Fiscal multipliers in a two-sector search and matching model

Transcription

1 Fscal mulplers n a wo-secor search and machng model Konsannos Angelopoulos Unversy of Glasgow We Jang Unversy of Ken James Malley Unversy of Glasgow and CESfo January 25, 25 Absrac Ths paper evaluaes he e ecs of polcy nervenons on secoral labour markes and he aggregae economy n a busness cycle model wh search and machng frcons. We exend he canoncal model by ncludng capal-skll complemenary n producon, labour markes wh sklled and unsklled workers and on-he-job-learnng (OJL) whn and across skll ypes. We rs nd ha, he model does a good job a machng he cyclcal properes of secoral employmen and he wage-skll premum. We nex nd ha vacancy subsdes for sklled and unsklled jobs lead o oupu mulplers whch are greaer han uny wh OJL and less han uny whou OJL. In conras, he posve oupu e ecs from cung sklled and unsklled ncome axes are close o zero. Fnally, we nd ha he secoral and aggregae e ecs of vacancy subsdes do no depend on wheher hey are nanced va publc deb or dsorng axes. Keywords: scal mulplers, secoral labour markes, search and machng JEL Class caon: E24, E32, J63, J64, J68 Correspondng auhor: w.jang@ken.ac.uk We would lke o hank Pedro Gomes, Amanda Goslng, Mguel León-Ledesma, Mahan Sach, Faben Posel-Vnay and parcpans a he Compung n Economcs and Fnance, 24 conference n Oslo and he Unversy of Ken Workshop on Labour Marke, Publc Polcy and he Busness Cycle for helpful commens and suggesons.

2 Inroducon Equlbrum unemploymen models wh search and machng frcons have been exensvely used n macroeconomc analyses of unemploymen (see e.g. Shmer (2) and Rogerson and Shmer (2) for an analycal overvew of hs research). Among oher exensons, hs leraure has consdered he mporance of boh d erences n workers sklls and he poenal for skll eroson due o unemploymen (see e.g. Cahuc e al. (26), Krause and Lubk (26 and 2), Dolado e al. (29), Hagedorn e al. (2), Doppel (24) and Laureys (24)). In hs paper, we conrbue o hs leraure by examnng unemploymen over he busness cycle n an economy wh fragmened labour markes for Unversy educaed (or sklled) and non- Unversy educaed (or unsklled) workers, when he producon srucure s characersed by capal-skll complemenary and workers producvy decreases wh unemploymen. Our neres n labour markes and unemploymen for sklled versus unsklled workers s movaed by emprcal evdence on he mporance of he d erences beween hese wo labour markes, regardng boh wage and employmen raes. We summarse some key d erences usng quarerly daa for he U.S. over he perod of for secoral employmen and for wage nequaly. Frs, he leraure on he skll premum has demonsraed ha here are sgn can d erences n he wages across he wo secors. In parcular, wage nequaly beween sklled and unsklled labour has ncreased n recen decades o s hghes levels n a cenury (see e.g. Goldn and Kaz (28) and Acemoglu and Auor (2) for a dscusson of longer me seres and hsorcal daa). Ths s demonsraed n he subplo (,) of Fgure, whch plos he skll premum, de ned as he rao of sklled o unsklled wage, usng he quarerly seres from Casro and Coen-Pran (28). Second, wage nequaly vares n busness cycle frequences, alhough less han oupu and s no srongly correlaed wh oupu (see also e.g. Lndqus (24) and Pourpourdes (2)). Ths s capured n subplo (2,) whch shows HP- lered oupu and he skll premum. In parcular, he relave o oupu volaly of he skll premum s :87 and s oupu The daa sources for he sklled and unsklled wage daa (979-23) are from he daase of Casro and Coen-Pran (28). Secoral employmen/unemploymen daa (992-2) are from he Curren Populaon Survey, Table A-4. We use daa on he employmen saus of he cvlan populaon 25 years and over by educaonal aanmen (see Fnally, per capa quarerly oupu daa (979-2) are from he U.S. NIPA. 2

3 correlaon s :86. [Fgure here] Thrd, employmen d ers sgn canly beween he wo secors. For example, he daa n subplo (,2), show ha unemploymen s wce as hgh for unsklled compared wh sklled workers. 2 Moreover, as demonsraed n subplo (2,2), unsklled employmen s more volale han sklled, alhough boh closely rack cyclcal oupu. In parcular, he volaly of he HP- lered unsklled employmen s :8 mes hgher han ha of sklled workers, whereas her oupu correlaons are abou :93. The leraure has documened furher d erences beween he sklled and unsklled secors. Cahuc e al. (26) nd ha sklled workers have hgher barganng power, whle Pssardes (994), Acemoglu (2) and Krause and Lubk (26 and 2) sugges ha he ow cos of posng a vacancy s hgher n good jobs. Moreover, n busness cycle frequences, here s no much movemen beween he sklled and unsklled secors. In lgh of he above, we buld a busness cycle model wh search and machng frcons ha lead o secoral unemploymen. To capure he above emprcal observaons, we rs assume ha unsklled workers canno become sklled. Insead, sklled workers work n sklled jobs and, f unemployed, search for employmen n he sklled secor. Smlarly, unsklled workers work n unsklled jobs, and f unemployed, search for employmen n he unsklled secor. Second, we assume ha he producon srucure allows for skllbased echncal change and, n parcular, s characersed by capal-skll complemenary. Ths seup has been shown o explan key characerscs of he skll premum n he daa, boh n erms of s evoluon over he pas several decades (see e.g. Kaz and Murphy (992), Krusell e al. (2) and He (22)) as well as over he busness cycle (Lndqus (24) and Pourpourdes (2)). The search and machng mechansm for employmen creaon ha we employ follows he benchmark Morensen-Pssardes framework wh he wage beng deermned va Nash barganng. Moreover, our seup allows for d erenaon beween he wo labour markes, such as d erences n relave barganng power, job separaon raes and job posng coss o re ec he emprcal observaons oulned above. Movaed by heores of labour augmenng echncal progress drven by on-he-job learnng (OJL), we allow sklled and unsklled workers producves o be posve funcons of employmen. Alernavely, snce he secoral producves are decreasng funcons of unemploymen, hey can equvalenly capure skll eroson due o no workng. We consder wo possbles 2 Fallck and Fleschman (24), Hagedorn e al. (2), and Plossoph (22) also documen hgher job separaon raes for unsklled versus sklled workers. 3

