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1 Edinburgh Research Explorer Job Reallocaion, Unemploymen and Hours in a New Keynesian Model Ciaion for published version: Hol, R 2007 'Job Reallocaion, Unemploymen and Hours in a New Keynesian Model' ESE Discussion Papers, no. 172, Edinburgh School of Economics Discussion Paper Series. Link: Link o publicaion record in Edinburgh Research Explorer Documen Version: Publisher's PDF, also known as Version of record Publisher Righs Saemen: Hol, R. (2007). Job Reallocaion, Unemploymen and Hours in a New Keynesian Model. (ESE Discussion Papers; No. 172). Edinburgh School of Economics Discussion Paper Series. General righs Copyrigh for he publicaions made accessible via he Edinburgh Research Explorer is reained by he auhor(s) and / or oher copyrigh owners and i is a condiion of accessing hese publicaions ha users recognise and abide by he legal requiremens associaed wih hese righs. Take down policy The Universiy of Edinburgh has made every reasonable effor o ensure ha Edinburgh Research Explorer conen complies wih UK legislaion. If you believe ha he public display of his file breaches copyrigh please conac openaccess@ed.ac.uk providing deails, and we will remove access o he work immediaely and invesigae your claim. Download dae: 01. Dec. 2018

2 Job Reallocaion, Unemploymen and Hours in a New Keynesian Model Richard W. P. Hol Economics, Universiy of Edinburgh This version Sepember 2007, Firs version Augus Absrac This paper focusses on he reallocaion of labour resources in a New Keynesian environmen wih labour marke search and endogenous separaions. We show ha inroducion of variaion in hours per worker alers he incenives for ineremporal subsiuion in a way ha generaes a more seeply downward sloping Beveridge curve and reduces he endency o synchronise gross job flows. This also enables he New Keynesian model o capure he ineracion of hours and employmen a business cycle frequencies. We show ha he impac of labour supply elasiciy on he slope of he Beveridge curve and he correlaion of gross job flows is deermined primarily by variaion in he response o moneary shocks. When hours variaion is suppressed he comovemen of job creaion wih job desrucion and of unemploymen wih vacancies are srongly posiive in response o moneary shocks. Whereas wih variaion in hours boh measures of reallocaion ake on he correc negaive sign. I am graeful o Jozsef Sakovics, Suar Sayer, Jonahan Thomas and seminar paricipans a he Cenre For Dynamic Macroeconomics Workshop 2005, Royal Economics Sociey Conference 2006, ESRC Money, Macroeconomics and Finance Research Group Conference 2006, Universiy of Birmingham and Universiy of Hull for advice and encouragemen. Earlier versions of his paper have circulaed under he iles "Unemploymen, Job Flows and Hours in a New Keynesian Model" and "Unemploymen, Labour Marke Dynamics and he New NeoClassical Synhesis". I am responsible for any errors. Correspondence should be addressed o Richard Hol, Economics, Universiy of Edinburgh, William Roberson Building, 50 George Square, Edinburgh, EH8 9JY, Scoland; Richard.Hol@ed.ac.uk.

3 1 Inroducion. Recen research exends he New Keynesian paradigm o incorporae labour marke search, moivaed by he explici accoun i provides of unemploymen. 1 Following Shimer (2005) and Hall (2005), who argue ha unemploymen dynamics are driven by flucuaions in ouflows no inflows, he bulk of he lieraure holds he inflow (separaion) rae consan. 2 Ye, subsequen empirical evidence indicaes ha inflows variaion also plays an imporan role in unemploymen dynamics. 3 This makes a model wih endogenous separaions a naural saring poin. The Morensen Pissarides (1994) feaures endogenous separaion of he leas producive maches o provide a Schumpeerian explanaion of microeconomic resrucuring and he reallocaion of resources o more producive aciviies ha underlies economic growh and flucuaions, see Davis e al. (1998). In he lieraure on resrucuring he slope of he Beveridge curve (correlaion of unemploymen wih vacancies) and he synchronisaion of gross job flows (correlaion of job creaion wih job desrucion) are criical measures of he reallocaion process. Boh are negaive in US daa. Suppressing inflow variaion, i.e. following Shimer (2005), avoids he synchronisaion of job flows and generaes a downward sloping Beveridge curve. However, i does so by assuming away reallocaion decisions. The Schumpeerian perspecive on business cycles suggess ha recessions are he bes ime o underake microeconomic resrucuring, since he opporuniy coss of doing so are low. Morensen and Pissarides (1993) show ha a emporary shock o profiabiliy produces srong posiive correlaion of gross job flows and an upward sloping Beveridge curve, in he conex of a (consrained) efficien equilibrium labour marke search model wih endogenous separaions (one which saisfies he Hosios condiion). 4 Krause and Lubik (2007) find a similar effec in a New Keynesian framework when, as in he Morensen-Pissarides model, labour inpu varies on 1 Auhors have examined he role played by labour marke search, and wage rigidiies in deermining dynamic behaviour of unemploymen, oupu and inflaion. In so doing hey address he amplificaion and persisence puzzles highlighed by Shimer (2005), Chari, Kehoe and McGraan (2000), Cogley and Nason (1995). 2 These include Moyen and Sahuc (2005), Chrisoffel and Linzer (2005), Chrisoffel e al. (2006), Jung and Kueser (2006), Trigari (2006), Faia (2007), Kueser (2007) and Thomas (2007). 3 Elsby e al. (2007) show ha Shimer s resul ha unemploymen inflows are invarian over he cycle is oversaed even using his own daa and mehodology. Davis e al. (2006) and Fujia and Ramey (2006) presen evidence from job flows and oher daa sources o suppor he view ha inflows are an imporan componen of unemploymen dynamics a business cycle frequencies, accouning for up o a hird of he variaion in unemploymen. 4 Suppressing inflow variaion avoids he synchronisaion of job flows and generaes a downward sloping Beveridge curve. However, i does so by assuming away reallocaion decisions. 1

