Regional Equilibrium Unemployment Theory at the Age of the Internet

Transcription

1 Rgional Equilibrium Unmploymnt Thory at th Ag of th Intrnt Vanssa Lutgn IRES - Univrsité catholiqu d Louvain Bruno Van dr Lindn FNRS, IRES - Univrsité catholiqu d Louvain, IZA and CESifo April 17, 2015 Abstract This papr studis quilibrium unmploymnt in a two-rgion conomy with matching frictions, whr workrs and jobs ar fr to mov and wags ar bargaind ovr. Job-skrs choos btwn sarching locally or sarching in both rgions. Sarchmatching xtrnalitis ar amplifid by th lattr possibility and by th fact that som workrs can simultanously rciv a job offr from ach rgion. Th rst of th framwork builds upon Mortti (2011). Incrasing th matching ffctivnss out of th rgion of rsidnc has an ambiguous impact on unmploymnt rats. Whil it rducs th probability of rmaining unmployd, it also dcrass labor dmand bcaus of a lowr accptanc rat. W charactriz th optimal allocation and conclud that th Hosios condition is not sufficint to rstor fficincy. A numrical xrcis indicats that th loss in nt output is non ngligibl and rising in th matching ffctivnss in th othr rgion. Kywords: Matching; Non-sgmntd labor markts; Spatial quilibrium; Rgional conomics; Unmploymnt diffrntials. JEL: J61, J64, R13, R23. W thank th ditor and two anonymous rfrs for usful commnts and suggstions, which hlpd us to improv th manuscript. W ar gratful to Pirr-Philipp Combs, David d la Croix, Bruno Dcrus, Philipp Kirchr, Patrick Klin, Etinn Lhmann, Frank Malhrbt, Ioana Marinscu, Enrico Mortti, Olivir Pirrard, Hnri Snssns, Matthias Wrd and Yvs Znou. W also thank participants to various workshops and confrncs for thir commnts on a prliminary vrsion of this papr (Th ARC workshop on mobility of factors hld in Louvain-la-Nuv, th Sarch and Matching workshops hld in Roun, in Aix-n-Provnc and in Louvain-la-Nuv, th Blgian Day for Labor Economists in Luvn, th 2013 EALE Confrnc in Turin, th 2014 Sarch and Matching Confrnc in Edinburgh and th 2014 EEA-ESEM Confrnc in Toulous). Th usual disclaimr applis. W acknowldg financial support from th Blgian Frnch-spaking Community (convntion ARC 09/ on Gographical Mobility of Factors ). Corrsponding author: vanssa.lutgn@uclouvain.b. Addrss: IRES, Univrsité catholiqu d Louvain, Plac Montsquiu 3, B-1348, Louvain-la-Nuv, Blgium. 1

2 1 Introduction Whil an abundant litratur in urban conomics addrsss unmploymnt issus within citis (s Znou, 2009, for a dtaild covrag), lss ffort has bn dvotd to analyz th causs of unmploymnt at th rgional lvl. Givn th larg gographical diffrncs in th prvalnc of unmploymnt obsrvd in th ral world, undrstanding spatial quilibrium whn th labor markt dos not instantly clar would appar to b of primary importanc. (Klin and Mortti, 2013, p. 239). Th main purpos of this papr is to contribut to this undrstanding. Th Intrnt allows both sids of th labor markt to find mor asily potntial partnrs, vn faraway, thanks to job boards and mta-sarch ngins. 1 Morovr, th rcruitmnt procss can now also b conductd onlin through virtual rcruiting tools. 2 Marinscu and Rathlot (2014) provid vidnc that th distanc btwn th job-skr and th job vacancy xcds 100 km (63 mils) in about 10% of th onlin applications on CarrBuildr.com. This suggsts that a non ngligibl shar of US job-skrs ramp up thir job sarch by xpanding it ovr long distancs. 3 Whil most of th litratur daling with rgional unmploymnt assums that popl nd to migrat bfor thy can start sarching locally, w rlax th assumption of sgmntd rgional labor markts. Our main contribution consists in dvloping a gnral quilibrium sarch-matching framwork whr job-skrs choos whthr thy sarch in thir rgion of rsidnc only or all ovr th country. To th bst of our knowldg, this has not bn don yt. In this stting, rgions ar strongly intrdpndnt and svral sourcs of infficincy xplaind latr ar prsnt. A numrical xrcis suggsts a non-ngligibl gap btwn th fficint and th quilibrium allocations. As a scondary contribution, our analysis also shds som light on a puzzl. Expctations that th Intrnt would improv th functioning of th labor markt by rducing sarch-matching frictions wr grat (s.g. Autor, 2001). A dcad latr, th vidnc is mixd. Som microconomtric valuations find that onlin job sarch shortns unmploymnt duration in th US (Kuhn and Mansour, 2014; Choi, 2011). For graduat studnts in Italy, Bagus and Labini (2009) conclud that th us of an onlin platform rducs th probability of unmploymnt and raiss gographical mobility. Howvr, via a diffrnc-in-diffrncs approach, Kroft and Pop (2014) find no vidnc that th rapid xpansion of a major onlin job board (during th yars ) has affctd city-lvl unmploymnt rats in th US. So, th rasons why improvmnts at th individual lvl disappar at a mor aggrgat lvl nd to b undrstood. This papr proposs an xplanation in a spatial conomy. 1 In 2010, 25% of th intrviwd Amricans who us th Intrnt dclard to do so to find a position (U.S. Cnsus Burau, 2012, Survy of Incom and Program Participation, 2008 Panl). In Europ, in 2005, among th unmployd workrs, 25% usd th Intrnt to sarch for a job. This shar has incrasd to 50% in 2013 (Eurostat, 2014, s ci_ac_i). 2 S.g. and th links on monstr.com/hr/hr-bst-practics/rcruiting-hiring-advic/acquiring-job-candidats/ virtual-rcruitmnt-stratgis.aspx. 3 Th vidnc is mor mixd for th UK (s Manning and Ptrongolo, 2011). 2

