Employment, hours and the welfare effects of intra-firm bargaining

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Employment, hours and the welfare effects of intra-firm bargaining

Maarten Dossche, Vivien Lewis, Céline Poilly

To cite this version:

Maarten Dossche, Vivien Lewis, Céline Poilly. Employment, hours and the welfare effects of intra-firm bargaining. Journal of Monetary Economics, Elsevier, 2019, 104, pp.67-84.

�10.1016/j.jmoneco.2018.09.002�. �hal-01995026�

Employment, hours and the welfare effects of intra-firm bargaining ^R

Maarten Dossche

^a

, Vivien Lewis

^b

, Céline Poilly

^{c, d, ∗}

aEuropean Central Bank, Sonnemannstrasse 20, Frankfurt am Main 60314, Germany

bResearch Centre, Deutsche Bundesbank, Wilhelm-Epstein-Str. 14, Frankfurt am Main 60431, Germany

cAix-Marseille Univ., CNRS, EHESS, Centrale Marseille, AMSE, France

dAMSE, 5 Boulevard Bourdet, Marseille 13001, France

JEL classification: D62, E24, E61, E64, H21

Keywords: Employment, Hours, Intrafirmbargaining, Overhiring

abstract

Bilateral bargaining between a multiple-worker firm and individual employees leads tooverhiring. With a concave production function,the firmcan reducethemarginalproductbyhiringanadditional worker,thereby reducingthe bargainingwagepaidtoall existingemployees.Weshowthatthisexternalityisamplified whenfirmscanadjusthours perworkeraswellasemployment.Firms keepdown workers’wagedemands byreducingthenumber of hours perworker and the resulting labor disutility. Our finding is particularlyrelevantforEuropeaneconomieswhere hoursadjustmentplaysanimportantrole.

1. Introduction

InEurope,firmsadjusttheirlaborinput toalargedegreeby varyinghoursperworker ratherthan theworkforce.This motivates ourresearch question: how doesallowing for variable hours per worker affectlabor market outcomes in the intrafirmbargainingmodel?

R An earlier version of this paper was circulated under the title ‘Employment, Hours and Optimal Monetary Policy’. We thank the associate editor, Antonella Trigari, and an anonymous referee for very useful comments and suggestions. We are grateful to Régis Barnichon, Björn Brügemann, Miguel Casares, Bruno Decreuse, Chiara Forlati, Christian Häfke, Martin Kliem, Keith Kuester, André Kurmann, Michael Krause, Luisa Lambertini, Branka Matyska, Stéphane Moyen, Carlos Thomas, Mirko Wiederholt and Raf Wouters for helpful discussions. Thanks also to seminar participants at Deutsche Bundesbank, EEA Congress 2013, Goethe University Frankfurt, KU Leuven, TU Dortmund, University of Lausanne, Vienna Macro Café. This paper was partly executed when Céline Poilly was at the University of Lausanne and when Vivien Lewis was at KU Leuven. The views expressed herein do not necessarily reflect those of the Bundesbank or the European Central Bank. All errors are ours.

∗ Corresponding author at: AMSE, 5 Boulevard Bourdet, Marseille 13001, France.

E-mail addresses: Maarten.Dossche@ecb.int (M. Dossche), Vivien.Lewis@bundesbank.de (V. Lewis), celine.poilly@univ-amu.fr (C. Poilly).

URL: http://sites.google.com/site/vivienjlewis (V. Lewis), http://sites.google.com/site/celinepoilly (C. Poilly)

Inanintrafirmbargainingframework,amultiple-workerfirmbargainswitheach ofitsemployeesindividuallyoverthe wagerate.Withaconcaveproductionfunction,hiringaworkerlowersthemarginalproductoflabor,therebyreducingthe bargainingwage.Sinceall workersareeffectivelymarginal,thewageoffered toall (alsotheinfra-marginal) workers, and thusthefirm’stotalwagebill,canbe reducedinthisway.Itisawell-knownresultthatfirmsexploitthisexternalityand overhire,i.e.theyhiretoomanyemployeesrelativetotheefficientallocation(Smith,1999;StoleandZwiebel,1996).

Ourcontributionistoanalyzethisphenomenon ina setupwhere,realistically,firmscanadjusttheirlabor inputalong twomargins,employmentandhoursperworker.Businesscyclemodelswithfrictionallabormarketshavelargelyneglected thehoursmargin.¹OurmodelwithbothlabormarginsisabletoreplicatekeystylizedfactsoftheEuroArealabormarket.

Weshowthatoverhiringismagnifiedinthepresenceofanhoursmargin.Thisisbecauseanexpansionoftheworkforce impliesareductioninthenumberofhoursworkedperemployee.This,inturn,reducesthemarginaldisutilityofworking and thus the wage that the firm pays all its employees. Especially iflabor disutility rises steeply in hours worked, the ensuingfallinthewagebill,andthereforetheoverhiringexternality,islarge.Theresultingmisallocationoflaboracrossthe twomargins,withemploymentbeingtoohighandhoursperworkerbeingtoolow,givesrisetoasizeablewelfareloss.

Ourpapercontributestothe normativeandthe positiveliterature onintrafirmbargaining(IFB,henceforth). Regarding thenormativeliterature,theseminalworkbyStoleandZwiebel(1996)showstheoverhiringresultunderIFBinpartialequi- librium.Smith(1999)andCahucandWasmer(2001)extendthisresulttoageneralequilibriumsearch-and-matchingmodel.

CahucandWasmer(2001)showthatoverhiringdisappearsifthereisanother,fullyflexible,productivefactor(e.g.capital), andtheproductionfunctionexhibitsconstantreturnstoscaleinallfactors.Theoverhiringdistortioncanberemovedwith collectivebargaining(BauerandLingens,2013) orlong-termwagecontracts(Hawkins,2015).Noneoftheaforementioned papersconsidershours perworkerasanadditionallabormargin,aswedointhispaper.KudohandSasaki(2011)discuss efficiencyin amodel withIFBandvariable hours. However, intheir modelvariant firmsandemployeesbargain over an earningsschedule;hoursarechosenbythefirmanddonotdependonthefirm’sworkforce.Inourpaper,thefirmusesits hiringdecisiontostrategicallyaffectthenumberofhoursperworker.Reducinghoursperworkerthroughoverhiringhelps thefirmtoreduceitswagebillevenmorethanintheconstant-hoursmodel.

