BrainDrainandFiscalCompetition: a Theoretical Model for Europe

Transcription

1 BrinDrinndFisclCompeiion: Theoreicl Model for Europe Pierpolo Ginnoccolo Absrc In his pper we sudy Brin Drin (BD) nd Fiscl Compeiion (FC) in unified frmework for he Europen Union (EU) specific conex. Poenil mobiliy of educed workers cn increse he degree of FC hrough xion or he provision of public educion. An increse in FC cn be cused by compeiion mong differen jurisdicions h im o rc educed workers. When he impornce of FC increses, hen he Europen Ses my employ FC s new policy ool. Firs, we nlyze FC nd BD wih reference o EU regions. In his insnce, he EU my find incenive o conrol he inercions beween BD nd FC in order o coordine fiscl policies nd/or he provision of public goods s educion. Second, we furhermore consider he enry of new se inside he EU. The bsence of coordinion implies h, in ddiion o he FC, migrion compeiion my be genered in EU, where he region inside he union ry o rc educed workers of he new enry. We derive he condiions which BD leds o decrese (increse) in welfre nd growh for new enry counry. EL clssificion: F02-F22-H30-H52-O15-O40 Keywords: Brin Drin; Fiscl Compeiion; Migrion Compeiion; Growh. I hnk for useful commens nd suggesions: Dvid Encou, nd presenions pricipns he groupe de rvil: economie de l Innovion EUREQu (Pris 1 - Sorbonne), he pricipns he Microeconomics workshop IDEA (UAB), he precipns he inernl seminrs Universiy of Souhmpon nd he pricipns he conference Economic Policies in he New Millennium (FEUC - Coimbr). I grefully cknowledge finncil suppor from Mrco Polo nd Mrie Curie grn. All errors re mine. Universiy of Bologn nd Universiy Colic of Miln. ginnoccolop@libero.i 1

2 1 Inroducion In he Europen Union (EU herefer) conex, mobiliy of Europen ciizens is free of insiuionl consrins so h culurl inegrion increses he probbiliy o migre inside he Union. For his reson, workers flows cquired relevn posiion in he EU reserch gend. The sudy of he Brin Drin (BD herefer) is linked wih he choice of educion for boh workers nd/or by governmens. If educion is public good, educed workers re free o migre, s side effec Fiscl Compeiion (FC herefer) cn rise. If governmens do no coordine xion nd provision of public goods, hen he economy my suffer srong negive exernliies. In fc, if he growh of he economies is ssocied wih educed workers, we my record lower xion, worse income redisribuion, nd lower provision of public goods. Furhermore, he sysem my record lower growh. A hough BD nd FC re conneced hrough gens mobiliy, he lierure sudied hem seprely due o he complexiy of join nlysis. In priculr, previous sudies developed wo sepre brnches for BD nd FC. The firs one focuses on BD in mcroeconomic seup nd sudies is impc on he growh of differen economies. The second one nlyses FC using microeconomic ools nd focuses on compeiive inercions beween workers nd urisdicions. Severl sudies re focused on exernliies semming ou from humn cpil migrion bu ll hese sudies nlyze only indirecly he inercion beween BD nd FC nd so hey re no deque o simule he new Europen frmework. In fc, in he ps his BD ws n unidirecionl flow of highly skilled lbor from hirdworld counries nd so he lierure hs explined he lower provision of humn cpil s negive fiscl exernliies due o migrion 1. More recenly, incresed inegrion in he lbor mrkes, especilly wihin he EU, hs drwn enion o problems h rise from bi-direcionl movemen of skilled lbor beween similrly developed counries. I is so necessry o define new BD ypology specific o he Europen conex where he FC cn be used s new policy ools by he regions. Furhermore, when we nlyze he enlrged EU hen we cn disinguish wo differen clubs of region, he former, wih higher growh, nd he new enries wih lower growh nd lbour produciviy. In his new conex new specificion of BD nd FC cn rise. The former regions cn compee o rc he educed workers of he new regions by use of he FC ool. This migrion compeiion, in he bsence of specific coordinion inside he EU, cn genere srong negive exernliies o he new enry. There 1 Berry nd Soligo (1969), for exmple, show h, s fr s he producion of humn cpil (i.e. schooling nd professionl or cdemic educion) is subsidized, he emigrion counry loses humn cpil when people wih humn cpil leve heir origin. Consequenly (nd ccording o he heory of public goods) he producion of humn cpil in he emigrion counries is oo low in comprison o world wihou migrion. Bhgwi (1976) shows he exisence of negive fiscl exernliy on he emigrion counry, if educion is publicly subsidized. If he economy wide educion is expnded in response o emigrion he governmenl defici increses ceeris pribus. Furhermore educionl subsidies cn be regrded s n invesmen of he old generion ino heir pension which is los in cse of permnen emigrion (Grubel & Sco, 1977). 2