3 W build upon th synthsis of Mortti (2011) who dvlops a two-rgion static spatial quilibrium modl à la Rosn (1979)-Roback (1982). Contrary to ths authors, Mortti (2011) assums that th supply of labor is not prfctly lastic. This proprty is obtaind by assuming that conomic agnts hav htrognous idiosyncratic prfrncs for rgions. Th aim of Mortti (2011) is to analyz how local shocks propagat in th long run to th rst of th conomy, with a focus on th labor markt. H discusss th cas whr agnts hav diffrnt skills, whil w kp labor homognous. Howvr, rgional unmploymnt disparitis ar not studid by Mortti. W introduc sarchmatching frictions and wag bargaining within this framwork (Pissarids, 2000) but w abstract from th housing markt. Contrary to most of th sarch and matching litratur, th spatial htrognity is xplicit in our framwork. In ach rgion, imprfct information and lack of coordination among agnts crat frictions summarizd by a constant-rturns-to-scal rgional-spcific matching function in which th numbr of job-skrs is a wightd sum of th rsidnts and of th non-rsidnts who dcid to sarch all ovr th country, both numbrs bing ndognous. W charactriz th quilibrium. W show how rgional unmploymnt diffrntials ar affctd by th partition of th population btwn th two rgions and btwn th statuss of national and rgional job-skrs. W also conclud that a ris in matching ffctivnss out of th rgion of rsidnc has an ambiguous ffct on th unmploymnt rats. It dcrass th probability of rmaining unmployd but it also rducs labor dmand through a lowr accptanc rat of job offrs. This ambiguity chos th main conclusion of Kroft and Pop (2014). In th standard sarch-matching litratur, frictions gnrat congstion xtrnalitis which ar not intrnalizd by dcntralizd agnts unlss th Hosios (1990) condition is mt. As soon as som workrs sarch all ovr th country, nw sourcs of infficincy aris. First, whn dcntralizd agnts dcid whthr to sarch nationally, thy look at thir privat intrst and ignor th consquncs of thir choics on job cration in all rgions. Scond, whn opning vacancis in a rgion, firms do not intrnaliz th changs in th matching probability and hnc in nt output in th rst of th conomy. W show that th Hosios condition is nvr sufficint to dcntraliz th constraind fficint allocation. W dvlop a numrical xrcis for a vry stylizd US conomy mad of two rgions that ar initially symmtric and whr th Hosios condition prvails. Th dcntralizd conomy appars to b far from fficint. For a vry wid rang of paramtrs, fficincy rquirs that nobody sarchs in th whol country whil 10% of th workforc dos it in th dcntralizd conomy. Furthrmor, th fficint unmploymnt rat lvl is lowr than th dcntralizd on. As this xrcis assums symmtric rgions, this conclusion is not in contradiction with th rcnt vidnc that gographical mismatch is ngligibl in th US (s.g. Sahin t al., 2014, Marinscu and Rathlot, 2014 and Nnov, 2014). Although a spatial quilibrium modl with gnuin unmploymnt has for long bn missing, som paprs hav rcntly partly filld th gap. Laving asid th litratur whr rgions ar so clos that commuting is an altrnativ to rlocation, th litratur about rgional unmploymnt diffrntials can b dividd in two groups according to th 3

4 typ of sarch: ithr on nds to mov bfor starting to sk a job in th rgion of rsidnc or on can sarch all ovr th country and thn mov if ndd. In th first cas, som paprs xtnd th island modl of Lucas and Prscott (1974) whos conomy is populatd by a larg numbr of sgmntd prfctly comptitiv labor markts whr only labor is mobil (workrs bing allowd to visit only on island pr priod). Lkhagvasurn (2012) adds sarch-matching frictions as wll as match-location spcific productivity shocks in an othrwis standard islands modl to rproduc th volatility of unmploymnt rats in th Unitd Stats. Focusing also on on (small) rgion out of many, Wrd (2014) studis th rlationships btwn wags, rnts, unmploymnt and th quality of lif in a dynamic framwork. H assums a standard sarch-matching framwork and analyzs how rgional amnitis affct unmploymnt and th quality of lif. Baudry t al. (2014) introduc sarch-matching frictions in a spatial quilibrium stting with wag bargaining, fr mobility of jobs, a vry stylizd housing markt, and amnitis with congstion xtrnalitis. In thir papr, with som xognous probability, th joblss population gts th opportunity to mov to anothr city in ordr to sk jobs, whil w lt agnts choos btwn two stratgis: rgional and national sarch. Furthrmor, Baudry t al. (2014) do not look at fficincy whil w do. Klin and Mortti (2013) dvlop a matching modl to charactriz th optimal (fixd) hiring cost and to look at th rational for plac-basd hiring subsidis. Finally, Klin and Mortti (2014) provid a two-rgion modl in which workrs hav idiosyncratic prfrncs for a rgion and ar mobil. Thy us th modl for policy purposs, namly to show that plac-basd policis ar not always wlfar improving for th whol country. This is du to th fact that taxs gnrat a dadwight loss. Scond, som rcnt paprs assum that workrs can sk a job in th whol country. In a stting with many rgions, Amior (2012) studis wags rsponss to a housing shock in th prsnc of skill htrognity. H assums national sarch in a sarchmatching framwork as wll as a random migration cost. Domingus Dos Santos (2011) builds a sarch-matching dynamic framwork with two rgions that ar ach considrd as a lin. Sh finds that incrasing sarch ffctivnss is bnficial for unmploymnt rats in both rgions. Howvr, sh kps wags xognous. Using a sarch-matching dynamic framwork with national sarch and ndognous wags, Antoun (2010) assums two typs of agnts who diffr in thir prfrnc for a rgion. H finds that a positiv productivity shock in on rgion dcrass unmploymnt locally but raiss it in th othr rgion. W xtnd ths modls by ndognizing th choic btwn rgional and national sarch undr wag bargaining. 4 Contrary to ths paprs, w also dvlop a normativ analysis by looking at fficincy. Howvr, w kp our framwork static whil thy all assum a dynamic stting. In th nw conomic gography litratur, Epifani and Gancia (2005) analyz th simultanous mrgnc of both agglomration conomis and unmploymnt rat dif- 4 Molho (2001) dvlops a partial quilibrium job-sarch framwork with both typs of sarch. Manning and Ptrongolo (2011) build a partial quilibrium framwork whr job-skrs choos thir sarch fild. S also Marinscu and Rathlot (2014). W shar a common intrst with ths paprs in a gnral quilibrium modl with ndognous wags and vacancis. 4

5 frntials. For this purpos, thy build a dynamic two-sctor two-rgion modl with transport costs and sarch-matching frictions. Thy assum a congstion ffct in th utility which could rflct th housing markt. Thy mphasiz th rol of migration following a productivity shock, which raiss th unmploymnt rat in th short run but dcrass it in th long run. Francis (2009) xtnds this framwork to ndognous job dstruction. Galnianos and Kirchr (2009) build a dirctd-sarch wag-posting modl whr workrs simultanously apply to a N > 1 jobs. Thy show that multipl applications lad to an infficint allocation whn a vacancy rmains unmatchd if th job applicant rfuss th offr (an assumption rvisitd by Kirchr, 2009). W stick to th random matching assumption with wag bargaining popularizd by Pissarids (2000). Mor importantly, by assumption in our papr, th sarch for an fficint allocation is constraind by th fr choic of agnts whn thy gt two job offrs. So, th possibility of unmatchd vacancis is common to both th dcntralizd quilibrium and th fficint allocation. Th rst of this papr is organizd as follows. Sction 2 dscribs th modl and its quilibrium. W first focus on a symmtric quilibrium, to highlight th main mchanisms. W thn turn to th cas of asymmtric rgions. Sction 3 studis fficincy. A numrical analysis is conductd in Sction 4. Sction 5 discusss som xtnsions to th framwork of Sction 2. Sction 6 concluds. 2 Th modl This sction dvlops a modl with two distant rgions. W first introduc th main assumptions and th matching procss. Thn, w dvlop a simpl vrsion of th modl with symmtric rgions and driv th symmtric quilibrium. W allow for diffrncs across rgions in a last subsction. W considr a static modl of an conomy mad of two larg rgions (i {1, 2}). Each rgion is a point in spac. Th distanc btwn th two rgions is such that commuting is ruld out, whil intr-rgional migration is allowd. Topl (1986) and Knnan and Walkr (2011) among othrs hav strssd th importanc of migration costs. As will soon b clar, idiosyncratic prfrncs for rgions will in our stup play th rol of individual-spcific rlocation costs. Th aggrgat national labor forc is mad of an xognous larg numbr N of homognous risk-nutral workrs. A workr living in rgion i supplis on unit of labor if th wag is abov th valu of tim if sh stays at hom, dnotd b i. Firms ar fr to locat in th rgion thy prfr. Thy us labor to produc a uniqu consumption good which is sold in a comptitiv markt and takn as th numrair. Workrs hav idiosyncratic prfrnc for rgions. Agnt j gts utility c ij from living in rgion i. As Mortti (2011), w assum that th rlativ prfrnc for rgion 1 ovr rgion 2, c 1j c 2j, is uniformly distributd on a givn support [ v; v], v > 0. Th prsnc of a distribution of rlativ prfrncs implis that th lasticity of intrrgional labor mobility is finit. A highr valu of v ntails lss intns rsponss to rgional diffrncs. 5