4 for OJL where boh skll ypes learn from her own and he oher skll ype. The rs follows he leraure ha proposes learnng-by-dong (LBD) as a propery of he producon echnology a he aggregae level and generaes knowledge spllovers. I hus works as an exernal e ec for he ndvdual (see e.g. Romer (986) and Barro and Sala--Marn (994)). In hs case he workers labour producvy depends on average employmen, so ha LBD s aken as gven a he level of he workers. The second follows more closely he leraure on skll eroson ha s due o unemploymen (see e.g. Laureys (24) and Doppel (24)). In hs nsance we allow he workers of each ype o nernalse he e ec of her own employmen on her labour producvy. However, we manan he assumpon ha he employmen of he oher worker ype s aken as gven. The model s calbraed o mach he seady-sae of aggregae and secoral labour marke daa n he U.S., followng he calbraon sraegy n Shmer (2). We nd ha he calbraon does a good job a machng he second momens n he secoral labour marke daa. In parcular, he model predcs a volaly for unsklled employmen ha s abou wce as bg as ha of sklled employmen. Moreover, predcs a seres for he skll premum whose volaly s less han oupu and s correlaon wh oupu s around zero. Conssen wh he resuls n Shmer, he model under predcs quanavely he volaly of employmen, bu he gap s no very bg. In parcular, he model varans consdered predc an employmen volaly whch ranges from 69% o 8% of he volaly of employmen n he daa, whereas n he canoncal models, e.g. Shmer (2), hs rao s ypcally abou 25%. Snce he model wh and whou OJL gves a relavely smlar o he daa, we presen resuls below for polcy nervenons usng boh spec caons. Our polcy analyss evaluaes he e ecs of emporary nervenons on he secoral labour markes and he aggregae economy. We consder vacancy subsdes and axes, snce as s demonsraed n Monacell e al. (2), pure scal spendng e ecs on oupu are rval and even negave n search and machng models. We nd scal mulplers on oupu from he subsdes o sklled and unsklled vacances, whch are greaer han uny when OJL s exernal and near uny when s nernal. These large mulplers are deermned by he crowdng-n of prvae nvesmen, whch follows he ncreases n employmen and labour producvy. Whou OJL, he labour producvy channel s absen and causes he oupu mulplers o fall o abou.6. In conras o he vacancy subsdes, he posve oupu e ecs from cung sklled and unsklled ncome axes are near zero, rrespecve of he presence, or model, of OJL. We nally nd ha he e ecs of he wo vacancy subsdes do no depend on wheher deb or dsorng axes 4

5 are used o nance hem. Ths s parcularly good news, and suggess ha hs ype of nervenon may be useful under he deb resrcons ha many governmens now face n he wake of he nancal crss. The res of he paper s organsed as follows. Secon 2 ses ou he model srucure. Secon 3 presens he calbraon and cyclcal properes of he model. Secon 4 underakes he scal mulpler analyss and Secon 5 concludes. 2 The model 2. Capal-skll complemenary There are N rms whch operae n compeve produc markes. To produce a sngle oupu, rms use capal, whch hey lease from he household, and sklled and unsklled workers. The producon echnology s characersed by capal-skll complemenary (see e.g. Goldn and Kaz (28) for hsorcal evdence on he emprcal relevance of hs echnology n he 2h cenury). In parcular, a represenave rm produces oupu y f, usng a consan elascy of subsuon (CES) spec caon followng e.g. Krusell e al. (2): y f = A l f;u + ( h ) k f + ( ) l f;s () where A > s he level of oal facor producvy (TFP); ; < are he parameers deermnng he facor elasces,.e. = ( ) s he elascy of subsuon beween capal and unsklled labour and beween sklled and unsklled labour, whereas = ( ) s he elascy of subsuon beween capal and sklled labour; and < ; < are he facor share parameers. In hs spec caon, k f s he quany of capal used by he rm, whereas l f;s and l f;u denoe he quanes of sklled and unsklled labour respecvely. 2.2 Sklled and unsklled workers There s a represenave household whose members nclude sklled or unsklled workers. The workers are dsnc uns and can o er eher sklled or unsklled servces n he labour markes. They can nd a job whn he skll secor n whch hey belong or reman unemployed. In he laer case, hey search for a job for he nex me perod whn her skll secor. In oher words, n busness cycle frequences workers canno change her skll saus. Ths assumpon s movaed by emprcal evdence suggesng ha over he 5

6 busness cycle, he share of college educaed or sklled populaon has low volaly and s e ecvely uncorrelaed wh oupu. In parcular, usng he daa n Acemoglu and Auor (2), we nd ha he sandard devaon of he cyclcal componen of he sklled populaon share, relave o ha of oupu, s.29, whle s correlaon wh oupu s Ths mples ha sklled workers can eher work n sklled jobs or reman unemployed (and search for sklled jobs), whereas unsklled workers can eher work n unsklled jobs or reman unemployed (and search for unsklled jobs). Followng he leraure on search frcons and unemploymen n macroeconomc DGE models snce Merz (995), we assume ha he head of he household makes all decsons on behalf of s members and provdes complee consumpon nsurance. In oher words, all workers consume he same amoun of goods, rrespecve of her labour marke saus,.e. regardless of wheher hey are employed n sklled or unsklled jobs or hey are unemployed. The numbers of sklled and unsklled members for he represenave household are denoed as N s and N u, respecvely. The oal sze of he household s normalsed o be N and s hus gven as: N = N s + N u. The respecve populaon shares of sklled and unsklled members whn a household are de ned as: n s = N s =N and n u = N u =N. We assume ha populaon and s composon reman consan. For each skll ype of household members, = s; u, he number of members/workers can be furher decomposed no employed and unemployed members, such ha: N = N ;e + N ;s (2) where = s; u for sklled and unsklled labour; and N ;e s he number of employed members and N ;s s he number of unemployed members, who are searchng for a job. By normalsng by N, we have: = e + s (3) where e N ;e s he employmen rae and s N N ;s N rae or he share of workers searchng for a job. s he unemploymen 2.3 Search and machng There are wo labour markes, for sklled and unsklled workers respecvely. Each unemployed worker needs o search for a job n he sklled or unsklled 3 Ths s obaned usng annual daa for he share of college educaed populaon measured n e cency uns, , from Acemoglu and Auor (2) and GDP per capa daa from he U.S. Naonal Income and Produc Accouns (NIPA). The cyclcal componen of he seres s obaned usng he HP- ler wh a smoohng parameer of. 6