4 he exensive margin(s) only. I is possible o reconcile models of reallocaion and resrucuring wih he facs while adhering o he Hosios condiion by incorporaing addiional feaures ino firms decision problems so as o aler he incenives for ineremporal subsiuion. For example, Morensen (1994) achieves his by inroducing on-he-job search and disinguishing beween job and worker flows, while Den Haan e al. (2000) do so by including capial accumulaion. This paper examines he reallocaion of labour resources in a New Keynesian environmen wih labour marke search and endogenous separaions. We show ha inroducion of variaion in hours per worker alers he incenives for ineremporal subsiuion in a way ha generaes a downward sloping Beveridge curve and reduces he endency o synchronise gross job flows. This also enables he New Keynesian model o capure he posiive comovemen of hours and employmen a business cycle frequencies. 5 The New Keynesian framework imposes he discipline of general equilibrium on our analysis. I inroduces fricions in price seing which permis meaningful discussion of he impac of boh produciviy and moneary shocks. 6 We show ha he effec of labour supply elasiciy on he slope of he Beveridge curve and he correlaion of gross job flowsisdeermined primarily by changes in he response o moneary shocks. Specifically, under inelasic hours he comovemen of job creaion wih job desrucion and of unemploymen wih vacancies are srongly posiive in response o moneary shocks, whereas under elasic hours boh measures of reallocaion ake on he correc negaive sign. The volailiy of unemploymen is relaively high wih or wihou hours variaion. The inroducion of hours variaion raises he variaion of unemploymen, bu reduces ha of vacancies. Variaion in hours reduces he variabiliy of wages, bu his remains oo high. Moreover, vacancies fail o exhibi sufficien persisence and are oo srongly correlaed wih job creaion. The mechanism by which hours per worker affecs reallocaion is relaively sraighforward. Variaion of hours per worker allows exising maches o adjus labour inpu o shocks by varying 5 Flucuaion in hours per worker accouns for a subsanial proporion of he variaion in labour inpu a business cycle frequencies, Cho and Cooley (1994). Despie is role in labour inpu variaion hours per worker is frequenly omied from models wih labour marke search, presumably on grounds of parsimony. 6 Shimer (2005) noes ha an upward sloping Beveridge curve arises when exogenous shocks o he (aggregae) job desrucion rae are permied - his may be inerpreed as an aggregae reallocaive shock. I is possible ha a suiable choice of correlaion of reallocaive shocks and produciviy shocks could generae a Beveridge curve wih he appropriae slope. However, Davis and Haliwanger (1999) are unable o find a clear and imporan role for such aggregae reallocaive shocks. For his reason i is more ineresing o examine wheher a negaively sloped Beveridge curve can arise when job desrucion varies endogenously in response o aggregae produciviy and money supply growh disurbances, which are sandard in a New Keynesian seing. 2

5 hours as well as hrough separaions - firms equae marginal coss of facor adjusmen across each margin. This exra flexibiliy alers rens, and as a consequence he incenives boh o dissolve exising maches and o creae new ones. Labour adjusmen on he exensive margin is aenuaed and he endency o synchronise creaion and desrucion aciviy is also reduced. 7 One poenial problem is ha variaion on he inensive margin may subsiue for variaion on he exensive margin and aenuae flucuaions in unemploymen - worsening he volailiy puzzle highlighed by Shimer (2005). In pracice, holding oher parameers consan, he effec on unemploymen is relaively small. In addiion, our calibraion sraegy adjuss parameers o mainain he sandard deviaion of job desrucion consan across experimens, so ha unemploymen variaion remains roughly consan regardless of he use of he inensive margin. However, he volailiy of job creaion and vacancies is reduced by realisic hours variaion. The sensiiviy of reallocaive measures o variaion in hours is enhanced by he presence of moneary shocks. I is cosly o dissolve exising maches or creae new maches in response o ransiory shocks, ye hours variaion wihin exising maches does no enail long-run consideraions. Since moneary shocks are less persisen han produciviy shocks, he inroducion of hours variaion has is greaes effec on he response o moneary shocks. Noneheless, even for produciviy shocks, realisic hours variaion alers he incenives for ineremporal subsiuion sufficienly o produce a negaive job flows correlaion. A small collecion of papers incorporae endogenous separaions ino he New Keynesian reamen of unemploymen, Krause and Lubik (2007), Trigari (2005), Braun (2006), Walsh (2005), Andres e al. (2006). These auhors all provide he same raionale and broadly address he same issues as considered in he lieraure which assumes a consan separaion rae (see foonoes (1) and (2)). For he mos par, hey do no address quesions on he iming of reallocaion ha we consider and ha a model wih endogenous job desrucion is designed o answer. In paricular, he join behaviour of he Beveridge curve and he correlaion of gross job flowsisconsidered only by Krause and Lubik (2007), who find boh measures of reallocaion o be posiive, in an 7 Wih he inensive margin suppressed, a rise in unemploymen occurs hrough a rise in job creaion and an even bigger rise in job desrucion. This reflecs he relaively low cos of alering he rae of job desrucion (relaive o ha of job creaion). I gives rise o a (counerfacual) posiive correlaion of job creaion and job desrucion. Since job creaion ends also o be high when he number of vacancies open is high, he Beveridge curve is flaer and may become posiive in he absence of hours variaion. 3