Regarding the positive literature, several papers investigate the goodness-of-fit of the intrafirm bargaining model.

Acemoglu and Hawkins (2014) study the cross-sectional implications of a model with IFB and heterogeneous firms.

KrauseandLubik(2013)showthatthetransmissionoftechnologyshocksisonlymarginallyaffectedbyIFB.Kim(2015)in- cludes variablehours inamodelwithIFB andstudies theimplied volatilityoflabor market variables.InKim (2015),the bargainingprotocolisdifferentfromours.Ifwagenegotiationsbreakdown,the remainingworkersbargain withthe firm collectively; their bargaining power dependson the business cycle. Here, asin the ‘Rolodex game’of (Brügemannet al., 2017),aworkerwhorejectsawageofferissenttothebackofthe‘bargainingqueue’andbilateralbargainingresumeswith thenext worker inline.Thisallows thefirm to treatall workersasmarginal.Kudohetal.(2017) assessthegoodness of fit ofa search-and-matching modelwithintrafirm bargainingandhours workedon Japanese data.Theyshow that their modelissatisfactoryinreplicatingfluctuationsinbothmarginsoflabor,especiallywhenhoursarechosenbythefirm.We showthatourmodelwithintrafirmbargainingandbargainedvariablehoursperformsratherwellintermsofmatchingkey businesscyclemomentsoftheEuroArea.

The remainder ofthe paperisstructured asfollows.Section 2 providesempirical evidenceto motivate ourmodelling choices.Section3presentsthemodel.Section4describesthedistortionarisinginthecompetitiveallocation,howitdepends onthe propertiesoftheproductionfunctionandtheFrischelasticityoflabor supply,andhowan appropriately designed unemployment insurancetransfer can remove it.In Section 5,we show how theoverhiring distortion isaffected by the presence ofhours asalabor adjustmentmargin andwe compute thewelfare lossesarisingfromintrafirmbargaining. In Section6,wedemonstratethattheproposedmodelperformswellempirically.Section7concludes.

2. Empiricalevidence

EuropeanandUSlabormarketsdifferinmanyrespects.Fig.1showsthatintheEuroArea,48%ofthevarianceintotal hours is accountedforby variations inhours per employee, whereas this isonly 6% forthe US.² The importance ofthe hours margin in France, Germany and Italy (vs. the US) is consistent with the findings of Llosa et al.(2012). The strict employmentprotectionlegislationinEurope,ascomparedwiththeUS,makeshours perworkerrelativelymoreattractive asanadjustmentmargin(see(OECD,2013)).Also,asmentionedinECB(2012),short-timeworkprogramshavebeenused moreextensively inEuropeancountriesthan inUSduringthe 2008crisis, whichmighthavelimited thedeteriorationof labormarketconditions.

At the same time, the evidence pointsto a gradual shift towards a production structure where individual European workersperformfewerhours onaverage.Rogerson(2006)documentsthat inFrance,GermanyandItaly, hoursperworker declined by morethan 30% since the 1950s, a development which wasnot observed inthe US. More recent data from OhanianandRaffo(2012)confirm thevirtual stabilityofhoursworkedperworkerintheUS,whereas inFrance,Germany andItalyhoursperworkerhavecontinuedtodeclineinthe2000s.Thiscannotbeascribedsolelytodifferencesinworkers’

1See, for example, ( Merz, 1995; Mortensen and Pissarides, 1994; Shimer, 2005 ) and ( Hall, 2006 ).

2Of the large Euro Area countries, Spain is an outlier; there, most of the adjustment of labor input happens via changes in the number of temporary workers.

Fig. 1. Total hours decomposition. Note: The contributions of hours/worker (logged) and employment (logged) to the variance of total hours worked (logged) have been computed as the variance (X)/(variance (X) + variance (Y) + 2covariance (X,Y)). All variables are hp-filtered with smoothing param- eter 1600. The decomposition for the euro area has been computed as an average of the contributions for Germany, France, Italy and Spain, weighted by their share in total Euro Area hours worked. The sample is 1999Q1-2015Q4. Data are from Ohanian and Raffo (2012) , updated to 2015Q4 with Eurostat and BLS data.

preferencesonthetwosidesoftheAtlantic.MoreEuropeansthanUScitizensreportthattheyworkpart-timeinvoluntarily, whichsuggeststhatlow hoursperworker inEuropemay– atleastpartially– betheoutcomeofadeliberatestrategy by firmstoreduce workerbargainingpower.IntheUSonaverageoverthebusinesscycle,only3% ofall workersreport that theyworkpart-timeinvoluntarily;intheEuroAreathisfigureisabout6%.³

Regardingwage determination,bilateral bargainingbetween afirm andan individual worker hasbecome much more prevalent todayas the importance ofcollective bargaining has steadilydecreased.⁴ Moreover, institutions such astemp- ingagencieshaveincreasinglybecomepartoftheworker-firmrelationship.FordeandSlater(2011)providesurveyevidence confirmingthattempingagenciesfacilitatebilateralwagebargainingbetweenaworkerandafirm.Whiletheirevidenceap- pliestotheUK,thefindingshouldarguablyextendtootherEuropeancountries.Thelegalframeworkismainlydetermined

3Data from the BLS and Eurostat, respectively.

4OECD measures of trade union density show a steady decline in most European countries.

Fig. 2. Wages and Hours per Worker across German Industries. Note: The data cover all available NACE sectors (excluding activities of households as employers), or 36 observations. Hours per worker is the average of annual hours worked per employee over the period 1995–2015. Wage per worker is the average of annual compensation per employee (in thousands of EUR) over the period 1995–2015. Data source: Eurostat.

by theEuropean UnionTemporaryAgencyWorkDirective, whichiscommontoall European countries.Tempingagencies allowfirmstoeasilybargainwithmanyindividualworkerssimultaneously.

