given hese funcional forms and sochasic processes for he shocks, o be explained laer, he shocks are uniquely ideni ed. Appendix A explains sep by sep how given daa on oupu, consumpion, invesmen, hours, unemploymen and vacancies one can recover he shocks. 1.4 Decomposing he Labor Wedge Here we explore how he hree labor marke shocks a ec he labor wedge and joinly deermine is behavior. Combining equaions (19) and (20) and subsiuing in he funcional forms we ge: 1 = V U (26) = U 0 U U + U 0 V V U 0 C M = MRS 1 B (1 ) 1 (27) Subsiuing hese ino equaion (18) moving everyhing excep he marginal produc o one side and rearranging we ge: 0 MP 1 UC 1 = E +1 MRS +1 1 L+1 MRS B (1 ) 1 U 0 C MRS B (28) Equaion (28) shows how he separaion, bargaining and maching shocks ogeher form a connecion beween he marginal produc of labor and he marginal rae of subsiuion beween leisure and consumpion: he labor wedge. Noe ha when he separaion shock is non-persisen and since only is expecaion a ecs he labor wedge, i canno play a signi can quaniaive role. This is rue because agens expec he separaion rae nex period o be in he neighborhood of he seady-sae. However, in general boh maching and bargaining shocks can have a signi can e ec on he labor wedge. Changes in he maching shock are always going o maer, while he imporance of he bargaining shock depends on he relaionship beween and he seady-sae value of. Noice ha an increase in he bargaining power of workers always leads o a corresponding decrease in he bargaining power of rms and he oal e ec of he erm (1 ) 1 depends on. 11

13 Equaion (29) saes ha if ss is equal o hen he e ecs of he bargaining shocks are (1 ) = = 1 (1 1 ) 1 = 0 (29) = This implies ha even when he Hosios condiion does no hold exacly, bu holds on average, changes in he bargaining power do no signi canly a ec he labor wedge. In fac he values of and ss have o be very far apar for he bargaining power o have a subsanial e ec on he labor wedge. Therefore i is naural o expec maching shocks o play a dominan role in deermining he behavior of he labor wedge. 2 Mehodology This secion describes he deails of he esimaion procedure. Our esimaion sraegy is di eren from ha of CKM in hree dimensions. Firs, because compuing he shocks exacly involves solving a complicaed xed poin problem, we apply he Kalman ler o a linearized version of he model o compue he shocks approximaely. Second, hough some of he parameers can be calibraed, ohers have no analogs in he lieraure. In order o le he daa choose appropriae values for hese parameers we apply a Bayesian esimaion sraegy, which uses he Kalman ler resuls from he rs sep and ieraes over he parameer space using Markov Chain Mone-Carlo (MCMC) mehods. Finally, esimaion resuls can depend signi canly (see Cogley and Nason (1995)) on he way he daa is pre- lered. In order o avoid cerain lering biases we minimize he exen o which we aler he daa. We embed he rends ino he model. We describe in deail a procedure of derending he model around a non-saionary sochasic rend, which we borrow from Fernandez-Villaverde and Rubio-Ramirez (2007). 2.1 Processes for he shocks In he daa real oupu, consumpion and invesmen are nonsaionary even wih respec o a log-linear rend. To make he daa comparable o he model, 12

14 he business cycle lieraure commonly uses he Hodrick-Presco (HP) ler. However, Cogley and Nason (1995) and Canova (1998) show ha he use of he ler inroduces signi can biases ino he daa by amplifying businesscycle frequencies even if i does no have any. To avoid pre lering he daa we follow he approach presened in Fernandez-Villaverde and Rubio-Ramirez (2007) and assume random walks for he wo processes ha are commonly hough o be exremely persisen: he TFP and invesmen shocks A and T 4. We denoe a ss he mean growh rae of TFP, and ss he mean growh rae of he invesmen-speci c echnology. We assume ha he res of he shocks follow rs-order auoregressive processes. We denoe he separaion rae in seady-sae Lss, he fracion of hours spen on head-huning in seady-sae B ss, he average bargaining power which deermines he vacancy-unemploymen raio ss, and he seady-sae fracion of GDP consumed by he governmen g ss. Also and denoe he sandard deviaions and auocorrelaions of he shocks. All innovaions are assumed o be sandard normal 5. We do no pu resricions on he correlaions of hese innovaions Derending Here we describe how o derend he model wih respec o a pair of nonsaionary rends. From he opimaliy condiions of he model we can see ha all variables excep capial grow a a facor (a ss ss) 1 1. Then, if we ake he rs di erences of he TFP and invesmen shocks by de ning a = A A 1 = a ss exp ( A " A ), = T T 1 = ss exp ( T " T ), we can derive an aggregae rend Z 1 = A T, which will be common o all he variables excep capial. Hence, we can de ne derended variables of he form: x = X Z 1 : Capial grows a a facor (a ss ss ) 1 1, so i is derended as follows: k +1 = K +1 Z T. Similarly o he resul of King e. al. (1988) given separabiliy of he uiliy funcion, we need o assume logarihmic uiliy for here o exis a balanced growh pah. This will no a ec our resuls since Hansen (1986) 4 In fac using an HP- ler does no change any of our main resuls. 5 We focus on realizaions of and L which are in he inerval [0,1]. 6 We esimaed he model under di eren correlaion srucures and his assumpion does no a ec any of our resuls. 13