6 environmen wih endogenous job desrucion and labour inpu variaion on he exensive margin only. 8 The combinaion of endogenous hours along wih endogenous job desrucion ha we discuss below was firs considered by Trigari (2005). Our principal conribuion is o demonsrae he effec of realisic hours variaion on measures of reallocaive aciviy. We also illusrae he role of differen shocks in deermining he effec of hours variaion on reallocaion. Trigari does no direcly consider he correlaion of gross job flows, nor does she aemp o mach he behaviour of unemploymen or vacancies. Noneheless her impulse response analysis is likely o be consisen wih he effecs of hours variaion on reallocaion ha we ouline here. 9 We use a simplified version of Trigari s model and use i o conras he behaviour of measures of reallocaion obained under differen assumpions abou labour supply elasiciy. Our focus on conrasing he impac of paricular (implici) assumpions on hours variaion in a relaively simple New Keynesian model leads us o calibrae raher han esimaing he elasiciy of hours direcly. I also means ha we ake a sand on he shocks ha affec he economy and mach uncondiional momens. Alhough no imune from criicism his sraegy facilaes comparison wih Krause and Lubik (2007) and Walsh (2005); Trigari (2005) considers he (condiional) response o moneary shocks alone. 10 Of he oher hree papers, Walsh (2005) was he firs o inegrae New Keynesian model wih labour marke search and endogenous job desrucion. Following from Den Haan e al. he does no allow variaion in hours per worker and does no consider measures of reallocaion a all. Andres e al. (2006) exend Walsh s model and use i o examine he variabiliy of unemploymen, vacancies and labour marke ighness. 11 Alhough hey do no compue he slope of he Beveridge curve, i can be inferred from he resuls hey presen. For he version of heir model (wihou capial and disorionary axaion) which mos closely approximaes ours, i is -0.08: negaive bu much smaller han in US daa, jus as we find when we suppress variaion in hours. They do no display 8 They find ha he inroducion of (complee) wage rigidiy can produce a downward-sloping Beveridge curve, bu hey are unable o avoid posiively correlaed gross job flows. Indeed, hey are unable o mach he Beveridge curve if wage rigidiy is se o mach observed wage variabiliy. 9 Trigari (2005) esimaes key parameers so as o mach he impulse responses of job flows, employmen, hours inflaion and oupu o an ineres rae shock. 10In principle one migh hen esimae he model using a Bayesian approach, as in he reamen of a model wih exogenous job desrucion provided by Jung and Kueser (2007). However, our calibraion suggess ha he presen model is oo simple o be aken o he daa in his way, so we leave his for fuure work. 11Their benchmark model also allows for habi persisence, capial accumulaion and disorionary axes. 4

7 any daa for he volailiy of gross job flows or relaed measures of reallocaion. 12 Our analysis suggess ha because hey suppress variaion in hours per worker he correlaion of gross job flows would be posiive bu ha his can be correced by allowing hours variaion. Andres e al. (2006) find a role for price rigidiy in deermining he variabiliy of unemploymen vacancies and labour marke ighness. Our analysis exends heirs o an environmen in which hours per worker can boh vary and allows for moneary as well as produciviy shocks. Braun (2006) applies Trigari s mehodology o worker flows raher han job flows; she considers a New Keynesian framework wih capial accumulaion. We consider a New Keynesian model wihou capial accumulaion. While capial can help discipline model calibraion, much work in he New Keynesian radiion boh for srucural modelling and policy analysis suppresses his margin, see Gali (2003), Woodford (2003), as do models wih unemploymen, Blanchard and Gali (2006), Chrisoffel and Linzer (2005), Faia (2007a), Trigari (2006). The main jusificaion (ofen implici) for his simplificaion appears o be he limied role played by capial accumulaion. 13 The omission of capial accumulaion and oher ineremporal feaures, such as habi persisence, serves o highligh he role of he inensive margin. 14 The model is oulined in Secion 2. Calibraion and soluion mehod are discussed in Secion 3. Secion 4 presens and discusses he resuls and assess he conribuion of various feaures in accouning for US business cycles facs. Secion 5 concludes. An appendix conains deails of he daa used, and he calibraion sraegy. 2 Model The economy conains four ypes of agen: inermediae good producers, final goods producers, households and a governmen. Producion of he inermediae good requires labour. Labour can 12In addiion, direc comparison beween our resuls and hose of Andres e al. (2006) is complicaed by he fac ha hey appear only o calibrae idiosyncraic shocks and he properies of he produciviy shock in he benchmark case, and proceed o allow he variabiliy of job desrucion and oupu o vary across experimens. 13In he lieraure on unemploymen dynamics, Hagedorn and Manowskii (2005) poin ou ha mach-level profis are an imporan deerminan of he ampliude of flucuaions in unemploymen and vacancies. Krussell e al. (2005), surveying developmens in he lieraure, commen ha he calibraion of his criical profi share parameer could be improved if capial accumulaion were incorporaed as a disciplining device, bu Jung (2005) demonsraes ha he inroducion of capial accumulaion does no overurn he insigh of Hagedorn and Manowskii (2005). 14An earlier version of his paper, Hol (2006), adoped a framework wih capial accumulaion. The resuls on he role of hours variaion are similar o hose displayed below. Hence a New Keynesian environmen capial accumulaion does no aler incenives sufficienly o generae a negaive job flows correlaion, conrary o he real business cycle based analysis of Den Haan e al (2000). 5

8 be varied on boh exensive and inensive margins. Hours are deermined hrough Nash bargaining raher han unilaerally by individual consumers. The srengh of variaion on he inensive margin is deermined by he elasiciy of labour supply (preferences over leisure). The model srucure is based on ha of Trigari (2005). We simplify her model in several ways in order o highlig he role of hours variaion and faciliae comparison wih he lieraure. We omi habi persisence o simplify he dynamic srucure of he model and hereby highligh he role of hours. We arge he BLS esmae of average unemploymen (6%) raher han he high (25%) unemploymen she uses, which acs o sabilise he unemploymen pool and hence vacancies in response o shocks. In he ligh of evidence of insabiliy of he Taylor rule over he sample period we follow Krause and Lubik (2007) and adop a money supply growh rule. We also adop idiosyncraic producion coss raher han idiosyncraic preference shocks, which is slighly more inuiive in he ligh of our ineres in reallocaion based on profiabiliy. Finally we specify preferences over leisure, following Andolfao (1995) raher han hours worked as is common in he New Keynesian lieraure. Below we discuss in urn he decision problem of households, he specificaion of goods and labour markes and he equilibrium characerisaion of he economy. 2.1 Households Assume ha he economy conains a coninuum of idenical households of uni mass. Each household is a family wih a coninuum of members. In equilibrium some members are employed while ohers are unemployed. Each member i, of household h has he following period uiliy funcion defined over consumpion, C, money balances M P and hours,h, C h i, 1 φ 1 φ + Υ M P 1 ξ Ã M h i, P! 1 ξ + 1 I U 1 H h 1 ϕ i, ΥH + I U 1 ϕ (1 e) 1 ϕ 1 ϕ Here Υ M P, he relaive weigh on money balances in he uiliy funcion, Υ H, he relaive weigh on leisure in he uiliy funcion, φ, he inverse of he elasiciy of ineremporal subsiuion, ϕ, he elasiciy of subsiuion of hours per worker, e ime spen underaking search and ξ are all posiive consans. I U is an indicaor funcion aking he value 1 if he individual is unemployed and zero oherwise. To avoid he disribuional issues ha arise hrough differing employmen hisories, we assume ha family members perfecly insure each oher agains (cross-secion) variaion in he 6

