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 January 8, 2014 Abstract This papr studis quilibrium unmploymnt in a two-rgion conomy whr homognous workrs and jobs ar fr to mov and th housing markt clars. Bcaus of th Intrnt, sarching for a job in anothr rgion without first migrating thr is nowadays much simplr than it usd to b. Sarch-matching xtrnalitis ar amplifid by this possibility and by th fact that som workrs can simultanously rciv a job offr from ach rgion. Th rst of th framwork builds on Mortti (2011). W study numrically th impacts of various local shocks in a stylizd US conomy. Contrary to what could b xpctd, incrasing matching ffctivnss in th othr rgion yilds growing rgional unmploymnt rats. W charactriz th optimal allocation and conclud that th Hosios condition is not sufficint to rstor fficincy. In th fficint allocation, th rgional unmploymnt rats ar much lowr than in th dcntralizd conomy and nobody sarchs in th othr rgion. Kywords: Matching; Sarch thn mov; Spatial quilibrium; Rgional conomics; Unmploymnt diffrntials. JEL: J61, J64, R13, R23. W ar gratful to Pirr-Philipp Combs, David d la Croix, Bruno Dcrus, Etinn Lhmann, Frank Malhrbt, Olivir Pirrard, Hnri Snssns, Matthias Wrd and Yvs Znou. W also thank participants of 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 SaM workshops hld in Roun and in Aix-n-Provnc, th Blgian Day for Labor Economists in Luvn and th 2013 EALE Confrnc in Turin). Th usual disclaimr applis. W acknowldg financial support from th Blgian Frnch-spaking Community (convntion ARC 09/14-019 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
1 Introduction Rgional diffrncs in unmploymnt rats ar larg and prsistnt. This holds tru whn controlling for usual obsrvd charactristics such as ducation and ag. S Klin and Mortti (2013) for th US and Ovrman and Puga (2002) and Elhorst (2003) for Europ. Som paprs xplain ths disparitis using a gnral quilibrium approach with intr-rgional migration. Migration occurs whn workrs ar unmployd and dcid to rlocat to sk jobs in an othr rgion. Nowadays 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 on-lin through virtual rcruiting tools. 2 W dvlop a gnral quilibrium sarch-matching framwork whr job-skrs can choos to sarch all ovr th country and only migrat in cas of succssful sarch. On could think that this opportunity of sarching all ovr th country would rduc rgional unmploymnt rats and hnc incras output. Our thortical and numrical analyss show that a ris in unmploymnt and a loss in fficincy ar instad vry plausibl. Our rsults shd 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. A rcnt microconomtric valuation finds that on-lin job sarch shortns unmploymnt duration (Kuhn and Mansour, 2011). Via a diffrnc-in-diffrncs approach, Kroft and Pop (2014) find howvr no vidnc that th rapid xpansion of a major onlin job board (during th yars 2005-2007) has affctd city-lvl unmploymnt rats. So, th rasons why improvmnts at th individual lvl disappar at an aggrgat lvl nd to b undrstood. This papr proposs an xplanation in a spatial conomy. 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 rgional supply of housing is not prfctly inlastic and th supply of labor not prfctly lastic. Th lattr 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 sarch-matching frictions and wag bargaining within this framwork (Mortnsn and Pissarids, 1999, Pissarids, 2000). 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 1 In 2010, 25% of th intrviwd Amricans who us th Intrnt dclard to do so to find a position. In Europ, in 2005, among th unmployd workrs, 25% usd th Intrnt to sarch for a job (U.S. Cnsus Burau (2012), Survy of Incom and Program Participation, 2008 Panl). This shar has incrasd to almost 50% in 2011 (Eurostat (2013)). 2 S.g. http://variousthingsliv.com/virtual-opn-hous/ and th links on http://hiring. monstr.com/hr/hr-bst-practics/rcruiting-hiring-advic/acquiring-job-candidats/ virtual-rcruitmnt-stratgis.aspx. 2
among agnts ar summarizd by a rgional-spcific matching function with constant rturns to scal. Bcaus of ths frictions, a ralizd match btwn a job skr and a job vacancy crats a pur conomic rnt. This rnt, also calld th surplus of th match, is shard through Nash bargaining. Frictions gnrat congstion xtrnalitis which ar not intrnalizd by dcntralizd agnts unlss th so-calld Hosios (1990) condition is mt. This gnral proprty in th sarch-matching litratur dos not hold in our stting whr job skrs living in a givn rgion can choos to sarch all ovr th country (with possibly a lowr fficincy out of th rgion of rsidnc). If thy do, rgions ar much mor intrrlatd, with a rang of consquncs. Mor spcifically, 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. In particular, national job-skrs can gt mor than on job offr but will only