Journal of Economic Dynamics & Control

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1 Journal of Economic Dynamics & Conrol 37 (213) Conens liss available a SciVerse ScienceDirec Journal of Economic Dynamics & Conrol journal homepage: Search fricions, real wage rigidiies and he opimal design of unemploymen insurance Julien Alberini a,b, Xavier Fairise c,n a THEMA, Universiy of Cergy-Ponoise, 33, boulevard du Por, 9511 Cergy-Ponoise Cedex, Sie des Chênes 1, France b CEPREMAP, France c GAINS-TEPP, Universié du Maine, Avenue Olivier Messiaen, 7285 Le Mans Cedex 9, France aricle info Aricle hisory: Received 21 July 29 Received in revised form 24 July 212 Acceped 25 March 213 Available online 31 March 213 JEL classificaion: E61 E65 J63 J65 Keywords: DSGE models Search and maching fricions Layoff ax Wage rigidiies absrac In his paper, we sudy he opimal unemploymen benefis financing scheme when he economy is subjec o labor marke imperfecions characerized by real wage rigidiies and search fricions. The US unemploymen insurance financing is such ha firms are axed proporionaely o heir layoffs o finance unemploymen benefis. Using DSGE mehodology, we invesigae how policy insrumens should inerac wih labor marke imperfecions. I is shown ha wage rigidiies in a search and maching environmen cause welfare coss, especially in he absence of an incenive-based unemploymen insurance. This cos is mainly due o he disoring effec of wage rigidiies which generae inefficien separaions. We show ha he opimal unemploymen benefis financing scheme corresponding o he Ramsey policy offses labor marke imperfecions and allows implemenaion of he Pareo allocaion. The second-bes allocaion brings he economy close o he Ramsey allocaion. The implemenaion of he opimal policies clearly highlighs he role of labor marke insiuions for shor-run sabilizaion. & 213 Elsevier B.V. All righs reserved. 1. Inroducion Wha is he opimal level of unemploymen benefis and how should hey be financed? This quesion is ofen discussed wihou aking ino consideraion labor marke rigidiies or he poenial role of labor marke insiuions for shor-run sabilizaion, leaving he job o macroeconomic policy, and especially moneary and budge policy. Indeed, he labor marke is characerized by search fricions and wage rigidiies which disor agens' job accepance behavior and firms' job posings. This may generae inefficiencies and affec labor marke performance as well as social welfare. Furhermore, i influences he response of macro-economic variables o aggregae shocks and can magnify flucuaion coss. The quesion of using labor marke policy o reduce flucuaion coss and o offse labor marke imperfecions naurally arises. Taking inspiraion from he US unemploymen insurance (UI hereafer) sysem, we wonder if firms should be axed in proporion o heir layoffs o finance he coss incurred by he unemploymen benefis fund. This paper invesigaes he opimal design of an unemploymen benefis financing scheme in a DSGE framework. While i is ofen argued ha labor marke insiuions can affec long-run labor marke performance, hey have received lile aenion in he field of shor-run sabilizaion. The marked inensiy of business cycles in he US and he high volailiy n Corresponding auhor. Tel.: addresses: [email protected] (J. Alberini), [email protected] (X. Fairise) /$ - see fron maer & 213 Elsevier B.V. All righs reserved. hp://dx.doi.org/1.116/j.jedc

2 J. Alberini, X. Fairise / Journal of Economic Dynamics & Conrol 37 (213) of unemploymen and vacancies highlighs he usefulness of sabilizaion policies based on he search for an opimal design of he UI. Moreover, he exisence of labor marke rigidiies gives rise o a complemenary moivaion. These rigidiies can be summed up as wo caegories: hose which limi quaniy adjusmens and hose which limi price adjusmens. Maching fricions ypically represen he firs caegory. They capure he ime-consuming search process and generae congesion exernaliies. They influence he average duraion of unemploymen and herefore he fiscal cos associaed wih a dismissal. The second caegory corresponds o real wage rigidiies. These have been poined ou by many auhors (Hall, 25; Shimer, 25; Chrisoffel and Linzer, 21) as a means of solving he unemploymen volailiy puzzle. They preven wages from adjusing insananeously o economic flucuaions. Consequenly, shocks ranslae ino quaniies such as employmen, job creaions and job desrucions. Furhermore, hey capure rigidiies arising from wage norms. They reduce he abiliy of firms and workers o use axes and benefis as a hrea in wage bargaining. Following Abbrii and Weber (28), he naure of adjusmens in he US labor marke may be linked o is insiuions which allow srong quaniy adjusmens associaed wih significan real wage rigidiies. 1 The idea ha maching fricions and wage rigidiies could inerac arises. These labor marke rigidiies inroduce some inefficiencies and leave room for policy insrumens o reduce inefficiencies and sabilize labor marke flucuaions. There is a growing lieraure abou flucuaion sabilizaion and labor marke imperfecions. However, mos of his lieraure is cenered on he design of he opimal moneary policy (see Chrisoffel and Linzer, 21; Blanchard and Gali, 21; Abbrii and Weber, 28). Zanei (211), Joseph e al. (24) and Faia (28) inroduce some labor marke insiuions such as unemploymen benefis, firing coss or a minimum wage in a DSGE model and sudy heir implicaions for business cycle dynamics. Krause and Lubik (27) show ha labor marke fricions and real wage rigidiies explain he negaive correlaion beween vacancies and unemploymen. However, heir conribuion in explaining inflaion dynamics is small. Campolmi and Faia (211) show ha labor marke insiuions, and especially he heerogeneiy of he replacemen raios, help o explain inflaion differenials beween he counries of he euro area. From a posiive poin of view, labor marke insiuions maer o explain labor marke dynamics and inflaion volailiy differences in he euro area. Krause and Lubik (27) and Campolmi and Faia (211) focus on moneary policy issues bu none of hem characerizes he opimal design of hese insiuions. The scope of our paper slighly depars from hese sudies as we only focus on unemploymen dynamics and on he design of unemploymen benefis financing. In he US, he rapid increase in unemploymen afer bad shocks can be relaed o he weakness of he employmen proecion legislaion and he exen o which firms can layoff workers a no cos. Inuiion suggess ha if firms do no pay he enire cos of heir dismissals, heir incenive o fire is higher. The role of unemploymen insurance in magnifying permanen and emporary layoffs has been illusraed by a large body of papers (Feldsein, 1976; Topel, 1983; Card and Levine, 1994). In his sysem based on he experience raing principle, individual employers' conribuion raes are varied on he basis of he firm's hisory of generaing unemploymen. Basically, more dismissals resul in a higher conribuion by firms o unemploymen insurance. Blanchard and Tirole (28) highligh he quesion of he design of unemploymen insurance and is link wih employmen proecion. The respecive levels of he layoff ax and of he unemploymen benefis may play a key role in achieving an opimal allocaion. In order o characerize he opimal UI financing scheme, we build a DSGE model wih search and maching fricions and where job creaions and job desrucions are boh endogenous. I has been widely recognized ha as firms bear a small share of he oal cos of job desrucions (due o imperfec experience raing), he UI induces oo many layoffs. Moreover he exisence of search fricions and wage rigidiies can srongly affec he firm's hiring and firing behavior. Then we wonder how much firms should be axed in proporion o heir layoffs o finance he fiscal cos induced by heir redundancies. Using he DSGE mehodology we compue he Ramsey policy. Under his seup we sudy (i) how he opimal policy can offse labor marke imperfecions (ii) how i reduces he welfare cos and (iii) how i affecs he business cycle. We show ha an opimal combinaion of unemploymen benefis and layoff axes is welfare-enhancing and can improve labor marke performance. Wage rigidiies in a search and maching environmen have a srong impac on he welfare cos, especially in he absence of an incenive-based unemploymen insurance. This cos is mainly due o he disoring effecs of wage rigidiies which generae inefficien separaions. The opimal unemploymen benefis financing scheme corresponding o he Ramsey policy allows he implemenaion of he Pareo allocaion. As he Ramsey policy may no be easily implemenable in he presen labor marke insiuions, we define a second-bes allocaion. The replacemen rae and an experience raing degree are chosen in order o minimize he welfare cos. This policy brings he economy close o he Ramsey allocaion. The implemenaion of he opimal policies clearly highlighs he role of labor marke insiuions for shor-run sabilizaion. The res of he paper is organized as follows. Secion 2 presens he model and he unemploymen insurance sysem. The equilibrium and he opimal policies are defined in Secion 3. Secion 4 is devoed o simulaion exercises and Secion 5 concludes. 2. The economic environmen and he model We build a discree ime DSGE model including a non-walrasian labor marke and endogenous job desrucions in he spiri of Morensen and Pissarides (1994) and Den Haan e al. (2). Workers may be employed or unemployed and endogenous separaions occur because of specific produciviy shocks o jobs. There are search and maching fricions in he labor marke. 1 Abbrii and Weber (28) esimae he degree of real wage rigidiy on OECD counry daa. Their esimaes sugges ha flexible labor markes are associaed wih a high degree of real wage rigidiy.