7 secor, gven her skll level, and can be mached wh a rm ha poss vacances n ha secor. As n he sandard search-and-machng leraure (see e.g. Pssardes (986) and Blanchard and Damond (989)), he machng echnology s represened by a Cobb-Douglas (CD) funcon for boh sklled and unsklled labour: M = S V where, M s he aggregae new maches a ; 4 S = N s denoes he aggregae number of unemployed searchng n labour marke ; V = Nv denoes he aggregae number of job vacances creaed by rms n labour marke ; > represens he consan e cency of machng for labour ype ; < < denoes he elascy of searches for labour ype. In addon, we de ne he vacancy-o-unemployed rao, z = V =S = v= (n s ), as he ghness of ype labour marke. The smaller he rao of z, he gher he labour marke and herefore he harder for unemployed workers o mach wh job vacances. The probably a whch aggregae job searches lead o a new job mach n ype labour marke s gven by: (4) p = M S = S V = z (5) and s nverse, =p, measures he duraon of ype search. The probably a whch a job vacancy can be mached wh an unemployed household member s calculaed by: q = M V = S V = z (6) and s nverse, =q, measures he duraon of ype job vacancy. 2.4 Household There s a represenave household comprsed of sklled and unsklled ndvduals whose head makes all decsons on behalf of s members by guaraneeng equal consumpon o each of hem, wh he objecve of maxmsng household welfare. 4 In wha follows, we use upper case leers for aggregae quanes and lower case leers for per capa quanes. 7

8 2.4. Problem The represenave household maxmses dscouned lfeme uly, U : U = E X = u (7) where E s he condonal expecaons operaor a perod ; and < < denoes he consan rae of me preference. The nsananeous uly funcon of he household s gven by (see e.g. Shmer (2)): u = ln(c ) n s e s n u e u (8) where > s he preference parameer ha measures he dsuly cos of employmen and c s he household s average (or per capa) prvae consumpon. As s common n he leraure, he dsuly cos capures he reducon n he me avalable for home producon when a member nds employmen. Hence, he spec caon n (8) assumes ha all members consume c and ha f a member s unemployed, her uly s gven by ln(c ), whereas f a member s employed, her uly s gven by ln(c ), so ha u measures average uly for he household. The budge consran of he household s: c + + b + = r k (r ) k + + +R b b + ( s ) n s w s e s Z s + ( u ) n u w u e u Z u (9) where s household s average prvae nvesmen; b + s he value of governmen bonds bough a perod ; r s he gross reurn o physcal capal; k s he ax rae on capal ncome; < < s he consan deprecaon rae of physcal capal; k s he average physcal capal held by he household a he begnnng of ; s average dvdends receved from he rms; R b = + r b s he gross reurn o bonds; s he labour ncome ax; w s he gross wage rae; and Z represens labour augmenng echnology drven by OJL. Ths echnology posvely depends on he level of employmen. Alernavely, Z can be nerpreed as a decreasng funcon of unemploymen and capures skll eroson due o no workng. 5 We allow for boh skll ypes o learn on-he-job from her own and he oher skll ype. We consder wo possbles for OJL. The rs follows he leraure ha proposes learnng-by-dong (LBD) as a propery of he producon echnology a he aggregae level. Ths seup generaes knowledge spllovers a he 5 See, for example, Davs and von Wacher (2) and Pollack (23) for he e ecs of unemploymen on labour producvy and Laureys (24) and Doppel (24) for search and machng models wh skll deprecaon due o unemploymen. 8

9 aggregae level whch work as an exernal e ec for he ndvdual (see e.g. Romer (986) and Barro and Sala--Marn (994)). In hs case, we allow he worker s labour producvy o depend on average employmen, so ha LBD (or, alernavely, skll eroson) s aken as gven a he level of he worker. Ths case s represened as follows: Z s Z s;x = s (e s ) s (e u ) s () Z u Z u;x = u (e s ) u (e u ) u () where a bar over a varable refers o average quanes; > are learnng producvy parameers; and < <, are he elasces of OJL wh respec o sklled employmen for sklled and unsklled workers. We wll proceed wh he model soluon below usng () and (). However, we wll also presen and dscuss resuls usng a second possbly for he deermnaon of Z, followng more closely he leraure on skll eroson ha s due o unemploymen (see e.g. Laureys (24) and Doppel (24)). Ths alernave assumes ha workers nernalse he e ec of employmen on her labour producvy. Hence, n hs case we allow he worker of each ype o nernalse he e ec of her own employmen on her labour producvy. However, we manan he assumpon ha he employmen of he oher worker ype s aken as gven: 6 Z s Z s;n = s (e s ) s (e u ) s (2) Z u The capal sock evolves accordng o: Z u;n = u (e s ) u (e u ) u. (3) k + = ( ) k + ~ A k (4) The capal evoluon equaon allows for an exogenous process, ~ A k, capurng an nvesmen-spec c echnologcal (IT) change, whch has been shown o conrbue o oupu ucuaons (see e.g. Greenwood e al. (2), as well as he changes n he skll premum (see e.g. Krusell e al. (2), Lndqus (24), and Pourpourdes (2)). The sochasc process for nvesmenspec c echnology, ~ A k s: ea k + = ea k A k A k A ek e " Ak + (5) where e A k > ; < A k < ; and " Ak + dn ; ( A k) 2. 6 Noe ha n boh spec caons examned, labour producvy s ncreasng and concave wh respec o employmen and bounded beween zero and, where = s; u. 9

10 By usng equaon (4) and de nng as A k ~ A k, we can rewre he budge consran of household: c + A k k + + b + = er k + + +R b b + ( s ) n s w s e s Z s + ( u ) n u w u e u Z u (6) where er = r k (r ) + A k ( ), s he ne reurn o physcal capal afer deprecaon and ax. Noe ha A k ~ A k measures he e ecve prce of nvesmen, snce A k uns of nvesmen are needed o creae one un of capal n he nex perod. Employmen for ype = s; u worker evolves accordng o: e + = p s + e (7) where < < s he rae of job separaon for ype labour. The sochasc process for he job separaon rae,, s: + = g g e "g + (8) h where > ; < g < ; and " g + dn ; 2 g. The household s opmzaon problem s o choose fc ; k + ; b + g = o maxmse (7) subjec o he consrans (3) and (6) akng facor prces fw s ; w u ; r ; rg b =; pro s f g = ; he evoluon of employmen fe g = ; he exogenous varables A k ; ; polcy varables k = ; s ; u and nal = condons for k ; b as gven Frs-order condons (FOCs) The recursve form of he household s problem s: V (k ; b ; e s ; e u ) = max f(ln c n s e s n u e u ) + c ;k + ;b + (9) +E V k + ; b + ; e s +; e u + g where V (:) s he value funcon. consran (6) gves: Replacng c makng use of he budge V (k ; b ; e s ; e u ) = max [ ln[er k A k k + b R b k + ;b + b + + ( s ) n s w s e s Z s + ( u ) n u w u e u Z u ] n s e s n u e u + +E V (k + ; b + ; e s +; e u +)]: (2) The envelope condon for capal sock, k s: V k (k ; b ; e s ; e u ) = er c (2)