6 Th indirct utility Vij of an mployd individual j living in rgion i is, as in Mortti (2011), assumd to b additiv and dfind by: V ij = w ij + a i + c ij (1) whr w ij rprsnts th wag arnd by agnt j in rgion i and a i is a masur of xognous local consumption amnitis in rgion i, such as th climat. Ths amnitis ar public goods and ar not affctd by th numbr of inhabitants in a rgion (no rivalry). 5 Similarly, th indirct utility V u of an unmployd prson j rsiding in rgion i is: ij V u ij = b i + a i + c ij. (2) In ach rgional labor markt, w assum a rgional-spcific random matching procss. Adopting a on-job-on-firm stting, firms dcid in which rgion thy opn at most on vacancy. Th cost κ i of opning a vacancy is constant, xognous and rgionalspcific. 6 Throughout th papr, w assum constant rturns in th production of th consumption good. 7 If th vacancy is filld, a firm producs y i > b i units of th consumption good. So, dpnding on th origin of th workr, a firm maks a profit J ij = y i w ij on a filld position. 2.1 Th timing of dcisions At th bginning of th uniqu priod, vrybody is unmployd, chooss in which rgion to rsid, and dcids to sarch for a job ithr rgionally or nationally (i.. ithr on only sarchs for a job in th rgion whr on livs or on sarchs in both rgions at th sam tim). Th rason why som workrs would only sarch in thir rgion of rsidnc rathr than nationally is intuitiv. If a workr has a sufficintly strong rlativ idiosyncratic prfrnc for hr rgion of rsidnc, sh will not accpt to migrat to tak a position. Sinc, following Dcrus (2008), w assum a small cost of rfusing a job offr, this individual will thn not tak part in th matching procss in th othr rgion. In a scond stp, firms opn vacancis and possibly mt a workr. This procss occurs simultanously in both rgions. If a vacancy mts a job-skr, this workr thn accpts or not th job offr. Whn a match is formd with a job-skr who dos not liv in th firm s rgion, this workr rlocats. Allowing unmployd workrs to rlocat at this stag would complicat th xposition without yilding mor insights. Aftr th 5 Contrary to what is somtims don in th litratur (s.g. Wrd, 2014 or Brucknr and Numark, 2014), amnitis a i do not affct th production function ithr. 6 Capital is assumd to mov frly across rgions through vacancy cration. Ignoring crdit markt imprfctions, ntrprnurs hav no problm financing thir vacancy cost κ i. 7 Although vry standard in th sarch-matching litratur, this assumption dos not account for an mpirical rgularity according to which firms ar mor productiv in largr citis. Th lasticity is quit small, spcially whn controlling for charactristics such as ducation, but diffrncs in population sizs can b substantial. In th US howvr, Baudry t al. (2014) find no significant vidnc of agglomration ffcts on productivity (ovr 10-yar priods). So, w think that our assumption is not too strong a simplification, at last in th US contxt. 6

7 rlocation stp, mployd workrs and firms bargain ovr wags. Fourth, production taks plac and good markts clar. Th momnt at which wags ar ngotiatd mattrs whn a rlocation of th workr is involvd. If this momnt occurs bfor th dcision to migrat is takn, through Nash bargaining, th workr will gt a partial compnsation for th diffrnc in th rgional non-wag componnts of utility a i + c ij. To implmnt this timing, on has to assum that th mployr is awar of th idiosyncratic prfrncs of th workr for both rgions. On can doubt that this information is availabl. 8 A survy conductd by CarrBuildr.com at th nd 2011 indicats that lss thn a third of mployrs ar rady to pay for rlocation costs of thir nw mploys. 9 This is casual vidnc in favor of th timing indicatd abov: Th bargaind wag will thn not compnsat th workr for th diffrnc in a i + c ij and hnc w ij can b writtn simply as w i. W will rturn to th timing of th wag bargain in Sction 3. Som additional notations hav to b introducd. Bfor th matching procss, N i agnts choos to rsid in rgion i (N i is calld th x-ant population in rgion i, with N 1 + N 2 = N). Population in rgion i is composd of Ni N agnts who sarch nationally and Ni R individuals who only sarch in thir rgion of rsidnc (N i = Ni N + Ni R ). For agnts locatd in rgion i, th notation i will dsignat th othr rgion. 2.2 Th matching procss W allow for distant sarch, maning that sarch in a rgion can b conductd whil living in th othr on. Th matching ffctivnss of thos living in th rgion whr vacancis ar opn is normalizd to on. For rsidnts of th othr rgion, this ffctivnss taks an xognous valu α with 0 α 1. Th main focus is hr on strictly positiv valus of α. 10 Th numbr of hirings in ach rgion is givn by a rgional-spcific matching function M i (, ) with: M i (V i, N i + αn N i) < min{v i, N i + αn N i}, i {1, 2}, (3) whr V i rprsnts th ndognous numbr of vacancis opnd in rgion i and N i +αn i N is th ndognous numbr of job-skrs masurd in fficincy units. As Molho (2001) and Manning and Ptrongolo (2011) do in a partial quilibrium framwork, w ndogniz sarch ffort by ltting job-skrs choos thir sarch fild. Following Pissarids (2000) and a larg mpirical litratur, th matching function has constant rturns to scal 11 and is incrasing and concav in both argumnts. Dfining tightnss in rgion i as V i θ i = N i + αn i N, 8 Notic that if th framwork was dynamic this timing would rais anothr issu. Undr th standard assumption of automatic rngotiation (Pissarids 2000, p. 15), th wag would b rvisd aftr th rlocation stp and would b chosn xactly as proposd in th timing of vnts w privilg. 9 S 1/18/2012&d=1/18/2099&sitid=cbpr&sc_cmp1=cb_pr677_. 10 Subsction 2.3 of Lutgn and Van dr Lindn (2013) discusss th cas α = 0 in dtail. 11 Manning and Ptrongolo (2011) provid rcnt vidnc at th local lvl for th UK. 7