3 1798 J. Alberini, X. Fairise / Journal of Economic Dynamics & Conrol 37 (213) Wages are deermined hrough a Nash bargaining process. Following Hall (25), we inroduce real wage rigidiies hrough awage norm consraining wage adjusmens. There are no oher marke imperfecions The labor marke The search process and recruiing aciviy are cosly and ime-consuming for boh firms and workers. The number of maches M is given by he following Cobb Douglas maching funcion: M ¼ χð1 N Þ ψ V 1 ψ wih ψ Š; 1½; χ 4 ð1þ wih V he vacancies and 1 N he unemployed workers. The labor force is normalized o 1, he number of unemployed workers saisfies U ¼ 1 N.Avacancyisfilledwihprobabiliyq ¼ M =V.Leθ ¼ V =ð1 N Þ be he labor marke ighness. The probabiliy of an unemployed worker finding a job is θ q ¼ M =ð1 N Þ. I is useful o rewrie hese probabiliies as follows: q ¼ χ 1 N ψ ð2þ V V 1 ψ θ q ¼ χ ð3þ 1 N A each dae, a job is characerized by is specific produciviy level ε drawn from he disribuion GðÞ defined on ½; εš. The job produciviy is also subjec o an aggregae produciviy shock z. The producion level is given by y ¼ z ε ð4þ The sequence of evens and he labor marke iming mainly follow (Zanei, 211). Employmen in period has wo componens: new and old workers. New employmen relaionships are formed hrough he maching process. Maches formed in period 1 conribue o period employmen. We assume new jobs begin wih he highes produciviy 2 level ε. Le N N ¼ M 1 denoe he new employmen relaionships. A he beginning of period, N 1 jobs are inheried from period 1. Separaions occur for wo reasons. Firsly, here are exogenous separaions such ha ρ x N 1 jobs are desroyed. Secondly, idiosyncraic shocks are drawn. If he job specific produciviy ε falls below an endogenous hreshold ε, he job is desroyed. A fracion ρ n ¼ Pðεoε Þ¼Gðε Þ of he remaining jobs ð1 ρx ÞN 1 is desroyed. Le n C ðεþ, wih ε ½ε ; εš, denoe he number of coninuing produciviy ε employmen relaionships. I saisfies n C ðεþ¼ð1 ρx ÞN 1 G ðεþ To sum up, he employmen law of moion is described by he following equaions: ð5þ N N þ1 ¼ M ð6þ N ¼ ε n C ðεþ dε þ NN ð7þ N ¼ð1 ρ x Þð1 ρ n ÞN 1 þ N N ρ n ¼ Gðε Þ ð8þ ð9þ 2.2. The large family program Each household may be viewed as a large family. There is perfec risk sharing, family members pooling heir incomes (labor incomes and unemploymen benefis), which are equally redisribued. We consider an economy wih maching fricions and wage rigidiies. Furhermore, here are unemploymen benefis financed by layoff axes. In order o allow a simple comparison wih wha is usually done, we focus only on he disoring impac of he UI sysem on he labor marke oucomes. The large family assumpion allows us o isolae he disoring effec of he UI variables on he labor marke oucomes and on welfare. 3 In oher words, axes and benefis do no aler labor marke oucomes hrough precauionary saving, leaving aside he delicae problem of opimal insurance (moral hazard). I should also be sressed ha Krusell 2 This assumpion follows ha of Morensen and Pissarides (1994). Assuming ha new jobs sar wih he highes produciviy may be a srong assumpion. According o Morensen and Pissarides (1994), exising firms creae mos of he new jobs over he cycle. In a given secor, exising firms are well informed abou he gains from he producion of a new differeniaed produc. Therefore, i is assumed ha new job produciviy is higher han ha of exising jobs. In addiion, his assumpion abou he produciviy of new jobs makes he analysis more racable. 3 If he uiliy funcion is linear, he model is compleely recursive. The labor marke dynamics do no depend on consumpion choices. In he case of a CRRA uiliy funcion, labor marke dynamics only depend on he marginal uiliy of consumpion, and hus, i follows ha saving choices do no direcly affec labor marke oucomes.

4 J. Alberini, X. Fairise / Journal of Economic Dynamics & Conrol 37 (213) e al. (21) show ha labor marke flucuaions are quaniaively very similar in heir model wih heerogeneous agens and precauionary saving and in large family models (Merz, 1995; Andolfao, 1996). The large family maximizes is lifeime uiliy. The large family's choice problem akes he following recursive form: ( ) V M ðθ Þ¼max fcg ðc þð1 N ÞhÞ 1 s þ βe V M ðθ þ1 Þ 1 s ð1þ 8 C þ R ε ε n C ðεþw ðεþ dε þ N N wn ðεþþð1 N Þb T þ Π ¼ ðλ Þ >< N þ R ε ε n C s:: ðεþ dε þ Nn ¼ ðμ1 Þ N N þ1 þ θ q ð1 N Þ¼ ðμ 2 >: Þ ð1 ρ x ÞN 1 G ðεþ n C ðεþ¼ ðμ ðεþþ; ε ½ε ; εš wih he sae vecor Θ ¼ðN 1 ; N N ; z Þ, aken as given he job finding probabiliy. β Š; 1½ is he discoun facor and s Š; 1½ Š1; ½ is he ineremporal elasiciy of subsiuion. h denoes unemployed workers home producion. Family consumpion is he sum of he oal home producion ð1 N Þh and of he marke consumpion goods C. b is he unemploymen benefi and w ðεþ denoes he wage associaed wih he produciviy level ε. The wage in a new job wries w N ðεþ. Finally, he large family receives insananeous profis for an amoun Π. The firs consrain is he budge consrain and he hree oher consrains describe he employmen moion. The consumpion opimaliy condiion wries ðc þð1 N ÞhÞ s ¼ λ ð11þ From he envelope condiions of he above problem, we deduce expressions providing he ne presen value of a new mached worker S M ðεþ and he ne presen value of an old mached worker SM ðεþ : S M ðεþ¼wn ðεþ b h θ q βe λ þ1 λ Z S M λ ε þ1 þ1 ðεþþð1 ρx ÞβE S M þ1ðεþ dgðεþ λ ð12þ ε þ1 S M ðεþ¼w ðεþ b h θ q βe λ þ1 λ Z S M λ ε þ1 þ1 ðεþþð1 ρx ÞβE S M þ1ðεþ dgðεþ λ ð13þ ε þ The large firm program A firm may be viewed as a large firm having many jobs and employing many workers. The large firm chooses a sequence of vacancies, hresholds and he number of coninuing employmen relaionships, ha is D ¼ðV ; ε ; fn C ðεþg ε ½ε ;εš Þ, maximizing he expeced discoun sum of insananeous profis subjec o a se of consrains describing he employmen moion. The large firm's choice problem akes he following recursive form ( V F ðδ Þ¼max D ε N 1 τ E þ τe z εn C ðεþ dε þ NN z ε ε n C ðεþ dε þ βe ε w ðεþn C ðεþ dε NN wn ðεþ κv λ þ1 V F ðδ þ1 Þ λ 8 N þ R ε ε n C ðεþ dε þ Nn ¼ ðλ1 >< Þ s:: N N þ1 þ q V ¼ ðλ 2 Þ >: ð1 ρ x ÞN 1 G ðεþ n C ðεþ¼ ðζ ðεþþ; ε ½ε ; εš ) ð14þ wih he sae vecor Δ ¼ðN 1 ; N N ; z Þ. The firm incurs a cos κ per vacancy posed. Old jobs, which are no exogenously desroyed, do no coninue if heir specific produciviy level is below he hreshold ε. The firm pays a ax τ E per separaion. The oal amoun of ax paid by he firm a dae is ½ρ x þð1 ρ x Þρ n ŠN 1τ E ¼ N R 1τ E ε τe ε n C ðεþ dε. The opimaliy condiions and he envelope condiions, afer some manipulaions, provide he ne presen value of a new job S F ðεþ, he ne presen value of coninuing jobs SF ðε Þ, he creaion and he desrucion condiions, ha is Z S F ðεþ¼z ε w N λ ε þ1 ðεþþð1 ρx ÞβE S F þ1 λ ðεþ dgðεþ βe λ þ1 τ E þ1 ð15þ ε þ1 λ