11 and he rs order condon for k + s: E V k k + ; b + ; e s +; e u + = A k c (22) whch equaes he dscouned expeced margnal bene o he margnal cos of nvesmen. Fnally, subsung he one-perod lead of he envelope condon (2) no he rs-order condon for capal (22) gves he consumpon Euler: E c er + = A k (23) c + whch shows ha he expeced, dscouned reurn on nvesng n capal mus equal s prce. Noe ha he reurn s dscouned usng he sochasc dscoun facor c c +. The envelope condon for governmen bonds, b s: and he rs order condon for b + s: V b (k ; b ; e s ; e u ) = Rb c (24) E V b k + ; b + ; e s +; e u + = c. (25) Subsung he one-perod lead of he envelope condon (24) no he rs-order condon for governmen bonds (25) gves he bonds Euler, whch has a smlar nerpreaon as he Euler for capal:! E c + r b + c + = (26) The FOCs for he household s problem are gven by (6), (23) and (26). These deermne he pahs for fc ; k + ; b + g = gven exogenous varables A k ; ; polcy varables k = ; s ; u ; nal condons, fk = ; b g; and he pahs for varables ha are exogenous o he household s problem,.e. hose deermned a he aggregae level, ; r ; r; b e + and by wage barganng, fwg = =. 2.5 Frms There s a represenave rm whch leases capal from he household and employs sklled and unsklled workers o produce a sngle good, wh he objecve of maxmsng pro s.

12 2.5. Problem To hre workers, he rm needs o pos vacances one perod before he jobs are requred. In parcular, he evoluon of he number of workers per sklled ype employed by he rm s gven by he job ranson funcon whch lnks he fuure number of lled jobs, l+, f; o he ne hrng, qv, plus he curren sock of lled jobs, ( ) l f; : l f; + = qv + l f; : (27) Gven ha posng vacances s cosly, he pro funcon of he rm s: f = y f r k f w s l f;s ( v;s ) ' s v s w u l f;u ( v;u ) ' u v u (28) where ' s ; ' u > sand for he consan resource coss of openng a new sklled and unsklled vacancy respecvely; and v; refer o he vacancy subsdes. The employmen evoluon equaons n (27) mply ha pro maxmsaon s neremporal, snce expendure on posng vacances oday wll ncrease pro s omorrow. Therefore, he objecve of he rm a me perod = s o maxmse he presen value of s lfeme pro s, whch s gven by: y f r k f wl s f;s ( v;s ) ' s v s wl u f;u ( v;u ) ' u v u + X Y +E er fy f r k f w s l f;s ( v;s ) ' s v s w u l f;u = = ( v;u ) ' u v u g (29) where y f and y f are gven by he CES producon funcon n () a me and respecvely. Snce pro s are reurned o he household, + reurns are convered o presen value erms by he sochasc dscoun facor from nhe household s o opmsaon problem, (23). For = s; u, he rm chooses k f ; v; l f; + = o maxmse (29) subjec o (27), akng facor prces fw; r g = ; machng probables fqg = ; exogenous job separaon raes f g = ; economc polcy v; ; and nal condons for flf; = g as gven. The varable, A s deermned by he followng sochasc process: A + = (A) A (A ) A e "A + (3) where A > ; < A < ; and " A + dn ; ( A ) 2. 2

13 2.5.2 Frs-order condons The rm s problem s wren n recursve form as: J l f;s ; l f;u = max [y f k f r k f w s l f;s ( v;s ) ' s v s ;vs ;vu w u l f;u ( v;u ) ' u v u ] + E er + ( s ) l f;s ; q u v u + ( u ) l f;u +J(q s v s + ) (3) where J(:) s he value funcon. The FOCs for k f, v s and v u are: r = h A l f;u + ( ) k f + ( ) l f;s ( ) h k f + ( ) l f;s k f ( v;s ) ' s = E er +q s J l f;s l+; f;s l f;u + ( v;u ) ' u = E er +q u J l f;u l+; f;s l f;u + mpl k (32) (33) (34) sang respecvely ha he margnal cos of capal s equal o s margnal bene and ha he margnal coss of creang sklled and unsklled vacances are equal o he expeced reurn of hrng one addonal sklled and unsklled worker nex perod. The envelope condon for sklled employmen, l f;s J l f;s l f;s ; l f;u where mpl s = A h ( ) + ( k f = mpl s w s + ( s ) E er +J l f;s l f;u for he connuaon value, er +J l f;s for v s n (33) hs condon becomes: J l f;s l f;s h + ( ) k f + ( ) l f;s ( ) l f;s s: l+; f;s l f;u + ) l f;s (35). Afer subsung l+; f;s l f;u +, usng he rs-order condon ; l f;u = mpl s w s + ( s) 's q s ( v;s ) : (36) Fnally, o oban he FOC for he rm, we rs lead (36) by one perod and subsue no (33) o oban: ( v;s ) 's = E q s er + mpl+ s w+ s + s ' s + q + v;s (37) + s 3

14 Workng, smlarly for unsklled employmen, we have: ( v;u ) 'u = E q u er + mpl+ u w+ u + u ' u + q+ u where mpl u = A l f;u + ( h ) k f + ( ) v;u + : (38) l f;s. l f;u These condons equae he margnal cos of posng a job vacancy o he expeced dscouned margnal bene for sklled and unsklled jobs respecvely. The bene s comprsed of wo elemens. Frs, he ncrease n pro s assocaed wh hrng an exra worker, mpl+ u w+, u and he savng assocaed wh no havng o pos a job vacancy n he nex perod, u ' u +. q+ u For = s; u, he FOCs for he rm s problemn are gven by o(27), (28), (32), (37) and (38), whch deermne he pahs for l+; f; f ; k f ; v, gven exogenous processes, fa ; g = ; varables ha are deermned a he aggregae level, fr ; qg =, or by wage barganng fw g = ; and nal condons for fl f; g. 2.6 Wage Barganng We assume ha once a worker/household member s mached wh a rm, he household and he rm bargan over he wage rae. The equlbrum wage s deermned by a Nash bargan. In parcular, he equlbrum wage rae maxmses he Nash produc: = h eve ew h e Jl f; ew (39) where measures he power of he household/worker relave o he rm n he Nash bargan; e V e ( ew ) s he value of a successful bargan a wage ew for he household and e J l f; ( ew ) s he value of a successful bargan a wage ew for he rm Household s valuaon of employmen The valuaon of he household for an addonal member beng employed a wage w s gven by he envelope condons of (2) for e s and e u respecvely: V e s (k ; b ; e s ; e u ) = ( s )ns w szs c n s + ( s p s ) (4) E V e s k + ; b + ; e s +; e u + 4