15 has shown ha quaniaively he degree of risk aversion has almos no e ec on he behavior of quaniies in real business cycle models. Appendix B shows he resuling derended equilibrium condiions of he model. This model possesses a unique seady-sae we describe he algorihm of compuing he seady-sae in Appendix B as well. We hen linearize he model around he seady-sae, compue he sae-space represenaion and esimae he parameers. 2.3 Daa We use six variables in our esimaion procedure: 1) real per capia GDP, 2) real per capia nondurable consumpion expendiures, 3) real per capia gross privae domesic invesmen (including durable consumpion), 4) an index of aggregae weekly per capia hours worked in privae indusries, 5) he unemploymen rae and 6) he Conference Board help-waned adverising index as a proxy for vacancies. All daa are seasonally adjused. Monhly daa is averaged o make i quarerly. We divide by populaion o obain per capia values. This corresponds o modeling he economy using a represenaive household/ rm. We remove an exremely low frequency rend from hours, unemploymen and vacancies, using an hp- ler wih a smoohing parameer (we follow Shimer (2005a)). This removes long-run secular rends, which are a resul of demographic and oher facors unrelaed o business cycles. We normalize he resuling derended indexes of hours and vacancies o one on average. We ake logs of GDP, consumpion and invesmen, and hen ake he rs di erence. All daa we use is for he period 1964:I-2007:III. To be able o esimae he model we need o add six measuremen equaions corresponding o he six variables ha we observe. Since he daa for real oupu, consumpion and invesmen are modeled as nonsaionary, we ake he rs di erences of he daa o make i comparable o he model. In addiion, he de niion of oupu in our model includes ime spen headhuning. In he real economy rms are paying head-huners a wage and i is measured as par of GDP. To accoun for his, we derive he price of ime spen head-huning, muliply i by he amoun of ime spen in his aciviy and include he produc in our de niion of GDP. Hours in our model correspond o he oal ime spen on he producive aciviy and head-huning. This index corresponds closely o oal employmen L + V, since mos of he cyclical variaion in hours is on he exensive 14

16 margin (see Gerler, Sala, Trigari (2008) and Hall (2005b)) 7. Due o he above correspondence beween hours and employmen, he ime spen by he represenaive agen searching for a job as a fracion of he oal ime spen U in he labor marke L +V +U corresponds o he number of people searching for a job as a fracion of people paricipaing in he labor marke he unemploymen rae. Changes in he help-waned adverising index proxy changes in he number of vacancies V posed by rms. 2.4 Calibraion and Esimaion Our model has 9 srucural parameers and 13 parameers ha characerize he shocks. The scale parameer L ss does no a ec he log-linearized represenaion of he model. There are hree parameers sandard o he business-cycle lieraure ha we calibrae. We se he share of capial in he Cobb-Douglas producion funcion o 0.34, he discoun facor o 0.99, he depreciaion rae K o 2.5% per quarer. We se he seady-sae value of he governmen shock o 22% of GDP, he average value in he daa. We also se he elasiciy of maches o unemploymen o 0.7, he value used by Shimer (2005a); his falls wihin he range of values plausible from a microeconomic perspecive repored by Blanchard and Diamond (1989). We calibrae his parameer because i is no well-ideni ed. We nd ha his value of helps us mach he volailiy of wages, a series which is no used in our esimaion procedure. From he average growh raes of invesmen, consumpion and oupu we infer he means of innovaions o TFP and invesmen shocks. We calibrae hem o be 0.16 percen and 0.12 percen per quarer respecively. Table 1 summarizes he calibraed parameers. We esimae he model using Bayesian mehods (see An and Schorfheide (2007)). Linearized equaions of he model combined wih he linearized measuremen equaions form a sae-space represenaion of he model. We apply he Kalman ler o compue he likelihood of he daa given he model and o obain smoohed esimaes of he innovaions o he shocks. We combine he likelihood funcion L Y Daa jp, where p is he parameer vecor, wih he priors 0 (p) o obain he poserior disribuion of he parameers pjy Daa = L Y Daa jp 0 (p). Draws from he poserior disribuion are 7 We have esimaed he model using daa on oal employmen insead of oal hours. Mos of our resuls remain unchanged. We prefer using hours so ha we can direcly compare our resuls o CKM. 15

17 K g ss a ss ss Table 1: Calibraed parameers Parameer Prior Poserior Disribuion Mean S.D. Mean [5% 95%] log Normal [2.97, 4.43] Lss Gamma [0.030, 0.043]! ss Gamma [0.49, 0.73] ss Bea [0.49, 0.64] Table 2: Prior and poserior disribuions of srucural parameers generaed using he Markov-Chain Mone-Carlo (MCMC) algorihm. We use he Random-Walk Meropolis-Hasings implemenaion. Table 2 repors he prior and poserior disribuions of each srucural parameer. The parameer! ss represens he seady-sae job nding rae. We esimae he elasiciy of he uiliy funcion wih respec o labor o be This high elasiciy leads o large variaions in he value of non-marke aciviy o be discussed laer. We esimae he seady-sae separaion rae o be 3.6%. This is much lower han Shimer s (2005a) quarerly esimae of he separaion probabiliy for employed workers. This di erence comes from he fac ha our separaion rae corresponds o he average fracion of jobs permanenly desroyed every quarer. In addiional o he permanen desrucion an esimae of he separaion rae would include a componen capuring shor-erm urnover beween employmen and unemploymen and a large job-o-job ransiion componen. Assuming (following Shimer) ha he average job nding rae is 40% per monh and he separaion rae o be 3% per monh, he e ecive number of people becoming and saying unemployed unil nex quarer should be around 2-3%, which is consisen wih our esimae. Our model implies a 61% average job- nding rae which is also comparable o Shimer s esimaes. We esimae he seady-sae bargaining power ss o be 0.56, which is close o he value of 0.5 common in he lieraure (see Morensen and Nagypal 8 We assume a log normal disribuion wih he suppor on [ 1; +1), which is equivalen o log ( + 1) being normally disribued. 16