5 18 J. Alberini, X. Fairise / Journal of Economic Dynamics & Conrol 37 (213) Z S F ðεþ¼z ε w ðεþþτ E λ ε þ1 þð1 ρx ÞβE S F þ1 λ ðεþ dgðεþ βe λ þ1 τ E þ1 ð16þ ε þ1 λ κ λ þ1 ¼ βe S F þ1ðεþ ð17þ q λ S F ðε Þ¼ ð18þ 2.4. Wage seing mechanism We have o disinguish wo wages, he wage of a new job and he wage of a coninuing job. The wo wages are deermined hrough an individual Nash bargaining process beween a worker and a firm who share he oal surplus. The bargaining process provides opimal rules for surplus sharing, ha is ξs M ðεþ¼ð1 ξþsf ðεþ and ξsm ðεþ¼ð1 ξþsf ðεþ, where ξ Š; 1½ and 1 ξ denoe he bargaining power of firms and workers respecively. The following expressions are obained for he wages: w N λ þ1 ðεþ¼ð1 ξþ z ε þ κθ βe τ E þ1 þ ξðb þ hþ ð19þ λ w ðεþ¼ð1 ξþ z ε þ κθ þ τ E βe λ þ1 τ E þ1 þ ξðb þ hþ ð2þ λ Boh equaions ake ino accoun he expeced firing coss. During he bargaining, firms inernalize he fac ha hiring a worker may be cosly if he job is desroyed. Eqs. (19) and (2) differ because of he layoff ax. Concerning an old job, firing coss should be paid in case of separaion. Each pary may use he cos of layoffs as a hrea. Real wage rigidiy: Following Shimer (25) and Hall (25), real wage rigidiies are inroduced. There exiss a wage norm ~w consraining wage adjusmen. The real wage paid for a given produciviy level job is a weighed average of he Nash bargaining process wage and he wage norm ~w. One has w N ðεþ¼ð1 γþwn ðεþþγ ~w ð21þ w ðεþ¼ð1 γþw ðεþþγ ~w ð22þ wih γ ½; 1Š. The higher he γ, he more he real wages are rigid. The wage norm ~w can be defined in differen ways. Following Krause and Lubik (27), we suppose he norm is equal o he seady sae value of he average wage ~w ¼ w. I should be sressed ha his form of rigidiy is a source of inefficiencies (see Krause and Lubik, 27). A he hreshold ε, he firm's surplus is equal o zero S F ðε Þ¼, whereas he worker's surplus is posiive SM ðε Þ4. As he job oal surplus is posiive, workers and firms could agree o reduce wages and a lower hreshold could be aained. This poin will be discussed laer Job creaion and job desrucion Job creaion and job desrucion are deermined by Eqs. (15) (18). Using he wage seing equaions (21) and (22), he job creaion and job desrucion condiions respecively wrie κ λ þ1 þð1 ð1 γþð1 ξþþβe ðz þ1 ε z þ1 ε q λ þ1 τ E þ1þ¼ ð23þ ð1 ð1 γþð1 ξþþðz ε þ τ E Þ ð1 γþð1 ξþκθ γ ~w ð1 γþξðb þ hþ þð1 ð1 γþð1 ξþþð1 ρ x λ þ1 ÞβE ðz þ1 ε z þ1 ε λ þ1 Þ dgðεþ ε þ1 λ þ1 ð1 ð1 γþð1 ξþþβe τ E þ1 λ ¼ ð24þ Eq. (23) says he expeced gain from hiring a new worker is equal o he expeced cos of search (which is κ imes he average duraion of a vacancy 1=q ). Eq. (24) is he job desrucion condiion, i eaches us ha he criical value of a job's produciviy depends on he reservaion wage and on firing coss The unemploymen insurance financing An unemployed worker receives a benefi b. Unemploymen benefis are financed hrough a layoff ax and a lump-sum ax paid by he large family. The layoff ax is paid by employers in case of separaion, eiher endogenous or exogenous.

6 J. Alberini, X. Fairise / Journal of Economic Dynamics & Conrol 37 (213) We impose he condiion ha unemploymen benefis may no be financed by deb. The UI fund budge consrain is hus balanced in every period: ð1 N Þb fflfflfflfflffl{zfflfflfflfflffl} Unemploymen benefis ¼ T {z} Lump sum ax þ½ρ x þð1 ρ x Þρ n ŠN 1τ E fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Layoff ax ð25þ The sequences followed by T, τ E and b may be chosen following differen mehods, provided hey saisfy he above budge consrain. Our aim is o evaluae an unemploymen benefis financing scheme close o he US labor marke insiuions and o sudy is opimaliy. We have o describe an insiuional rule seing axes and unemploymen benefis level which is close o he US sysem. An unemployed worker receives a benefi b equal o a proporion of he average wage w, ha is b ¼ ρ R w ρ R o1 is he average replacemen rae. The average wage of he economy w is given by w ¼ NN w N N ðεþþn 1 ð1 ρ x Þ w ðεþ dgðεþ ð27þ N ε Using Eqs. (19) and (2), he above equaion may usefully be rewrien as follows: λ þ1 N w ¼ N ð1 γþξðb þ hþþn ð1 γþð1 ξþ κθ βe ( Z ) ε þð1 γþð1 ξþ N N z ε þ N 1 ð1 ρ x Þz ε dgðεþ ε λ τ E þ1 ð26þ þn 1 ð1 ρ x Þð1 ρ n Þð1 γþð1 ξþτe þ N γ ~w ð28þ Concerning he seing rule of he layoff ax τ E, he US unemploymen insurance is characerized by a sysem known as experience raing. We adop a simplified represenaion based on Cahuc and Malherbe (24). The layoff ax is proporional o he expeced fiscal cos of an unemployed worker Q þ1. Le e4 be he experience raing index (ERI), he firing ax τ E saisfies τ E ¼ eq ð29þ where Q ¼ b þ βe fθ q þð1 θ qðθ ÞÞQ þ1 g ð3þ The above equaion recursively deermines he expeced cos of an unemployed worker. The layoff ax is proporional o he expeced fiscal cos of an unemployed worker paid by he firm. If e¼1, firms incur he full expeced fiscal cos of an unemployed worker. The above represenaion allows us o approximae in a simple way he US unemploymen insurance sysem. 3. The equilibrium and he unemploymen benefi financing policies This secion is devoed o he definiion of he equilibrium and o he descripion of he unemploymen insurance financing policies The aggregae resource consrain The aggregae oupu Y is obained hrough he sum of individual producions: Y ¼ð1 ρ x ÞN 1 z ε dgðεþþn N z ε ð31þ ε The aggregaion of he individual profis provides he amoun of profis Π received by he large family, ha is Π ¼ Y w N κv τ E ð1 ρx Þρ n N 1 The above equaion ogeher wih he large family budge consrain (program (1)) and he governmen budge consrain (Eq. (25)) give he aggregae resource consrain: Y ¼ C þ κv ð32þ

7 182 J. Alberini, X. Fairise / Journal of Economic Dynamics & Conrol 37 (213) Definiion of he equilibrium Our aim is o compare he allocaion implied by he insiuional unemploymen benefis financing scheme (described in Secion 2.6) o he ones implied by any oher ax process. Two definiions of he equilibrium, corresponding o he differen financing schemes, need o be given. Definiion 1. For given lump sum ax T and firing ax τ E processes, and for a given exogenous sochasic process z, he compeiive equilibrium is a sequence of prices and quaniies N, N N, C, V, ε, θ, λ, q, w, Y, ρ n, M and b saisfying Eqs. (1) (3), (6), (8), (9), (11), (23) (25), (28), (31) and (32). Finally, if axes and benefis are se according o Eqs. (26), (29) and (3), he equilibrium definiion wries as follows: Definiion 2 (Experience raing sysem). For given parameers ρ R and e and for a given exogenous sochasic process z, he compeiive equilibrium is a sequence of prices and quaniies N, N N, C, V, ε, θ, λ, q, w, Y, ρ n, M, b, τ E, T and Q saisfying Eqs. (1) (3), (6), (8), (9), (11), (23) (25), (28), (31), (32), (26), (29) and (3) The Pareo allocaion The Pareo allocaion could be derived from he cenral planner's problem. However, i can be direcly deermined. To do his, consider Eqs. (23) and (24). Suppose he Hosios condiion is saisfied, here are no wage rigidiies and unemploymen benefis and axes are equal o zero. One ges he following creaion and desrucion condiions: κ V λ þ βe fλ þ1 ðz þ1 ε z þ1 ε 1 ψ M þ1 Þg ¼ ð33þ ( λ ðz ε hþ κ ψ Z!) V ε 1 ψ 1 N λ þ βð1 ρ x ÞE λ þ1 ðz þ1 ε z þ1 ε þ1 Þ dgðεþ ¼ ε þ1 We can give he exensive definiion of he Pareo allocaion (or equivalenly he firs-bes), ha is Definiion 3 (The Pareo allocaion). For a given exogenous sochasic process z, he Pareo allocaion is a sequence of quaniies N, N N, C, V, ε, λ, Y, ρ n, M saisfying Eqs. (1), (9), (6), (8), (11), and (31) (34) The unemploymen insurance policies If he Hosios condiion is no saisfied or if he real wage is rigid, he decenralized equilibrium of he economy wihou unemploymen benefis and axes is no opimal. In his subsecion, we define he wo unemploymen insurance policies. Firsly, we define he Ramsey allocaion. We show ha his policy allows he implemenaion of he Pareo allocaion. Secondly, we define a second-bes allocaion The Ramsey allocaion The Ramsey policy is he axaion policy under commimen maximizing he ineremporal welfare of he represenaive household. Definiion 4 (The Ramsey allocaion). The Ramsey equilibrium is a sequence of prices, quaniies and axes N, N N, C, V, ε, θ, λ, q, w, Y, ρ n, M, b, T, τ E maximizing he represenaive agen's life-ime uiliy: E j ¼ β j ðc þj þð1 hþn þj Þ 1 s 1 s subjec o he equilibrium condiions (1) (3), (6), (8), (9), (11), (23) (25), (28), (31) and (32) and given he exogenous sochasic process z The Pareo allocaion and he equivalence wih he Ramsey allocaion ð34þ Proposiion 1. The Ramsey policy (Definiion4) subjec o he wage norm 4 ~w ¼ w implemens he Pareo allocaion. Proof. See Appendix A. Proposiion 1 provides a simple way o deermine he axes and unemploymen benefi processes implemening he Pareo allocaion. The equilibrium values of N, N N, C, V, ε, θ, λ, q, Y, ρ n and M are deermined using Eqs. (1), (9), (6), (8), (11), and (31) (34), he exogenous sochasic process z being given. The processes followed by he average wage, he axes and unemploymen benefis T, τ E and b are hen easily deduced from Eqs. (28), and (23) (25). Finally, w N is provided by Eq. (19). 4 This resul can easily be exended o he case ~w ¼ w 1. Numerical invesigaions show ha resuls do no significanly differ from he case ~w ¼ w.