9 marginal uiliy of consumpion. Separabiliy of he individual s uiliy in consumpion, money balances and leisure ensures ha family members have idenical consumpion and money holdings. Under hese simplifying assumpions, household member s decisions can be analysed in erms of a represenaive household. 15 The represenaive household chooses consumpion and money balances o maximise expeced uiliy of is members over heir lifeimes: E 0 " X =0 " β (C ) 1 φ 1 φ + Υ M P 1 ξ µ M P 1 ξ Z ## 1 U (1 H i, ) 1 ϕ (1 e) 1 ϕ + Υ H di + U. 1 ϕ 1 ϕ Here β, is he discoun facor and U represens he fracion of he household membership which is unemployed (we suppress he household superscrip for convenience). Hours of work are deermined hrough bargaining beween he individual worker and he firm raher han being unilaerally deermined by he household. 0 Households own all reail and wholesale firms. They can save by holding 1-period ineres bearing bonds, or non-ineres bearing money balances. The represenaive household maximises expeced lifeime uiliy subjec o he following sequence of consrains P C + R n B + M = I + B + M 1 + P T, 0. (1) Here B represens holdings of a nominal 1 period bond, and R n represens he gross nominal ineres rae on his bond. M represens holdings of nominal money balances a he end of period, P T represens lump-sum nominal ransfers. C is a composie index of final goods consumpion. I is he household s nominal income (household labour income, plus he household s share of firms profis ne of expendiures on vacancies). 16 The soluion o he represenaive household s problem is characerised by firs-order condiions for bond holdings, B, consumpion C, money balances M. Subsiuing he firs order condiion for he shadow value of wealh using he marginal uiliy of consumpion in he remaining condiions we have: 1=βR n E " P µ C P +1 C +1 φ #. (2) 15This sor of assumpion is a common simplificaion in he lieraure on business cycle flucuaions under labour marke search designed o faciliae racabiliy, see e.g. Andolfao (1996), Merz (1995). 16Given he represenaive family assumpion, all families hold he same share of firms profis, so in equilibrium his share is one a all daes. 7

10 Υ M P µ M P ξ C φ = βe C φ P P (3) 2.2 Goods and Labour Markes Labour Marke Flows The mach specific producion, bargaining and separaion decisions described below depend on he probabiliy ha unemployed workers find jobs and he probabiliy ha vacancies are filled. Here we define hese probabiliies and he associaed labour marke flows. Define he number of maches a he beginning of period as N [0, 1]. Following he lieraure, we allow some job desrucion in he form of quis which are aken as exogenous and independen of he mach-specific profiabiliy. We capure his by allowing a fracion, λ x,of maches o separae prior o he realisaion of period shocks. Subsequenly, shocks are realised, including an idiosyncraic cos shock, X, drawn from disribuion F (X) and a mach may choose o break up if he value of he mach surplus is negaive. Le he X denoe he hreshold value of he cos shock, so ha higher realisaions of idiosyncraic coss cause maches o separae. Endogenous separaion hus occurs wih probabiliy λ n R X X =1 df (X) =1 F X, where df ( ) is he probabiliy densiy funcion over X. The overall separaion rae in period is λ = λ x +(1 λ x ) 1 F X. (4) We model maching fricions using an aggregae maching funcion. Maching occurs a he same ime as producion. Assume ha here is a coninuum of poenial firms, wih infinie mass, and a coninuum of workers of uni mass. Unmached firms choose wheher or no o pos a vacancy and incur a cos κ per period. Free enry of unmached firms deermines he size of he vacancy pool. Define he mass of firms posing vacancies in period as V. Le he mass of searchers, unmached workers, be U. All unmached workers may ener he maching marke in period - even if heir mach dissolved a he sar of period, so U =1 (1 λ ) N. (5) New maches in dae beginproducionindae +1, while unmached workers remain in he 8

11 worker maching pool. The flow of successful maches creaed in period is given by he consan reurns maching funcion M = MU γ V 1 γ. (6) where γ (0, 1) and M > 0. The number of employed workers a he sar of period +1is N +1 =(1 λ ) N + M. (7) Denoe he probabiliy ha a vacancy is filled in dae as p V = M V, (8) and he probabiliy ha an unemployed worker eners employmen in period as p U = M U. (9) The gross job desrucion rae is he number employmen relaionships ha separae less exogenous separaions ha remach wihin period as a fracion of curren employmen JD = λ N p V λ x N N = λ p V λ x. (10) Gross job creaion is he flow of new maches (as a fracion of exising employmen) less maches due o firms filling vacancies ha resuled from exogenous separaions JC = M p V λ x N N = M N p V λ x. (11) The Inermediae secor Producion Producion of inermediae goods akes place in mached firm-worker pairs - or, for noaional ease, maches. Each mach consiss of one worker and one firm, who ogeher engage in producion unil he employmen relaionship is erminaed. By assumpion, boh firms and workers are resriced o a single employmen relaionship a any given ime. Maches are subjec o aggregae produciviy and idiosyncraic cos shocks, Z and X respecively. 17 Following Den Haan e al. (2000) assume ha idiosyncraic cos disurbances are serially uncorrelaed. Dae 17Cos shocks are a naural way o model heerogenous produciviy underlying he process of creaive desrucion a he hear of he model. Trigari (2005) adops an formally equivalen bu arguably less inuiive approach in which he idiosyncraic disurbances affec he uiliy derived from leisure. 9