18 Parameer Prior Poserior Disribuion Mean S.D. Mean [5%, 95%] S Bea [0.65, 0.80] M Bea [0.84, 0.88] B Bea [0.96, 0.99] G Bea [0.87, 0.92] A IGamma [0.0062, ] T IGamma [0.0067, ] S IGamma [0.154, 0.212] M IGamma [0.070, 0.099] B IGamma [0.038, 0.055] G IGamma [0.022, 0.028] Table 3: Prior and poserior disribuions of shock parameers (2007) and Hall (2005b)). The esimaes of he wo parameers! ss and ss joinly imply, ha he average reservaion uiliy is approximaely 80% of he worker s marginal produc. This moves in he direcion of Hagedorn and Manovsky s (2008) calibraion of he value of non-marke aciviy (0.95) and is higher han he calibraion of Hall (0.4). Our esimae of he parameer! ss also pins down he raio of ime spen head-huning o ime spen in he producion aciviy which urns ou o be 4%. Taking ino accoun he proximiy of he shadow prices of di eren allocaions of ime, his mimics closely Hagedorn and Manovsky s esimae of he cos of vacancies being 3-4.5% of he quarerly wage. However, unlike heir model, mos of he variaion in he bargaining se comes from variaions in he value of nonmarke aciviy, no he marginal produc. Table 3 repors he prior and poserior disribuions of he persisence and variance parameers of he shocks. The separaion rae is he leas persisen wih a quarerly auoregressive parameer equal o The maching and governmen shocks are more persisen, bu sill signi canly less persisen han a random walk. The persisence of governmen consumpion is 0.90 exacly like in he daa. The bargaining shock is close o a random walk. See Figures in Appendix C o compare he prior and poserior disribuions of he parameers. Our model explains 100% of he variaion in he six variables and hus provides a decomposiion we need for he business cycle accouning exercise. 17

19 3 Resuls We divide our resuls ino hree pars. In he rs par we show ha he labor wedge can be inerpreed as he insananeous welfare gain from a new mach. This surplus shrinks in good imes and expands in recessions. The second par consiues he descripive core of our resuls. We describe he behavior of he recovered shocks, measure heir conribuions and evaluae heir e ecs on he labor wedge, oupu and unemploymen. We address he debae beween Fujia and Ramey (2007) and Shimer (2005a) on wheher job desrucion or job creaion is more imporan for ucuaions in unemploymen and oupu. The las par of his secion consiues he analyic core of our resuls. I describes he mechanisms driving our resuls. We show he prediced wage rae, compare i o he daa and explain why he model can solve Shimer s puzzle. 3.1 Behavior and Inerpreaion of he Labor Wedge Following mos of he lieraure, we de ne he labor wedge as he raio of he marginal rae of subsiuion beween leisure and consumpion (MRS) and he marginal produc of labor (MP). Figure 1 depics he behavior of hese wo deerminans of he labor wedge. The shaded verical areas correspond o he o cial recession periods according o NBER. The picure forces one o conclude ha mos of he volailiy of he labor wedge comes from variaions in he marginal rae of subsiuion, raher han he marginal produc. Though we esimae he elasiciy of he uiliy funcion o be high, his resul is rue for mos values of he elasiciy used in he macro lieraure. 9 In he conex of our model, he labor wedge has a new inerpreaion. The MRS represens he reservaion value (ouside opion) of workers when bargaining over he wage, which implies ha he di erence beween he MP and he MRS represens he insananeous welfare gain of a new mach. 10 I is clear from Figure 1 ha he bargaining se narrows in good imes and widens signi canly in recessions. Thus, in bad imes he labor wedge widens, re ecing an increase in he value of new maches and vice versa. 9 If we x he elasiciy a lower values and re-esimae he model, he resuls are very similar. 10 The behavior of he insananeous gain is very similar o ha of he lifeime gain. 18

20 3.2 Behavior of he Underlying Shocks Given he esimaed parameer values, we can compue he underlying shocks using he Kalman ler. Figure 2 describes he behavior of he recovered shocks over he whole fory- ve year period. Noe ha TFP and invesmen shocks are random walks wih drifs, while he res of he processes are saionary. We nd ha oal facor produciviy slows down a he beginning of each recession. The invesmen speci c echnology ends o increase in recessions and has a negligible e ec on oupu and he labor marke variables. This suppors he main nding of CKM, ha he invesmen wedge plays only a eriary role in U.S. business cycles. The governmen shock as well as he invesmen shock, only a ecs consumpion and invesmen. Because we are primarily ineresed in he behavior of oupu, hours, unemploymen and vacancies, for he res of he exposiion we absrac from he behavior of invesmen and governmen shocks. Insead we focus on echnological shocks and shocks ha make up he labor wedge. Figure 1: The Labor Wedge Figure 2 demonsraes ha he separaion rae ends o be high a he beginning of each recession. 11 The wave of separaions ypically sars earlier 11 As a consisency check noe, hese spikes in he separaion rae ypically coincide wih 19

21 Figure 2: The six shocks han he recession iself and dies ou quickly wihin a year afer he sar of a recession. Noice as well he large decreases in he maching e ciency a laer sages of recessions which lead o declines in he number of new maches and, hence, cause he amouns of hours worked o fall. While he ouside opion of he worker ends o decrease in recessions because of a decrease in heir marginal disuiliy of work, an increase in he bargaining power ends o leave wages largely unchanged. Thus, our nding ha bargaining power of workers increases signi canly during recessions is consisen wih a view of wage sickiness as a major source of ine ciency in he labor marke. Le us now ake a closer look a he iming of shocks. From Figure 2 i is clear ha declines in TFP slighly precede increases in he separaion rae. An increase in he separaion rae is ypically followed by an increase in he bargaining power of workers which precedes or coincides wih a decrease in he maching shock (see also cross-correlaions in Appendix D). This implies ha shocks o he separaion rae are imporan a early sages of recessions, and bargaining and maching shocks come ino play laer. spikes observed in Shimer s daa. 20