8 J. Alberini, X. Fairise / Journal of Economic Dynamics & Conrol 37 (213) Corollary 1. The opimal policy obained by solving he Ramsey allocaion problem is ime-consisen. Proof. See Appendix A. The Ramsey problem which commis o long-run plans and chooses he fuure policies oday is equivalen o he policy inervenions decided on a period-by-period opimizaion. In oher words, he governmen has no ineres in going back on is word and rying o deviae from he opimal policy rules chosen in he Ramsey problem, over ime. Proposiion 2. If he Hosios condiion (ξ ¼ 1 ψ) holds bu wages are rigid (γ 4) he compeiive equilibrium will no implemen he Pareo allocaion. Furhermore, he opimal layoff ax and unemploymen benefis implemening he firs-bes allocaion are no equal o zero. Proof. Firsly, compare (23) and (24) o (33) and (34) and se ξ ¼ 1 ψ. I immediaely follows ha efficiency canno be achieved. Secondly, using Eqs. (23), (24), (33) and (34) aken a he seady sae, he opimal layoff ax and unemploymen benefis (τ EO and b O respecively) are obained: τ EO ¼ κ V ψ ð1 γþð1 ξþ ð35þ β M ð1 ð1 γþð1 ξþþð1 ψþ b O ¼ 1 ð1 γþξ κv ψ ð1 γþð1 ξþ 1 ψ 1 β β 1 M þ 1 þ γðh wþ 1 N I is easy o see ha he usual Hosios condiion does no imply τ E ¼ and b¼ when oγ o1. In our economy, here is no reason why he equilibrium should be a Pareo opimum. This occurs if here is real wage rigidiy or, wihou wage rigidiy, if he Hosios condiion is no saisfied, ha is if 1 ψ ξ. ð36þ The second-bes allocaion The replacemen rae ρ R and he index e are se by he policy maker. Thereafer, quaniaive evaluaions are made using a benchmark calibraion based on US daa, bu here is no reason for hese wo parameers o be opimal. Consider he iniial condiions N 1 and N N prevailing a dae ¼. Saring from his sae, each se of parameers ðe; ρ RÞ characerizes an equilibrium allocaion (Definiion 2). We denoe respecively by ~ C and ~ N he consumpion and employmen equilibrium allocaion. The associaed condiional welfare wries ~Wðρ R ; e; N 1 ; N N Þ¼E ¼ β ð ~ C þð1 ~ N ÞhÞ 1 s 1 s The ask of he policy maker is o choose parameers maximizing condiional welfare. We define he second-bes allocaion as follows: Definiion 5 (Opimal values of ðe; ρ R Þ and second-bes allocaion). Le us denoe by ðn 1 ; N N Þ he sae of he economy a ¼. We suppose he economy is iniially a is seady sae, so ha N 1 ¼ N and N N ¼ NN. Opimal values for ρ n R and en are obained solving he following problem: ðρ n R ; en Þ¼arg max ρ R ;e ~Wðρ R ; e; N; N N Þ The second-bes allocaion is he compeiive economy equilibrium allocaion (Definiion 2) obained by imposing ρ R ¼ ρ n R and e ¼ e n The condiional welfare coss Consider he compeiive economy equilibrium (Definiion 2) and suppose we are iniially (¼) a is seady sae (N 1 ¼ N and N N ¼ NN ). Saring from his poin, le C a and N a denoe he compeiive economy allocaion and W a ðn; N N Þ he associaed condiional welfare. Finally, le W n ðn; NN Þ be he condiional welfare under he Ramsey allocaion. In oher words, i is he condiional welfare ha would be aained if, saring from N 1 ¼ N and N N ¼ NN, he Ramsey allocaion were implemened. 5 The welfare cos Ψ is obained by solving he following equaion: W n ¼ E ¼ β ½ð1 þ ΨÞðCa þð1 Na ÞhÞŠ1 s 1 s Ψ can be obained as follows: ð37þ 1=ð1 sþ Ψ ¼ Wn W a 1 wih W a ¼ E ¼ β ðca þð1 Na ÞhÞ1 s 1 s 5 Welfare comparisons are made for economies in which he iniial endowmens are he same.

9 184 J. Alberini, X. Fairise / Journal of Economic Dynamics & Conrol 37 (213) Ψ is he fracion of consumpion sream from he compeiive allocaion o be added o achieve he Ramsey allocaion welfare. Ψ is numerically compued using a second-order approximaion Quaniaive evaluaion of he model 4.1. Calibraing and solving he model The benchmark economy is calibraed according o quarerly frequencies o mimic firs and second order momens of he US economy over he period 1951Q1 24Q4 (see Table 1). We se he discoun facor o.99, which gives an annual seady sae ineres rae close o 4%. The risk aversion coefficien s is se o 2. The aggregae produciviy shock follows a firs-order auoregressive process: log z þ1 ¼ ρ z log z þ ε z þ1, where ρ z he auocorrelaion coefficien is equal o.95. ε z þ1 N ð; s2 z Þ is a random variable whose sandard deviaion is se o mach oupu volailiy (one ges s z ¼ :9). The disribuion GðÞ of idiosyncraic produciviy shocks is i.i.d. and log-normal wih mean zero. Is upper bound is locaed a 95 percenile of he disribuion, as in Zanei (211). The probabiliy of being unemployed is 3.51% on average in he US. We suppose, as in Den Haan e al. (2), Zanei (211), and Algan (24), ha exogenous separaions are wo imes higher han endogenous ones. Consequenly, ρ x ¼ :236 and ε is fixed in such a way ha a he seady sae ρ n 1=2ρ x ¼ :118. Following he sandard approach in he search lieraure (Perongolo and Pissarides, 21), he elasiciy of he maching funcion wih respec o unemploymen (ψ) is se o.5. We impose he condiion ha workers' bargaining power be equal o.5 (Hosios condiion). Our sraegy (ξ ¼ 1 ψ) allows us o offse he search exernaliies 7 generaed by he violaion of he Hosios condiion and o focus on he exernaliies coming from wage rigidiies. I should be sressed ha, as wage rigidiies affec he firms' and workers' search behavior, hey creae search exernaliies. The equilibrium unemploymen rae U is calibraed o 5.5%. A he seady sae, he number of maches mus be equal o he number of separaions: M ¼ ρn. Following Den Haan e al. (2), he rae a which a firm fills a vacancy is.71. We can deduce he number of vacancies V¼M/q and a job finding probabiliy of abou.65. χ is calculaed in such a way ha M ¼ χð1 NÞ ψ V 1 ψ. Saisics from he Census Bureau of Labor exhibi an average ERI across saes and over he period of abou.65. According o he OECD, he average US ne replacemen rae is.32. The remaining parameers κ and h are given by solving he sysem of Eqs. (23), (24) and (27) afer eliminaing w. In his way, he expeced cos of a vacan job κv=y is abou 1.4%, which is broadly consisen wih empirical findings. Finally we se s ε and he rigidiy wage parameer γ o cach up wih he observed cyclical properies 8 of he labor marke. We focus on he relaive sandard deviaion of he vacancies (s θ =s Y ) and unemploymen (s U =s Y ) wih respec o he ne oupu. The obained γ is.6 and s ε ¼ :23. Some US business cycle properies are shown in Table The opimal labor marke policy Opimal policy under he baseline paramerizaion: Simulaions are shown in Table 2. 9 Compared o he benchmark, oupu and employmen in he Ramsey allocaions are increased by around 2% and 3% respecively while wages are reduced by 1%. Vacancies increase and he sharp decline in he unemploymen rae ighens he labor marke. I resuls in a higher job finding rae. This effec is similar in he second-bes allocaion. The separaion rae is srongly reduced in he Ramsey and he second-bes allocaion. There are almos no endogenous firings a he seady sae. As menioned by Algan (24), L'Haridon and Malherbe (29), Joseph e al. (24) and Zanei (211), urnover coss inroduce a labor hoarding phenomenon. As long as firing is cosly, firms are willing o coninue he relaionship wih a low produciviy level raher han pay for he layoff ax. They cu back he reservaion produciviy o reduce endogenous separaions. The reservaion produciviy falls o a poin where endogenous separaions are close o zero. Then, ins and ous of employmen are almos only governed by exogenous separaions. The second-bes allocaion eliminaes mos of he welfare cos and makes he economy close o he Ramsey allocaion. 6 Following Kim and Kim (23), a firs-order approximaion may lead o inaccurae welfare approximaions. Firs-order approximaions do no ake ino accoun uncerainy effecs and significan approximaion errors may occur. To avoid he spurious welfare reversals phenomenon underlined by Kim and Kim (23), we use a perurbaion mehod (Schmi-Grohé and Uribe, 24) o compue second-order approximaions of he policy rules. This approximaion mehod allows us o capure uncerainy effecs and avoids spurious welfare reversals. 7 In he sandard search and maching model wih a Nash wage bargaining, firms and workers do no inernalize he effecs heir search decisions have on ohers. I resuls in search exernaliies. An appropriae bargaining power is able o make agens inernalize heir search decisions and allows he implemenaion of he firs-bes allocaion. The condiion saisfied by he bargaining power is ξ ¼ 1 ψ and is called he Hosios condiion. Then, search exernaliies exis if he Hosios condiion is violaed. On he oher hand, here are no search exernaliies if he Hosios condiion is me. In he res of he paper, we coninue o refer o he Hosios condiion concerning ξ ¼ 1 ψ, even if (due o he wage rigidiies) i does no achieve opimaliy. 8 We solve he model wih a second-order perurbaion mehod (see Schmi-Grohé and Uribe, 24). Sae variables are N ; N N and z. Changing parameers leads o a new seady sae. I is calculaed wih a Newon algorihm. To evaluae inegrals we use Gauss Legendre quadraures. 9 The Ramsey program direcly provides axaion and unemploymen compensaion rules as a funcion of he sae variables. These rules canno be direcly compared o he insiuional rules used in he benchmark economy. In order o make a comparison, we evaluae he seady sae experience raing index and replacemen raio implied by he Ramsey allocaion.