accpt on and this gnrats a loss of rsourcs. In addition, 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. Thrfor, as soon as som workrs sarch all ovr th country, w can show that th Hosios condition is nvr sufficint to dcntraliz th (constraind) fficint allocation (which maximizs nt output subjct to th matching constraints). Whn rgions ar symmtric, w also show that fficincy rquirs that ithr nobody sarchs all ovr th country or all job skrs do. Nxt, w dvlop a numrical xrcis for a stylizd US conomy whr rgions ar initially symmtric and th Hosios condition prvails. Augmnting th ffctivnss of sarch out of th rgion of rsidnc incrass th unmploymnt rat vrywhr bcaus th inducd ngativ ffct on vacancy cration outwighs th dirct favorabl ffct on th risk of unmploymnt. This drop in labor dmand is a rspons to lowr accptanc rats of job offrs whn additional job-skrs gt mor than on offr. A ris in productivity in on rgion sharply rducs th local unmploymnt rat and incrass it lswhr. Rising th cost of vacancy cration in a rgion is dtrimntal to this rgion and favorabl to th rst of th conomy. A chang in amnitis in a rgion turns out to hav ngligibl ffcts on unmploymnt rats. Finally, th dcntralizd conomy is far from fficint. Th fficint unmploymnt rats amount to 1% whil thy ar qual to 7% in th dcntralizd conomy. For a vry wid rang of paramtrs, fficincy rquirs that nobody sarchs in th whol country whil 65% of th workforc dos it in th dcntralizd conomy. To our knowldg, fficincy has not oftn bn studid in modls with imprfct labor markt and intr-rgional migration. Boadway t al. (2004) build a static modl with sarch-matching frictions, immobil workrs but mobil firms. Thy aim at studying policis to rstor fficincy whn thr is an infficint distribution of firms bcaus of agglomration ffcts in matching and production. Within th limit of this papr, w do not look at policy intrvntions to improv fficincy. 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
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 (2012) 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. Inspird by Ortga (2000) s intrnational migration modl, Fonsca (2003) dvlops a two-rgion modl. Ignoring th cost of sarch, sh xplains th link btwn rgional migration and unmploymnt rats whn a productivity shock ariss. Th modl is thn calibratd using Spanish data and simulatd. A highr productivity in on rgion rducs its unmploymnt rat, a fact that has bn widly documntd. Building upon Baudry t al. (2012), 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. W hav ths faturs in common. 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. Thy dvlop a thorough mpirical analysis whil w calibrat and simulat our modl. 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. 3 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 diffrntials. For this purpos, thy build a dynamic two-sctor two-rgion modl with 3 Molho (2001) dvlops a partial quilibrium job-sarch framwork with both typs of sarch. W xtnd his approach by intgrating it in a gnral quilibrium modl with ndognous wag. 4
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. Th rst of this papr is organizd as follows. Sction 2 dscribs th modl and its quilibrium. Sction 3 studis fficincy. A numrical analysis is conductd in Sction 4. Sction 5 concluds. 2 Th modl This sction dvlops a modl with two distant rgions. It first proposs a bnchmark cas undr prfct comptition on th labor markt. Thn, it introducs sarch-matching frictions in th labor markt. To start with, it is assumd that a match can only b formd if th vacant job and th unmployd workr ar locatd in th sam rgion. This standard assumption is rlaxd in a last subsction whr vacancis and job skrs locatd in diffrnt rgions ar allowd to mt. 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 mobility through migration is allowd at no cost. 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 4 and takn as th numrair. Thr is also a prfctly comptitiv housing markt. Each individual consums on unit of housing. 5 All th housing stock is ownd by absnt landlords. As in Mortti (2011), th housing supply curv in rgion i is xognous and linar: r i = r i + s i N P i s i 0, i {1, 2} (1) whr r i is th rnt paid for a unit of housing and Ni P dnots housing dmand, i.. th population living in rgion i whn th housing markt clars (w call it th x-post population). Th lvl of paramtr s i dpnds on gographic and institutional rgional spcific faturs. At th limit, whn this paramtr bcoms hug, th housing supply is clos to vrtical (a fixd supply bing th standard assumption in th Rosn-Roback framwork). Whn s i is finit, th housing markt crats a congstion ffct: On additional rsidnt in a rgion riss th cost of living of all inhabitants. 6 4 It is asily shown that ach rgional good markt is balancd. S Appndix A for a proof. 5 W howvr nglct th amount of land occupid by plants. To introduc firms dmand for land would complicat th modl without yilding mor insights. 6 This vry simplifid way of modling th housing markt is similar in spirit with th approach adoptd by Baudry t al. (2014) and, as far as supply is concrnd, with Amior (2012). Glasr (2008) modls th us of local inputs in th housing production function. Notowidigdo (2011) taks 5