Taken together,thesefacts suggest that intrafirmbargainingandvariable hours now belong tothe salientfeatures of Europeanlabormarkets.

Afinal pieceofevidence thatwe wantto highlightisshowninFig.2,which plotssectoraldataforGermany.We see thatsectorswithlowerhours peremployeealsohavelower wages.Whilethefigureshowsper-personwages,thepositive correlationholdsalsoforhourlywages.OurmodelwithIFBandhourswillbeabletocapturethiscorrelation.

3. Model

Our model features search-and-matchingfrictions in the labor market à la (Mortensen and Pissarides, 1994). A firm canemploymultipleworkersandwagesare setthroughbilateralbargaining,asinStoleandZwiebel(1996)orCahucand Wasmer(2001),amongothers.Thefirmcanadjustits workforceaswell ashoursperemployee.Hours,likewages,areset throughbargainingbetweenthefirmandeachindividualworker.⁵

Thetiming ofeventsisthefollowing.Theperiodis splitintotwosubperiods:thebargainingstage earlyintheperiod and thehiring stage atthe end ofthe period. At the hiring stage, currentemployment atthe firm is given, production andwage paymentshave alreadytakenplace.Let xbe the setof exogenousvariables that define the aggregate state,in particular,technologyAandgovernmentspending G.Tolightenthenotation,wedonot usetime subscripts.Letasymbol withoutaprimedenoteavariableinthecurrentperiodandletasymbolwithaprimedenoteavariableinthenextperiod.

Wethereforewritethecurrentvalueofanyvariableyasyandnext period’svalueasy.Thesteady-statevalueofvariable yiswritteny¯.⁶

3.1. Unemploymentandmatching

Firms post vacancies and unemployed workers search for jobs. Let M=M₀u^η

v

^1−η denote the number of successful matches,whereuistheunemploymentrate,

v

istheaggregatenumberofvacancies,

η

∈(0,1)istheelasticityofthenum- berofmatchestounemploymentandM₀>0denotesthematchingtechnology.Theprobabilityofavacancybeingfilledis

5In Germany, it is common that wages and working time, i.e. hours, are set as part of the same bargaining process, see ( Bispinck et al., 2010 ).

6The online appendix contains detailed model derivations.

q≡M/

v

_,wheretheratioofvacanciestounemployedworkers,

θ

≡

v

_/u,isameasureoflabormarkettightness.Thejobfind- ingrateisdenotedp≡M/u.Firmsandworkerstaketheprobabilityoffillingavacancyqandthejobfindingratepasgiven.

Theemployment rateis n=1−u. Aconstant fraction

λ

∈(0, 1) ofemployment relationshipsare exogenouslyterminated eachperiod.Newlyhiredworkersjointheworkforceonlyinthenextperiod,suchthat

n=

(

¹⁻

λ )

^n+q

v

(1)

describestheevolutionofthefirm’sworkforce.Givenaunitmeasureofidenticalfirms,nand

v

denoteaggregateaswellas firm-levelemploymentandvacancies,respectively.

3.2.Hiring

LetV^v(ⁿ,x) ^{and Sf(n, x) denote,}respectively, the value to the firm ofposting a vacancy andthe value to thefirm of successfullyforming a matchinthe labor market. Inthe next period, avacancy is filledwith probability q, yielding the value S^f(n, x), andremains open otherwise, inwhich casethe value ofthe vacancyisV^v(ⁿ,x)^{.The value of postinga} vacancyisminustheper-periodcostofpostingavacancy,c,plustheexpectedcontinuationvalue,

V^v

(

n,x

)

=−c+

β

^E

{ (

x,x

)

^[qSf

(

ⁿ,x

)

+

(

^1−q

)

^Vv

(

ⁿ,x

)

^]

}

^{, (2)}

whereE(·) is the expectations operator. The variable ^{(x, x) captures the} household’sstochastic discount factor andis definedbelow.Firmsareownedbythehouseholdsandthereforeuse^(x,x)fordiscounting.The firmpostsvacanciesas longasthevalueofavacancyisgreaterthanzero.Inequilibrium,V^v(ⁿ,x)=0andsothevacancypostingconditionis

c

q=

β

^E

{ (

^x,x

)

^Sf

(

ⁿ,x

) }

^{. (3)}

At the optimal hiring rate, the cost of hiring a worker, given by the vacancy posting cost, c, multipliedby the average durationofavacancy,1/q,equalsthepresentdiscountedvalueofthefirm’smatchsurplusinthenextperiod.

3.3.Firm’smatchsurplus

Eachfirmontheunitintervalemploysnworkers.Worker j∈[0,n]earnsawagew(ⁿ,x;j)^{forworkingh(n,x;j)hoursat} thefirm.Thevariableswandhareindexedbyjtoindicatethatthewagerateandhoursperworkeraresetforeachworker individuallythrough bilateralbargaining. Theirdependenceonn reflectsthestrategic effectthefirm’shiringdecisionhas onwagesandhoursthroughintrafirmbargaining.

Supposethefirmhasformedamatchwithaworker,withmatchvalue denotedV^f(n,x).Inthenextperiod,theworker remainsemployedbythefirmwithprobability1−

λ

,inwhichcasethematchvalueisV^f(n,x),ortheemploymentrelation isdissolvedwithprobability

λ

^{,thenthevalueofthematchtothefirminthenextperiodiszero.Thefirm’scurrentmatch}

valueisgivenbyitsrevenue,minusthewagebill,minusthecostofpostingvacancies,plustheexpectedcontinuationvalue,

V^f

(

n,x

)

=A

_n

0

h

(

n,x;j

)

^dj

α

− n

0

w

(

n,x;j

)

^dj−c

v

+

(

¹⁻

λ ) β

^E

{ (

x,x

)

^Vf

(

ⁿ,x

) }

^{, (4)}

whereAisaneconomy-widetechnologyindexand

α

∈[0,1)^{capturesthedegreeofconcavityoftheproductionfunctionin} totalhours.