15 V e u (k ; b ; e s ; e u ) = ( u )nu w uzu c n u + ( u p u ) (4) E V e u k + ; b + ; e s +; e u + : We nex consder he margnal value o a household of allowng a small number of s members, s >, o be pad an arbrary wage, ew s, n perod, assumng ha he wage revered o he equlbrum wage w+ s from nex perod. In hese crcumsances he value funcon of household n equaon (2) becomes: bv ( ew s ; s ) = max f ln er k A k k + b R b k + ;b + b + + ( s ) n s w s e s Z s + ( s ) n s ew s s Z s + ( u ) n u w u e u Z u ) n s (e s + s ) n u e u g + E V fk + ; b + ; [p s ( e s s ) + + ( s ) (e s + s )]; [p u ( e u ) + ( u ) e u ]g: (42) D erenang V b ( ew s ; s ) wh respec o s and evaluang he dervave a s = o derve he margnal value of a sklled worker employed a an arbrary wage, ew s : bv s ( ew s ; ) = ( s )ns ew szs c n s + ( s p s ) (43) E V e s k + ; b + ; e s +; e u + : If we combne he expresson for e V e s ( ew s ) b V s ( ew s ; ) wh he envelope condon for e s n equaon (4) we oban: ev e s ( ew s ) = ( s ) n s c ( ew s w s ) Z s + V e s (k ; b ; e s ; e u ) : (44) Equvalenly, we can derve he margnal value of an unsklled worker employed a an arbrary wage, ew u : ev e u ( ew u ) = ( Frm s valuaon of employmen u ) n u c ( ew u w u ) Z u + V e u (k ; b ; e s ; e u ) : (45) We work smlarly o oban he rm s valuaon of agreeng o employmen a a wage ew. Assume ha he rm pays a small fracon, s >, of employed workers an arbrary wage ew s a me perod, and ha he wage rae wll reurn o he equlbrum wage w+ s from he nex perod. The value funcon of rm, (3) can be mod ed o: bj ( ew s ; s ) = maxfy f v s r k f w s l f;s + ew s s ( v;s ) ;vu ' s v s w u l f;u ( v;u ) ' u v u + E er +J([q s v s + ( s ) (46) (l s;f + s )]; [q u v u + ( u ) l f;u ])g: 5

16 We d erenae J b ( ew s ; s ) wh respec o s and evaluae a s = o ge he margnal pro of employng a sklled worker a ew s : bj s ( ew s ; ) = A f l f;u + ( ) [ k f + ( ) h l f;s ] g ( ) k f + ( ) l f;s (47) ( ) l f;s ew s + ( s ) E er +J l l f;s f;s +; l f;u + : We hen combne hs wh he envelope condon for l f;s n (36) o ge he margnal pro of employng a sklled worker a an arbrary wage, ew s, a me, and he equlbrum wage hereafer: ej l f;s ( ew s ) = w s ew s + J l f;s l f;s ; l f;u (48) where J e l f;s ( ew s ) J b s ( ew s ; ). Smlarly, we can derve he respecve condon for unsklled workers: ej l f;u ( ew u ) = w u ew u + J l f;u l f;s ; l f;u : (49) (Nash) equlbrum wage The rs-order condon of he Nash bargan (39) wh respec o ew s s: = s h eve s ( ew s ) s h ejl f;s ( ew s ) h + ( s s h ) eve s ( ew ) s ejl f;s ( ew s ) e Ve s ( ew s ) ew s J e f;s ( ew l ew s : (5) Subsung he dervaves of (44) and (48) wh respec o ew s as well as he expressons for V e e s ( ew s ) and J e l f;s ( ew s ) from (44) and (48) respecvely no (5) and evaluang a w s = ew s gves: s ( s ) n s c J l f;s l f;s ; l f;u Z s = ( s ) V e s (k ; b ; e s ; e u ) : (5) Workng as descrbed n deal n Appendx A, we can derve he wage equaons (A3) - (A4), whch can alernavely be wren as: h ( s ) Z s w s = s f( s ) Z s mpl s + ( s ) 's ( v;s q s ) (52) ( s p s ) E + s Z s + A k ( v;s )g + ( s ) c : ' s q s 6

17 ( u ) Z u w u = u f( u ) Z u ( u p u ) E u + Z u + A k h mpl u + ( ' u q u ( v;u u ) 'u ( v;u q u ) (53) )g + ( u ) c These equaons are generalsaons of wage equaons under Nash barganng obaned n he leraure (see e.g. Shmer (2)). For = s; u, he reurn of an addonal worker o he household s gven by ( ) Zw,.e. he afer-ax e ecve (or producvy-adjused) wage. In equlbrum, hs s equal o a weghed average of he e ecve h margnal produc of labour under search and machng,.e. ( ) Z mpl + ( ) ' v; q ( p ) E + Z + A k ' v; q, and he margnal rae of subsuon beween consumpon and lesure, MRS,.e. c, wh he weghs gven by he barganng power of he worker. The MRS follows he common de non of he rao of he margnal uly of lesure,, over he margnal uly of consumpon, =c. The e ecve margnal produc of labour measures he addonal afer-ax producvy-adjused oupu generaed by movng a worker from unemploymen o employmen. I s comprsed of () he drec afer-ax ncrease n oupu provded by an addonal sklled worker, mpl; () he addonal savngs n erms of resources ha would be requred o pos a vacancy f he mached job survves, ( ) ' v; q, where ( ) s he probably ha a worker wll reman n place n he nex perod and ' v; q s he cos per job posng mulpled by he duraon ha he job needs o be posed, ; 7 () he ncrease n job-posng q coss for he rm mpled by he decrease n fuure successful maches due o he ncrease n employmen, ( p ) E + Z + A k ' v; q. Noe ha an ncrease n curren employmen ncreases fuure unemploymen (and hus he requremen for he rm o pos a vacancy o ll he los job) = ( p ), because here s reducon n he number of workers who search for jobs. Furhermore, noe ha hese coss need o be dscouned by he prce of ransferrng resources beween perods, A k, whch equals, from (23), expeced fuure reurns o nvesmen dscouned by he sochasc dscoun facor. The above wage equaons hold when here s no OJL and under purely exernal OJL. If we employ he alernave OJL mechansm whch nernalses own employmen on labour producvy, he rgh hand sde of he 7 Noe ha from (37) - (38), ' q rm from posng a job. v; s also equal o he expeced bene o he 7

18 above equaons respecvely are mulpled by he erm: 2 3 Z ; = s; s Z = s s (e s ) s (e u ) s u = ( u ) u (e s ) u (e u ). These exra erms: () collapse o uny under exernal OJL,.e. = () are less han one, 8 mplyng ha nernalsng OJL creaes a channel ha ends o reduce he Nash barganed wage, relave o he cases of no or exernal OJL. When he workers nernalse he e ec of employmen on her producvy and hus on her reurns, hey are wllng o work for a lower wage rae. 2.7 Governmen budge and marke clearng The governmen budge consran s: g + v;s ' s v s + v;u ' u v u + Rb b = = b + + k (r ) k + s n s w s e s Z s + u n u w u e u Z u (54) where g s he per-capa governmen consumpon. The capal markes clear when he supply s equal o he demand for capal per capa: k = k f : (55) In he sklled and unsklled labour markes, he equaly of per capa labour supply and demand s gven by: n s e s Z s = l f;s (56) and n u e u Z u = l f;u : (57) Moreover, dvdends pad o he household mus equal pro s: = f : (58) Fnally, n he goods markes, he economy s per capa resource consran s sas ed: y f = c + A k k + A k ( ) k + g + ' s v s + ' u v u : (59) 8 To see hs, rs noe ha e Z Z >, + Z e > ) + Z e Z e > ) ez Z e > ) >, whch s rue. Hence, 8 >. Then, noe ha h + e Z <.