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 in, say, th ral wag. 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 r i (2) 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). 7 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 r i. (3) 2.1 Th prfctly comptitiv labor markt bnchmark Bfor introducing sarch-matching frictions, it is usful to prsnt th quilibrium undr prfct comptition in th labor markt. Throughout th papr, w assum constant rturns in th production of th consumption good. 8 Lt y i > b i b th constant ral marginal product of labor in rgion i. Th whol labor forc in this rgion is mployd in quilibrium and is paid at th marginal product: w ij = w i = y i. Workrs ar fr to locat in th rgion thy prfr. Agnt j chooss a dwlling in rgion 1 if V1j V 2j, othrwis sh dcids to liv in rgion 2. Hnc, thr xists a valu of th rlativ prfrnc c 1j c 2j, say x, such that th workr is indiffrnt btwn th two rgions: x = y 2 y 1 + a 2 a 1 + r 1 r 2 with r 1 = r 1 + s 1 N1 P and r 2 = r 2 + s 2 N2 P. Individuals whos rlativ prfrnc for rgion 1 ovr rgion 2 is highr than x locat in rgion 1, whil in th opposit cas, thy into account th limitd dprciation of housing and studis th impact of positiv and ngativ labor dmand shocks. In a rcnt papr, Karahan and Rh (2013) analyz th link btwn labor and housing markt outcoms through a dirctd sarch modl on th housing markt. Th housing markt is oftn tratd in th urban conomics litratur through bid-rnt offrs. S Znou (2009). Not that what w call th housing markt could actually rprsnt any sourc of congstion rlatd to th rgional numbr of inhabitants. 7 Contrary to what is somtims don in th litratur (s.g. Wrd, 2012 or Brucknr and Numark, 2011), amnitis a i do not affct th production function. 8 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
dcid to liv in 2. Thrfor, providd that v < x < v, th x-post population sizs ar rspctivly N1 P = v x N and N 2 P = v+x N. If th condition is not mt, on gts a cornr solution. Substituting Ni P in th rnts (1), th thrshold x is xplicitly givn by x = [ y 2 y 1 + a 2 a 1 + r 1 r 2 + s 1 s 2 2 N ] (4) + (s 1 + s 2 )N which dtrmins th partition of th population and hnc of mploymnt if th quilibrium is intrior. This holds tru undr conditions on th paramtrs immdiatly drivd by substituting (4) into th inqualitis : v < x < v. Condition 1. A ncssary and sufficint condition for an intrior quilibrium is that v > 0 vrifis th two following conditions: v > y 2 y 1 + a 2 a 1 + r 1 r 2 s 2 N (5) v > y 1 y 2 + a 1 a 2 + r 2 r 1 s 1 N (6) If y 2 y 1 + a 2 a 1 + r 1 r 2 > 0, (5) can b binding if its RHS is positiv, condition (6) bing thn ncssarily mt sinc v is positiv. In th opposit cas, (6) can put a constraint on v, but thn not (5). In this modl, th impact of various local shocks is asy to driv. As an xampl, considr first an incras in productivity in rgion 1. As th wag rat riss, this rgion bcoms mor attractiv. So, workrs with a lowr idiosyncratic rlativ prfrnc c 1j c 2j mov to rgion 1, i.. x gos down and th population in rgion 1 gos up. Howvr, as rlativ rnts r 1 r 2 ris as wll, th dcras in x is lss than proportional: 0 > dx/dy 1 = /[ + (s 1 + s 2 )N] > 1. (7) It is asily vrifid that th highr v, th lowr th marginal impact of y 1 on x-post population sizs in absolut valu. As scond xampl, if rlativ amnitis a 2 a 1 go up, rgion 2 attracts mor inhabitants. Wags ar unaffctd by this chang bcaus of constant rturns to scal, whil rlativ rnts r 1 r 2 dcras. So, as on can vrify from (4), th ris in th x-post population siz in rgion 2 is again lss than proportional. For a thorough analysis of th comptitiv bnchmark cas (undr dcrasing rturns to scal), th radr is rfrrd to Mortti (2011). 2.2 Introducing labor markt frictions To gnrat a stting whr unmploymnt is an quilibrium phnomnon, w now introduc sarch-matching frictions (Mortnsn and Pissarids (1999); Pissarids (2000)). 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 7