10 J. Alberini, X. Fairise / Journal of Economic Dynamics & Conrol 37 (213) Table 1 Baseline parameers. Variables Symb. Value Discoun facor β.99 Sd. aggregae shock s z.9 Risk aversion coef. s 2 Maching elasiciy ψ.5 Exo. separaion rae ρ x.236 Experience raing index e.65 Maching funcion scale χ.65 Home producion h.429 Auocorr. coefficien ρ z.95 Sd. idiosyncraic shock s ε.23 Upper bound ε 1.46 Firm bargaining power ξ.5 Replacemen rae ρ R.32 Vacancy cos κ.3 Wage rigidiy γ.6 Table 2 Opimal labor marke policy. Oupu, wages, employmen, welfare and vacancies have been sandardized. e and ρ R have been recalculaed when we compue he Ramsey. Percenage welfare coss are relaive o he Ramsey allocaion. Variables Benchmark economy Firs bes Second bes allocaion Pareo Ramsey Layoff ax Unemploymen benefis Experience raing index Replacemen rae Ne oupu Average wages Employmen Vacancies Unemploymen (%) Job finding rae (%) Separaion rae (%) Welfare cos (%).9.51 The Ramsey and he second-bes allocaion share he same paern: (i) an increase of he layoff ax, (ii) a srong fall in he unemploymen benefis. Indeed, when ξ ¼ 1 ψ ¼ :5, here is only one inefficiency (wage rigidiies). The exisence of a wage norm ends o reduce wage dispersion. Compared o he case wihou any wage rigidiies, wages in new jobs are reduced while wages in low-produciviy jobs become higher. The impac on he average wage is ambiguous bu we show ha he average wage in he benchmark is increased. 1 Oher hings being equal, he marginal value of a new job o a large firm is increased (Eq. (15)). I leads o more vacancies. Higher wages in low-produciviy jobs lead o a rise in he hreshold produciviy and he job desrucion rae. Therefore, wage rigidiies generae inefficien separaions. 11 The overall urnover rae is increased. A layoff ax is necessary o reduce inefficien layoffs, which are caused by he wage norm. The fac ha he layoff ax makes i less aracive for firms o creae vacancies calls for negaive unemploymen benefis, 12 which enail lower wages and hus reduce he drop in he number of vacancies induced by he layoff ax. The workers' ouside opion should decrease o offse he low araciveness of new jobs and he wage pressure. The negaive value of b O can be inerpreed as a decrease of he workers' ouside opion o reduce he wage pressure. Recall ha even if b O becomes a lile bi negaive he workers' ouside opion (h þ b o ) remains posiive as hey benefi from he home producion. 13 A higher layoff ax 14 and lower unemploymen benefis joinly resore efficiency. 1 Wihou wage rigidiy (γ ¼ ) he minimum wage is wðεþ¼:6251, he average wage is w ¼ :9887 and he maximum wage is wðεþ¼1:59. The benchmark (γ ¼ :6) involves a minimum wage equal o.98, an average wage equal o.998 and a maximum wage equal o Inefficien separaions are discussed in deail laer. 12 As we will see laer, he negaive impac of he layoff ax on vacancies can be reduced if he firm's bargaining power is sufficienly higher han he level sipulaed by he Hosios condiion. There is no need for unemploymen benefis o be negaive. 13 h is equal o The opimal experience raing index is srongly negaive. This is because of he insiuional rule of he fiscal cos (Eq. (3)) which expresses e O as a funcion of he unemploymen benefis level. Indeed, e O akes an exreme value because b O is a lile bi negaive. When he opimal level of unemploymen benefis is equal o zero or negaive he experience raing index is no suiable for measuring he conribuions charged o he firms. In his case, he level of

11 186 J. Alberini, X. Fairise / Journal of Economic Dynamics & Conrol 37 (213) The opimal level of aggregae employmen is higher han he benchmark one. Implemening he opimal financing scheme reduces inefficien separaions and increases vacancies which in urn raises employmen and gross oupu. Since he cos of vacancies increases less han gross oupu, ne oupu (consumpion) rises. How do policy insrumens reduce he welfare coss of labor marke exernaliies? We now perform a sensiiviy analysis by varying 15 ξ from.4 o.7 and γ from.2 o.8. The difference beween ξ and 1 ψ allows us o inroduce addiional search exernaliies ino he model, ha is he search exernaliies of he sandard search and marching model. Wage rigidiies are capured by γ. We consider he economy wih and wihou UI. The resuls are presened in Table 3. In he hree cases of Table 3, he welfare cos is increasing wih he degree of real wage rigidiies, whaever he value of ξ. In our baseline se up (ξ ¼ :5 and γ ¼ :6), he welfare cos is equal o.9% relaive o he Ramsey allocaion. Wihou an UI sysem, i is equal o 2.36%. Baseline values for axes and benefis are no opimal bu conribue o reduce he welfare cos. A decline of he wage rigidiy makes he economy wihou UI closer o he Ramsey allocaion (Hosios). An increase of γ can lead o disproporionae welfare coss (more han 25%). The wors siuaion is a high degree of wage rigidiies and a high level of workers' bargaining power. Our numerical invesigaions are made for given parameers of he UI. Then, he UI may eiher offse exernaliies coming from search behaviors and real wage rigidiies or amplify hem (recall he UI is disorive). The resuls show ha, excep for low levels of ξ and γ and high levels of ξ and γ, he UI sysem is preferable o an economy wihou UI. However, in he benchmark allocaion, he unemploymen insurance is clearly no opimal. Indeed, he alernaive opimal policy (secondbes) displays a very weak welfare cos for differen values of ξ and γ. Then, he second-bes allocaion approximaes o he Ramsey prey well. Whaever he degree of deparure from he Hosios condiion 16 and he level of wage rigidiies in he economy, he experience raing is welfare improving bu can be reformed o make he economy close o he Ramsey allocaion. How should policy insrumens inerac wih labor marke exernaliies? The resuls are given in Table 4. In he Ramsey allocaion he opimal seady sae layoff ax increases wih he degree of real wage rigidiies and he firms' bargaining power. The opimal unemploymen benefis level falls a lile bi wih γ and srongly increases wih ξ. The second-bes opimal policy shares he same paerns. The experience raing index slowly increases wih γ bu sharply varies around he Hosios condiion. If ξ is higher han 1 ψ (ξ ¼ :7), he bargaining process is in he firms' favor. I allows hem o ge a higher share of he surplus and o lower wages. In he Ramsey allocaion, he experience raing index is greaer han one if γ 4. Firms should pay more o he UI han he expeced fiscal cos induced by heir firing acions. Recall ha more real wage rigidiies are likely o increase job separaions. The opimal layoff ax rises and becomes greaer han he expeced fiscal cos. 17 Our resul is similar o ha obained by Cahuc and Malherbe (24) under a minimum wage assumpion. As he wage norm ends o reduce wage dispersion, i plays a similar role o ha of a minimum wage. For high levels of he firms' bargaining power, he workers' ouside opion mus be higher o compensae for low wages. I resuls in a siuaion where unemploymen benefis are posiive o offse he negaive iner-group exernaliies and o resore efficiency. This condiion is reversed when ξo1 ψ. The layoff ax falls (bu always remains posiive) while unemploymen benefis can become negaive. Workers exer oo high a wage pressure, which resuls in few job vacancies. The layoff ax and unemploymen benefis fall o reduce workers' ouside opion and o resore efficiency. Policy implemenaion: In erms of welfare cos, he second-bes allocaion appears o be a good approximaion of he Ramsey allocaion. The second-bes provides a good way o implemen a labor marke reform hrough a change in parameers e and ρ R enhancing he welfare. Inefficien separaions: In he absence of wage rigidiies, he firm's and worker's surplus are equal o zero if ε ¼ ε. The oal surplus is coninuously equal o zero and no inefficien separaion occurs. As menioned by Krause and Lubik (27), our form of wage rigidiy creaes inefficiencies. Compared o he case wihou rigidiy, he wage norm ends o increase wages in low-produciviy jobs. A he hreshold ε, he firms' surplus is equal o zero (S F ðε Þ¼) whereas he workers' surplus is sricly posiive (S M ðε Þ4). If i were possible, workers migh accep wage reducions and a lower hreshold could be aained o coninue he employmen relaionship. One quesion naurally arises: Wha is he conribuion of inefficien separaions in he welfare cos evaluaion? To answer his, we consider ha he wage norm can be violaed. More precisely, if he oal surplus is posiive whereas he firm's surplus (under he rigid wage) is negaive, we allow he wage o be se hrough he sandard Nash bargaining. There are now wo hresholds, bε and ε obε.ifε ½bε ; εš, he wage norm applies whereas if ε ½ ε ; bε ½, wages are deermined by he Nash bargaining process. These hresholds saisfy he following condiions : S F ðbε Þ¼ and S F ðε Þ¼SM ðε Þ¼. (foonoe coninued) he layoff ax is more relevan o compare he opimal policies. In his case, he governmen has fiscal benefis from he layoff. Bu he fiscal revenues are redisribued hrough a non-disorionary (negaive) lump-sum ax. In oher words, he fiscal revenues are no pure wase. 15 For his sensiiviy exercise, we resric ξ o be beween.4 and.7 because he seady sae of he model does no exis ouside of his consellaion. The resuls are valid for his se of consellaions. Usually, he search and maching lieraure considers ha ξ lies in his range of parameers (Perongolo and Pissarides, 21). 16 The difference jξ ð1 ψþj. 17 When γ ¼ he Ramsey experience raing index is equal o 1. In ha case he firm is fully responsible for is firing decisions.