Forthefirm,thesurplusfromemployingamarginalworker,definedasS^f(ⁿ,x)≡^∂Vf_∂⁽_n^n,x),isgivenby:

S^f

(

n,x

)

=

χ (

n,x

)

+

(

¹⁻

λ ) β

^E

{ (

x,x

)

^Sf

(

ⁿ,x

) }

^{, (5)}

where

χ

^{(n,x)capturestheshadowvalueofamarginalworker,}

χ (

n,x

)

≡

α

^A

_n

0

h

(

n,x;j

)

^dj

α−1

h

(

n,x;n

)

+ n

0

hn

(

n,x;j

)

^dj

−

w

(

n,x;n

)

+ n

0

wn

(

n,x;j

)

^dj

. (6)

Themarginal worker isidentified by settingtheindexj to its upperbound, j=n,such that hours andthewagerateof themarginalemployeeareh(n,x;n)andw(ⁿ,x;n),respectively.Thefirsttermontherighthandsideoftheshadowvalue (6)is theoutput produced when hiringan additionalworker workingh(n, x; n) hours, whereh_n(ⁿ,x;j)≡_∂^∂_nh(ⁿ,x;j)^is theeffectofamarginalworkeronthenumberofhoursworkedbyemployeej.Thesecondtermontherighthandsideof (6)capturesthe reductioninthewagebillduetoan additionalemployee,wherewn(ⁿ,x;j)≡_∂^∂_nw(ⁿ,x;j)^{istheeffectof} themarginalworkerontheequilibriumwageofemployeej.Thehiringdecisiongivesrisetoanexternalityinthatitaffects

bothhoursandwagesofallexistingworkers.Rewritingtheshadowvalueasfollows,

χ (

^n,x

)

⁼

α

^A

_n

0

h

(

^n,x;j

)

^dj

α−1

h

(

^n,x;n

)

^−w

(

^n,x;n

)

+

α

^A

_n

0

h

(

n,x;j

)

^dj

α−1 n

0

hn

(

n,x;j

)

^dj− n

0

wn

(

n,x;j

)

^dj

IFBeffect

, (7)

we canidentifythe intrafirmbargainingeffectasthe second linein(7). Inamodelwithout IFB,theshadowvalue ofan extraworkercorresponds tohismarginalrevenue product,netofhiswage.Thetwotermshn(n,x;j)andwn(ⁿ,x;j)^arise throughintrafirmbargaining.Inthemultiple-workerfirmmodelwithouthours,theIFB-effectisgivenbywn(ⁿ,x;j)^.See,for instance,(KrauseandLubik,2013).Here,duetothepresenceofvariablehours perworker,intrafirmbargainingintroduces an additional channelcaptured by thetermhn(n,x; j). The dependenceofthe numberofhours worked onemployment comes fromthesubstitutabilitybetweenemploymentandhours peremployee;it isonlypresentwhen thefirm hasboth labormarginsatitsdisposal.Itappearsneitherintheone-workerfirmsetupwithhoursofTrigari(2006),norinthelarge firmmodelwithouthours,seee.g.(KrauseandLubik,2013).WedemonstrateinSection5howtheintroductionofvariable hoursaffectstheequilibriumoutcome.

Combiningthefirm’ssurplus(5)inthenext periodwiththevacancypostingcondition(3),we obtainthejobcreation condition,

c q=

β

^E

(

^x,x

)

χ (

ⁿ,x

)

+

(

¹−

λ )

_q^c

. (8)

A firm posts vacanciesuntil the cost of hiringa worker equals the expecteddiscounted futurebenefits from employing thisextraworker. The benefitsofhiringa worker arehisshadowvalue, plus thevacancypostingcosts savedincasethe employmentrelationshipcontinues.

3.4. Worker’smatchsurplus

LetW(n,x)andU(x)denote,respectively,thevaluetothehouseholdofthemarginalhouseholdmemberbeingemployed vs.unemployed.

Ahouseholdmemberthatisemployedreceivesthewagew(ⁿ,x;n)^{forworkingh(n,x;n)hours.Sincethefirmbargains} withallworkersindividually,eachworkeriseffectivelymarginal,suchthatweset j=ninthebargainingproblembelow.

Inthenext period,theworker iseitherstill employedwithprobability1−

λ

,inwhichcasehisvalue tothehouseholdis W(n,x),or theemployment relationis dissolvedwithprobability

λ

^{, thenthe} corresponding value inthenext periodis U(x).Thecurrentvalueofbeingemployedistherealwage,minusthedisutilityofemployment(dividedbythehousehold’s shadowvalueofconsumption,^{,toconvertutilsinto}consumptiongoods),plusnextperiod’sexpectedvalue,

W

(

ⁿ,x

)

=w

(

ⁿ,x;n

)

−

ζ

^h

(

n,x;n

)

^1+σ

1+

σ

^{1 +}

β

^E

(

^x,x

)

(

¹−

λ )

^W

(

ⁿ,x

)

+

λ

^U

(

^x

)

. (9)

In(9),

ζ

>0istheweightonlaborinhouseholdutilityand

σ

>0istheinverseFrischelasticityoflaborsupply.

An unemployed household member produces b units of market output andreceives lump-sum transfers T^b from the government.Inthenextperiod,heeitherfindsajobwithprobabilityp,inwhichcasehis valuetothehouseholdisW(n, x), orheremains unemployedwith probability 1−p,then the corresponding value isU(x). The currentvalue of being unemployedisthus

U

(

^x

)

=

(

^b+T^b

)

+

β

^E

(

x,x

)

pW

(

ⁿ,x

)

+

(

^1−p

)

^U

(

^x

)

. (10)

Definingthesurplusfromemploymentforaworkerasthedifferencebetweentheemploymentandunemploymentvalues, S^w(ⁿ,x)≡W(ⁿ,x)−U(^x),wecansubtract(10)from(9)towrite

S^w

(

ⁿ,x

)

=w

(

ⁿ,x;n

)

−

ζ

^h

(

n,x;n

)

^1+σ

1+

σ

^{1 −}

(

^b+T^b

)