3 exis numerously exmples of he FC nd he migrion compeiion in EU. For exmple, recen Swedish policy reduces he x burden for high level reserchers going o Sweden for hree yers. Similr iniiives re being implemened in Denmrk. Furhermore, scruiny of he work permi sysem of mos Europen ses indices clerly h professionl, mngeril nd echnicl consiue he bulk of hose cceped: Germny hs inroduced Green Crd sysem o rc 20,000 IT workers o fill shorges, lhough here re sill difficulies in finding enough poenil migrns wih he necessry skills. The UK governmen hs lso doped more posiive iude owrds skilled lbor migrion, mking chnges o he work permi sysem which re designed o increse he inflow of rnge of skilled occupions, including IT nd medicl personnel 2. Finlly, much of he discussion of he migrion of highly skilled hs focusedonhepoenilbdfromesowes.sisics 3 show migrion of scieniss from Esern Europe nd he former Sovie Union o Wesern Europe. This pper is orgnized s follows. In Secion 2 we describe he srucure of he model. In Secion 3 we solve he model in n urchic conex where here is no migrion. In Secion 4 we describe he full mobiliy cse where here is migrion of educed worker beween wo regions. In Secion 5 we describe he pril mobiliy cse beween wo regions. In Secion 6 we nlyse he pril mobiliy cse beween hree regions. 1.1 Survey of he lierure In his survey we nlyse he BD s lierure nd is link wih he FC s lierure. Le us sr by give definiion of he Brin Drin: Brin Drin is n expression of Briish origin commonly used o describe one of he mos sensiive res in he rnsfer of echnology. I refers o skilled professionls who leve heir nive lnds in order o seek more promising opporuniies elsewhere. Cuses Migrion of his ype hs been linked o severl possible cuses. The mos frequenly cied re he lck of employmen opporuniies for reurning grdues, lower slry levels in he indigenous counry, preference of grdues o live brod, symmeric informion in he lbor mrke 4,differen fiscl nd 2 See Sl (2001); Buer nd Zimmermnn (1999) Sl, Compon, Denshm, Hogrh, nd Schmid, (1999); Dudersäd (2001); Srubhr (2000). 3 See, for exmple, Wolburg (1996, 1997) nd Wolburg & Woler (1997) 4 Kwok Viem (1982) suggess s cuse for he exodus of foreign -rined sudens: symmeric informion in he lbor mrke. Th is, employers in he counry rining he sudens hve more ccure (bu no necessrily more opimisic) judgmen of he rue produciviy of sudens hn hve employers in he sudens nive counry. This symmery resuls from foreign employers fmiliriy wih heir own cdemic sysem nd wih he curricul offered by differen schools; heir ps experience in hiring lrge numbers of boh foreign nd domesic grdues of heir universiies; nd he in-deph inerviews which re regulr pr of he employmen process in mny Wesern counries, nd priculrly in he Unied Ses. He lso shows h he grdues who do reurn end o be hose of lesser produciviy hn hose who remin brod. 3