8 m i (θ i ) dsignats th probability M i /V i that a vacancy in rgion i mts a workr, with 0 < m i (θ i ) < 1 by th inquality in (3) and m i (θ i) < 0 bcaus of sarch-matching congstion xtrnalitis. So, unfilld jobs find a partnr mor asily in a rgion abl to attract mor job-skrs. Th probability that an unmployd workr living in i mts a firm locatd in rgion i is p i (θ i ) = θ i m i (θ i ), with 0 < p i (θ i ) < 1. Job-skrs find a job mor asily in a thickr local labor markt: [p i (θ i )] > Th probability that an unmployd workr sarching nationally and living in i mts a firm sttld in rgion i is αp i (θ i ). In cas of national sarch, for somon living in i, th probability of gtting an offr in i and no offr from th othr rgion is p i (θ i )(1 αp i (θ i )). Th probability of th opposit vnt is αp i (θ i )(1 p i (θ i )). Th probability of gtting an offr from ach rgion is αp i (θ i )p i (θ i ). In this cas, th workr accpts th position that offrs th highst indirct utility lvl. Finally, this workr living in i facs a probability (1 p i (θ i ))(1 αp i (θ i )) of rmaining unmployd. 2.3 A modl with symmtric rgions Bfor discussing th gnral cas of asymmtric rgions, lt us considr th cas of th symmtric quilibrium. Th main ffcts of a ris in matching ffctivnss out of th rgion of rsidnc alrady appar in this framwork (in which, whn not ncssary, w drop th rgion subscript i) Wag bargaining Individual Nash bargaining taks plac x-post, onc th cost of opning a vacancy is sunk. So, whn a vacancy and a job-skr j hav mt, th wag solvs th following maximization: max(v w j Vj u ) β (J V ) 1 β j whr V is th valu of an unfilld vacancy and β [0, 1) dnots th bargaining powr of a workr. Th first-ordr condition can b rwrittn as: w j = w = βy + (1 β)b βv. (4) Hnc, th wag is indpndnt of th workr s prfrncs. As w > b, undr fr-ntry, workrs always tak th position Accptation dcision A workr sarching locally always accpts a job offr, as Vj > V j u in a fr-ntry quilibrium with Nash bargaining. Similarly, a workr sarching nationally who only gts a job offr from a firm locatd in th rgion whr sh livs always taks th position. In cas this workr only rcivs a job offr from a firm sttld in th othr rgion, sh always 12 As is standard, w assum Inada conditions: lim m(θ) θ 0 = 1; lim p(θ) θ 0 = 0; lim θ + = 0; lim p(θ) = 1. θ + 8

9 accpts th job, as sh dcidd to sarch for a job thr (as shown in Appndix A). In cas a workr gts two job offrs, sh dcids which position to tak to maximiz hr utility in mploymnt (1). Whn rgions ar symmtric, th only variabls that mattr ar prfrncs. On thus gts: Lmma 1. Accptanc dcisions in cas of symmtric rgions Whn rciving two job offrs, a workr j accpts th position in rgion 2 if c 1j c 2j < 0, taks th job in rgion 1 othrwis Opning of vacancis Th xpctd valu of a vacant position V is qual to κ + πm(θ)(y w). π corrsponds to th probability that mting a workr lads to a filld vacancy. Firms opn vacancis frly until this valu V is nil. Anticipating corrctly th outcom of th wag bargain, th fr-ntry condition bcoms: κ = πm(θ). (5) (1 β)(y b) Th probability of filling a vacancy πm(θ) incrass with th (x-post) surplus of a match y b and dcrass with th cost of opning a vacancy κ and workrs bargaining powr β Sarch and location dcisions Appndix A shows that taking sarch and location dcisions simultanously or choosing first th location and thn th sarching ara is quivalnt (th proof is givn for th gnral cas of asymmtric rgions). Thrfor, to as th xposition, th prsntation blow opts for th scond timing. Sarch dcision Whn dciding whthr to sarch or not in th othr rgion, a workr maximizs hr xpctd utility as rgional or national job-skr. Lt p i b a short notation for p i (θ i ). Th xpctd utility if th agnt locatd in i sarchs nationally is p i (1 αp i )Vij+αp i (1 p i )V ij+αp i p i (max { V ij; Vij } ε)+(1 pi )(1 αp i )Vij u, (6) whr ε stands for th small cost of rfusing a job offr.th xpctd utility of a jobskr living in rgion i and sarching for a job in this rgion only is p i V ij + (1 p i )V u ij. (7) Whn th small cost of rfusing a job offr ε tnds to zro, somon sarchs nationally if hr rlativ prfrnc for rgion i ovr rgion i, c ij c ij, is abov a thrshold 9

10 z qual to w b. Othrwis, sh dcids to look for a job in rgion i only. Prfctly anticipating th wag (4), undr fr-ntry, lt zβ(y b). Dfining th following lmma applis. z 1 = z and z 2 = z, (8) Lmma 2. Whn α > 0 and th cost of rfusing an offr ε 0. Assuming that both z i s li in ( v, v), If c 1j c 2j < z 1, agnt j sarchs in rgion 2 only; If z 1 c 1j c 2j z 2, agnt j sarchs nationally; If c 1j c 2j > z 2, agnt j sarchs in rgion 1 only. Th numbr of rgional job skrs in ach rgion is givn by: N R = v z N (9) By comparing thir xpctd utility in cas of rgional and national sarch, unmployd workrs turn out to compar th utility lvls whn thy ar actually mployd in th othr rgion and whn thy rmain unmployd in thir rgion of rsidnc. Ths utility lvls ar not in xpctd trms and so sarch dcisions ar indpndnt of probabilitis to gt a job offr. 13 Thrfor, th numbr of workrs who sarch nationally is indpndnt of th matching ffctivnss α > A highr wag rlativ to hom production yilds a highr thrshold z, implying that a lowr numbr of workrs sarch for a job rgionally. Location choic As an unmployd workr who dcids to look for a job rgionally only locats in hr rgion of sarch, w hav to compar th xpctd utility of an agnt j who sarchs nationally whil bing locatd in rgion 1 or in rgion 2. Ths xpctd utility lvls ar givn by (6), for i = {1, 2}. As rgions ar symmtric, th problm simplifis to c 1j c 2j. On gts th following rsult: Lmma 3. Location choic whn rgions ar symmtric If c 1j c 2j < 0, thn agnt j locats in rgion 2; Els, workr j sttls in rgion 1. Th siz of th population is thrfor qual in both rgions. 13 If ε was non ngligibl, ths probabilitis would howvr play a rol in th dfinition of th z i s. 14 This would still b tru if ε was non ngligibl. Whn α is nil, sarching all ovr th country cannot b ruld out but th probability of finding a job in th othr rgion is zro. So, vry job-skr sarchs locally. 10