+

(

¹−

λ

−p

) β

^E

(

^x,x

)

^Sw

(

ⁿ,x

)

. (11)

3.5. Hoursandwages

Aworkerandthefirm Nash-bargainover theindividualworker’s wagew(ⁿ,x;n)^{andhours h(n,x;n),giventhefirm’s} employmentlevelnandthemacroeconomicstatex.⁷ Intheabsenceofacommitmenttechnologyforlaborcontracts,bar- gainingstartsaneweachperiod.Thepartiesdividethematchsurplusaccordingtotheirrespectivebargainingweightsgiven

7( Brügemann et al., 2017 ) show that the Stole–Zwiebel bargaining protocol needs to be amended to ensure that the wage is symmetric and equal to the marginal worker’s wage: a worker who rejects a wage offer is sent to the end of the queue of workers waiting to bargain with the firm. That threat weakens his bargaining position and makes him accept the wage offer corresponding to the marginal worker’s bargaining wage.

by

γ

^and1−

γ

,

maxw,hS^w

(

n,x

)

^γSf

(

n,x

)

^1−γ. (12)

The first order conditions for wages and hours are, respectively, (¹−

γ

)^Sw(ⁿ,x)=

γ

^Sf(ⁿ,x), and

ζ

^h(ⁿ,x;n)^σ/=

α

^A(0ⁿh(ⁿ,x;j)^dj)^{α−1. Next, imposing symmetry on wages and hours by setting h}(ⁿ,x;j)=h(ⁿ,x;n)=h(ⁿ,x) ^and w(ⁿ,x;j)=w(ⁿ,x;n)=w(ⁿ,x)^{forany j}∈[0,n),theoptimalityconditionforhoursbecomes:

ζ

^h

(

^n,x

)

^σ

⁼

α

^A

(

^nh

(

^n,x

))

^{α−1. (13)}

Theleft handside of(13)isthemarginaldisutility ofhours worked, dividedbytheshadowvalue ofconsumption,repre- sentingthemarginalrateofsubstitutionofworkintoconsumptionforanindividualworker.Therighthandsideof(13)is theadditionaloutputproduced,perworker,byanadditionalhourworked.

Equation(13)determines hoursper employeeasa functionofemployment; itsderivative, hn(ⁿ,x)=−^h(nn^,x),implies anelasticityofhourstoemploymentgivenby−,where:

≡ 1−

α

σ

⁺¹⁻

α

^∈

⁽

^0,1

⁾

^{. (14)}

Twopropertiesoftheelasticitydefinedin(14)areworthnoting,

∂

∂α

⁼⁻

σ

( σ

⁺¹⁻

α )

^{2 <0,}

∂

∂σ

⁼⁻

1−

α

( σ

⁺¹⁻

α )

^{2 <0. (15)}

First,thegreater theconcavity inthe productionfunctionwithrespectto totalhours (the lower is

α

^{),all else equal,the}

moresubstitutablearethetwolaborinputs(thegreateris^{).Second,thehigheristheFrischelasticityoflaborsupply(the} loweris

σ

^),themoresubstitutablearethetwolaborinputs(thegreateris^{).Consideraproductionfunctionthatislinear} intotalhours,such that

α

=1,oradisutilityfunction withazeroFrischelasticity,suchthat

σ

→∞.Ineithercase, hours areinvarianttoemployment,=0.Afirm’sintensiveandextensivelabormarginsareinsteadsubstitutesiftheproduction functionisconcaveintotalhours,i.e.

α

<1,andthelabordisutilityfunctionisconvexinhoursworked,i.e.

σ

^isfinite.

Usingtheelasticityofhourstoemployment(14),theworker’sshadowvalue(6)becomes:

χ (

n,x

)

=

(

¹⁻

) α

^A

(

^nh

(

n,x

) )

^α−1h

(

n,x

)

−[w

(

n,x

)

+wn

(

n,x

)

^{n]. (16)}

Thefirsttermin(16) showsthat theshadowvalue is reducedwhen anadditionalworker lowersaveragehours per em- ployee,>0.However, weshowbelowthathiringalsoreducesthewagebill, suchthattheoverallexternaleffectonthe shadowvalueispositive.Theequilibriumwagecanbeshowntosatisfy

w

(

n,x

)

=

(

¹⁻

γ )

ζ

^h

(

^n,x

)

^1+σ 1+

σ

1

⁺

(

^b+T^b

)

+

γ

^·

α

^A

(

^nh

(

n,x

))

^α−1h

(

n,x

)

+c

θ

, (17)

where

≡ _{1−γ (}^1−γ _{1+σ )}≥1.Eq.(17)hasasimilarstructureasthebargainingwageintheone-worker-firmmodel.Thewage perworkerisaconvexcombinationoftwo components,wheretheweightsare givenby thebargainingshares.Onecom- ponentisthehousehold’smarginal rateofsubstitutionbetweenanadditionalworkerandconsumption,plustheworker’s outsideoption.Thesecondcomponentisthemarginalproductofan additionalworker, plusthecostofpostingavacancy.

Eq.(17)onlydiffersfromtheequilibriumwageintheone-worker-firmmodelofe.g.(Trigari,2006)throughthecoefficient ϰ≥1,whichinturndependsonthedegreeofsubstitutabilitybetweentheextensiveandtheintensivemarginsoflabor, ^. Differentiatingthe wage(17)withrespecttoemployment n,we seethat hiringanadditionalworker reducesthetotal wagebill,

wn

(

^n,x

)

^n=−[

(

¹⁻

γ )

⁺

γ (

¹⁺

σ )

^]

^·

α

^A

(

^nh

(

^n,x

))

^α−1h

(

^n,x

)

^{. (18)}

Then,combiningEqs.(16),(17)and(18),andaftersomealgebra,theshadowvaluecanbewrittenas

χ (

^n,x

)

⁼

(

¹⁻

γ )

^·

α

^A

(

^nh

(

^n,x

))

^α−1h

(

^n,x

)

⁻

ζ

^h

(

^n,x

)

^1+σ 1+

σ

1

⁻

(

^b+Tb

)

−

γ

^c

θ

^{. (19)}

Wewillcompare(19)withitscounterpartintheefficientallocation.