22 3.3 Decomposiion of he Labor Wedge To measure he conribuion of each shock o a given variable, we shu down he innovaions o i and simulae he model. We obain pahs of oupu, hours, unemploymen and vacancies, which would have aken place, if only his disorion were absen. This allows us o compare he acual pahs of variables o heir hypoheical pahs in a world where one of he imperfecions is absen. Shuing down innovaions o each one of he labor marke shocks reveals a sriking picure. Figure 3 shows ha absence of separaion and bargaining shocks leaves he labor wedge essenially unchanged, while he absence of shocks o maching e ciency produces an essenially consan labor wedge. Thus, mos of he ucuaions in he labor wedge are explained by maching shocks alone. Figure 3: The decomposiion of he labor wedge Remember ha when he Hosios condiion holds on average, changes in he bargaining power do no a ec he labor wedge. Our calibraion of of 0.7 as suggesed by Shimer and our esimae of he seady-sae value of he bargaining power of 0.56 are no ha far. 12 Hence, our resul ha he 12 The e ecs of he di erence beween and ss are second order, so a di erence of 20% would accoun for 0:2 2 =2 = 2% of variaions in he labor wedge. 21

23 Figure 4: Oupu wih all bu one shock bargaining shock has lile or no e ec on he labor wedge is no all ha surprising. 3.4 Decomposiion of Oupu and Unemploymen To analyze in deail he e ecs of each shock on oupu and unemploymen and he iming paerns, we focus on he 2001 recession episode, which is he las recession in our sample. We use his recession o illusrae our resuls as i is easier o see he resuls in a more deailed graph han i is o see hem in a graph conaining he whole period. A he end of he secion, we show ha he resuls hold for all recession periods in he sample. As in he previous subsecion, we compare he acual pah of GDP wih pahs i would have aken if we eliminae e ecs of jus one of he shocks. Alhough only maching shocks maer for he labor wedge, he impac on oupu is no so clear-cu. Figure 4 illusraes he e ecs of shocks o TFP, he separaion rae, he bargaining power of workers and he maching e ciency on oupu. The verical axis measures percenage deviaions from he pah ha oupu would have followed, if all he shocks were consan (he random walks would preserve heir drifs, bu innovaions are shu down). The solid line 22

24 depics he acual pah of oupu in he daa. The res of he lines depic he pahs of oupu if we shu down innovaions o jus one of he shocks, eliminaing is e ec on he economy. Figure 4 shows ha if here were no change in oal facor produciviy, he recession probably would no have sared. If here were no separaion shocks a he early sage of he recession, oupu would have fallen by half as much. The bargaining shocks added o he deph of he recession, while shocks o maching e ciency are key o undersanding he slow recovery: in he absence of he adverse maching shocks, he economy would have fully recovered by summer of Thus, Figure 4 demonsraes ha alhough maching shocks explain mos of he dynamics of he labor wedge, hey can only accoun for a fracion of oupu dynamics. While shocks o TFP and he separaion rae sar recessions by accouning for he iniial slowdown in oupu, he role of bargaining and maching shocks is o deepen he recession and delay he recovery. Figure 5 depics a similar decomposiion of unemploymen. I follows from Figure 5 ha separaion shocks are responsible for he iniial increase in unemploymen. Increases in he bargaining power of workers sar playing a role only once he economy is already in a recession, reinforcing his iniial increase in unemploymen. Declines in maching e ciency leave unemploymen a a high level for a longer period of ime afer he o cial recession has already ended, hus accouning for he so-called jobless recovery. Therefore, afer some rms in he economy have become less producive, he role of he separaion shock is o creae he iniial pool of unemployed people. As he number of unemployed goes up, he reservaion value of workers goes down signi canly hey are willing o work a a lower wage. The sluggish response of wages drives up he bargaining power of he workers, while he rm is now in a worsened posiion. As a resul rms sar posing less vacancies, and here are more and more unemployed in he marke. Consisen wih his explanaion, he sharp increase in he bargaining power of workers accouns for he bulk of changes in unemploymen and vacancies in he second phase of he recession. As he number of workers seeking jobs is high and he number of vacancies is low, he maching e ciency goes down, hus causing oupu o fall deeper and he recession o las longer. Figure 5 con rms ha if here were no decline in maching e ciency, he recovery from he recession would have been much faser. Hence, he so-called "jobless recovery" is due mainly o maching shocks. We aribue his o some form of congesion, which sill 23

25 Figure 5: Unemploymen wih all bu one shock requires an explanaion. I can also be some form of disorganizaion, when he leas e cien and more specialized workers become desperae o nd a job and wai unil beer imes, consisen wih he idea of res unemploymen. 3.5 Decomposiions: Summary Noe ha he saemens made regarding he las recession hold more generally over he period of ineres. A similar decomposiion of he previous four recession episodes shows ha he emphasized paern holds more generally: separaions creae he iniial pool of unemployed, and adverse maching shocks slow down he recovery. To summarize conribuions of each shock o each variable of ineres, we se all he oher shocks o heir seady-sae values and simulae he model. We obain pahs of oupu, hours, unemploymen and vacancies, which would have aken place if all he oher disorions excep one were absen. Table 4 repors fracions of sandard deviaions of oupu, hours, unemploymen and vacancies, ha can be explained by each one of he shocks. The conribuion of he "labor wedge" is measured by hiing he economy by hree shocks: separaion, bargaining and maching a he same ime. Earlier we have shown ha in our model he "labor wedge" is an 24