4 socil pckges 5 nd he incenive o finnce educion 6. Welfre nd growh effecs TheBDlierureislinkedoheconcep of humn cpil nd is mesuremen hs been developed by Schulz (1960) nd Becker (1964). Posiive echnologicl exernliies of immigrion rise by he ddiionl cpil h is vilble o he hos economy. The heoreicl rgumen goes bck o he developmen lierure of 50 s (Hirschmn, Myrdl, Perroux, Wllersein) They hve seen revivl in he mid-1980 s wih he birh of he so clled New Growh Theory. Sring wih Pul Romer (1986, 1987, 1990) nd Rober Lucs (1988) he immigrion of skilled migrns hs been evlued s simuling for he dynmics of economic growh. The possibiliy h he welfre of hose remining in he LDCs could be reducedbynouflow of educed mnpower hd been recognized in he lierure s well. From he work of Grubel nd Sco, Berry nd Soligo, nd Hrry ohnson in he 1960s, he min conclusion ws h welfre of non-migrns would fll only if he migrns conribuion o nionl oupu were greer hn heir income (or consumpion in sic model). For number of resons he lierure believes h he condiions for BD o be welfre-deerioring re ofen verified. Differenly from he sndrd resuls, Mounford (1997) find some condiions in which BD generes posiive exernliies for he regions where some educed workers migre 7. Similr resuls re in Srk nd Wng (2002), Srk e l. (1997) nd Srk (2004). 5 When he choice is mong counries, rher hn mong municipliies, mobiliy is much less, nd he fiscl nd socil pckges cn be, nd re, much more differen. Bu he bsic poin [bsed on he model of Chrles Tiebou s (1956) which explins how poliicl jurisdicion scnoffer quie differen pckges of services nd x res, nd where individuls voe wih heir fee o find he pckges mos suiing heir ses nd vlues] remins hose who move fce, no only differen xes res bu differen perns nd ypes of public services, s well. Perhps even more relevn o he sudy of migrion of he well- educed nd welloff counries differ, no only in heir verge xes res nd in he size nd efficiency of heir public services nd rnsfer pymens, bu lso in he disribuion of coss nd benefis mong differen groups of xpyers nd beneficiries. Among hose who do migre wheher domesiclly or brod, he highly educed re over-represened, prly becuse hey re more likely o posses skills h re in demnd, bu lso becuse hey re more likely o hve concs in nd knowledge bou possible plces o move. To exen h migrion of he highly skilled my o be riggered by differen fcors, survey d repored by Grubel nd Sco (1966, 1976) suggess h job opporuniies nd chllenges re even more imporn o he highly educed. I is lso rue h for mny such workers, priculrly in helh cre, educion, nd governmen-suppored fundmenl reserch, he 1990s hve seen lrge cus in governmen spending induced by budge pressures. For exmple he pre x nd pos x disribuions of he income hve become more unequl in he US relive o Cnd. All of hese fcors my hve incresed he ne rcion of migrion for he beer-educed. [Helliwell 1999] 6 Beyond he overll pckge of xes nd public services, specil enion hs been given, especilly in he conex of BD discussions, o he srucure of educion finnce. Mny commenors hve rgued h becuse BD migrns ke heir xpyer-suppored educionl cpil wih hem, hey should fce n exi x or n educionl lon h is forgiven only for hose who sy nd work where hey cquired heir subsidized educion. 7 He shows h when migrion is no ceriny, BD my increse verge produciviy ndequliyinhesourceeconomyevenhoughvergeproduciviyisposiivefuncionof he ps verge levels of humn cpil in n economy. 4