vacancy. Th cost κ i of opning a vacancy is constant, xognous and rgion spcific. 9 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. 2.2.1 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 to th matching procss in th othr rgion. In a scond stp, firms opn vacancis and possibly mt a workr. 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 rlocation stp, mployd workrs and firms bargain ovr wags. Fourth, production taks plac, th housing 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 r i ). 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. 10 So, w prfr th timing indicatd abov: Th bargaind wag will thn not compnsat th workr for th diffrnc in a i +c ij r 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). 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 ). Aftr th rlocation stag, Ni P workrs liv in rgion i. For agnts locatd in rgion i, th notation i will dsignat th othr rgion. 9 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. 10 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. 8
2.2.2 Th matching procss Following Molho (2001), Antoun (2010) and Domingus Dos Santos (2011), and in th sam spirit as Amior (2012), workrs can sarch for a job in th whol country. W allow for distant sarch, maning that sarch in a rgion can b conductd whil living in th othr on. Compard to th altrnativ (i.. on nds to mov in a rgion bfor bing abl to look for a job in it), this assumption looks to us mor in accordanc with currntly availabl communication tchnologis. Th sarch-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 numbr of hirings in ach rgion is givn by a rgional-spcific matching function M i (, ) with: M i (V i, N i + αn i) N < min{v i, N i + αn i}, N i {1, 2}, (8) whr V i rprsnts th numbr of vacancis opnd in rgion i and N i + αn i N is th numbr of job skrs masurd in fficincy units (whr N i N stands for th national job skrs locatd in th othr rgion). Th uppr-bound in (8) is ndd in a static framwork to guarant that both sids of th rgional labor markt will b rationd in quilibrium. 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 θ i = V i N i + αn i N, 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 (8) 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 )] > 0. 12 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 bst opportunity for hr, i.. th position that offrs hr 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 gtting no offr at all and of rmaining unmployd. 11 For a rcnt vidnc at th lvl of local labor markts in th UK, s Manning and Ptrongolo (2011). 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 + 9
2.3 A modl with rgional sarch only Bfor considring th gnral cas of Subsction 2.2.2 with α > 0, lt us brifly considr th standard assumption according to which an unmployd can only sarch in th rgion whr sh livs (th so-calld mov thn sarch cas). So, w impos α = 0. W solv th problm by backwards induction. Th housing supply is givn by quation (1). Th housing dmand is rprsntd by th population living x-post in th rgion, N P i. 2.3.1 Individual wag ngotiation Individual Nash bargaining taks plac x-post, onc th cost of opning a vacancy is sunk. So, whn a vacancy and a job skr hav mt, th wag solvs th following maximization: max(v w ij Vij u ) β i (J ij V i ) 1 β i (9) ij whr V i is th valu of an unfilld vacancy and β i [0, 1) dnots th bargaining powr of a workr in rgion i. Th first-ordr condition can b rwrittn as: w ij = β i y i + (1 β i )b i β i V i. (10) Hnc, th wag is indpndnt of th location of th unmployd and can thrfor b dnotd by w i. It is lowr than th marginal product y i. As w i > b i, undr fr-ntry, workrs always tak th position. 2.3.2 Opning of vacancis Th xpctd valu of a vacant position V i is qual to κ i + m i (θ i )(y i w i ). Firms opn vacancis frly until this valu V i is nil in both rgions. Anticipating corrctly th outcom of th wag bargain, th fr-ntry condition bcoms: κ i (1 β i )m i (θ i ) = y i b i, i {1, 2}. (11) Th probability of filling a vacancy m i (θ i ) incrass with th (x-post) surplus of a match y i b i and dcrass with th cost of opning a vacancy κ i and workrs bargaining powr β i. 2.3.3 Location choic Agnts dcid in which rgion to locat in ordr to maximiz thir xpctd utility. Thy thus compar th xpctd utility of living in rgion 1, p 1 (θ 1 )V1j + (1 p 1(θ 1 ))V1j u, with th xpctd utility of living in rgion 2, p 2 (θ 2 )V2j + (1 p 2(θ 2 ))V2j u. Th workr whos rlativ prfrnc for rgion 1 ovr rgion 2 is abov b 2 b 1 + a 2 a 1 + r 1 r 2 + p 2 (θ 2 )(w 2 b 2 ) p 1 (θ 1 )(w 1 b 1 ) chooss to liv in rgion 1, whil if thir rlativ prfrnc is blow this thrshold, workrs rsid in rgion 2. Lt us dfin 1 = b 2 b 1 + a 2 a 1 + r 1 r 2 and 2 = b 2 b 1 + a 2 a 1 + r 1 r 2. W gt th following lmma: 10