26 Shock TFP Inves- Labor Separaion Maching Bargaining Oupu Hours Unemp Vacancies Table 4: Raios of sandard deviaions explained by each shock over he whole period exac combinaion of jus hese hree shocks (see equaion (28)). We hus provide an exac decomposiion of he labor wedge ino job desrucion and job creaion shocks. These resuls address he debae beween Fujia and Ramey (2007) and Shimer (2005b) on wheher job desrucion or job creaion is more imporan for ucuaions in unemploymen and oupu. We nd ha alhough shocks o job creaion are more imporan for he behavior of oupu and unemploymen, shocks o job desrucion canno be ignored. Changes in he separaion rae accoun for a signi can fracion of ucuaions and explain he iniial increase in unemploymen and decrease in oupu. Essenially, hese shocks sar he recession. Thus, even hough heir conribuion is relaively small, wihou job desrucion shocks recessions migh no have happened in he rs place. 3.6 Wages and Shimer s Puzzle In an in uenial paper, Shimer (2005a) shows ha a sandard Morensen- Pissarides-ype model, when hi by produciviy shocks of plausible magniude, predics wages o be more volaile han in he daa, while generaing relaively small variaions in unemploymen and vacancies. This is known as Shimer s puzzle. Our model s he volailiy of unemploymen and vacancies by consrucion. Figure 6 depics he behavior of wages prediced by he model and compares i o daa (adjused for he sochasic rend). The model predics wages as volaile as in he daa, and he correlaion beween he wo is high (0.56). I is imporan o consider ha we are no using observed wages in he esimaion procedure. The abiliy of he model o generae a wage series ha close o he observed one is remarkable. The prediced wage level splis he insananeous value of he mach be- 25

27 Figure 6: Wages: model versus daa ween he worker and he rm in he proporion of heir bargaining weighs, as illusraed by Figure 7. Figure 7 demonsraes ha while he reservaion value of workers falls in recessions, wages say fairly consan, hus indicaing ha he bargaining power of workers increases in recessions. This resul suppors wage sickiness as a mechanism behind he large changes in he bargaining power of he workers. However, unlike previous models of Hall (2005a) and Farmer and Hollenhors (2006), where increases in he bargaining power in recessions were a resul of declines in he marginal produc combined wih wage sickiness, in our model hey are a consequence of declines in he reservaion value (MRS) ogeher wih wage sickiness. Thus, allowing for changes in he marginal rae of subsiuion beween consumpion and leisure and, consequenly, for changes in he reservaion value of workers, our model boh maches he volaile behavior of unemploymen and vacancies, and predics an absence of signi can ucuaions in wages, jus as in he daa. Hence, by allowing for variaions in he ouside opion of workers our model provides a mechanism, which can solve Shimer s puzzle. Why do we ge hese large swings in he marginal rae of subsiuion, in he maching e ciency and in he bargaining power? We hink ha his 26

28 should be a general resul in models where agens decide on he margin. More precisely, we argue ha models, where workers and rms equalize bene s and coss of searching for a job and opening a vacancy, would predic sizable changes boh in shocks and incenives. Firs, noice ha when workers choose wheher o search for a job (equaion (14)), hey equae he cos of searching for a job which is equal o he MRS in our model wih he poenial bene s of forming a mach imes he probabiliy of nding a job. The bene s are equal o he presen discouned value of he wages minus he cos of working, which is also equal o MRS: Figure 7: Variaions in he bargaining se MRS = M = P V (W MRS ) M (30) U U Given ha in he daa he job nding probabiliy M U declines signi canly in recessions (documened by Shimer (2005a)) and he wage is fairly consan, equaion (30) implies ha he MRS has o fall by a fair amoun. In he model he large swings in he MRS are due o he esimaed elasiciy of he uiliy funcion of 3.54 a signi canly bigger number han 0.5, ypically assumed in he RBC lieraure. This resul leads us o obain a much more volaile series for he labor wedge, which, noneheless, maches he behavior 27

29 of previous esimaes very well. For a comparison of our labor wedge wih he labor wedge of CKM, see Appendix D. Secondly, noice ha when rms choose wheher o open a new vacancy, hey also equae he compeiive salary hey pay o a head-huner wih he poenial bene s of forming a mach imes he probabiliy of nding a worker o ll he vacancy. The bene s are equal o he presen discouned value of he marginal produc minus he wage ha hey pay o he worker: M MRS = (1 ) = P V (MP W ) B V U V (31) Given ha we have already esablished he signi can decreases in he MRS in recessions, and aking ino accoun he fac ha in he daa unemploymen increases, while he number of vacancies falls and boh he wage and he marginal produc are no very volaile, equaion (31) implies ha he maching e ciency has o fall signi canly in recessions. Combining equaions (30) and (31) one can nd ha he bargaining power of he workers is direcly pinned down by he marke ighness: 1 = V U (32) Thus, when unemploymen increases and here are fewer vacancies, he bargaining power of workers has o increase by a comparable amoun. Variaions in he separaion rae ha we esimae are a residual of he labor accumulaion equaion in he producive secor. To summarize, for a model, where boh households and rms decide on he margin how much ime o spend searching for each oher, o mach aggregae daa, one needs o generae large changes in boh he marginal rae of subsiuion beween consumpion and leisure and he bargaining power of workers. Procyclical reservaion values, along wih counercyclical bargaining power of workers, help mach he volaile behavior of unemploymen and vacancies and predic absence of signi can ucuaions in wages, jus as in he daa. This mechanism provides a soluion o Shimer s puzzle. 4 Conclusion Moivaed by he fac ha variaions in he labor wedge accoun for a large fracion of business cycle ucuaions, we look a he labor wedge hrough 28