12 producion occurs afer realisaion of he dae shocks. A dae an ongoing mach (one facing idiosyncraic shock X < X )produces Y I (X )=AZ H (X )+F X unis of inermediae good. 18 The parameers A and F are posiive consans. Maches are price akers and sell heir homogeneous inermediae oupu a (nominal) price P I. The formal separaion of he job-desrucion and price-seing decision problems is mainained for racabiliy, bu is consisen wih he view ha prices are no se a he level of an individual mach. Curren profis of an ongoing mach are Π I (X ) AZ H (X ) µ + F X µ W (X ) H (X ) P, (12) where µ = P P I is he markup of he index of final goods prices over he price of he inermediae good (he reciprocal of marginal cos) and W (X ) is he mach specific (nominal) wage. Value Funcions Nex we describe he value funcions for firms and workers decision problems. In equaion (13) V U, he dae value of unemploymen, expressed in final goods, comprises he consumpion value of uiliy from search, he discouned presen value of ongoing unemploymen nex period, V U +1, andhedifference beween he value of employmen, V W (X), and ha of unemploymen in he even ha he worker maches his period (wih probabiliy p U )andhe mach survives o produce nex period (wih probabiliy (1 λ x ) F X+1 ): " " V U (1 e)1 ϕ C φ Z ## X +1 = 1 ϕ Cφ V + βe V+1 U + p U (1 λ x ) W (X) V+1 U df (X). (13) C φ +1 Maching and producion occur simulaneously, so ha a mach which is formed in period canno produce unil period +1, afer aggregae and idiosyncraic shocks have been realised. As a resul a new mach survives wih probabiliy (1 λ x ) F X+1. Le V W (X ) denoe he dae value, expressed in erms of consumpion goods, o a worker of employmen in an ongoing mach wih idiosyncraic cos shock X. V W (X ) = W (X ) H (X ) (1 H (X )) 1 ϕ + Υ H P 1 ϕ " " C φ +βe V+1 U +(1 λ x ) C φ +1 18An addiive idiosyncraic shock avoids wide variaion of hours across maches, Cooley and Quadrini (1999). C φ Z ## X +1 V W (X) V+1 U df (X). (14) 10

13 The worker supplies H (X ) hours of labour o he firm for real hourly wage W (X) P. Boh wage and hours are oucomes of a bargaining process - described below. Hours worked generaes income, bu hours spen in he workplace reduce uiliy. These concerns are capured in he firs wo erms in (14). The remainder of he dae value o an employed worker from he ongoing mach is C he discouned presen value, βe φ, of unemploymen plus he difference beween he C φ +1 V U +1 value of employmen, V W (X), and ha of unemploymen in he even ha he mach coninues o produce nex period (where we sum across values of X which do no lead o erminaion prior o dae +1producion). The dae value, V J (X ),ofafirm ha forms par of an ongoing mach wih curren mach specific shockx, consiss of curren profis plus he appropriaely discouned value o he firm of he sum of a dae +1vacancy, V V +1, in he even ha he mach erminaes prior o producion in period +1(where erminaion occurs wih probabiliy λ +1 = λ x +(1 λ x ) 1 F X+1 ) and he expeced value in he even ha he mach coninues o produce in +1; " " Z ## X+1 V J (X )=Π I (X )+βe λ +1 V+1 V +(1 λ x ) V J (X) df (X). C φ C φ +1 We assume vacancy posing coss κ per period. Then he value in dae of a firm wih an unfilled vacancy, V V,reflecs he cos of posing ha vacancy plus he value of firm, V+1, V inheeven ha he firm fails o fill he vacancy by dae +1or else he even ha he vacancy is filled bu he mach is erminaed prior o producion in period +1(his occurs for a sufficienly adverse realisaion of he idiosyncraic shock), plus he value V J (X) in he even ha he vacancy is filled and he period +1idiosyncraic cos shock akes a value X, ha does no lead o erminaion V V = κ + βe " C φ C φ +1 " 1 p V (1 λ x ) F Z ## X +1 X+1 V V +1 + p V (1 λ x ) V J (X) df (X). The free enry condiion on vacancies drives he value of a vacancy o zero, V V =0,, sohe Bellman equaions for V J (X ),andv V become V J (X )=Π I (X )+(1 λ x ) βe " κ = p V (1 λ x ) βe " C φ C φ +1 C φ C φ +1 Z X +1 V J (X) df (X)# (15) Z X+1 V J (X) df (X)#. (16) 11

14 Moreover, using (15), we can re-wrie (16) as a Bellman equaion for p V : κ p V = β (1 λ x ) E " C φ C φ +1 Z X +1 Π I (X)+ κ p V +1 # df (X). (17) Bargaining: Hours and Wages We assume ha for each mach engaged in producion, he firm and worker adop Nash bargaining over hours worked and he hourly wage. Given he full consumpion insurance agains unemploymen risk provided by our family srucure some care is required o ensure ha his problem is well defined. We discuss his issue firs before urning o he oucome of he bargaining process. Assume ha workers evaluae he consequences of heir acions on he basis of he conribuions hese make o heir family s lifeime uiliy. Then he worker s surplus from employmen, V W (X ) V U is he same as he value (in erms of consumpion goods) of he change in he family s uiliy from having one more addiional member in employmen, Ω (1 U ) Cφ. Tha is V W (X ) V U = Ω (1 U ) Cφ,whereΩ is he represenaive family s value funcion. To check his noe ha we can wrie Ω recursively as Ω = (C ) 1 φ 1 φ + Υ M P 1 ξ µ M P 1 ξ Z 1 U (1 H (X i, )) 1 ϕ (1 e) 1 ϕ +Υ H di+u 1 ϕ 1 ϕ +βe Ω+1 X X +1 0 subjec o he dae consrain in (1), and he evoluion equaion for he number of individuals engaged in producion 1 U +1 =[1 λ +1 ][1 U ]+p U [1 λ +1 ] U. Compuing he derivaive wih respec o (1 U ) we find Ω (1 U ) Cφ = W (X )H(X ) P + 1 p U β (1 λ x ) E Υ H (1 H(X )) 1 ϕ 1 ϕ C φ C φ +1 C φ + (1 e)1 ϕ 1 ϕ Cφ R X +1 V W (X) V U +1 df (X). Using equaions (13) and (14) we find ha his equals V W (X ) V U, as required. Thus he worker s hrea poin in he bargaining process is clearly defined in erms of household welfare. Given ha he bargaining problem for he worker is well defined, he division of he mach surplus S (X )=V W (X ) V U + V J (X ) V V = V W (X ) V U + V J (X ), (18) 12