12 J. Alberini, X. Fairise / Journal of Economic Dynamics & Conrol 37 (213) Table 3 Welfare cos (in percenages) relaive o he firs-bes allocaion. γ ¼ :2 γ ¼ :6 γ ¼ :8 Benchmark wih unemploymen insurance ξ ¼ : ξ ¼ : ξ ¼ : Benchmark wihou unemploymen insurance ξ ¼ : ξ ¼ : ξ ¼ : Second-bes allocaion ξ ¼ :4 5: ξ ¼ :5 2: ξ ¼ : Table 4 Opimal policies. Ramsey and second-bes allocaion. Ramsey Second-Bes γ ¼ :2 γ ¼ :6 γ ¼ :8 γ ¼ :2 γ ¼ :6 γ ¼ :8 Opimal layoff ax τ EO ξ ¼ : ξ ¼ : ξ ¼ : Opimal unemploymen benefi b EO ξ ¼ : ξ ¼ : ξ ¼ : Opimal ERI e O ξ ¼ : ξ ¼ : ξ ¼ : Opimal replacemen rae ρ O R ξ ¼ : ξ ¼ : ξ ¼ : Table 5 Welfare coss comparison. Models comparison Baseline calibraion Alernaive calibraion ξ ¼ :7 e¼.65 e¼ e ¼ 9:65 e¼.65 e¼ e¼1.16 ρ R ¼ :32 ρ R ¼ ρ R ¼ :35 ρ R ¼ :32 ρ R ¼ ρ R ¼ :31 Benchmark model vs No inefficien separaions No inefficien separaions vs Ramsey Benchmark model vs Ramsey Table 5 presens welfare cos evaluaions. We compare he benchmark economy (under he baseline calibraion) o he Ramsey allocaion, he benchmark economy o he economy wihou inefficien separaions, and he economy wihou inefficien separaions wih he Ramsey allocaion. We also consider hree unemploymen insurance financing policies. If here is no unemploymen insurance sysem (e¼ and ρ R ¼ ), our simulaions provide an evaluaion of he welfare cos induced by inefficien separaions. Recall ha for low-produciviy level jobs, workers would agree o accep wage cus. I follows ha some maches wih a non-negaive surplus do no lead o an employmen relaionship. Allowing firms o discard he wage norm significanly reduces he welfare cos (by around 2.35 percenage poins). Consider now he benchmark UI sysem (e¼.65 and ρ R ¼ :32). Two hirds of he welfare cos is caused by inefficien separaions. Compared o he previous case he difference arises from he UI. I parly reduces exernaliies coming from inefficien separaions.

13 188 J. Alberini, X. Fairise / Journal of Economic Dynamics & Conrol 37 (213) The welfare cos wih respec o he Ramsey allocaion is negligible. I should be noed ha inefficien separaion coss are reduced hrough an increase of he layoff ax. Finally, if he second-bes policy is implemened, here is no significan difference beween he benchmark model and he model wihou inefficien separaions. To complee our discussion, we also evaluae he welfare cos if we depar from he Hosios condiion. 18 The bargaining power akes now he value of ξ ¼ :741 ψ ¼ :5. Once again, under he benchmark insurance sysem (e¼.65 and ρ R ¼ :32), mos of he welfare coss are caused by inefficien separaions and under he second-bes policy, here is no significan difference beween he benchmark model and he model wihou inefficien separaions. As in he baseline calibraion case, he UI (wih e¼.65 and ρ R ¼ :32) reduces he welfare cos. However, increasing he bargaining power of firms changes he disribuion of welfare coss when here is no UI. The difference beween he economies wih and wihou inefficien separaions becomes very small (he welfare cos is equal o.58). All hings being equal, under he baseline calibraion, he UI sysem ends o increase wages. The removal of he unemploymen insurance combined wih a bargaining power in favor of firms leads o a wage reducion and narrows inefficien layoffs. Under our calibraion, inefficien layoffs become very rare. The Campolmi and Faia (211) wage norm: The wage norm may be modeled in a differen way. To provide some insigh, we also evaluae he Campolmi and Faia wage norm 19 (see Tables A.7 and A.8). This wage norm avoids inefficien separaions. Numerical experimens are made using he parameer values of he benchmark model calibraion (see Table 1). Under he benchmark calibraion (ξ ¼ 1 ψ ¼ :5), he main source of inefficiency emanaes from he UI. 2 The welfare cos is equal o.347, which is very close o is counerpar in he no inefficien separaions case, ha is.2917 (see Table 5). The second-bes allocaion involves a similar welfare cos in using he Krause and Lubik wage norm. We also evaluae he effec of a bargaining power in favor of firms, ha is ξ ¼ :7. As his wage norm does no imply a reducion of he wage dispersion (no minimum wage effec), he opimal experience raing index is equal o 1 while he opimal replacemen rae remains virually unchanged. Since he opimal UI is close o heir baseline counerpar, he welfare coss in he benchmark economy and in he second-bes allocaion are weak and very close. This resul confirms ha inefficien separaions maer in welfare evaluaion Business cycle analysis We show in his secion how he opimal unemploymen benefis financing scheme impacs labor marke flucuaions. We analyze he impulse response funcions, second-order momens, auocorrelaions and correlaions. We compare he benchmark economy o he Ramsey allocaion and he second-bes allocaion. The resuls are shown in Table 6 and Fig. A1. We simulae a 1% negaive produciviy shock and invesigae how labor marke policies affec he way variables respond o he shock. The recessionary shock leads o a fall of oupu and a reducion of job posing aciviy. I increases he separaion and unemploymen raes, as is sandard in maching models. The wo opimal policies srongly reduce he response of unemploymen, new maches and he job finding rae. The volailiy of he separaion rae in Table 6 confirms he labor hoarding phenomenon induced by layoff axes over he cycle. 21 I reduces he incenive of firms o adjus employmen hrough separaions. In he secondbes allocaion, firms cu back on vacancies during he recession, leading o a higher volailiy han in he Ramsey allocaion. The opimal policies lead o an imporan fall in oupu, ighness and unemploymen volailiy. However, in he Ramsey allocaion he sabilizaion effecs are quie srong. The Ramsey policy makes he persisence of oupu and unemploymen 1% and 6% lower respecively. The second-bes policy involves similar effecs. The impac of he recessionary shock on he financing scheme depends on he sensiiviy of unemploymen. In he benchmark economy, he unemploymen compensaion is reduced bu he probabiliy of finding a job falls, leaving he overall effec on he expeced fiscal cos of an unemployed worker (Eq. (3)) undeermined. Simulaions show ha he increase of he average duraion of unemploymen has a higher impac on he fiscal cos Q 22 han he decrease of benefis per unemployed worker. Consequenly, he layoff ax jumps above is seady sae level o finance he oal cos of unemploymen benefis. In he second-bes allocaion, he increase of he layoff ax is reduced because he rise in unemploymen is lower. In he Ramsey allocaion, i is shown ha he layoff ax jumps in he opposie direcion. One can explain his by he reduced flucuaions of unemploymen associaed wih a large fall in unemploymen benefis. As a consequence, he unemploymen insurance expendiures go down following he (negaive) shock. Despie he difference beween he wo opimal policies in erms of shock propagaion and adjusmen of he financing scheme, he sabilizaion effecs are roughly similar, especially concerning unemploymen and he job separaion rae. These resuls confirm he efficiency of hese policy insrumens in reducing he welfare cos and labor marke flucuaions. 18 In he baseline calibraion ξ and ψ are boh se o he convenional value of.5. For his exended simulaion exercise we op for a calibraion ha violaes he Hosios condiion. Lubik (211) esimaes he worker's bargaining power and he elasiciy of he maching funcion w.r.. unemploymen over differen subperiods in he US. He finds ha ξ is always higher han 1 ψ, which implies an average difference of abou.25. We follow him and explore he case ξ41 ψ. This is also consisen wih he calibraion of Moyen and Sähler, 212 (see also Zanei, 211 for he UK). For his ask we se 1 ψ o.5 and vary he firm's bargaining power (as we have done previously) in order o violae he Hosios condiion. ξ is equal o.7 which involves a difference beween ξ and 1 ψ of he same order as hose of he papers previously menioned. 19 Tha is, w ðεþ¼ð1 γþw ðεþþγwðεþ. 2 Under his wage norm, imposing e¼ end ρ R ¼ leads o a welfare cos of.53, which is a lile bi more han he value obained in he secondbes allocaion, ha is.51 (see Table A1). 21 Due o he very small level of he seady sae, he response of he separaion rae in percenage deviaion is quie large in he Ramsey and he second-bes allocaion. We do no display his response as i makes he reading unclear. 22 Measured by he IRF of he layoff ax which is proporional o Q.