Lmma 1. Givn that agnts prfctly anticipat th wag w i dfind in (10), thr is a thrshold x = 2 + p 2 (θ 2 )β 2 (y 2 b 2 ) p 1 (θ 1 )β 1 (y 1 b 1 ), (12) assumd to b in ( v, v), such that A job skr dcids to liv in rgion 1 if c 1j c 2j x; Els, sh dcids to rsid in rgion 2. Whn unmployd workrs nd to mov bfor starting to sarch for a job, a highr (rspctivly, lowr) tim valu of bing unmployd in rgion 2 (rsp., 1) or highr rlativ lvls of amnitis a 2 a 1 as wll as lowr rlativ rnts r 2 r 1 induc a highr thrshold x, maning that mor workrs locat in 2. A ris in tightnss in rgion 1 has an unambiguous ngativ ffct on x: as th probability of gtting a job in rgion 1 incrass (and thrfor th probability of rmaining unmployd gos down), mor workrs dcid to locat thr. A similar clar-cur conclusion holds if θ 2 incrass. 2.3.4 Equilibrium Lt u i dsignat th unmploymnt rat in rgion i at th nd of th matching procss. Dfinition 1. Whn α = 0, an intrior quilibrium is a vctor {w i, θ i, u i, N i, Ni P } i {1,2} and a scalar x whr w i is givn by (10), in which undr fr-ntry V i = 0, θ i is fixd by (11), u i = 1 p i (θ i ), N 1 = N1 P = v x N, N 2 = N2 P = v+x N, and x = [ 1 + s 1 s 2 2 N + p 2 (θ 2 )β 2 (y 2 b 2 ) p 1 (θ 1 )β 1 (y 1 b 1 ) ] (13) + (s 1 + s 2 )N assumd to b in ( v, v). Th quilibrium is rcursiv. Onc tightnss is fixd in ach rgion by th fr-ntry condition, th quilibrium valu of x is known and population sizs ar dtrmind as wll. Th quilibrium unmploymnt rat in a rgion is only affctd by th dtrminants of tightnss in this rgion. By looking at (11), ths dtrminants ar rgional-spcific. So, a chang in say th marginal product of labor in a rgion has no spill-ovr ffct on th quilibrium unmploymnt rat in th othr rgion. Condition 2. A ncssary and sufficint condition for an intrior quilibrium is that v > 0 vrifis th two following conditions: v > 1 s 2 N + β 2 p 2 (θ 2 )(y 2 b 2 ) β 1 p 1 (θ 1 )(y 1 b 1 ) (14) v > 1 s 1 N + β 1 p 1 (θ 1 )(y 1 b 1 ) β 2 p 2 (θ 2 )(y 2 b 2 ) (15) whr quilibrium tightnss lvls θ i ar a function of paramtrs only thanks to (11). A commnt similar to th on mad aftr Condition 1 could b rplicatd hr. 11
2.3.5 Comparison with th frictionlss cas Lt us considr th sam marginal shocks as in th comptitiv cas. Whn productivity riss in rgion 1, wags incras lss than undr prfct comptition (s (10)). So, firms profit y i w i incrass and hnc mor vacancis ar cratd in this rgion. By totally diffrntiating (11), it can b shown that dθ 1 θ 1 = dy 1 η 1 (y 1 b 1 ) > 0 and du 1 dy 1 = p 1(θ 1 ) y 1 b 1 1 η 1 η 1 < 0 (16) whr η 1 m (θ 1 )θ 1 m(θ 1 ) is th lasticity of th probability at which vacancis ar filld in rgion 1 with rspct to θ 1 in absolut valu. Sinc th prospct of gtting a job in rgion 1 gts bttr and w 1 riss, mor workrs locat in rgion 1. Th total impact on th location of workrs is on th on hand lss important than in th comptitiv cas bcaus th ris in th wag is lss important. On th othr hand, an inducd ffct appars in th frictional cas only: A ris in rgional productivity improvs th chancs of finding a job in this rgion. Undr som conditions th nt ffcts can b rankd: Proposition 1. Undr th Hosios condition (β 1 = η 1 ) or if th workrs bargaining powr is infficintly low (β 1 < η 1 ), th chang in th population sizs inducd by a ris in productivity in rgion 1 is smallr in absolut valu in th frictional cas with α = 0 than in th comptitiv cas. A corrsponding proposition can b xprssd for a ris in productivity in rgion 2. Proof. Diffrntiating (13) with rspct to y 1 and taking into account th adjustmnt of tightnss (16), on can chck that in th frictional cas dx dy 1 = + (s 1 + s 2 )N β 1 η 1 p 1 (θ 1 ) which has to b compard with (7) in th comptitiv cas. Undr th so-calld Hosios condition, i. β 1 = η 1, it is wll-known that sarch-matching xtrnalitis ar intrnalizd by dcntralizd agnts (s.g. Pissarids, 2000). Thn, as p 1 (θ 1 ) < 1, on gts th announcd proprty. This conclusion is rinforcd if β 1 < η 1 so that too fw vacancis ar cratd in quilibrium. No conclusion can b drawn if instad β 1 > η 1. Furthrmor, by (16), w now hav that an incras in th productivity in i yilds a lowr unmploymnt rat in th rgion. As tightnss in th othr rgion rmains constant, th unmploymnt rat thr is not affctd. Th impact of a variation in amnitis is th sam whthr th labor markt is frictional or prfctly comptitiv. For, consumption amnitis do not impact th frntry conditions, which dtrmin th lvls of labor markt tightnss. Thrfor, if rgion 2 bcoms rlativly mor attractiv, agnts rlativ prfrnc c 1j c 2j should b highr to locat in rgion 1 (x incrass), but unmploymnt rats rmain constant in both rgions. 12