11 2.3.5 Accptanc probability, vacancy cration and unmploymnt rats A dtaild xplanation is providd in Appndix C. Th numbr of job-skrs for a firm is v + αz in fficincy units. As w hav sn, a workr locatd in a rgion always accpts th position thr. Th workrs locatd in th othr rgion accpts th position if thy did not gt an offr from a firm locatd in thir rgion of rsidnc (probability p). Dnoting by π th accptanc rat conditional on th mting, on gts π = 1 αpz v + αz Whn workrs cannot sarch out of thir rgion of rsidnc (i.. α = 0), th accptanc rat is 1. Whn th probability p of mting a vacancy incrass, th accptanc rat shrinks. Th sam holds for a ris in α. In both cass, th incras lads to a highr numbr of workrs that gt two job offrs. Bcaus ths workrs hav to rfus on of th two, th accptanc rat gos down. Combining (5) with (10) lads to th following fr-ntry condition: κ (1 β)(y b) (10) v + α(1 p)z = m(θ) (11) v + αz Bcaus α affcts ngativly th accptanc rat, a ris in th matching ffctivnss lads to a lowr quilibrium tightnss. So dos a ris in κ. Th unmploymnt rat in a rgion can b writtn as: u = (1 p)(v αpz) v To construct this rat, on nds to notic that bcaus rgions ar symmtric, th population laving x-post in a rgion is N/2. Rgarding th numbr of unmployd, it is qual to (1 p)n R + (1 p)(1 αp)n N, with (12) N N = N/2 N R = (z/v)(n/2) (13) A highr tightnss or a highr matching ffctivnss out of th rgion of rsidnc α yilds, othr things qual, lowr unmploymnt rats Equilibrium and impact of a shock to th matching ffctivnss Dfinition 1. An intrior quilibrium is a vctor {w, θ, u, N R, N N } and a scalar z that satisfis quations (4) undr fr-ntry V = 0, (8), (9), (11),(12) and (13). This quilibrium xists and is uniqu whnvr v > β(y b) (s Appndix E). It is dtrmind rcursivly. Onc w and z ar computd, tightnss θ is dtrmind using th fr-ntry condition (11), which allows to st uniquly u. Proposition 1. Whn rgions ar symmtric, a ris in th matching ffctivnss in th othr rgion α has an ambiguous ffct on th quilibrium unmploymnt rats. 11

12 π α, θ m(θ) κ 1 β (y b) 1 m(θ) θ 1 θ 0 u(θ, α) π α, θ m(θ) θ u(θ, 0) u 0 u 1 1 u(θ, α) Figur 1: Th symmtric quilibrium: α = 0 (π = 1; th black curvs) vrsus α > 0 (π < 1; th rd curvs). Th consquncs of incrasing th matching ffctivnss can b shown graphically in this simpl conomy. In th uppr part of Figur 1, w draw th lft-hand sid of (11) whn α = 0 (in black) and whn 0 < α 1 (in rd). A ris in α inducs a lftward shift of th curv. Th quilibrium lvl of tightnss thrfor dclins bcaus th accptanc rat π shrinks with α. Th lowr part of th figur draws (12) and illustrats th favorabl partial ffct of a ris in α on th unmploymnt rat conditional on tightnss. Dpnding on th importanc of th shifts of th two curvs th quilibrium unmploymnt rat can vary in both dirctions. If α = 0 (rsp., α > 0) can b intrprtd as a world without (rsp., with) th modrn communication tchnologis using th Intrnt, Figur 1 illustrats that th introduction of ths tchnologis can hav ambiguous ffcts on th rgional quilibrium unmploymnt rats. This is a way of rationalizing th rsults of Kroft and Pop (2014). 2.4 A modl with asymmtric rgions This sction highlights th main diffrncs with rspct to th symmtric cas and shows how th main conclusions from a shock on α ar affctd. W solv th problm by backwards induction. 12

13 Th gnralizd Nash bargaining procss is now givn by max(v w ij Vij u ) β i (J i V i ) 1 β i. i Th first-ordr condition can b rwrittn as: w i = β i y i + (1 β i )b i β i V i. (14) Th conclusions drawn in th prvious subsction still holds: wags ar indpndnt of workr s prfrncs and ar a positiv function of th workr s bargaining powr and productivity. Wags in rgion i do not dpnd on th conditions in th othr rgion Accptanc of a job offr If a workr sarching nationally gts two offrs, on from ach rgion, sh rjcts on of thm (incurring an arbitrary small cost ε) and accpts th othr on. To tak this dcision, th unmployd workr compars V1j with V 2j. Th agnt whos rlativ prfrnc c 1j c 2j is abov th thrshold w 2 w 1 + a 2 a 1 chooss to work in rgion 1 rathr than in rgion 2. Lt = b 2 b 1 + a 2 a 1. Lmma 4. Whn a job-skr sarching nationally has on job offr from ach rgion, thr is a thrshold ˆx = + β 2 (y 2 b 2 ) β 1 (y 1 b 1 ), (15) assumd to b in ( v, v), such that sh accpts th job offr in rgion 2 if c 1j c 2j < ˆx. Othrwis, sh accpts th job offr in rgion 1. Th highr th workr s shar of th surplus in rgion 2 or th highr, th mor popl will accpt th position in rgion 2 whn gtting two job offrs Vacancy cration Firms opn vacancis in rgion i until th xpctd gain V i is nil (i {1, 2}). Combining (14) and th fr-ntry condition yilds κ i (1 β i )m i (θ i ) = π i(y i b i ), i {1, 2}, (16) whr π i is th conditional accptanc rat in rgion i. Th rat at which vacancis ar on avrag filld, π i m i (θ i ), varis as in th prvious subsction with th paramtrs in i Sarch dcision and location choic Sarch dcision An individual j living in rgion 2 dcids to sarch rgionally or nationally by comparing th xpctd utility in both cass (Equations (6) and (7) for i = 2). Whn th small cost of rfusing a job offr ε tnds to zro, somon sarchs nationally if hr rlativ prfrnc for rgion 1 ovr rgion 2, c 1j c 2j, is abov a 13

14 thrshold z 1 = b 2 w 1 + a 2 a 1. Othrwis, sh dcids to look for a job in rgion 2 only. This is shown in Appndix A (th comparison of cass and f). A similar dvlopmnt is conductd for a workr sttld in rgion 1. A job-skr living in rgion 1 whos rlativ prfrnc for rgion 1 ovr rgion 2 is highr than a thrshold z 2 = w 2 b 1 + a 2 a 1 sarchs in rgion 1 only. Blow this thrshold, th workr looks for a job in th whol country (s th comparison of cass a and c in Appndix A). Undr prfct anticipation of bargaind wags and fr-ntry, lt now th z i thrsholds bcom: z 1 = β 1 (b 1 y 1 ) + and z 2 = β 2 (y 2 b 2 ) + (17) Thn, Lmma 2 applis. Th shars of ths thr groups in th total population ar v+z 1 for rgional job-skrs in rgion 2, z 2 z 1 for national ons and v z 2 for rgional job-skrs in 1. Rmmbring (15), it is asily sn that z 1 ˆx z 2. A ris in rlativ amnity lvls in rgion 2, a 2 a 1, lads to mor rgional job-skrs in rgion 2, lss rgional job-skrs in rgion 1 but lavs th numbr of national ons unchangd. A ris in th xpctd surplus of a match in rgion 1 dos not affct th dcision to sarch or not in rgion 2 (th thrshold z 2 rmains constant)but implis lss rgional job-skrs living in 2. Finally, a ris in α still has no impact on th z thrsholds. Location choic Th sam comparison as in th symmtric cas lads to th following lmma. Lmma 5. If rlativ idiosyncratic prfrnc c 1j c 2j < x, agnt j chooss to rsid in rgion 2, whr x = + 1 α 1 α 1 αp 1 αp 2 + αp 1 p 2 (p 2 β 2 (y 2 b 2 ) p 1 β 1 (y 1 b 1 )), (18) with 0 1 αp 1 αp 2 +αp 1 p 2 < 1 and, by (17), z 1 x z 2. Othrwis agnt j sttls in rgion 1. Th shar of th population living x-ant in rgion 2 (rspctivly, 1) is thn v+x (rspctivly v x ). Compard to th symmtric cas whr x was qual to 0, w s that rgional diffrncs mattr. Notic that whn sarch is qually fficint whrvr on looks for a job (i.. α = 1), diffrncs in wags and lvls of tightnss do not impact th workr s location dcision. An incras in rlativ amnitis in rgion 2, a 2 a 1, as wll as a ris (rspctivly, a drop) in th valu of hom production in rgion 2 (rsp., 1) induc mor workrs to locat in 2 x-ant. A ris in α attracts mor workrs in rgion 1 if p 2 β 2 (y 2 b 2 ) p 1 β 1 (y 1 b 1 ) > 0 and convrsly. 15 An incras in tightnss in rgion 1 has svral ffcts if 0 < α < 1. First, if on livs in rgion 1, th incras in th probability of bing mployd in this rgion quals th dcras in th probability of 15 Whn this inquality holds, a ris in α augmnts th chancs of bnfiting from th bttr mploymnt prospcts in rgion 2 whil staying in th first on. 14