3.6.Consumptionandsaving

Thereexists a unit mass ofidenticalinfinitely-lived households, each withalarge number ofmembers.In the repre- sentativehousehold, afractionn∈(0,1) ofmembers areemployed inthe marketeconomy. Thehousehold’semployment ratencorrespondstoaggregateemployment.ThehouseholdmayconsumeCorsaveintermsofrisklessbondsBthatcost oneunit ofthe finalgood andyielda returnof(¹+r) ^{unitsone periodlater. LetW}(^x)^{denotethe valuefunction ofthe} representativehousehold.Themaximizationproblemisexpressedas

W

(

^x

)

=max

C,B

lnC− n

0

ζ

^h

(

n,x;j

)

^1+σ

1+

σ

^dj+

β

^E

{

^W

(

^x

) |

^x

}

, (20)

where

β

∈(0,1)isthehousehold’sdiscountrate,consumptionisthesumofgoodsproducedinthemarket,C^m,andhome- producedgoods,C=C^m+(¹−n)^b,subjecttothebudgetconstraint,

C^m+B+T≤ _n

0

w

(

ⁿ,x;j

)

^dj+

(

¹−n

)

^Tb+

(

¹+r

)

^B+D, (21) where T are lump-sum taxes andD are firm profits,both ofwhich are takenas given by the households. The first or- der conditionfor theoptimal consumption-savingsdecisionyields 1=(¹+r)

β

^E

{

(^x,x)

|

^x

}

, where(^x,x)=C/C. As in Andolfatto(1996)and(Merz,1995), thereexistsaninsurancetechnologyguaranteeingcompleteconsumption risksharing betweenhouseholdmembers,such that Cdenotes consumption by amemberaswell asoverall householdconsumption.

Giventhatallhouseholdsareidenticalinequilibrium,Calsorepresentseconomy-wideconsumption.

3.7. Marketclearingandequilibrium

We aggregate the budget constraint(21) over households andimpose zeronet supply ofbonds, B=B=0. Then, we combinetheaggregatehouseholdbudget constraintwiththegovernmentbudgetconstraint, T=G+(¹−n)^Tb,andaggre- gateprofits,D=A(^nh)^α−wn−c

v

,toobtaintheaggregateaccountingidentity,

A

(

^nh

)

^α+

(

¹−n

)

^b=C+G+c

v

. (22)

Totaloutput,i.e.thesumofmarketoutputandhome-producedoutput,mustequalprivateconsumption,governmentcon- sumptionandtheresourcesusedupforpostingvacancies.

Definition1. Acompetitiveallocationisaset

{

^h,

v

_,n,C

}

^{atwhich,givenaninitialemploymentleveln,employmentfollows}

thelawofmotion(1),householdsmaximizeutility,firmsmaximizeprofits,goodsandbondmarketsclear.

4. Overhiringandoptimalunemploymentinsurance

In the following, we characterize the steady-state distortions arising in our model. We investigate the role of hours forthesedistortions inthenext section. Wefirstderive theefficient allocation,thenwe calibratethemodelandprovide a graphical illustration of the overhiring result. We also show how the distortion can be removedwith an appropriate unemploymentinsurancescheme.Thestarredvariablesdenotetheallocationsthatsolvetheplannerproblem.

4.1. Labormarketdistortion

Thesocialplannermaximizeshouseholdutilitysubjecttotheevolutionofemploymentandtheresourceconstraint.

Definition 2. Anefficient allocationisa set

h^∗,

v

^∗,n^∗,C^∗

which, givenaninitial employment leveln^∗, maximizesutility (20),subjecttotheemploymentdynamics(1)andtheresourceconstraint(22).

Theefficientallocationischaracterizedbythesamefirstorderconditionforhoursasthecompetitiveallocation(13),and thesamejobcreationcondition(8),wheretheefficientshadowvalueofanextraworkeris

χ

^∗=

(

¹⁻

η )

α

^A

(

^n∗h∗

)

^{α−1h∗−}

ζ (

^h∗

)

^1+σ 1+

σ

^C∗−b

−

η

^c

θ

^{∗. (23)}

Eq.(23)istheefficientcounterpartof(19).Itstatesthatanewworkeraddsanamount

α

^A(^n∗h∗)^{α−1h∗togoodsproduced} inthe market andgeneratesthree costs:a utility costof working, ^{ζ (h}₁^∗₊⁾_σ^1+σC^∗, foregone homeproduction, b,andvacancy postingcosts.

Proposition1. Suppose thattheworkerbargaining powerandthematching functionelasticity areequal,

γ

=

η

^{.Then, in the}

absenceofemploymentinsurance,T^b=0,adistortioninemploymentandhoursworkedarisesiftheproductionfunctionisnon- linear in total hours,

α

=1.Inparticular, decreasingreturnsto total hoursin production,

α

<1,imply thatemploymentis too highandhoursperworkeraretoolowinthecompetitiveallocation.

Proof. Tosee that thecompetitive allocation isdistorted when

γ

=

η

^andTb=0, comparethe shadowvalue inthe de- centralized allocation (19) with its counterpart in theefficient allocation (23). Giventhat ϰ>1, we see that, for a given marginalrateofsubstitution(fromtheviewpointofthehousehold)betweenanadditionalworkerandconsumption, ^ζ₁₊^h1+_σ^σC, the shadowvalue ofamarginal worker ishigherinthe decentralizedallocationthan inthe efficientallocation,implying thatfirmshaveanincentivetooverhire.Asaconsequence,hoursperworkeraretoolowinthecompetitiveallocation,see (13).

Notethatweabstractfromanotherdistortionthatariseswhentheelasticityofmatchestounemployment

η

^isdifferent

fromtheworkerbargainingweight

γ

^.When

η

^{ishigh,afirmthatpostsavacancyincreasesvacancydurationforallother}

Table 1

Externally calibrated parameters.