30 he lens of a search and maching model. Using a model ha feaures imevarying search and maching fricions in he spiri of Morensen and Pissarides (1994), Shimer (2005a) and Hall (2005a) we decompose he labor wedge ino hree broad classes of fricions capured by separaion, bargaining and maching shocks. Using a business cycle accouning mehodology similar o ha of Chari, Kehoe and McGraan (2007), we idenify he sources of variaions in he labor wedge and assess he imporance of job desrucion (separaion) and job creaion (bargaining and maching) fricions for business cycles. We nd ha imperfecions in he job desrucion and bargaining processes are no helpful in explaining he labor wedge which is mainly driven by maching shocks. This implies ha heories emphasizing wage sickiness and endogenous job desrucion are no very useful for explaining he behavior of he labor wedge. Insead, according o our resuls, more aenion should be devoed o sudying fricions equivalen o he maching shock in our model, for example, fricions ha lead o cyclical variaions in job creaion coss, search e or, or coordinaion problems. More speci cally, one poenial microfoundaion for he maching shock in our model is proposed by Leser (forhcoming), who shows ha when rms have he abiliy o pos muliple vacancies hen he e ciency of he maching process depends on he disribuion of vacancies among rms, increasing in concenraion of vacancies. Alhough only maching shocks a ec he labor wedge, boh fricions in job creaion and job desrucion play an imporan role in oupu and unemploymen ucuaions. We nd ha job desrucion and job creaion shocks play a role a di eren poins in ime. In a recession, separaion shocks accoun for he iniial increase in unemploymen, bargaining shocks help amplify he increase in unemploymen, while maching shocks are responsible for he slow recovery. Finally, our resuls also provide a poenial soluion o Shimer s puzzle. We nd ha inroducing variaions in he reservaion value of workers is a feaure worh exploring in search and maching models in order o mach he observed volailiy of unemploymen, vacancies and wages. References Andolfao, David Business Cycles and Labor-Marke Search. American Economic Review, 86(1):

31 An, Sungbae, and Frank Schorfheide Bayesian Analysis of DSGE Models. Economeric Reviews, 26(2-4): Blanchard, Oliver Jean, and Peer Diamond The Beveridge Curve. Brookings Papers on Economic Aciviy, 20(1989-1): Canova, Fabio Derending and Business Cycle Facs. Journal of Moneary Economics, 41(3): Chari, V. V., Parick J. Kehoe, and Ellen R. McGraan Business Cycle Accouning. Economerica, 75(3): Cogley, Timohy, and James M. Nason E ecs of he Hodrick- Presco Filer on Trend and Di erence Saionary Time Series: Implicaions for Business Cycle Research. Journal of Economic Dynamics and Conrol, 19(1-2): Cole, Harold L., and Lee E. Ohanian The U.S. and U.K. Grea Depressions Through he Lens of Neoclassical Growh Theory. American Economic Review, 92: [826]. Farmer, Roger E. A., and Andrew Hollenhors Shooing he Aucioneer. NBER Working Paper, Fernandez-Villaverde, Jesus, and Juan F. Rubio-Ramirez Esimaing Macroeconomic Models: A Likelihood Approach. Review of Economic Sudies, 74(4): Fujia, Shigeru, and Garey Ramey Reassessing he Shimer Facs. Federal Reserve Bank of Philadelphia, Working Paper, Gerler, Mark, Luca Sala, and Anonella Trigari An Esimaed Moneary DSGE Model wih Unemploymen and Saggered Nominal Wage Bargaining. Journal of Money, Credi and Banking, 40(8): Hagedorn, Marcus, and Iourii Manovskii The Cyclical Behavior of Equilibrium Unemploymen and Vacancies Revisied. American Economic Review, 98(4): Hall, Rober E. 2005a. Employmen Flucuaions wih Equilibrium Wage Sickiness. American Economic Review, 95(1):

32 Hall, Rober E. 2005b. The Labor Marke and Macro Volailiy: A Nonsaionary General-Equilibrium Analysis. NBER Working Paper, Hansen, Gary D Growh and Flucuaions. Universiy of California a Sana Barbara, Unpublished. Hosios, Arhur J On he E ciency of Maching and Relaed Models of Search and Unemploymen. Review of Economic Sudies, 57(2): King, Rober G., Charles I. Plosser, and Sergio T. Rebelo Producion, Growh and Business Cycles : I. The Basic Neoclassical Model. Journal of Moneary Economics, 21(2-3): Leser, Benjamin. forhcoming. Direced Search wih Muli-Vacancy Firms. Journal of Economic Theory. Lubik, Thomas A Esimaing a Search and Maching Model of he Aggregae Labor Marke. Economic Quarerly, 95(2): Merz, Monika Search in he Labor Marke and he Real Business Cycle. Journal of Moneary Economics, 36(2): Morensen, Dale, and Eva Nagypal More on Unemploymen and Vacancy Flucuaions. Review of Economic Dynamics, 10(3): Morensen, Dale T, and Chrisopher A Pissarides Job Creaion and Job Desrucion in he Theory of Unemploymen. Review of Economic Sudies, 61(3): Shimer, Rober. 2005a. The Cyclical Behavior of Equilibrium Unemploymen and Vacancies. American Economic Review, 95(1): Shimer, Rober. 2005b. The Cyclicaliy of Hires, Separaions, and Job-ojob Transiions. Federal Reserve Bank of S. Louis Review, 87(4): Shimer, Rober Labor Markes and Business Cycles. Princeon Universiy Press. 31

33 A Appendix. A.1 Appendix A In he following subsecion we show he soluion o he same model as in he main body of he paper, bu when solved as a social planner s problem. The decenralized version of he model has a missing equilibrium condiion ha is ypically replaced wih a Nash bargaining condiion o x he real wage. We ake advanage of his missing condiion, and by comparing he social planner s soluion wih he decenralized version of he model, we consruc a ime varying bargaining shock, which implicily deermines he wage rae. A.1.1 The Social Planner s Problem To compare compeiive allocaions wih an e cien one we solve he social planning problem. The social planner maximizes he discouned presen value of he uiliy funcion: subjec o max E fc ;L ;V ;U ;;K +1 g 1 X =0 U (C ; L ; V ; U ; ) (33) C + 1 T (K +1 (1 K )K ) + G A F (K ; L ) (34) L = (1 L )L 1 + M (35) The opimaliy condiions of he planner are given by: 0 1 UC = E +1 A T U 0 +1 F 0 K +1 (K ; L ) + 1 (1 K ) C T A F 0 L (K ; L ) + U! 0 0 L UC = U 0 E +1 C U 0 +1 (1 L ) C (36) (37) U 0 V U 0 (38) U 0 U U 0 (39) 32