15 bing unmployd. As th individual stays in th sam rgion, th nt gain is proportional to w 1 b 1. Scond, if on livs in rgion 2, th incras in th probability of bing mployd in rgion 1 quals th dcras in th probability of staying unmployd in rgion 2. This ffct is proportional to V1j V 2j u. Third, th dclin in th probability of bing mployd in 2 is th sam whrvr on livs. So, this ffct cancls out. Th first and th scond ffcts push th diffrnc in idiosyncratic prfrnc of th indiffrnt agnt, x, in opposit dirctions so that th nt ffct is ambiguous. This conclusion also holds if θ 2 incrass. Howvr, th first ffct dscribd just abov dominats if α is sufficintly small. 16 Proposition 2. Whn 0 < α < 1, a tightr labor markt in rgion i inducs mor popl to rsid thr undr th following sufficint condition: α < 1 (β 1(y 1 b 1 ) β 2 (y 2 b 2 )) 2 (β 1 (y 1 b 1 ) + β 2 (y 2 b 2 )) 2. (19) Whn α = 1, th lvls of tightnss do not influnc th choic of rsidnc any mor. Th proof is providd in Appndix B. Rcalling (18), Condition (19) is asy to intrprt: Th highr th intr-rgional diffrnc in workrs shars in th surpluss, 17 th lowr α should b in ordr to gt th intuitiv rlations btwn th lvls of tightnss and x Summary of th accptanc, sarch and location dcisions Combining Lmmas 2 and 5, th total labor forc is mad of four groups with distinct bhaviors: Proposition 3. Partition of th population If c 1j c 2j < z 1, agnt j locats in rgion 2 and sarchs thr only; If z 1 c 1j c 2j < x, agnt j sttls in rgion 2 and sarchs in th whol country; If x c 1j c 2j z 2, agnt j locats in rgion 1 and looks for a job nationally; If c 1j c 2j > z 2, agnt j sttls in rgion 1 and looks for a job in rgion 1 only. Figur 2 illustrats this partition of th total population if v < z 1, z 2 < v. In gnral, on cannot rank thrshold valus x and ˆx sinc x varis with th lvls of tightnss. Whn α = 1 and rgions ar asymmtric, th comparison of th thrsholds is obvious sinc x = thn : ˆx x β 2 (y 2 b 2 ) β 1 (y 1 b 1 ) W thank on of th anonymous rfrs who suggstd to study this condition. 17 Compard to th sum of ths shars in th 2 rgions. 15

16 Figur 2: Th partition of th population in cas of an intrior solution Accptanc probability and vacancy cration A dtaild xplanation is providd in Appndix C. Considr again a vacant position in rgion 1. Th mass of job-skrs sarching for a job in 1 is now [v x+α(x z 1 )][N/] in fficincy units. Conditional on mting on of ths unmployd workrs, all thos whos rlativ prfrnc c 1j c 2j lis abov ˆx accpt for sur an offr from rgion 1. For thos btwn z 1 and ˆx, this is only th cas if thy gt no offr from rgion 2. So, conditional on a contact btwn a vacancy in rgion 1 and a job-skr, th accptanc probability π 1 is (with a corrsponding xprssion for π 2 ): αp 2 (ˆx z 1 ) π 1 = 1 v x + α(x z 1 ) αp 1 (z 2 ˆx) π 2 = 1 v + x + α(z 2 x) It is asily chckd that th highr ˆx, th mor workrs accpt job offrs in rgion 2 and so th lowr th accptanc rat in 1. Th sam is tru for x. Th highr th numbr of workrs sarching in rgion 2 only (an incrasing function of z 1 ), th highr th accptanc rat in 1. Finally, an incras in th probability of gtting a job offr in rgion 2 dcrass th accptanc rat in rgion 1. Similarly, π 2 incrass with ˆx and x and dcrass with z 2 and p 1. Th impact of th matching ffctivnss α should b strssd: A highr α lads to a lowr conditional accptanc probability, as th probability of gtting two job offrs incrass. Convrsly, lim α 0 π i = 1. Combining (16) with (20)-(21) lads to th following fr-ntry conditions: κ 1 (20) (21) (1 β 1 )m 1 (θ 1 ) = v x + α(x z 1) αp 2 (θ 2 )(ˆx z 1 ) (y 1 b 1 ) v x + α(x z 1 ) (22) κ 2 (1 β 2 )m 2 (θ 2 ) = v + x + α(z 2 x) αp 1 (θ 1 )(z 2 ˆx) (y 2 b 2 ) v + x + α(z 2 x) (23) 16