15 is deermined on a period by period basis as: max W (X ),H(X ) V W (X ) V U η V J (X ) V+1 V 1 η. The firs order condiions for hours and wages respecively are " # ηv J W (X ) (1 H (X )) ϕ (X ) Υ H P C φ = Opimal hours worked are hus (1 η) V W (X ) V U i h AZ µ W (X ) P, (19) ηv J (X )=(1 η) V W (X ) V U. (20) Υ H (1 H (X )) ϕ C φ = Υ H (1 H ) ϕ C φ = AZ µ X X. (21) Equaion (21) says ha, under Nash bargaining, he marginal rae of subsiuion beween consumpion and hours worked is equal o he marginal produc of labour. Hours per worker in ongoing maches are decreasing in he markup, bu increasing in aggregae produciviy. Variaion in hours perworkerisdecreasinginϕ, sochoiceofϕcanbeusedoshudownheinensivemargininour experimens. Hours per worker are independen of he mach specific shock:h (X )=H. Recall ha he worker s surplus from employmen is V W (X ) V U = W (X )H P + 1 p U β (1 λ x ) E + Υ H (1 H ) 1 ϕ 1 ϕ C φ C φ +1 C φ (1 e)1 ϕ 1 ϕ Cφ R X +1 V W (X) V U +1 df (X). Using (20) and (16) i follows ha V W (X ) V U = W (X ) H P Lasly, combining (15) and (16) (1 H ) 1 ϕ + Υ H C φ (1 e)1 ϕ 1 ϕ 1 ϕ Cφ + η 1 p U κ 1 η p V. V J (X )= AZ H + F X W (X ) H + κ µ µ P p V. So he opimal wage for a mach wih idiosyncraic cos realisaion X becomes W (X ) H AZ H = η + F X " + κ pu (1 e) 1 ϕ (1 H ) 1 ϕ P µ µ p V +(1 η) 1 ϕ Cφ Υ H 1 ϕ Define aggregae labour income as W H R X P = H W (X ) P df (X). Then AZ η H µ + 1 X XdF(X) µ F F( X + κ pu ) p V W H P = +(1 η) h (1 e) 1 ϕ 1 ϕ Cφ (1 H Υ ) 1 ϕ H 1 ϕ C φ i C φ #. F X. (22) 13

16 The firs erm wihin he firs square brackes on he righ hand side of equaion (22) represens he workers share of he marke value of producion, he second erm reflecs he marke value of idiosyncraic coss (relaive o seady sae), and he hird erm reflecs he impac of labour marke ighness. The remaining erm reflecs he worker s reservaion wage. Separaion For values of he idiosyncraic cos shock above a cerain hreshold level, X,separaion occurs. The condiion S X =0, pins down his hreshold value of he mach specific shock. Combining (18) and (20), V J (X )=(1 η) S (X ).So X is deermined by he condiion V J X =0: AZ H + F X W X H + κ µ µ P p V =0. This equaion indicaes ha a job is desroyed when coss are sufficienly high ha he value of producion ne of idiosyncraic cos shock and wage equals he (expeced) cos of posing a vacancy. Subsiuing for he mach specific wage, he hreshold value X is deermined by (1 η) " AZ H + F X " (1 e) 1 ϕ (1 H ) 1 ϕ µ µ 1 ϕ Cφ Υ H 1 ϕ Final Goods Secor C φ ## ηκ pu p V + κ p V =0. (23) Assume ha here is a coninuum of final goods producers, wih uni mass. Final good firm z acquires he wholesale good a price P I and coslessly ransforms i ino he divisible final good z ³ R 1 which is hen sold direcly o households a price p (z). Define P = 0 p (z) 1 ε 1 1 ε dz as he uiliy based price index associaed wih he consumpion composies. The marke for final goods is characerised by monopolisic compeiion - ε represens he elasiciy of subsiuion across varieies of final good. Aggregae demand for he final good z in period is y (z) =c (z), where c (z) represens consumpion demand for final good z oupu. The opimal choice of consumpion ³ p (z) ε expendiures on final good z is hen c (z) = P C, where aggregae consumpion, C = ³ R 1 0 c (z) ε 1 ε ³ 1 ε R ε 1 dz and aggregae final good oupu Y = 0 y (z) ε 1 ε 1 ε ε dz are composie indices of final goods. Final goods prices exhibi nominal rigidiies which follow a hybrid Calvo-syle adjusmen scheme. Wih probabiliy (1 ω) a final good producer can se he price of is oupu in period 14

17 . This probabiliy be independen of when he firm las adjused price. Then he average price for final goods producers who do no adjus heir price is simply P 1. Define he average price se by firms who do adjus price as p. Since pure forward-looking price adjusmen schemes seem no o accoun adequaely for observed inflaion dynamics, we employ a hybrid scheme (following Gali and Gerler (1999)). Assume ha a fracion (1 τ) of he final goods producers are forwardlooking and se prices opimally (o maximise expeced discouned profis given he probabiliy of fuure adjusmen). Define he price se by forward-looking producer z a dae as p (z). Since all forward-looking firms seing price a dae face he same expeced fuure demand and cos condiions hey choose he same price, so p (z) =p,where p = P ³ ε E s=0 ωs β s Cφ p 1 ε C φ P Y+s +s P+s I +s 1 ε E P s=0 ωs β s Cφ C φ +s ³ p P +s 1 ε Y+s (24) The remaining fracion, τ, offirms which rese price in period are assumed o se a price equal o he average of he prices rese in he previous period, correced for inflaion, π 1 : p b = p 1 π 1. (25) The average price se in period is p = h (1 τ)(p ) 1 ε + τ i 1 p b 1 ε 1 ε 1, and he aggregae reail price index evolves according o P 1 ε =(1 ω)( p ) 1 ε + ωp 1 ε 1. (26) 2.3 Moneary and Fiscal Policy and Exogenous Driving Processes We se governmen spending o zero and assume ha he governmen mainains a balanced budge by rebaing seigniorage revenues o households in he form of lump-sum ransfers. The governmen budge consrain is hus P T = M M 1,where M is he aggregae money sock. Moneary policy is specifed by M = M 1 e υ (27) where υ evolve according o he AR(1) process υ = ρ υ υ 1 + ε υ,. (28) 15