2.4 Rgional and national sarch To captur som of th possibilitis cratd by th Intrnt, this sction lts workrs sarch simultanously in both rgions (0 < α 1) if thy prfr to do so. Thn, if thy mt a vacancy and accpt a job offr in th othr rgion, thy migrat at no cost (th so-calld sarch thn mov cas). Appndix B shows that taking sarch and location dcisions simultanously or choosing first th location and thn th sarching ara is quivalnt. Thrfor, to as th xposition, th prsntation blow opts for th scond timing. To prsnt th various stags of th modl w mov backwards. Th housing markt is takn into account as in Sction 2.3. At th tim of individual bargaining in any rgion i, a workr migrating from i has alrady movd in rgion i and thus has th sam fall back position as a workr sttld in rgion i from th start. Th gnralizd Nash bargaining procss is thrfor (9). Th wag is still givn by (10) and dnotd w i. 2.4.1 Accptanc of a job offr A workr sarching in hr rgion only always accpts a job offr, as Vij > V ij u in a frntry quilibrium. 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 accpts th job, as sh dcidd to sarch for a job thr (as shown in Appndix B). Howvr, 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 th indirct utility sh gts from working in rgion 1, V1j, with th on from working in rgion 2, V 2j.13 Th agnt whos rlativ prfrnc c 1j c 2j is abov th thrshold w 2 w 1 + a 2 a 1 + r 1 r 2 chooss to work in rgion 1 rathr than in rgion 2. So, Lmma 2. As agnts prfctly anticipat th wag w i dfind by (10), thr is a thrshold assumd to b in ( v, v), such that ˆx = 2 + β 2 (y 2 b 2 ) β 1 (y 1 b 1 ), (17) Whnvr a job skr sarching nationally has on job offr from ach rgion, sh accpts th job offr in rgion 2 if c 1j c 2j < ˆx; Els, sh accpts th job offr in rgion 1. A highr rlativ wag in rgion 2, w 2 w 1, or highr rlativ amnitis in rgion 2, a 2 a 1, as wll as a highr rlativ rnts in rgion 1, r 1 r 2, obviously induc mor workrs to choos to work in rgion 2 whnvr rciving two job offrs. 13 Rcall that th wag bargain taks plac aftr migration, if any. So, somon sarching in all th country who gts two offrs slcts on of thm, thn migrats if th vacant position is in th othr rgion, and finally bargains ovr th wag. At that momnt, th job offr prviously rjctd cannot b rcalld. Hnc, th workr s outsid option is th valu in unmploymnt whr sh now livs and th wag is givn by (10). 13
2.4.2 Vacancy cration Firms opn vacancis in rgion i until th xpctd gain V i is nil (i {1, 2}). This condition is now m i (θ i )π i (y i w i ) = κ i, whr π i is nw and dsignats th conditional probability that th mting lads to a filld vacancy (s Sction 2.4.4 for mor dtails). Combining (10) and th fr-ntry condition yilds κ i (1 β i )m i (θ i ) = π i(y i b i ), i {1, 2}. (18) Th rat at which vacancis ar on avrag filld, π i m i (θ i ), varis with th paramtrs xactly as in paragraph 2.3.2. 2.4.3 Sarch dcision and location choic As announcd abov, w hr trat ths dcisions squntially. Sarch dcision Lt p i b a short notation for p i (θ i ). An individual j living in rgion 2 dcids to sarch rgionally or nationally by comparing th xpctd utility in both cass. Th xpctd utility if th agnt locatd in 2 sarchs nationally is p 2 (1 αp 1 )V 2j +αp 1 (1 p 2 )V 1j +αp 1 p 2 (max { V 1j; V2j } ε)+(1 p2 )(1 αp 1 )V u 2j. (19) Th xpctd utility of a job skr living in rgion 2 and sarching for a job in this rgion only is p 2 V 2j + (1 p 2 )V u 2j. Whn th small cost of rfusing a job offr ε tnds to zro, th individual whos rlativ prfrnc for rgion 1 ovr rgion 2, c 1j c 2j, rachs a lvl abov b 2 w 1 +a 2 a 1 +r 1 r 2 sarchs nationally. Othrwis, sh dcids to look for a job in rgion 2 only. This is shown in Appndix B (th comparison of cass and f). A similar dvlopmnt is conductd for a workr sttld in rgion 1. Th comparison of xpctd utility btwn national and rgional sarch bcoms: p 1 (1 αp 2 )V 1j + αp 2 (1 p 1 )V 2j + αp 1 p 2 (max { V 1j; V 2j} ε) +(1 p 1 )(1 αp 2 )V u 1j p 1 V 1j + (1 p 1 )V u 1j c 1j c 2j w 2 b 1 + a 2 a 1 + r 1 r 2 whn ε tnds to zro. A job skr locatd in rgion 1 whos rlativ prfrnc for rgion 1 ovr rgion 2 is highr than w 2 b 1 + a 2 a 1 + r 1 r 2 sarchs in rgion 1 only. If agnt s rlativ prfrnc is blow this thrshold, th workr looks for a job in th whol country (s th comparison of cass a and c in Appndix B). Undr prfct anticipation of bargaind wags, th following lmma dfins two thrsholds and thr typs of job skrs: 14
Lmma 3. Whn α > 0, lt z 1 = β 1 (b 1 y 1 ) + 2 (20) z 2 = β 2 (y 2 b 2 ) + 2 (21) with z 1 < z 2. Givn ths thrsholds assumd to b 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 shars of ths thr groups in th total population ar rspctivly v+z 1 v z 2. Rmmbring (17), it is asily sn that z 1 ˆx z 2., z 2 z 1 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 thus sarch dcisions ar indpndnt of probabilitis to gt a job offr. Thrfor, th numbr of workrs who sarch nationally is indpndnt of sarch ffctivnss α. A ris in 2 shifts z 1 and z 2 upwards, whil kping z 2 z 1 unchangd. Hnc, mor unmployd workrs sarch in rgion 2 only and lss do so in 1 only, whil th shar of th population sarching nationally rmains constant. If th valu of tim b i incrass or th productivity y i dcrass (which yilds a drop in w i ), workrs prfr this rgion rlativly mor to sarch for a job thr (th numbr of rgional job skrs in i gos up). 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 rspctivly p 1 (1 αp 2 )V 1j + αp 2 (1 p 1 )V 2j + αp 1 p 2 (max { V and (19), as shown in Appndix B. Lmma 4. Lt with 0 x = 2 + 1j; V2j and } ε) + (1 p1 )(1 αp 2 )V u 1j (22) 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 )), (23) 1 α 1 αp 1 αp 2 +αp 1 p 2 < 1 and, by (20) and (21), z 1 x z 2. If c 1j c 2j < x, thn agnt j locats in rgion 2; Els, workr j sttls in rgion 1. 15