19 2.8 Decenralzed equlbrum Gven he pahs of he exogenous varables A ; A k ; s ; u and nal = condons for fk ; b ; e s ; e u g, a decenralzed equlbrum s de ned as a seres of prces, w s ; w u ; r ; r b, machng probables, = fps ; p u ; q s ; q u g n o = and allocaons, c ; ; k + ; b + ; e s +; e u +; f ; k f ; v s ; v u ; l+; f;s l f;u +, such ha = () households and rms underake her respecve opmzaon problems, akng aggregae oucomes and economc polcy as gven, under search and machng n he labour marke as oulned above; () wage raes for boh ypes of labour are deermned by a Nash bargan for mached household members and rms; () all budge consrans are sas ed; and (v) all markes clear. Fnally noe ha n equlbrum, we have e s = e s and e u = e u. Usng Walras law we drop he household s budge consran, so ha he DE consss of he followng equaons: () he search and vacancy machng probables n (5) and (6); () he consumpon and bonds Euler equaons (23) and (26); () he rm s opmaly condons gven by (27) for ( = s; u), (28), (32), (37) and (38); (v) he wage equaons (A3) and (A4); and (v) he marke clearng condons n (55), (56), (57), (58) and (59). 9 3 Quanave mplemenaon In he followng secon we rs dscuss he model calbraon followed by he quanave predcons of he model regardng he seady-sae and near seady-sae dynamcs. We consder hree model varans, dependng on he assumpons we make regardng he labour producvy echnology, as capured by Z, for = s; u. In parcular, snce we wan o conexualse he poenal mporance of OJL, we rs consder a benchmark case whou, so ha Z = =. We hen choose n he cases of OJL ha we consder (where he employmen e ecs are purely exernal, OJL x, and where he own e ec s nernalsed, OJL n ) so ha he level of labour producvy n he seady-sae, Z, s he same across all hree model varans. Ths furher mples ha he models whou OJL and OJL x have dencal seady-saes, whereas OJL n s re-calbraed followng he same sraegy as he oher wo models so ha s seady-sae s e ecvely he same. 9 Noe ha when he marke clearng condons (56) and (57) and he machng probables n equaons (5) and (6) are mposed on he employmen evoluon equaons (7) and (27) he laer become dencal. Hence, we drop he employmen evoluon equaons (7) from he household s problem from he DE. 9

20 3. Model Calbraon Table repors he values for he srucural parameers of he model based on a quarerly calbraon. The able ndcaes how each parameer s obaned by referrng o varous sources. Ths ncludes calculaons usng: () he daa; () esmaes and assumpons from oher sudes n he leraure; and () calbraon o arge seady-sae values for he relevan endogenous varables of he model. As explaned above, hese refer o he model varans whou and wh purely exernal OJL. We summarse a he end s hs sub-secon he changes n parameers requred for he OJL n model. Table : Model Parameers Parameer Value De non Source < n s <.45 populaon share of sklled workers daa k <.36 ax rae on capal ncome esmae s <.35 ax rae on sklled labour ncome esmae u <.25 ax rae on unsklled labour ncome esmae g >.425 per-capa governmen consumpon calbraon < <.99 me dscoun facor calbraon.22 deprecaon rae of capal sock calbraon.669 capal o sklled labour elascy esmae.669 capal o unsklled labour elascy esmae < <.493 share of compose npu o oupu calbraon < <.82 share of capal o compose npu calbraon >. dsuly cos of employmen calbraon < s <.28 sklled job separaon rae calbraon < u <.45 unsklled job separaon rae calbraon < s <.6 elascy of sklled search assumpon < u <.5 elascy of unsklled search assumpon < s <.6 barganng power of sklled workers assumpon < u <.5 barganng power of unsklled workers assumpon ' s >.9 un cos of posng sklled job calbraon ' u >.82 un cos of posng unsklled job calbraon < v;s ; v;u <. job vacancy subsdy assumpon s >.8 sklled machng e cency calbraon u >.6 unsklled machng e cency calbraon < s ; u <.5 elascy of learnng assumpon Noe ha, where possble, we follow Shmer (2, see Appendx A) n he choce of me perod (959-27). Noe however, ha he secoral daa employed below are only avalable from 992:-2:4. 2

21 3.. Populaon shares, polcy, dscoun and deprecaon raes We use daa from Acemoglu and Auor (2) for he perod (963-28) o calculae he populaon share of sklled workers, n s = :45. Conssen wh he range used n he leraure, he me dscoun facor, = :99, s se o gve an annual reurn o capal, ne of deprecaon, of abou 4%. Smlarly he deprecaon rae, = :22, s calbraed o arge a quarerly seady-sae capal o oupu rao of abou 8 whch on an annual bass s conssen wh a rao of around 2. Followng Uhlg (2) we se he ax rae on capal ncome o 36%. Moreover, we choose he wo labour ncome ax raes o be s = 35% and u = 25%, whch mply a weghed average close o he 28% labour ncome ax rae used n Uhlg (2). The level of governmen spendng s se so ha he deb o oupu rao s :63 or n quarerly erms 2:52 (as n Uhlg (2)) Producon The elasces of subsuon beween sklled labour and capal and beween unsklled labour and capal have been esmaed by Krusell e al. (2). We use her esmaes, so ha = :495 and = :4. To ensure he skll premum and labour share n ncome are conssen wh he daa, and respecvely are calbraed o :493 and :82 (see, e.g. Lndqus (24), He and Lu (28), Pourpourdes (2) and He (22) who use a smlar approach o calbrang he producon funcon). The arge value for he skll premum of approxmaely :68 s obaned from Acemoglu and Auor (2) for he perod (963-28). We measure he labour ncome share usng daa from Naonal Income and Produc Accouns Table., , whch gves a value of approxmaely.66. Fnally, he parameers capurng seady-sae TFP and nvesmen-spec c echncal change,.e. A and A k are normalsed o uny Uly funcon and job separaon raes Followng Shmer (2) we se he dsuly of employmen parameer, = :, o mply an aggregae unemploymen rae of abou 5%. Also noe, ha Shmer (25) repors ha an average employmen ex probably of :34. Gven hs and he assumpon ha sklled labour has a lower job separaon rae (see, e.g. Fallck and Fleschman (24), Hagedorn e al. (2), and Plossoph (22)) we se he job separaon raes, s = :28 and u = :45, o approxmaely mach he secoral unemploymen raes of 2