18 The logarihm of aggregae produciviy also follows an AR(1) process: ln Z = ρ Z ln Z 1 + ε Z, (29) where ε ν, and ε Z, are independen mean zero processes. 2.4 Equilibrium Under he represenaive consumer framework, household choices (superscrip h) are common across householdsand in equilibrium M h = M ec, in (1) o (3). Aggregae income, I comprises labour income, plus profis of final goods producers, plus profis of inermediae goods producers ne of vacancy posing coss I =(1 λ x ) N W H + P Π F + P Π I. Here, nominal final goods profis are P Π F = R p (z) y (z) dz P I R y (z) dz = P Y P I Y I,and Y I Z X =(1 λ x ) N [AZ H + F X] df (X) κµ V (30) 0 denoes aggregae inermediae oupu ne of vacancy posing coss. 19 Nominal inermediae good producers profi can be wrien as he sum of oupu ne of vacancy coss, less aggregae wage paymens: P Π I = P I Y I (1 λ x ) N W H. Using hese insighs and cancelling erms we find I = P Y. In equilibrium, when combined wih he governmen budge ideniy, he household budge consrain reduces o he aggregae (final) goods marke equilibrium condiion Y = C (31) Thus he sysem of equaions governing equilibrium in he economy consiss of he numbered equaions (1) - (12), (17) and (21) - (31). 3 Calibraion & Model Soluion Mehod We log-linearise he model abou is (zero-inflaion, zero growh) seady sae and use dynamic simulaions o ease ou he dynamic srucure of he economy. Model soluion requires choice of several parameers governing seady sae values of labour and goods marke variables; nominal rigidiy, and household preferences. We also specify he processes governing idiosyncraic coss, 19Noe Y I = 1 0 y (z) dz. Using he demand funcion for final good z: y (z) = p(z) ε P Y, we have Y I = 1 p (z) ε ε 0 Y dz = P Y,where P P = 1 P 0 p (z) ε dz, is an auxilliary price index. 16

19 aggregae produciviy and money supply growh. The parameer values are summarised in Table 1, Appendix A conains discussion of he raionale for hese choices. 4 Resuls Table (1) here. In his secion we discuss evidence on he impac of variaion in hours per worker on he srengh and iming of reallocaive aciviy and on oher sandard macroeconomic aggregaes. We conras he behaviour of a model varian in which hours variaion is suppressed (which represens he sandard approach in models of labour marke search wih endogenous job desrucion) wih an equivalen se up in which he elasiciy of labour supply, governed by he parameer ψ, is seleced o mach he variaion of hours in he daa. To provide a fair basis of comparison across varians of he model, we hold consan across experimens boh he sandard deviaion of simulaed oupu and he sandard deviaion of simulaed job desrucion relaive o simulaed oupu. To do his, we adjus he sandard deviaion of produciviy shocks o allow he sandard deviaion of (Hodrick-Presco filered) model oupu o mach he variabiliy of oupu in US daa (Hodrick-Presco filered US NIPA GDP), which is 1.69%. We hold he auocorrelaion of produciviy shocks, boh sandard deviaion and serial correlaion of money supply growh shocks consan across hese experimens. The sandard deviaion of idiosyncraic cos shocks is varied in order o mach he volailiy of job desrucion relaive o ha of GDP. These dimensions canno be used for falsificaion. Insead, we examine he abiliy of he model o capure wo key aspecs of he srengh and iming of reallocaive behaviour: i) he Beveridge curve and ii) he correlaion of gross job flows. We also consider he operaion of he labour marke as capured by he correlaion of hours wih employmen, he behaviour of job creaion and vacancies and sandard macroeconomic aggregaes such as unemploymen and inflaion. We explain he mechanism by which variaion in hours per worker improves he reamen of labour reallocaion, consider he role played by differen shocks and examine he robusness of he resuls o plausible variaion in labour supply elasiciy. 17

20 4.1 Reallocaion and Hours Variaion Table (2) illusraes he role of labour supply elasiciy (variaion in hours) in deermining he naure of reallocaion. Column (1) of Table (2) displays properies of US Daa. The oher columns of Table (2) correspond o a paricular model varian. 20 For column (X >1), he enry in he row labelled oupu indicaes he variabiliy of oupu in column (X) relaive o he variabiliy of oupu in US daa. The oher enries in column (X) (excep he final 8) correspond o he variabiliy relaive o ha of oupu generaed by model (X). The final wo enries in each column are serial correlaion saisics for oupu and inflaion, while he penulimae six enries are simple correlaion saisics capuring aspecs of labour marke aciviy. Table (2) here. Labour supply elasiciy, H,is 1 ψ 1 H H = 2 ψ, since in seady sae H =1/3. In he limi as ψ, H 0, and variaion in hours is eliminaed. Column (2) repors resuls for he model where labour supply elasiciy is se o a low value, 0.01, using ψ = 200. This suppresses hours variaion allowing our model o approximae he framework used by Krause and Lubik (2007). Column (5) displays simulaion resuls when labour supply elasiciy is se o mach he variabiliy of hours observed in US daa. This enables our model o approximae he model of Trigari (2005). The oher columns are discussed in Secion (4.3). Wihou hours variaion i is no possible o generae he paerns of reallocaion found in he daa. The Beveridge curve is almos fla and he correlaion of gross job flowsisposiive. This mirrors he finding of Krause and Lubik (2007). This is exacly he effec ha one would expec, in he ligh of he wider lieraure on reallocaion under socially efficien search, Morensen and Pissarides (1993). 21 Once realisic variaion in hours is permied he model is much beer able o capure he direcion of reallocaion. However, he srengh of he relaionship beween unemploymen and vacancies is no capured as he Beveridge curve, while downward sloping, is 20All saisics (for model simulaions and daa) are compued from Hodrick-Presco filered daa, expressed as percenage deviaions from seady sae (or rend in he case of he daa). The business cycle saisics for model varians are compued by averaging across 200 simulaions. Each simulaion conains 250 daa poins bu he firs 50 are omied when underaking derending and compuing momens. 21I appears o confirm he difficulies of allowing for endogenous job desrucion oulined by Shimer (2005) even hough here movemens in job desrucion are driven by aggregae produciviy and moneary disurbances raher han he reallocaive shocks o exogenous job desrucion ha he discusses. 18