Th shar of th population living x-ant in rgion 2 (rspctivly, 1) is thn v+x (rspctivly v x ). Compard to (12) whn α was assumd to b nil, incrasing th diffrntial in xpctd rnts p 2 β 2 (y 2 b 2 ) p 1 β 1 (y 1 b 1 ) has now a lss positiv ffct on th numbr of popl choosing to locat in rgion 2 sinc thr is th opportunity of sarching nationwid whrvr on livs. At th limit, if sarch is qually fficint whrvr on looks for a job (α = 1), this diffrntial dos not affct th location choic any mor. Whn α > 0, an incras in rlativ amnitis in rgion 2, a 2 a 1, or a dcras in rlativ rnts in rgion 2, r 2 r 1, as wll as a ris (rspctivly, a drop) in th valu of hom production in rgion 2 (rsp., 1) still induc mor workrs to locat in 2 x-ant. Howvr, an incras in tightnss in rgion 1 has svral ffcts. 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 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. So, a first major diffrnc with th cas whr popl only sarch in thir rgion of rsidnc is that a ris in th numbr of vacancis in a rgion has no clar-cut impact on th location choic any mor. 2.4.4 Summary of th accptanc, sarch and location dcisions Equations (20), (21) and (23) provid th thrshold valus for th sarch and location dcisions. Sinc z 1 x z 2, Lmma 5. Givn (20), (21) and (23), sarch and location dcisions vrify th following conditions: 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 1 illustrats this partition of th total population if v < z 1, z 2 < v. Comparing thrshold valus x and ˆx, on cannot rank thm sinc x varis with th lvls of tightnss. Whn rgion ar fully symmtric howvr, ths 2 thrsholds ar qual to zro. 16
Figur 1: Th partition of th population in cas of an intrior solution. 2.4.5 Accptanc probability and vacancy cration A dtaild xplanation is providd in Appndix C. Considr a vacant position in rgion 1. Th mass of job skrs sarching for a job in 1 is 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 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) (24) (25) 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 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 dcrass with z 2 and p 1. Th impact of sarch-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. Combining (18) with (24)-(25) lads to th following fr-ntry conditions: κ 1 (1 β 1 )m 1 (θ 1 ) = v x + α(x z 1) αp 2 (ˆx z 1 ) v x + α(x z 1 ) κ 2 (y 1 b 1 ) (26) (1 β 2 )m 2 (θ 2 ) = v + x + α(z 2 x) αp 1 (z 2 ˆx) (y 2 b 2 ) (27) v + x + α(z 2 x) 17
Through th ndognous accptanc rat, vacancy cration in any rgion is now affctd by charactristics of th othr rgion, namly paramtrs (lik th productivity and th valu of tim) and tightnss. This is a scond major diffrnc with th cas whr popl only sarch in thir rgion of rsidnc. 2.4.6 Populations dfinitions and unmploymnt rats In Subsction 2.2.1, w dfind th x-ant populations N i, split into Ni N workrs sarching in all th country and Ni R workrs sarching in thir rgion of rsidnc only, as wll as th x-post populations Ni P. Sinc z 1 x z 2, an intrior solution is such that: N R 1 = (v z 2) N R 2 = (z 1 + v) N N1 N = (z 2 x) N (28) N N2 N = (x z 1) N (29) with N 1 = N N 1 + N R 1 = v x N, N 2 = N N 2 + N R 2 = v+x N, N 1 + N 2 = N, and N P 1 N P 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) N (30) = (v + x) + αp 2(1 p 1 )(z 2 x) αp 1 (1 p 2 )(x z 1 ) αp 1 p 2 (x ˆx) N (31) with N1 P + N 2 P = N. 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 got 2 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 partial ffct of a ris in tightnss in a rgion is to incras th siz of th x-post population in this rgion and to dcras it in th othr on. A ris in th thrshold x rducs N1 P and augmnts N 2 P. Sinc th impact of tightnss on x is ambiguous, th total ffct of a ris in tightnss on th rgional x-post population sizs is ambiguous as wll. This third diffrnc with rspct to th cas whr α = 0 is a consqunc of th first on. A ris in any of th thrsholds z 1, z 2 or ˆx lowrs N1 P and augmnts N2 P. 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 unmploymnt rats which ar th ratio of th numbr of (x-post) unmployd workrs ovr th (x-post) population, can b writtn as (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) (32) (33) 18