22 3% and 7% respecvely New maches and barganng power The values used for he elasces of new maches wh respec o search me, s = :6 and u = :5, are whn he range of economerc evdence repored n Perongolo and Pssardes (2). To ensure ha he Hosos (99) condon s sas ed we se he relave barganng power of worker n he sklled and unsklled secors respecvely o s = :6 and u = :5 (see, Cahuc e al. (26) who nd ha sklled workers have hgher barganng power) Job posng coss and subsdy Pssardes (994), Acemoglu (2) and Krause and Lubk (26 and 2) sugges ha he ow cos of posng a vacancy s hgher n good jobs. Followng hese sudes, we assume ha he job posng for sklled s greaer han ha for unsklled labour,.e. ' s > ' u. These parameers are calbraed o ensure aggregae job coss as a share of GDP of abou 2.5% whch coheres wh Arseneau and Chugh (22) and aggregae labour marke ghness of abou uny whch s he value used n Pssardes (998) and Campolm and Gnocch (24). Also followng Campolm e al. (2) we se he vacancy subsdy rae o % Machng e cency and OJL Conssen wh an aggregae unemploymen rae of 5% and an average employmen ex probably of :34, Shmer (2, see p. 67) mples a job ndng probably abou :65. Followng hs approach for each labour marke gves us arge probables of p s = :828 and p u = :59 whch we oban by calbrang s = :8 and u = :6. The ndng probables n urn mply unemploymen duraons of abou :2 and :69 quarers for sklled and unsklled respecvely. The calbraon also suggess ha he job llng rae s hgher for he sklled versus he unsklled conssen wh Krause and Lubk (26 and 2). As explaned above, we presen he models resuls below boh whou and wh learnng. In he former, Z s = Z u = n (-). In boh forms of laer (.e. OJL x and OJL n ) we se he exponens s = u = :5 and calbrae s and u so ha n he seady-sae The secoral employmen and unemploymen daa are from he monhly Labor Force Sascs, Curren Populaon Survey for perod (992:-2:4). I repors daa for cvlan non-nsuonal populaon 25 years and over by educaonal aanmen. Sklled workers are hose wh a Bachelor s degree and hgher. 22

23 he Z funcons are equal o uny as under no learnng. Ths requres ha s = u = : Seady-sae The seady-sae mpled by he above calbraon s repored n Table 2 for he models whou and wh purely exernal learnng. These resuls show ha grea raos are well n lne wh he U.S. daa. Moreover, he remanng values cohere wh he arges dscussed n he calbraon above. For he hrd model varan, he resuls are quanavely smlar. To ensure ha he model under OJL n mples an analogous seady-sae wh he remanng wo model-varans, we re-calbrae = :5, ' s = :6, ' u = :52, and g = :4, followng he same calbraon sraegy oulned above. Noe ha as dscussed n Secon 2.6.3, when workers nernalse OJL, barganed wages end o be lower and hus unemploymen lower. Therefore, o manan he same level of unemploymen and labour marke ghness n he seady-sae, job-posng coss need o ncrease. c y k y g y Table 2: Seady-sae b y we y v s s s u s y w s w u er r b z s z u p s p u q s q u Sochasc processes When underakng he model smulaons we draw he four processes dscussed above from a mulvarae normal dsrbuon, denoed x = N (x; ; ) where x = [" A ; " Ak ; " gs ; " gu ]; x s he vecor of means and s he varancecovarance marx of shocks. The parameers of sochasc processes drvng he model are repored n Table 3. The auocorrelaon parameer of TFP s se equal o.95, followng Gerler and Trgar (29), and Arseneau and Chugh (22). As n he leraure, he volaly parameer, A, s calbraed o mach he sandard devaon of HP- lered oupu,.. Re- 2 Gven he lack of daa for exponens n he learnng funcons, we expermen wh some alernave combnaons. For example, we place more wegh on he own-elascy for he sklled,.e. s = :8 and ( s ) = :2 whle a he same me keepng wegh of he own-elascy for he unsklled: () he same, u = :5 and ( u ) = :5; () hgher, u = :8 and ( u ) = :2; and () lower, u = :2 and ( u ) = :8. We nd ha he resuls repored below, ncludng seady-sae, second-momens, mpulse responses and scal mulplers, are robus hese o alernave parameersaons. Ths apples o boh he OJL x and OJL n seups. 23

24 gardng nvesmen-spec c echncal change, we use he esmaes from Pourpourdes (2), whch mples seng A k, o.65 and A k, o.47. Gven he lack of secoral daa for he job separaon raes, we apply he same quarerly auocorrelaon, s and u, and sandard devaon, s and u, parameers for sklled and unsklled usng daa from he Job Openngs and Labor Turnover Survey (JOLTS) for he perod 2Q-24Q2. Fnally, he correlaon beween job separaon shocks, cor(" gs ; " gu ), s calbraed o mach he correlaon beween HP- lered sklled and unsklled employmen/unemploymen raes n he daa. 3 Table 3: Sochasc processes Parameer Value De non Source A.8 SD of TFP calbraon A.95 AR() coe cen of TFP assumpon A k.47 SD of IT esmae A k.65 AR() coe cen of IT esmae s.73 SD of sklled separaon rae daa s.74 AR() coef. of sklled separaon rae daa u.73 SD of unsklled separaon rae daa u.74 AR() coef. of unsklled separaon rae daa cor(" gs ; "gu ).98 Job separaon rae shock correlaon calbraon 3.4 Soluon and second momens Followng Shmer (2), we presen resuls under shocks o TFP and he job separaon raes bu we also consder nvesmen-spec c echnologcal change, gven he mporance aached o skll-based echncal change n explanng he behavour of he skll premum n he leraure. The resuls for he secoral varables dscussed n he Inroducon are presened n Table 4. To oban hese resuls we rs solve a rs-order approxmaon of he dynamc sysem of equaons characersng he DE around he seady-sae, by mplemenng he perurbaon mehods n Schm-Grohé and Urbe (24). We hen smulae me pahs under shocks o oal facor producvy, he job separaon raes and nvesmen-spec c echnologcal change, as ndcaed. We conduc, smulaons of 8 perods (.e. 992Q-2Q4) o mach he secoral employmen and unemploymen daa and perods (.e. 979Q-23Q4) o mach he skll premum daa, nalsed from he seady-sae n Table 2. For each smulaon, we HP- ler he logged seres and hen compue he requred momens and repor he means of hese 3 Noe ha no allowng for hs correlaon only a ecs hs arge. 24