17 Through th ndognous accptanc rat, vacancy cration in any rgion is now affctd by th lvl of tightnss and th valu of paramtrs in th othr rgion Populations and unmploymnt rats Sinc z 1 x z 2, and assuming an intrior solution, th sizs Ni N of th workforc sarching nationally and Ni R of thos sarching in thir rgion of rsidnc only ar givn by: N R 1 = (v z 2) N R 2 = (z 1 + v) N N1 N = (z 2 x) N (24) N N2 N = (x z 1) N (25) so that th total x-ant population sizs ar N 1 = N1 N + N 1 R = v x N and N 2 = N2 N + N 2 R = v+x N, with N 1 + N 2 = N. Th unmploymnt rats which ar th ratio of th numbr of (x-post) unmployd workrs ovr th (x-post) population, ar: (1 p 1 )(v x αp 2 (z 2 x)) u 1 = v x αp 2 (1 p 1 )(z 2 x) + αp 1 (1 p 2 )(x z 1 ) + αp 1 p 2 (x ˆx) (1 p 2 )(v + x αp 1 (x z 1 )) u 2 = v + x + αp 2 (1 p 1 )(z 2 x) αp 1 (1 p 2 )(x z 1 ) αp 1 p 2 (x ˆx) (26) (27) Th numbr of (x-post) unmployd workrs in, say, rgion 1 is composd of th agnts living x-ant in 1 who did not gt a job offr in rgion 1, (1 p 1 ) v x N, to which w subtract th workrs who did not gt an offr from rgion 1 but wll from rgion 2 (αp 2 (1 p 1 ) z 2 x N). Th dnominator corrsponds to th population living x-post in th rgion (up to a N/ trm). Ex-post, th numbr of inhabitants in, say, rgion 1 is th sum of 4 trms. Th first trm rprsnts th population living x-ant in rgion 1. Th scond trm corrsponds to th workrs who wr living x-ant in 1 and who lav rgion 1 as thy only gt a position in rgion 2. Th third trm is composd of th agnts who livd x-ant in rgion 2 and who mov as thy only gt an offr from rgion 1. Finally, th fourth trm rprsnts th numbr of workrs who gt two offrs. This trm is positiv whnvr x > ˆx, maning that som mor workrs living in 2 x-ant accpt a position in rgion 1. Th following lmma provids th signs of th partial drivativs of (26) and (27) with rspct to th othr ndognous variabls and α. 18 Lmma 6. Whn α > 0 and rgions ar asymmtric, th unmploymnt rat u i dcrass with th lvl of tightnss θ i in th sam rgion. A ris in th matching ffctivnss α lowrs both rgional unmploymnt rats. Furthrmor, (i) Th sign of th cross-drivativs u i / θ i varis with th sign of p 2 β 2 (y 2 b 2 ) p 1 β 1 (y 1 b 1 ) (mor on this blow). (ii) A ris in th thrshold x (i.. a biggr workforc N 2 living x-ant 18 Th proof, which is availabl upon rqust, consists in rorganizing ths partial drivativs. 17

18 in rgion 2) dcrass th unmploymnt rat in rgion 1 whil it incrass it in rgion 2. (iii) An incras in th numbr of rgional job-skrs incrass th unmploymnt rat in both rgions. (iv) Th unmploymnt rat in rgion 1 incrass with ˆx, whil th opposit holds for th unmploymnt rat in rgion 2. An incras in rgion i s labor markt tightnss θ i boosts th probability that a workr living in rgion i finds a job thr and it riss th probability that a workr locatd in th othr rgion gts a position in rgion i (which incrass th labor forc living in rgion i). Consquntly, th unmploymnt rat in rgion i gos down. Turning to Proprty (i) of Lmma 6, augmnting θ i riss th probability of laving rgion i and this rducs both th numbr of unmployd workrs and th siz of th labor forc in rgion i. Th nt ffct on th unmploymnt rat dpnds on th intr-rgional diffrnc in th gains p i β i (y i b i ). Mor prcisly, Appndix D shows that p 2 β 2 (y 2 b 2 ) < p 1 β 1 (y 1 b 1 ) u 1 θ 2 < 0, p 2 β 2 (y 2 b 2 ) > p 1 β 1 (y 1 b 1 ) u 2 θ 1 < 0, u 2 u 2 lim > 0 and lim < 0, (28) α 0 + θ 1 α 1 θ 1 u 1 u 1 lim > 0 and lim < 0. (29) α 0 + θ 2 α 1 θ 2 Consquntly, th cross-partial drivativs ar both ngativ whn α is big nough. Whn α = 0, th unmploymnt rat in a rgion dos not dpnd on tightnss in th othr. Howvr, for sufficintly small positiv valus of α, on cross-drivativ u i / θ i is positiv. As x gos up (Proprty (ii)), th numbr N 1 of agnts living x-ant in rgion 1 shrinks whil N 2 incrass. Ths population sizs ar (up to a N/ trm) prsnt in th numrators and th dnominators of (26) and (27). In addition, a ris in x affcts th numbrs of national job-skrs who dplts th rgional workforcs if thy ar rcruitd in th othr rgion. All in all, a ris in x rducs (rsp., incrass) th unmploymnt rat in rgion 1 (rsp., 2). A corollary of Proprty (iii) is that mor workrs sarching all ovr th country rducs th unmploymnt rats in both rgions. Ths proprtis as wll as th favorabl rol of α on th unmploymnt rats ar conditional on th othr ndognous variabls. In a standard Mortnsn-Pissarids stting (whr gographical htrognitis ar concald in an aggrgat matching function), th siz of th labor forc dos not affct th quilibrium unmploymnt rat (as vntually th numbr of vacancis riss proportionatly, laving th quilibrium lvl of tightnss unaffctd). This quilibrium proprty is not diffrnt hr (N plays no rol in (26)-(27)). Howvr, Proposition 4. If α > 0 and rgions ar asymmtric, th quilibrium unmploymnt rats ar affctd by th partition of th population btwn th two rgions and btwn th two statuss of national vrsus rgional job-skrs Equilibrium Dfinition 2. Whn 0 < α 1 and rgions ar asymmtric, an intrior quilibrium is a vctor {x, ˆx, z 1, z 2 } assumd to b in ( v, v) and a vctor {w i, θ i, u i, Ni N, Ni R} i {1,2}, 18

19 solving (14) undr fr-ntry V i = 0, (15), (16), (17), (18), (20), (21), (24), (25), (26) and (27). As w alrady know that z 1 ˆx, x z 2, an quilibrium is intrior if v < z 1 and z 2 < v. From th dfinition of and Lmma 2, ths conditions bcom: Condition 1. Ncssary and sufficint conditions for an intrior solution ar v > β 1 (y 1 b 1 ) (b 2 b 1 + a 2 a 1 ) and v > β 2 (y 2 b 2 ) + (b 2 b 1 + a 2 a 1 ) (30) Th systm of quations charactrizing an quilibrium is block rcursiv. Th thrsholds z i s and ˆx bing xplicit functions of th paramtrs only, th cntral ndognous variabls ar {θ 1, θ 2, x}. Thy ar dtrmind by th systm (18)-(22)-(23). Onc this systm is solvd, all othr ndognous variabls gt uniqu valus. Undr Condition (19), Dfinition (18) is an implicit rlationship which is dcrasing in θ 1 and incrasing in θ 2 (unlss α = 1 in which cas x is constant). Substituting this rlationship into th fr-ntry conditions (22)-(23) yilds: κ 1 (1 β 1 )m 1 (θ 1 ) π 1(θ 1, θ 2 )(y 1 b 1 ) = 0 (31) 0 <0 κ 2 (1 β 2 )m 2 (θ 2 ) π 2(θ 1, θ 2 )(y 2 b 2 ) = 0, (32) <0 whr th inquality signs undr th θ i s dsignat thos of th partial drivativs. Th fr-ntry condition in rgion 1, (31), can b sn as an implicit raction function θ 1 = Θ 1 (θ 2 ). Similarly, (32) dfins an implicit rlationship θ 2 = Θ 2 (θ 1 ). Figur 3 draws ths functions for th limit cass, namly α = 0 (th two rlationships bing thn orthogonal) and α = 1 (th dottd curvs), 19 and for an intrior valu α. By looking at (20) and (21), it is asily sn that th valus of Θ i (0) ar indpndnt of th magnitud of α. Th nt ffct of a chang in a rgional gap y j b j on th quilibrium lvls of tightnss is hard to sign bcaus it affcts almost all thrsholds prsnt in th accptanc rats π i, i {1, 2}. A ris in α has a clar ffct on on of th accptanc probabilitis but an ambiguous on on th othr. So, th impact on both quilibrium lvls of tightnss is hard to sign (this ffct was ngativ in th symmtric cas). Th cost of opning a vacancy is a dtrminant of unmploymnt diffrntials mphasizd in th urban sarch-matching litratur (s.g. Coulson t al., 2001). A ris in th cost of opning a vacancy in, say, rgion 1 only shifts th Θ 1 (θ 2 ) function to th lft, lading to a lowr quilibrium valu of θ 1 and, mor intrstingly, to a ris in θ 2 (s th rd curv and compar quilibrium E and E in Figur 3). According to Lmma 6, th dirct implications ar a ris (rsp., a dclin) in th unmploymnt rat in rgion 1 (rsp., 2). Thr ar howvr also 19 If α = 1, π i is a function of tightnss in th othr rgion only. θ 1 = Θ 1(θ 2) and θ 2 = Θ 2(θ 1) ar ngativly slopd. It can asily b chckd that + > lim Θ1(θ2) > lim Θ1(θ2) > 0 and + > θ 2 0 θ 2 + lim Θ2(θ1) > lim Θ2(θ1) > 0. Furthrmor, th Θi (θ i) functions ar convx if th matching function θ 1 0 θ 1 + is a Cobb-Douglas. With mor gnral matching functions, th sam conclusions hold if th matching rats m i(θ i) ar sufficintly convx. 0 19