Lmma 6. As in th cas whr α = 0, th unmploymnt rat u i dcrass with tightnss in rgion i, θ i. Th following partial ffcts ar nw. Tightnss in th othr rgion θ i and th thrshold x hav ambiguous ffcts on u i. An incras in th numbr of rgional job-skrs incrass th unmploymnt rat in both rgions. Th unmploymnt rat in rgion 1 incrass with ˆx, whil th opposit holds for th unmploymnt rat in rgion 2. Finally, in a symmtric quilibrium, a ris in sarch ffctivnss α or in th common tightnss valu lowrs rgional unmploymnt rats. Ths proprtis ar asily drivd by diffrntiating (32) and (33). An incras in rgion i labor markt tightnss boosts th probability that a workr living in rgion i finds a job 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. A ris in tightnss in th othr rgion i has an ambiguous impact on th unmploymnt rat in rgion i. Th probability of laving rgion i incrass. Both th numbr of unmployd workrs and th siz of th labor forc go down, lading to an ambiguous impact on u i. Unmploymnt rats also vary in an ambiguous way with th thrshold x. As x gos up, th numbr N 1 of agnts living x-ant in rgion 1 shrinks whil N 2 incrass. Th lvls of rgional unmploymnt, hnc th numrators of (32) and (33), chang in th sam way. Th x-post population sizs, which ar th dnominator of (32) and (33), vary in th sam dirctions: N1 P dcrass and N 2 P incrass. Hnc, w do not gt clar-cut conclusions. Som partial ffcts hav howvr a clar sign. Mor workrs sarching for a job in thir rgion of rsidnc only (i.. an incras in z 1 or a dcras in z 2 ) riss th unmploymnt rat in both rgions. Mor workrs sarching all ovr th country thrfor rducs th unmploymnt rats in both rgions. 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 (32)-(33)). Howvr, if α > 0, 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. 2.4.7 Equilibrium Dfinition 2. Whn 0 < α 1, 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, N i P } i {1,2}, solving (10), in which undr fr-ntry V i = 0, (17), (18), (20), (21), (23), (24), (25), (28), (29), (30), (31), (32) and (33) with: r 1 = r 1 + s 1 N (v x αp 2(1 p 1 )(z 2 x) + αp 1 (1 p 2 )(x z 1 ) + αp 1 p 2 (x ˆx)) r 2 = r 2 + s 2 N (v + x + αp 2(1 p 1 )(z 2 x) αp 1 (1 p 2 )(x z 1 ) αp 1 p 2 (x ˆx)) 19
W now considr conditions for an intrior quilibrium. As w alrady know that z 1 ˆx, x z 2, w nd to guarant that v < z 1 < z 2 < v: Condition 3. Sufficint conditions for an intrior solution ar Proof S Appndix D.1. v > β 1 (y 1 b 1 ) 1 + s 2 N (34) v > β 2 (y 2 b 2 ) + 1 + s 1 N (35) Whn α 0, th xistnc and uniqunss of th quilibrium can only b shown analytically whn rgions ar fully symmtric (s Appndix D.2 for a proof). 2.4.8 Th symmtric quilibrium Whn rgions ar symmtric, as xplaind in Appndix D.2, th uniqu symmtric quilibrium tightnss and unmploymnt rat ar charactrizd by th following systm: π(θ, α)m(θ) = κ (1 β)(y b) u(θ, α) = (1 p(θ))(v αp(θ)z 2) v whr π(θ, α) = v + α(1 p(θ))β(y b) v + αβ(y b) In th uppr part of Figur 2 w draw th lft-hand sid of (36) 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 (37) and illustrats th nd of Lmma 6, namly 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. 3 Efficincy This sction studis th fficincy of th laissz-fair 14 dcntralizd quilibria introducd in th prvious sction. W first driv th optimal allocation whn workrs can only sarch in th rgion whr thy liv (α = 0) and compar it with th dcntralizd quilibrium drivd in Sction 2.3. It will turn out that th dcntralizd quilibrium is fficint whn th Hosios condition is satisfid. In a scond stag, w driv th optimal allocation whn α > 0 and analyz th diffrncs btwn this allocation and th dcntralizd quilibrium charactrizd in Sction 2.4. Th Hosios condition is thn not sufficint to guarant fficincy of th dcntralizd quilibrium. 14 This xprssion is addd sinc thr is no public intrvntion in Sction 2. (36) (37) 20
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 1 u 0 u(θ, 0) 1 u(θ, α) Figur 2: Th symmtric quilibrium: mov thn sarch (π = 1; th black curvs) vs sarch thn mov (π < 1; th rd curvs). 3.1 Th cas whr α = 0 Th cntral plannr s chooss th lvls of tightnss and th thrshold x to maximiz nt output subjct to th sam matching frictions as dcntralizd agnts. Nt output is th sum of output producd, hom production, amnitis and agnts idiosyncratic prfrncs, nt of vacancy costs. W also add th total surplus cratd by th housing markt. 15 W writ this problm as 16 : max y 1L 1 + y 2 L 2 + b 1 (N1 P L 1 ) + b 2 (N2 P L 2 ) + a 1 N1 P + a 2 N2 P κ 1 V 1 θ 1,θ 2,x κ 2 V 2 + N [ x ] c 2j dj + N [ v ] c 1j dj ( r 1 + s 1N1 P )N1 P ( r 2 + s 2N2 P 2 2 v x )N P 2 (38) 15 Bcaus th dmand for housing is vrtical, th consumr surplus and hnc th total surplus on ach housing markt is infinit. It can howvr b shown that th total surplus masurd at th lvl of th country is mad of an infinit constant minus th costs of production of housing. This constant trm dos not mattr for th optimal allocation. 16 In xprssion x c2jdj and v c1jdj, thr is a slight abus of notation sinc v and x ar valus for v x 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. 21