Preferences and production

β 0.99 Discount factor; 4% average annualized real interest rate

σ 3 Inverse Frisch elasticity of labor supply; ( Keane and Rogerson, 2012 ) α ^0.60 Production elasticity to labor

Y¯ 1 Steady state output; normalization

h¯ 0.30 Steady state hours; One third of total time for working

G ¯/ Y¯ 0.21 Steady state share of government consumption in GDP; Euro Area data Labor market

u¯ 0.096 Steady state unemployment rate; Euro Area data

η ^0.60 Elasticity of matches to unemployment; ( Petrongolo and Pissarides, 2001 ) γ ^0.60 Workers’ bargaining power; ( Petrongolo and Pissarides, 2001 )

¯

q 0.70 Steady state vacancy filling rate; ( Christoffel et al., 2009 ) λ ^0.03 Job separation rate; ( Christoffel et al., 2009 )

c v¯/ Y¯ 0.01 Steady state share of total vacancy posting cost in GDP; ( Andolfatto, 1996 )

Table 2

Implied parameters.

Preferences and production

A¯ 2.18 Steady state technology ζ ⁹⁹ Weight on labor disutility M 0 0.41 Scale parameter matching function C ¯/ Y¯ 0.82 Steady state share of consumption in GDP Labor market

¯

n 0.904 Steady state employment

¯

p 0.28 Steady state job finding rate θ¯ ^0.40 Steady state labor market tightness c 0.25 Cost of posting a vacancy v¯ 0.04 Steady state number of vacancies

¯

w 0.84 Steady state wage rate b 0.50 Home production

firms,creatingacongestioneffect.Thiseffecthastobeoffsetbygivingmorebargainingpowertoworkers,whichdiscour- agesfirmsfrompostingvacancies.See(Pissarides,2006).Inordertoisolatetheintrafirmbargainingdistortion,weassume thattheso-calledHosiosconditionissatisfiedbysetting

γ

=

η

,see(Hosios,1990).

Under IFB, the returns to total hours in the production function have key implications for the firm’s optimal hiring decision. When the productionfunction is linear intotal hours (

α

=1), labor productivityaswell as the wageschedule are independent of the number ofemployees, as inthe one-worker-firm model à la (Mortensen andPissarides, 1994).⁸ Incontrast,afirmemployingseveralworkerscanstrategicallyaffectthemarginalproduct,andthusthebargainingwage, throughitshiringchoiceswhentheproductionfunctionexhibitsdecreasingreturnstototalhours.Indeed,underaconcave productionfunction,themarginal productdependsnegatively onthenumberofemployeeswithinthefirm. Anadditional hirereducesthemarginalproductofaworker.Thatlowersthebargainingwagepaidtoallexistingworkers, suchthatthe wagebill is reduced, w_n(ⁿ,x;j)<0. Thisleads firmsto hire a suboptimally highnumber ofworkers – this is the well- knownoverhiringresultintheintrafirmbargainingliterature(see(AcemogluandHawkins,2014;CahucandWasmer,2001;

StoleandZwiebel,1996).Now,whenfirmsareallowedtoadjusttheirlaborinputthroughthehoursmargin,anewworker reduceshoursworkedofthefirm’sotheremployeesbyshiftingproductionfromtheintensivetotheextensivemargin,such thathn(n,x;j)<0.Thissubstitutioneffect,capturedbythetermhn(n,x;j),isabsentintheintrafirmbargainingmodelthat abstractsfromhours.

Theelasticityofhourstoemployment,measuredby^{,drivesthestrengthoftheoverhiring}externality.Since0≤

α

≤1, 0≤

γ

≤1and

σ

≥0,thewagedependsnegativelyonemployment.Noticethattheslopeofthewagecurve(17)withrespect ton,dependsontheIFBexternalitythrough^{.Withalinearproductionfunction(}

α

=1)orinelasticlaborsupply(

σ

→∞), there isno substitution between employment andhours (=0) and thebargained wage is invariant to the numberof employeeswithin thefirm(w_n(ⁿ,x)=0).Instead,themoresubstitutablearethetwolabormargins(thehigheris^),the moreanadditionalworkerreduceshoursworkedand,inturn,theequilibriumrealwage.

4.2.Calibration

Themodel parametersare calibratedata quarterly frequencyfortheEuro Area. Table1lists theexternally calibrated parameters and Table 2 describes the remaining variables implied by the model’s steady state, given these parameters.

8( Cahuc and Wasmer, 2001 ) highlight the conditions under which a multiple-worker firm model with constant returns to scale is equivalent to the standard one-worker-firm model.

All externally calibrated parameters in Table1 are either set to valuescommonly accepted inthe literature or to target empiricalevidencefromtheEuroArea.Thediscountfactorinhouseholdpreferencesissetto

β

=0.99,implyinga steady state annualizedrealinterestrateof4%. WesettheinverseFrischelasticityoflaborsupplyto

σ

=3,whichcorresponds tothe intermediatevaluessuggestedby KeaneandRogerson(2012).Sincethewageisnot equalto themarginalproduct oflaborinasearch-and-matchingmodel,thevalueoftheproductionelasticityoflabor

α

^{isnotexactlyequaltothelabor}

share.Weset

α

=0.6,whichisclosetotheconventionalvalue inthefrictionlessmodel.Thisvalue impliesasteadystate laborshareof75%inourbaselinecalibration.Itislargerthantheaveragelaborshare(55%)intheEuroArea.⁹Steadystate TFP, A¯,is deducedfromthe productionfunction withmarketoutput normalizedto unity(Y¯=1). The scaleparameter in labordisutility,

ζ

^{,iscalibratedtomeetthetargeth¯}=0.3.Thistargetmeansthatroughlyonethirdofthetimeendowment (normalizedtoone)isspentworking.Theshareofgovernmentspending inGDP,G¯/Y¯,issetto21%,whichcorrespondsto theaverageshareofpublicspendingintheEuroArea.