20 θ 2 Θ 1 θ 2, α = 0 Θ 1 θ 2 α = 1 Θ 1 θ 2 dκ 1 > 0 Θ 1 θ 2 0 < α < 1 Θ 2 θ 1 α = 0 E E Θ 2 θ 1 0 < α < 1 Θ 2 θ 1 α = 1 θ 1 Figur 3: Th quilibrium lvls of tightnss for various valus of α. inducd ffcts in various dirctions. Givn Proposition 2, mor popl dcid to rsid in rgion 2 (x riss). Thn, by Rsult (ii) of Lmma 6, th unmploymnt rat in rgion 1 (rsp., 2) shrinks (rsp., incrass). In addition th cross-ffcts u i / θ i can rinforc or not th dirct ffcts (s (28) and (29)). Th nt ffct on th unmploymnt rats is thrfor ambiguous, contrary to th symmtric cas whr th nt ffct was ngativ. 3 Efficincy This sction studis th fficincy of th laissz-fair 20 dcntralizd quilibria introducd in th prvious sction. W first driv th symmtric optimal allocation and analyz th diffrncs btwn this allocation and th dcntralizd quilibrium charactrizd in Sction 2.3. W xplain why th Hosios condition is not sufficint to guarant fficincy of th dcntralizd quilibrium. W nxt turn to th analysis of th asymmtric conomy and show that additional infficincis aris. In a dirctd sarch framwork with multipl applications, wag posting with commitmnt and no rcall of applications, Galnianos and Kirchr (2009) conclud that th dcntralizd quilibrium is infficint bcaus firms cannot influnc th probability of rtaining applications whn thy gt mor than on. By allowing firms to rcall all th applicants thy rciv, Kirchr (2009) shows that fficincy is rstord. In our framwork, firms cannot rciv mor than on application whil som job-skrs can gt two job opportunitis. Consquntly, in th dcntralizd quilibrium, th probability that a vacancy is rjctd is not nil. This fatur, which is also prsnt in Galnianos and Kirchr (2009), is as such a sourc of fficincy that could b avoidd in a dynamic modl in continuous tim. W rturn to this in Sction 5. Hr, w show that thr ar 20 This xprssion is addd sinc thr is no public intrvntion in Sction 2. 20

21 othr sourcs of infficincy that ar spcific to our rgional modl. 3.1 Th symmtric cas In our static framwork, w will first look at th cas whr th constraind social plannr only chooss th lvls of tightnss. All thrsholds ar dtrmind as in th dcntralizd conomy. In this nvironmnt, th probability of rjcting an offr will b xactly th sam in th dcntralizd and th fficint conomis if th corrsponding tightnss lvls ar qual too. Nxt, w will lt th plannr choos in addition all th thrshold valus but w will impos that th arrival of two offrs cannot b avoidd by th plannr if it is fficint that som workrs sarch all ovr th country. If w assum two symmtric rgions, th fficint allocation is symmtric as wll. W can dnot th thrsholds z 2 = z 1 = z [0, v] and th probability of bing rcruitd p 1 = p 2 = p. Th plannr s objctiv function masurs nt output adjustd to tak account of amnitis and idiosyncratic prfrncs for rgions: 21 2 ((y b)l κv) + (b + a)n + N 0 v c 2jdj + N v 0 c 1jdj + N αp(θ)(1 p(θ)) [ z 0 (c 1j c 2j )dj ] + N αp(θ)(1 p(θ)) [ 0 z (c 1j c 2j )dj whr mploymnt L = (N/) [p(θ)v + αp(θ)(1 p(θ))z] and th numbr of vacancis V = θ(n/) [v + αz]. Hnc, up to a constant trm, th objctiv can b rwrittn as: N v [(y b)p(θ)(v + αz(1 p(θ))) κθ(v + αz) αp(θ)(1 p(θ))(z2 /2)]. (33) To start with, lt z b qual to its dcntralizd valu (8). Thn, th optimal tightnss vrifis: κ + αz(1 2p(θ))) α(1 2p(θ)) z 2 = (y b)(v (34) (1 η)m(θ) v + αz v + αz 2 In a dcntralizd quilibrium, tightnss solvs (5), whr th accptanc rat π vrifis (10) and hnc is similar to but diffrnt from th trm that multiplis y b in (34). In addition, th last trm in th lattr xprssion is not prsnt in (5). So, two sourcs of infficincy aris. First, th plannr rcognizs that an additional vacancy in rgion i rducs th chancs of a match btwn rsidnts of rgion i and vacancis in rgion i, whil firms in rgion i hav no rason to shar th sam concrn. By subtracting th nonngativ trm (y b)p(θ)αz/(v + αz), 22 th plannr intrnalizs an inducd ffct on xpctd output in th othr rgion and this pushs optimal tightnss downwards. Scond, whn choosing th numbr of vacancis in ach rgion, th plannr taks into account th impacts of a ris in th numbr of vacancis on th valu of th idiosyncratic 21 In xprssions x c2jdj and v c1jdj, thr is a slight abus of notation sinc v and x ar valus v x for th diffrnc c 1j c 2j. This notation is quivalnt to assuming a bijctiv rlationship btwn th idntifir of workrs, j, and thir rlativ prfrnc for rgion 1, c 1j c 2j. 22 Mrgd with th probability of rjcting an offr, it lads to th trm 1 2p(θ) in (34) whil it is only 1 p(θ) in (11). ] 21