34 Togeher wih equaions (7)-(8) hey describe he allocaions a social planner would choose. is he Lagrange muliplier associaed wih he labor accumulaion consrain. Given ha T, A, L, G and B are exogenous, we have a sysem of eigh equaions and eigh unknowns fk +1 ; L ; C ; M ; Y ; V ; U ; g. A.1.2 Consrucing a Time Varying Bargaining Shock By comparing he social planner s opimaliy condiions wih hose of he decenralized problem, we can nd he necessary assumpions o make he decenralized problem e cien. M By puing equaions (12) and (13) ogeher we ge ha = U 0 V, C and if we compare his expression wih equaion (38), and equaion (14) wih equaion (39), we need U 0 V M (40) M (41) so ha he opimaliy condiions on vacancies and unemploymen are he same in he decenralized and planner s problem. Furhermore, if we assume ha he maching funcion has consan reurns o ; V U ; V V = M(U ; V ) (42) hen = + and he decenralized oucome is Pareo-opimal. Hence, he Hosios condiion for e ciency is given by: = = As an wih condiions (43) and (44) give V M (43) U M (44) V M = (1 ), which ogeher and = (1 ) (45) 33

35 = (46) Noice ha if we replace equaions (45) and (46) in equaions (10) and (11) and sum hem up, we ge equaion (37), hence he opimaliy condiions for labor in he decenralized version become equal o he opimaliy condiion for labor in he planners problem. Furhermore, if we divide equaion (10) by equaion (11) and use equaions (45) and (46) we ge w + U 0 L U 0 C A F 0 L (K ; L ) w = 0 U C E +1 U 0 +1 (1 L+1 ) C E U 0 C +1 U 0 C +1 (1 L+1 ) = 1 (47) Given ha w is he wage earned by he worker, and U 0 L U 0 C is his reservaion uiliy, he erm w + U 0 L represens he insananeous bene from U 0 C he mach earned by he worker. Since he bargaining power of he worker is consan and equal o, he opimal wage rae sais es!! w + U 0 L U 0 C = A F 0 L (K ; L ) + U 0 L U 0 C (48) where A F 0 L (K ; L ) + U 0 L is he di erence beween he marginal produc of U 0 C labor and he marginal disuiliy of labor. This erm represens he insananeous marginal value of he mach, and a fracion goes o he worker. To inroduce he ime varying bargaining shock we build on his resul, re-parameerize and subsiue by. is ime varying and follows an exogenous auoregressive process. Noice ha replacing by implies ha allocaions are subopimal whenever 6=. Equaions (45) and (46) are replaced by = (1 ) (49) = (50) Once again if we subsiue equaions (49) and (50) in (10) and (11) we ge equaion (37) so i is sill rue ha he opimaliy condiions for labor of he 34

36 decenralized version imply he opimaliy condiion for labor of he planner s problem. Dividing equaion (10) by equaion (11) and using equaions (49) and (50) we ge w + U 0 L U 0 C A F 0 L (K ; L ) w = hence, he opimal wage rae sais es w + U 0 L U 0 C! = 0 U C E +1 U (1 L+1 ) C (1 ) E U 0 C +1 U 0 C (1 +1 ) +1 (1 L+1 ) 0 U C E +1 0 U C 1 E +1 U 0 +1 C U 0 C A F 0 L (K ; L ) + U 0 L U 0 C! (51) (52) where +1 = +1 (1 L+1 ) and can be inerpreed as a sochasic discoun facor for labor. A.1.3 Ideni caion In his secion we show how given daa on allocaions: oupu, invesmen, consumpion, employmen, vacancies and unemploymen one can solve for he shocks. Le us rs rewrie he equaions of he model given he parameric assumpions and funcional forms used in he paper: Y = A K L 1 (53) X = K +1 (1 K ) K (54) C + X T + G = Y (55) L = (1 L ) L 1 + B U V 1 (56) 1 = E T C C +1 Y +1 + (1 K) K +1 T +1 (57) 35

37 = w C (L + U + V ) + E C C (1 L+1 ) (58) = (1 ) Y L w + E C C (1 L+1 ) (59) C (L + U + V ) U = B (60) 1 C (L + U + V ) V = B (61) U = 1 V (62) Now we shall describe a mechanism o recover he shocks given parameers and funcional forms. Given daa on consumpion C (or governmen spending G ), oupu Y, invesmen X, employmen L + V, number of vacancies V and he unemploymen rae U L +V +U, one can uniquely recover he ime pah for he variables of ineres L,V,U. Then equaion (54) uniquely pins down he pah for capial given he iniial level K 0, equaion (53) pins down he e ciency shock A, equaion (55) pins down consumpion or governmen spending, equaion (57) can be solved forward o obain he pah for he invesmen shock as in CKM. From equaions (60) and (61) i follows ha U = V. Then summing up equaions (58) and (59) one obains: = 1 + V U C (L + U + V ) + (1 C E C V U +1 ) Y L = (1 L+1 ) (63) Using equaion (60) he Lagrange muliplier can be expressed as a funcion of he maching shock B : = (L + U + V ) B V U 1 (64) 36