21 no as seep as ha in he daa, see Columns (1), (2) and (5). I is worh spending some ime rying o undersand he mechanism by which variaion in hours alers he incenives o creae and desroy jobs. To do so we conras he model s response o a shock ha raises unemploymen under inelasic hours and hen under elasic hours. Firs noice ha from an accouning viewpoin, a rise in unemploymen can be achieved in a variey of ways including a rise in job desrucion and a fall in job creaion, or by one of hese in isolaion wih no change in he oher, by a fall in job creaion combined wih a smaller fall in job desrucion or even by a rise in job creaion combined wih a larger rise in job desrucion. The firs of hese would end o give rise o a negaive conemporaneous correlaion of job creaion and job desrucion. The firs case describes he daa, as is well documened Davis e al. (1998). I also applies o he case wih elasic hours. The firs case describes he daa, as is well documened Davis e al. (1998). I also applies o he case wih elasic hours. The las (wo) of cases would end o produce a posiive conemporaneous correlaion of job creaion wih job desrucion. Under inelasic hours a rise in unemploymen occurs hrough a rise injobcreaioncombinedwiha larger rise in job desrucion. Second, since he correlaion of gross job flowsisaflow measure of reallocaion while he Beveidge curve is (a leas in par) is a sock measure, i is no clear ha here should be a srong associaion beween he correlaion of gross job flows and he slope of he Beveridge curve. The relaionship beween he wo will depend on he exen o which high levels of vacancies are srongly associaed wih periods of above average job creaion and o he exen ha above average job desrucion is associaed wih periods of above average unemploymen. Consider he case where variaion in hours is suppressed. Separaions are efficien and job desrucion faciliaes he socially efficien creaion of jobs, so he opimal ime o creae jobs is a he poin a which he opporuniy cos of doing so is a is lowes, namely when mach level profis (rens) are low. This makes job creaion and job desrucion move ogeher, Column (2). As Column (2) shows, job creaion and vacancies also move ogeher very closely wih hours variaion suppressed - whereas in US daa hese variables appear virually uncorrelaed, so a decline in rens for ongoing maches (following a shock) drives up job desrucion, which is posiively correlaed wih unemploymen, alhough he correlaion is weaker han in US daa. Such a shock also 19

22 leads o a sharp rise in job creaion, which is srongly associaed wih a rise in vacancies. So wih hours variaion suppressed unemploymen and vacancies are less srongly associaed han in he daa because in he model job desrucion is relaively weakly associaed wih changes in unemploymen, despie he srong correlaion of vacancies wih job creaion, and of job creaion wih job desrucion. This demonsraes he difficulies of accouning for reallocaive behaviour under he sandard (implici) assumpion ha labour inpu can only be varied on he exensive margin. I also provides and example of an environmen in which he correlaion of gross flows akes a differen sign o he slope of he Beveridge curve. Nex consider he environmen in which ψ is se o mach he variabiliy of hours in US daa, σ H /σ Y (o achieve his we se ψ =2.25 as in Column (5)). Wih realisic variaion in hours, he model generaes a posiive correlaion beween hours per worker and employmen (albei weaker han ha in he daa). So hours per worker will be above average in an expansion, as unemploymen rises, and below average in a recession as unemploymen falls. Variaion in hours per worker reduces he exen o which rens vary in response o shocks (as a resul of he convexiy of he mach level rens in hours per worker. Increased variaion in hours per worker is likely o reduce he variaion on he exensive margin. The response of job desrucion o shocks will, oher hings equal, be more mued when hours can vary and insulae he economy from he full reallocaive effecs of any shock. As a direc resul of his reduced response of job desrucion, he incenives for vacancy and job creaion (in response o a shock ha raises unemploymen) will be aenuaed for wo reasons. Firsly, he reduced response of job desrucion will leave a larger number of ongoing maches, which will reduce he poenial rens available o new maches and consequenly reduce job creaion and vacancy creaion. Secondly he response of job creaion will be aenuaed because he probabiliy of filling an open vacancy will fall, due o he reducion in he size of he pool of unemployed workers (which follows from he more mued response of job desrucion). Finally he job creaion response is likely o be aenuaed, independenly of any variaion in job desrucion, since he flexibiliy of hours allows ongoing maches o respond o improved condiions (as he economy moves back owards seady sae following a shock). 20

23 Now, in pracice, our calibraion sraegy holds he variabiliy of job desrucion consan across experimens a he value observed in US daa, σ JD /σ Y. This places greaer emphasis on he final effec described in he previous paragraph. Columns (2) and (5) confirm ha greaer elasiciy of hours reduces he variabiliy of job creaion as would be expeced under he laer effec. By aenuaing he use of he exensive margin (job creaion) in response o shocks, hours variaion alers he correlaion of gross job flows. So, in response o a shock ha leads o an increase in unemploymen, job creaion rises when job desrucion falls, as in he daa, despie he fac ha we require he economy o saisfy he Hosios condiion. Vacancies are less variable when variaion in hours is inroduced which is consisen wih he idea ha variaion in hours aenuaes use of he exensive margin. Vacancies remain posiively correlaed wih job creaion when hours can vary (Column (5)) bu he correlaion is weaker han wih hours variaion suppressed (Column (2)). By conras, he correlaion of job desrucion wih unemploymen is sronger under he variable hours environmen han wih hours variaion suppressed. I is he combinaion of he sronger correlaion beween job desrucion and unemploymen, and he reducion in he correlaion of job creaion wih vacancies ha permis he negaive correlaion of job creaion and job desrucion ogeher wih a negaively sloped Beveridge curve. Alhough, he Beveridge curve remains shallower han required by he daa. While a model wih hours variaion offers an improved reamen of many aspecs of reallocaion, he join behaviour of vacancies wih job creaion is one area in which he variaion of hours does no really ge close o he daa. In US daa here is virually no relaionship beween he job creaion and he number of open vacancies, ye wih realisic hours variaion he model generaes a posiive correlaion beween vacancies and job creaion (he associaion is even greaer when hours variaion is suppressed). This reflecs he lack of persisence in vacancies (no displayed in Table (2)): in US daa he firs order serial correlaion coefficien for vacancies is 0.92, while in he model wih hours variaion i akes he value