TherestoftheexternallycalibratedparametersinTable1arespecifictothesearch-and-matchingliterature.Thesteady state unemploymentrate isset to9.6%, whichcorresponds to the averageunemploymentrate inthe Euro area between 1999and2015.Theelasticityofthenumberofmatchestounemploymentissetto

η

=0.6,whichisthemid-pointofthe range0.5–0.7 proposed by Petrongolo andPissarides (2001).The worker’s bargainingweight,

γ

^{,is alsoset to0.6, which}

isclosetothe valuesuggestedby Christoffeletal.(2009).¹⁰ Following(Christoffel etal.,2009), wesetthevacancy filling rate,q¯,to0.7andthejob separationrate,

λ

^{,to0.03,inlinewithEuroarea dataonjob flows.Totalvacancycosts amount}

to1%ofGDP(c

v

¯/Y¯=0.01);thisislargerthanthevalue inChristoffeletal.(2009)butitisinlinewithAndolfatto(1996); GertlerandTrigari(2009)and(Sunakawa,2015).

Giventhesecalibratedvalues,wecandeducetheremainingparametersimpliedbythemodel’ssteadystate(seeTable2).

Homeproductionaccountsfora smallproportionofmarketoutput,bu¯/Y¯=0.05(forb=0.5)^{assuggestedbyGertlerand} Trigari(2009).Interpretingthehomeproductionparameterbasanunemploymentbenefit,wehaveareplacementrate, _w^b_¯, of0.60,whichisclosetoChristoffeletal.(2009)valueof ^b

¯

wh¯ =0.65,wheretheirw¯,however,denotesthehourlywagerate.

Ourimplied steadystatejobfindingrateis p¯=0.28andthelabormarkettightness,

θ

¯,is0.4,bothvaluesbeingveryclose toChristoffeletal.(2009),whoalsocalibratetheirmodeltotheEuroArea.

ThevaluesweobtainfortechnologyA¯,governmentspendingG¯,theweightonlaborinutility

ζ

^{,thematchingtechnol-}

ogyM₀,homeproductionbandvacancypostingcostcarethenkeptfixedacrossallocations.

4.3. Illustration

Fig.3displaysthecompetitiveandefficientallocationsinthelabormarket.Intheupperpanel,westudytheemployment margin by plotting,in (^u,

v

)^{-space, the}downward-sloping BeveridgeCurve together withtheupward-sloping competitive andefficientjobcreationconditions(JCC).TheBeveridgeCurveisthelawofmotionforemployment(1)rewrittenas:

v

=

(

¹−u

) λ

M₀u^η

1−¹η

. (24)

Thecompetitive andefficientJCCcurves,i.e.Eq.(25)and(26)respectively, areobtainedby usingtheoptimality condition forhours(13)intherespectiveexpressionsfortheshadowvalue,(19)and(23),suchthat:

1−

β (

¹⁻

λ ) β

c M0

v

u

η

=

(

¹⁻

γ )

1

1−

γ (

¹+

σ )

^·

σ

1+

σ α

^A

⁽⁽

^1−u

⁾

^h

⁾

^α−1h−b

−

γ

^c

v

u, (25)

1−

β (

¹⁻

λ ) β

c M0

v

^∗

u^∗

η

=

(

¹⁻

η ) σ

1+

σ α

^A

⁽⁽

^1−u∗

⁾

^h∗

⁾

^{α−1h∗−b}

−

η

^c

v

^∗

u^∗. (26)

Comparing(25)and(26),where[1−

γ

(¹+

σ

)^]−1>1,weseethatinthecompetitiveallocation,vacanciesrespondmore to unemploymentthan in theefficient allocation. The competitive JCCcurve issteeper than the efficient JCCcurve. This illustratestheoverhiringdistortion;theequilibriumunemploymentrate– attheintersectionbetweentheBeveridgeCurve andthecompetitive JCC– islower thantheefficientunemploymentrate.The lowerpanel inFig.3showsthathours per employeearetoolowinthecompetitiveallocation.Lookingatthehoursdecision(13),inefficiencyintheintensivemargin in labor onlycomes fromthe misallocation inemployment. Put differently,since hours andemployment are substitutes, overhiringgoeshandinhandwithsuboptimallylowhours.Withsubstitutabilitybetweenhoursandemployment,hiringa workerallowsthefirmtoreduce hoursperworker.InTrigari(2006)’sone-worker-firmsetupwherebothwagesandhours

9We use data from Eurostat, over the sample 1995q1-2015q4, the labor share is measured as the ratio between total compensation and gross value added. We provide robustness exercises for the normative analysis in Section 5 and in the online appendix for the positive analysis.

10Christoffel et al. (2009) suggest γ^{= 0 . 5} for the Euro Area. We prefer the value γ⁼ ηin the spirit of the so-called ( Hosios, 1990 ) condition. As shown by Cahuc et al. (2008) and in Proposition 1 , this condition is not sufficient to ensure an efficient level of employment in our model. However, γ⁼ η^ensures that the inefficiency in the extensive margin comes from the concavity in the production function only, and thus we are able to study the IFB distortion in isolation.

Fig. 3. Employment and hours distortion. Note: In the upper panel, the solid line displays the Beveridge curve, the dotted line displays the competitive JCC and the dashed line displays its efficient counterpart. In the lower panel, the dotted line ‘MRS’ is the marginal rate of substitution, ^ζh(n,x)σ; the solid line displays the competitive marginal product of hours α^A (nh (n, x ))^α−1and the dashed line is its efficient counterpart.

are determined throughNash bargaining, hours are set efficiently.This is what (Trigari, 2006) calls ‘efficient bargaining’.

Here,duetotheIFBexternality,hoursperworkerarenotefficientdespitebeingdeterminedbyNashbargaining.Theinef- ficiencyintheextensivemargin carriesovertotheintensivemargin.InSection5,weinvestigatehowtheinclusionofthe intensivemarginaffectsthisoverhiringbehavior.

4.4.Optimalunemploymentinsurance

Wehaveshownthatinourmodel,firmstendtohiretoomanyworkersandeachoneofthemworkstoofewhours.How canpolicy address thisdistortion? Giventhat thedistortion worksonlythrough the shadowvalue ofa marginalworker, any instrumentthat lowers this shadow value can be used to restore efficiency. An additional worker should be made