14 J. Alberini, X. Fairise / Journal of Economic Dynamics & Conrol 37 (213) Table 6 Cyclical properies. The model is simulaed 5 imes over a 12 quarer horizon. The resuls are shown in logs as deviaions from an HP rend wih smoohing parameer 16 and ignoring he firs 1 observaions. All sandard deviaions are relaive o ne oupu (excep ne oupu). US economy Benchmark Ramsey Second bes Sandard Deviaions Ne oupu Employmen Average wage Unemploymen Vacancy Tighness Job finding rae Separaion rae Auocorrelaion (1) Oupu Unemploymen Job finding rae Separaion rae Correlaion U ; V Conclusion In his paper, he properies of he opimal unemploymen benefis financing schemes are sudied using a search and maching DSGE model calibraed on he US economy. There exiss an unemploymen insurance sysem approximaing o ha of he US. Our crucial assumpion is he exisence of real wage rigidiy. In he absence of search fricions, we evaluae how he disorive effec of wage rigidiy may be offse by means of an appropriae unemploymen insurance policy. Furhermore, hrough a sensiiviy analysis, we evaluae how search fricions impac he design of he UI financing scheme. Our numerical invesigaions show ha wage rigidiies are disorive and generae imporan welfare coss. Indeed, he wage norm reduces wage dispersion and increases he average wage. I resuls in inefficien separaions characerized by oo high a separaion rae and oo heavy wage pressure. The opimal unemploymen benefis financing scheme corresponding o he Ramsey policy allows he policy maker o offse his disorion and o implemen he Pareo allocaion. The opimal layoff ax should be posiive o reduce he urnover rae. When he Hosios condiion is saisfied (search exernaliies of he sandard maching model wih a Nash wage bargaining are eliminaed) he opimal unemploymen benefis end o be a lile bi negaive o reduce he wage pressures. This can be inerpreed as a decrease of he workers' ouside opion (he sum of he home producion and he unemploymen benefis). We also invesigae a paricular case in which he Hosios condiion does no hold (search exernaliies characerisic of he sandard maching model are inroduced). If he bargaining power is in favor of firms and if here are real wage rigidiies, i is opimal ha he firms pay an amoun greaer han he expeced fiscal cos of an unemployed worker. In his case, unemploymen benefis are posiive and in he neighborhood of heir insiuional counerpar. We provide a way o implemen a reform of he unemploymen insurance. Numerical experimens sugges he second-bes allocaion appears o be a good approximaion of he Ramsey allocaion. An appropriae modificaion of he experience raing degree and he replacemen rae brings he economy closer o he opimum. Real wage rigidiies magnify he flucuaions of employmen and unemploymen over he cycle. An opimal design of he unemploymen insurance allows he policy maker o reduce labor marke flucuaions. As a whole, an opimal combinaion of unemploymen benefis and layoff axes is welfare-enhancing and improves labor marke performance. Our sudy may be exended in several direcions. Firs, hroughou he paper we assume ha incomes pooled by family members are equally redisribued. The assumpion of perfec risk sharing limis he resuls. Indeed, considering individual risks would magnify welfare coss and give o he policy insrumens a more imporan role. An exension in a heerogeneous agen framework would be worhwhile. Second, our sudy shows ha he design of he unemploymen insurance can conribue o flucuaion sabilizaion, which complees he resuls obained in he opimal moneary policy lieraure. In a more general model, oher sources of disorions, such as imperfec compeiion and nominal rigidiies, could be inroduced. A greaer number of disorions would require he use of oher policy insrumens. In his case, he opimal join design of unemploymen insurance and moneary policy could be sudied. Acknowledgmens We are graeful o he edior, Michel Juillard and he wo anonymous referees for numerous and valuable commens. We hank seminar and conference paricipans (Le Mans, Evry, T2M, EEA, PET) for improving an earlier draf of his paper. We are also graeful o François Lango for his useful recommendaions and Sébasien Langlois for his re-reading of he aricle. Any errors and omissions are ours.

15 181 J. Alberini, X. Fairise / Journal of Economic Dynamics & Conrol 37 (213) Appendix A. Proof of resul 1 Le us wrie he Lagrangian of he Ramsey allocaion problem aken a a dae : " L ¼ E β j ðc þj þð1 N þj ÞhÞ 1 s 1 s j ¼ þω 1 þj κ V þj Ω 1 þj 1 M ð1 ð1 γþð1 ξþþλ þjðz þj ε z þj ε þj τ E þj Þ þj þω 2 V þj þj ð1 γþð1 ξþκλ þj þð1 γþξðb 1 N þj þ hþλ þj þj ð1 ð1 γþð1 ξþþðz þj ε þj þ τ E þj Þλ þj þ γwλ þj Z! ε Ω 2 þj 1 ð1 ð1 γþð1 ξþþð1 ρ x Þλ þj z þj ðε ε þj Þ dgðεþ ð1 ð1 γþð1 ξþλ þj τ E þj ε þj þω 3 þj N þj w þj λ þj þ N þj ð1 γþξðb þj þ hþλ þj þ N þj ð1 γþð1 ξþκ V þj λ 1 N þj þj ( Z ) ε þð1 γþð1 ξþ N N þj z þjε þ N 1þj ð1 ρ x Þz þj ε dgðεþ λ þj ε þj þn 1þj ð1 ρ x Þð1 ρ n þj Þð1 γþð1 ξþτe þj λ þj þ γn þj wλ þj Ω 3 þj 1 N 1þjð1 γþð1 ξþλ þj τ E þj % dev. 8 Unemploymen Q1 Q2 Q3 % dev Vacancies 8 Q1 Q2 Q3 15 % dev. 2 4 Job finding rae % dev. 1 5 Endogenous separaion rae 6 Q1 Q2 Q3 Q1 Q2 Q3 % dev..2.4 Employmen Q1 Q2 Q3 % dev. Maches 2 2 Q1 Q2 Q3 % dev. % dev. Oupu.5 1 Q1 Q2 Q3 Layoff ax Q1 Q2 Q3 Diff. % dev. 1 2 Unemploymen benefis Lump sum ax 1 Q1 Q2 Q3 Fig. A1. Impulse response funcions. We simulae a 1% negaive aggregae produciviy shock. Benchmark: no marker, Ramsey: downward-poining riangle markers, 2n bes: poin markers. For unemploymen benefis, we display he difference from he seady sae (b b) insead of he deviaion from he seady sae. In opimal policies he seady sae is negaive.

16 J. Alberini, X. Fairise / Journal of Economic Dynamics & Conrol 37 (213) þλ 1 þj ðy þj C þj κv þj Þ þλ 2 þj ð NN þ1þj þ M þjþþλ 3 þj ð N þj þð1 ρ x Þð1 ρ n ÞN 1þj þ N N þj Þ þλ 4 þj ð M þj þ χð1 N þj Þ φ V 1 φ þj ÞþΛ5 þj ρ n þj þ þj dgðεþ þλ 6 þj Y þj þð1 ρ x ÞN 1þj z þj ε þj ε dgðεþþn N þj z þjε þλ 7 þj ððc þj þð1 N þj ÞhÞ s λ þj Þ i þλ 8 þj ð ð1 N þjþb þj þ T þj þðρ x þð1 ρ x Þρ n þj ÞN 1þjτ E þj Þ! As is usual in his class of problem, he mulipliers associaed wih he forward dynamic consrains have iniial values equal o, ha is Ω 1 1 ¼ Ω2 1 ¼ Ω3 1 ¼. This opimizaion problem poenially has a ime-inconsisen soluion. This occurs because of he forward dynamic consrains. Firs, consider he iniial period, ha is ¼. The opimaliy condiions wih respec o T, b and τ E wrie T ¼ Λ 8 ¼ ða:1þ Table A1 Opimal labor marke policy (ξ ¼ :5). Oupu, wages, employmen and vacancies have been sandardized. e and ρ R have been recalculaed when we compue he Ramsey. Percenage welfare coss are relaive o he Ramsey allocaion. Wage seing (see Campolmi and Faia, 211) : w ðεþ¼ð1 γþw ðεþþγwðεþ. For a given produciviy level ε, he wage norm corresponds o he seady sae wage. Variables Benchmark economy Firs bes Second bes allocaion Pareo Ramsey Layoff ax Unemploymen benefis Experience raing index Replacemen rae Oupu Average wages Employmen Vacancies Job finding rae (%) Separaion rae (%) Welfare cos Table A2 Opimal labor marke policy (ξ ¼ :7). Oupu, wages, employmen and vacancies have been sandardized. e and ρ R have been recalculaed when we compue he Ramsey. Percenage welfare coss are relaive o he Ramsey allocaion. Wage seing (see Campolmi and Faia, 211) : w ðεþ¼ð1 γþw ðεþþγwðεþ. For a given produciviy level ε, he wage norm corresponds o he seady sae wage. Variables Benchmark economy Firs bes Second bes allocaion Pareo Ramsey Layoff ax Unemploymen benefis Experience raing index Replacemen rae Oupu Average wages Employmen Vacancies Job finding rae (%) Separaion rae (%) Welfare cos