25 momens across he smulaons n Table 4. Table 4: Second momens Shocks o A, Shocks o A, A k, Daa no OJL OJL x OJL n no OJL OJL x OJL n (e s ; e u ) (e s ; y) (e u ; y) (w s =w u ; y) (e s )=(y) (e u )=(y) (w s =w u )=(y) As can be seen n Table 4, all hree model varans predc secoral labour marke quanes ha have qualavely smlar cyclcal properes wh he daa. In parcular, he model predcs posve correlaons beween oupu and employmen (alhough no so srong as n he daa) and a very low correlaon beween oupu and he skll premum. I s noable ha he oupu correlaons of employmen are very low whou OJL, bu ncrease and are qualavely closer o he daa under OJL. Furhermore, he models predc a sandard devaon for de-rended skll premum and oupu ha s lower han ha of oupu, conssen wh he daa. Quanavely, he volaly of he skll premum s mproved when shocks o A k are ncluded n he se of exogenous processes. All models under-predc, quanavely, he volaly of employmen n he wo secors, whch s expeced gven he resuls n Shmer (25 and 2). However, he predced volaly of employmen n he secoral model s sgn canly mproved compared wh he canoncal one-secor model. In parcular, he model varans predc an employmen volaly whch ranges from 69% o 8% of he volaly of employmen n he daa, whereas n he canoncal models n e.g. Shmer (2), hs rao s ypcally abou 25%. 4 Moreover, he models do capure a key propery of secoral employmen, n parcular ha unsklled employmen s nearly wce as volale as sklled employmen. These model-generaed second momens mach he characerscs of secoral employmen and wage nequaly ha were dscussed n he Inroducon. The resuls n Table 4 furher sugges ha s no easy o dsngush he model varans based on her ably o he secoral second momens. Whle qualavely hey are all smlar, each model has relave 4 Shmer (2, p. 95) descrbes sandard devaons of employmen relave o oupu of a magnude of abou 45%, o " nally generae neresng ucuaons n employmen". 25

26 successes compared wh he ohers. However, he models wh OJL are clearly preferable o he model ha does no allow for OJL regardng he oupu correlaons for employmen. Therefore, o analyse he e ec of polcy below we presen resuls for all hree model varans. Ths allows us o provde a range for he lkely sze of he scal mulplers arsng from argeed labour marke nervenons and hus o quanavely evaluae he mporance of OJL or skll eroson for he e ecs of polcy nervenons. 4 Fscal nervenons n he labour marke We nex consder he e ec of argeed labour marke polces n he form of job-posng subsdes and ncome ax cus. 5 In parcular, we focus he followng polcy expermens. Frs, ha he governmen ncreases v;s emporarly by one percenage pon and les governmen deb absorb he scal mplcaons of he shock. Second, we examne he e ecs of a smlar ncrease n v;s under xed governmen deb, so ha he polcy nervenon s nanced by a proporonal ncrease n he labour and capal ncome ax raes. 6 Ths budgeary resrcon s parcularly relevan gven he curren, pos nancal crss economc realy ha severely lms he use of deb o nance scal nervenons n mos advanced economes. We hen repea he same expermens for an ncrease n v;u by one percenage pon. Fnally, we dscuss he mplcaons of labour marke nervenons n he form of cus n he labour ncome axes by one-percenage pon. 7 To mplemen he above expermens, we assume ha when emporarly shocked, vacancy subsdes and ncome axes follow sandard AR() processes. 8 We rs dscuss he mpulse responses of he rs polcy expermen n deal, o analyse he channels va whch vacancy subsdes a ec employmen n he wo secors and oupu. We focus on he wo models ha have he same calbraon and seady-sae,.e. whou OJL and under OJL x, snce 5 See Campolm e al. (2) and Faa e al. (23) who conduc smlar analyss n he conex of a search and machng model wh endogenous parcpaon and a labour selecon model wh urnover coss and Nash barganed wages, respecvely. 6 In hs case, we x he level of governmen deb o s seady-sae level, le one of he ncome axes o be he resdual polcy nsrumen n he governmen budge consran, and x he remanng wo ncome axes o reman n he same proporon wh he resdual ncome ax as n he seady-sae. 7 Noe ha normalsng he ax cus o have he same scal mplcaons as he vacancy subsdy nervenons yelds e ecvely he same resuls. 8 Followng Campolm e al. (2) we se he AR() parameers for he vacancy subsdes o.9 and followng Angelopoulos e al. we se he AR() parameers on he sklled and unsklled ncome axes o.95 and

27 he e ecs are drecly comparable, and hus we can race he workngs of he OJL mechansm followng he polcy nervenon. We hen summarse he mplcaons for oupu and unemploymen n each case by calculang he scal mulplers for he remanng labour marke nervenons ha we consder. Followng Leeper e al. (29) we calculae scal mulplers as follows: F M k = kp Q j j= kp Q j j= R + = R + = y +j x +j (6) where R + ( k )(r ); y = (y y) and x = (x x). For he vacancy subsdy mulplers x = e v; e v;, where = s; u and e v; = ' v v; gves he cos of he subsdy. Fnally for he ax mulplers x = e e, where e = n we (whou OJL) and e = n we Z (under OJL x and OJL n ) represen he respecve coss of he ax reducon. 4. Impulse responses o v;s In Fgure 2 we presen he mpulse response funcons of a emporary ncrease n v;s by one percenage pon under exble deb. The man resuls from an ncrease n he subsdy o he vacancy posng coss for sklled workers s ha employmen for sklled workers ncreases along wh her wages and he skll premum, whle, a he aggregae level, nvesmen and oupu ncrease. However, he magnudes of hese aggregae e ecs depend on he assumpons regardng on-he-job learnng, whch s also crcal n deermnng he e ecs of hs polcy on aggregae consumpon and he unsklled labour marke. [Fgure 2 here] An ncrease n v;s mples ha he cos for posng sklled vacances s reduced, so ha he number of vacances, v s, ncreases and employmen follows. These e ecs are quanavely very smlar for all model varans rrespecve of OJL, snce job posng decsons of he rm are no drecly a eced by OJL. Wages ncrease for sklled workers when employmen ncreases, snce he workers can bargan for a share of he hgher expeced rm pro s from a successful mach under reduced unemploymen. Noe ha boh p s and =q s ncrease when unemploymen s reduced (and/or vacances ncrease), so ha from (52) w s ncreases. The rse n employmen crowds n nvesmen and consumpon (a leas afer some perods whou OJL) for wo reasons. Frs, he margnal produc of capal, mpl k, has ncreased, snce s a posve funcon of employmen 27