38 Also he separaion rae is conneced o he maching shock hrough he labor accumulaion equaion (56): (1 L+1 ) = L +1 B +1 U+1V +1 1 (65) L Then subsiuing equaions (64) and (65) ino equaion (63) we obain: V U 1 V U C 1 + +E C +1 V +1 U +1 1 V+1 U +1 1 C 1A L = B L+1 + U +1 + V +1 L + U + V (1 ) Y C (L + U + V ) + (66) L+1 U B +1V Equaion (66) provides a forward-looking equaion for he maching shock B +1 as a funcion of B. Solving his equaion recursively given some iniial value B 0 we can recover he whole pah for he maching shock. Then equaion (65) allows us o back up he separaion rae, equaions (61) and (60) allow us o calculae he Lagrange mulipliers and. Then from equaion (62) we can compue he bargaining shock. Alogeher equaions (54-62) describe a one-o-one mapping beween he daa and he underlying shocks. However he algorihm described here is hard o implemen direcly for wo reasons. Firs, he equaions are forward looking and can only be solved under cerain assumpions abou expecaion formaion. Second, many of he parameers of he model are unknown and canno be simply calibraed from microeconomic daa. Tha is he reason why we posulae sochasic processes for he shocks, linearize he model around a seady-sae o compue an approximae soluion and use Kalman ler o recover he underlying processes for he shocks. A.2 Appendix B A.2.1 The Derended Model Once we derend all he variables of he model, we come o he following represenaion: 37

39 E +1 y +1 1 K = 1 k y = a k L 1 = c + z k +1 (1 K ) k + g = y (1 ) y + E L (1 L ) B U V 1 = (V + U ) V = (1 ) U L = (1 L ) L 1 + B U V 1 z 1 = a m = (1 L ) = c 1 c 1 1 z 1 = c (L + U + V ) U q = (1 ) B V 1 E m (w ) = E m (1 ) y L a = a ss exp ( A " A ) 38

40 = ss exp ( T " T ) log L = (1 S ) log Lss + S log L 1 + S " S log B = (1 M ) log b ss + M log B 1 + M " M log = (1 B ) log ss + B log 1 + B " B log g = (1 G ) log g ss y ss + G log g 1 + G " G d log GDP = log d log Inv = log y + q V y 1 + q 1 V 1 z 1 d log Cons = log c c 1 z 1 k +1 z (1 K ) k k z 1 1 (1 K ) k 1 z 1 1 Hours = Unemp = L + V L ss + V ss U L + V + U : HW an = V V ss A.2.2 Compuing he Seady-Sae. Choose a value of L ss 1) z ss = (a ss ss) 1 1 2) Denoe ' = z ss + 1 K ss =a ss 1 1 3) k ss = 'L ss y ss = a ss ' L ss i 4) c ss = h(1 g ss ) a ss ' (1 z K ) ss ss ' 5) B ss = 1! ss ss 1 ss 1 39 L ss

41 6) U ss =! ss L L ss V ss = 1 ss ss 7) = yss 1 L ssc ss 1+!ss L)) ss (1 zss (1 U ss 8) We have assumed a normalizaion = 9) ss = c ss m ss = 1 L 10)! ss = c ss ss ss 11) w ss = ss (1 ) yss L ss (1 ss ) ss 12) ss = z ss q ss = ss (L ss+u ss+v ss) 40

42 A.3 Appendix C Figure 8: Prior (grey) and poserior (black) disribuions of parameers 41

43 Wedge E ciency Invesmen Labor Oupu Table 5: Raios of sandard deviaions explained by each wedge. Source: Chari, Kehoe and McGraan (2007) A.4 Appendix D A.4.1 Comparison o CKM Figure 9: Comparison of he Labor Wedge o he Esimae of CKM A.4.2 Full Decomposiions and Correlaion Srucure Table 5 repors he numbers from he original paper by Chari, Kehoe and McGraan (2007). Comparing he second row of Tables 5 and 6 one can verify ha our decomposiions are comparable wih hose of CKM since he di erence in he conribuions of TFP, Invesmen and Labor shocks is insigni can. Table 6 also gives a clearer picure of he relaive conribuions of he labor shocks. Table 7 repors he same fracions of sandard deviaions as Table 4, bu averaged over a selecion of recession periods. I demonsraes ha during recessions he labor wedge and TFP play a slighly more imporan role in business cycles han in normal imes, while he conribuion of invesmen shocks is negligible boh in recessions and overall. 42

44 Shock TFP Invesmen Governmen Labor Oupu Consumpion Invesmen Hours Unemploymen Vacancies Shock TFP Separaion Maching Bargaining Oupu Consumpion Invesmen Hours Unemploymen Vacancies Table 6: Raios of sandard deviaions explained by each shock over he whole period ( ) Shock TFP Invesmen Governmen Labor Oupu Consumpion Invesmen Hours Unemploymen Vacancies Shock TFP Separaion Maching Bargaining Oupu Consumpion Invesmen Hours Unemploymen Vacancies Table 7: Raios of sandard deviaions explained by each shock averaged over 5 recessions (70,75,82,91,01) 43

45 Correlaion of X wih Y a lag k Shocks (X,Y) TFP, Invesmen TFP, Governmen Invesmen, Governmen TFP, Separaion TFP, Maching TFP, Bargaining Separaion, Bargaining Separaion, Maching Bargaining, Maching Table 8: Cross Correlaions of Shocks and Their Lags Table 8 repors cross correlaions of shocks a di eren lags con rming he picure of TFP and separaion shocks saring recessions and bargaining and maching shocks coming ino play only laer on. A.5 Appendix E Figure 10 demonsraes he emphasized decomposiion of unemploymen for previous four recession episodes: separaions creae he iniial pool of unemployed, and adverse maching shocks slow down he recovery. Figure 11 shows ha if here was no change in he labor wedge, he recession would have been much shorer (if a all noiceable) and half as severe. If here was no change in oal facor produciviy, he recession probably wouldn have sared. Absence of invesmen shocks would have almos no e ec on he pah of oupu. Thus he TFP shock is a work mosly a he sar of he recession of The labor wedge explains he bulk of ucuaions in oupu afer he recession has sared. Figure 12 shows ha maching shocks are he main source of declines in he number of hours worked. Figure 13 demonsraes ha bargaining shocks are he main source of declines in he number of vacancies during he 2001 recession. 44

46 Figure 10: E ecs of separaion and maching shocks on unemploymen Figure 11: Oupu wih all bu one shock 45