17 1812 J. Alberini, X. Fairise / Journal of Economic Dynamics & Conrol 37 (213) b ¼ Ω 2 ð1 γþξλ Λ 8 ð1 N ÞþΩ 3 N ð1 γþξλ ¼ ¼ Ω 2 ð1 ð1 γþð1 ξþþλ þ Λ 8 ðρx þð1 ρ x Þρ n ÞN 1 τ E þ Ω 3 N 1ð1 ρ x Þð1 ρ n Þð1 γþð1 ξþλ ¼ ða:2þ ða:3þ Consider now he case wih 1, he opimaliy condiions wih respec o T, b and τ E wrie T ¼ Λ 8 ¼ b ¼ Ω 2 ð1 γþξλ Λ 8 ð1 N ÞþΩ 3 N ð1 γþξλ ¼ ða:4þ ða:5þ ¼ Ω 1 1 ð1 ð1 γþð1 ξþþλ Ω 2 ð1 ð1 γþð1 ξþþλ τ E þ Ω 2 1 ð1 ð1 γþð1 ξþþλ þ Ω 3 N 1ð1 ρ x Þð1 ρ n Þð1 γþð1 ξþλ Ω 3 1 N 1ð1 γþð1 ξþλ þ Λ 8 ðρx þð1 ρ x Þρ n ÞN 1 ða:6þ The opimaliy condiion wih respec o w, for all wries ¼ Ω 3 w N λ ¼ I follows from Eqs. (A.1) (A.7) ha Λ 8 ¼ Ω2 ¼ Ω3 ¼ and Ω1 ¼,. Consequenly, he forward dynamic consrains vanish and he opimizaion problem is hus ime-consisen. The oher opimaliy condiions may be wrien as follows: ¼ðC þð1 N ÞhÞ s Λ 1 C Λ7 sðc þð1 N ÞhÞ s 1 ¼ λ ¼ Λ 7 ¼ Y ¼ Λ 1 Λ6 ¼ ða:7þ ða:8þ ða:9þ ða:1þ V ¼ Λ 1 κ þ Λ4 χð1 N Þ φ ð1 φþv φ ¼ ða:11þ ρ n ¼ Λ 3 ð1 ρx ÞN 1 Λ 5 ¼ ða:12þ N ¼ Λ 3 þ βe Λ 3 þ1 ð1 ρx Þð1 ρ n þ1 Þ N N þ1 Λ 4 χφð1 N Þ φ 1 V 1 φ þ βe Λ 6 þ1 ð1 ρx Þz þ1 ε dgðεþ ε þ1 þλ 7 shðc þð1 N ÞhÞ s 1 hðc þð1 N ÞhÞ s ¼ ¼ Λ 2 þ βe Λ 3 þ1 þ βe Λ 6 þ1 z þ1ε ¼ M ¼ Λ 2 Λ4 ¼ ε ¼ Λ 5 Λ6 ð1 ρx ÞN 1 z ε ¼ ða:13þ ða:14þ ða:15þ ða:16þ The sysem formed by Eqs. (A.8) (A.16) seems inracable. However, i can easily be shown ha i reduces o he equaions sysem defining he Pareo allocaion. I immediaely follows from Eqs. (A.9), (A.8) and (A.1) ha Λ 7 ¼, Λ1 ¼ðC þð1 N ÞhÞ s ¼ λ and Λ 6 ¼ λ. From Eqs. (A.11), (A.12), (A.15) and (A.16), i is easily deduced ha Λ 4 ¼ κ V λ 1 φ M

18 J. Alberini, X. Fairise / Journal of Economic Dynamics & Conrol 37 (213) Λ 5 ¼ð1 ρx ÞN 1 λ ε Λ 3 ¼ λ z ε Λ 2 ¼ Λ4 Subsiuing in Eqs. (A.13) and (A.14) provides κ V λ βe fλ þ1 ðz þ1 ε z þ1 ε 1 φ M þ1 Þg ¼ λ ðz ε hþ κ φ V λ 1 φ 1 N ( Z!) ε þβð1 ρ x ÞE λ þ1 z þ1 ðε ε þ1 Þ dgðεþ ¼ ε þ1 The above equaions are exacly Eqs. (33) and (34). We have verified ha he Ramsey allocaion corresponds o he Pareo one. References Abbrii, M., Weber, S., 28. Labor marke rigidiies and he business cycle: price vs. quaniy resricing insiuions. HEI Working Paper no: 1/28. The Graduae Insiue of Inernaional Sudies, Geneva. Algan, Y., 24. La proecion de l'emploi : des verus sabilisarices? Cahiers de la Maison des Sciences Economiques v419, Universié Paris 1 Panhéon- Sorbonne. Andolfao, D., Business cycles and labor marke search. American Economic Review 86, Blanchard, O., Gali, J., 21. Labor Markes and Moneary Policy: A New-Keynesian Model wih Unemploymen. American Economic Journal, Macroeconomics 2 (2), 1 3. Blanchard, O., Tirole, J., 28. The join design of unemploymen insurance and employmen proecion. A firs pass. Journal of European Economic Associaion 6, Cahuc, P., Malherbe, F., 24. Unemploymen compensaion finance and labor marke rigidiy. Journal of Public Economics 88, Campolmi, A., Faia, E., 211. Labor marke insiuions and inflaion volailiy in he euro area. Journal of Economic Dynamics and Conrol 35, Card, D., Levine, P., Unemploymen insurance axes and he cyclical and seasonal properies of unemploymen. Journal of Public Economics, 53, Chrisoffel, K., Linzer, T., 21. The role of real wage rigidiy and labor marke fricions for inflaion persisence. Journal of Money, Credi and Banking 42 (7), Den Haan, W., Ramey, G., Wason, J., 2. Job desrucion and propagaion of shocks. American Economic Review 9, Faia, E., 28. Opimal moneary policy rules wih labor marke fricions. Journal of Economic Dynamics and Conrol 32, Feldsein, M., Temporary layoffs in he heory of unemploymen. Journal of Poliical Economy 84, Hall, R., 25. Employmen flucuaions wih equilibrium wage sickiness. American Economic Review 95, Joseph, G., Pierrard, O., Sneessens, H., 24. Job urnover, unemploymen and labor marke insiuions. Labor Economics 11, Kim, J., Kim, S., 23. Spurious welfare reversals in inernaional business cycle models. Journal of Inernaional Economics 6, Krause, M., Lubik, T., 27. The (ir)relevance of real wage rigidiy in he New-Keynesian model wih search fricions. Journal of Moneary Economics 54, Krusell, P., Mukoyama, T., Sahin, A., 21. Labour-marke maching wih precauionary savings and aggregae flucuaions. Review of Economic Sudies 77, L'Haridon, O., Malherbe, F., 29. Employmen proecion reform in search economies. European Economic Review 53, Lubik, T., 211. The Shifing and Twising Beveridge Curve: An Aggregae Perspecive. Mimeo. Federal Reserve Bank of Richmondy. Merz, M., Search in he labor marke and he real business cycle. Journal of Moneary Economics 36, Morensen, D., Pissarides, C., Job creaion and job desrucion in he heory of unemploymen. Review of Economic Sudies 61, Moyen, S., Sähler, N., 212. Unemploymen insurance and he business cycle: Prolong benefi enilemens in bad imes? Macroeconomic Dynamics, Published online by Cambridge Universiy Press, hp://dx.doi.org/1.117/s , 2 Ocober 212. Perongolo, B., Pissarides, C., 21. Looking ino he black box: a survey of he maching funcion. Journal of Economic Lieraure XXXIX, Schmi-Grohé, S., Uribe, M., 24. Solving dynamic general equilibrium models using a second-order approximaion o he policy funcion. Journal of Economic Dynamics and Conrol 28, Shimer, R., 25. The cyclical behavior of equilibrium unemploymen and vacancies. American Economic Review 95, Topel, R., On layoffs and unemploymen insurance. American Economic Review 73, Zanei, F., 211. Labor marke insiuions and aggregae flucuaions in a search and maching model. European Economic Review 55,