5 Furhermore here re differen sudies bou he BD 8 nd considerble enion hs been given o proposl of Bhgwi s for brin drin x which would reduce he incenives for such migrion o ke plce 9. Finlly here re differen mehodologies o compue hese benefis nd coss. For exmple Usher (1977) suggess h n ssessmen of he coss nd benefis of migrion need ke ccoun of he fc h lrge porion of counry s propery is publicly owned, so h migrn on going from one counry o noher mus s rule bndon his shre of publicly owned propery of origin nd cquire shre of publicly owned propery in his counry of desinion. The emigrn exchnge his righ o send his children o school in his counry of origin for he righ o send his children in his counry of desinion, reducing he need for new school building in he former counry nd incresing i ccording in he ler. Grubel nd Sco (1976) poin ou h since our concern is wih he gins o he Unied Ses, i is pproprie o use U.S. prices, so h our compuions moun o esiming wh i would hve cos o bring nive Americn o he level of educion held by he verge immigrn he ime he rrives. The effecs of provision of public goods If one ssumes h he llocion o humn cpil invesmens mde by he region (e.g., locl expendiures or se suppor for educion in he nionl frmework nd nionl invesmen oulys in he inernionl seing) depends on he reurns expeced o ccrue inernlly (s he individul invesmen decisions re ssumed o be deermined by expeced prive reurns), he exisence of exernl benefis from invesmens mde by region will cuse subopiml llocion judged from mrginl produciviy rules 10. The cos of educion would be irrelevn o he ssignmen of gins nd losses from migrion if ech mn pid he full cos of his educion, bu i becomes imporn when educion is subsidized or provided free of chrge by he se. I is someimes supposed h here is n implici conrc beween he suden nd he se in which he ler supplies educion lower hn cos on he undersnding h he ne income of educed lbor will one wy noher, be lower hn is mrginl produc. The immigrion of educed lbor generes he benefis of his rrngemen wihou he cos See Bhgwi nd Hmd (1974); Bhgwi nd Rodriguez (1975; 1975b); McCulloch nd Yellen (1975); Blomqvis (1986); Bodenhofer (1967); Sjsd (1962); Rodriguez (1975); Romns (1974); Edding nd Bodenhofer (1966); ohnson (1965); Kesselmn (2000). 9 Bhgwi nd Dellfr (1973), Bhgwi (1975,1976, 1976b) nd Hmd (1977). 10 Thereislrgelierureonheefficency properies of sysem of compeing regionl jurisdicions. One srnd is he fiscl exernliy lierure. The sndrd conclusion in his lierure is h here is n exernliy ssocied wih n individul s migrion h generlly leds o n inefficien disribuion of populion cross region. 11 Educion in generl ccouns for s much s of 5% of GNP, nd 10% or more of public spending in dvnced indusrilized counries, wih public funding covering, on verge, lmos 90% of educion coss in hese counries. Higher educion ypiclly ccouns for 15-20% of overll educion expendiures. Migrion of skilled lbor implies h hose who py he bill for public higher educion my find i difficul o fully cpure is benefis. 5

6 2 The model The model we nlyze in his pper is bsed on Mounford (1997) 12. The lierure on BD idenifies negive exernliy of BD on regions growh. Differenly from sndrd resuls, he Mounford s model finds some condiions in which BD generes posiive exernliies for he regions where some educed workers migre 13. This ineresing resul opens he wy for beer idenificion of he negive effec genered by he muul inercion of FC nd BD. We exend Mounford s model in differen direcions. Firs, we inroduce role for he governmen in he educionl decisions of gens hough he inroducion of educionl subsidies nd xion. Second, we sudy he specific cse in which he region nlyzed is member of he Europen Union where he mobiliy of workers is freely llowed 14. The model nlyses smll open economy, under perfec cpil mobiliy, wih only one good produced under consn reurns o scle by wo fcors, cpil nd efficiency unis of lbor. There is coninuum of gens wihin ech generion 15. The educion decision is ssumed o be discree choice: gens cn choose eiher o be educed or no be educed. Le us define K o be he ol moun of cpil in ime period nd L o be he efficiency unis of lbor. The produciviy of lbor (or he se of echnology) in period is given by λ. Producion is genered by consn reurns o scle producion funcion. The oupu produced ime, Y,is Y = F (K, λ L )=f(k )λ L, where k = K λ L. We mke he sndrd ssumpions bou his funcion, nmely nd he Ind condiions f(k) > 0,f 0 (k) > 0,f 00 (k) < 0 k >0 lim k 0 f(k) =0,limf(k) = nd lim f(k) =0. k 0 k 12 This model is simple version of Miygiw (1991) sudiing of he model of he brin drin nd humn cpil formion. 13 See noe (7) 14 In his nlysis we do no ke in ccoun redisribuion policies of he governmens. Even if pril redisribuion of income derives from he progressive xion uses o finnce he educionl coss. If we ke in ccoun he redisribuion policies we ccenue he negive effecs of he FC. According o he lierure we will obin less redisribuion nd less provision of public good wih respec o he efficien vlue (which could be obined in he bsence of mobiliy or in he presence of coordinion mong jurisdicions). In Ginnoccolo (2003) we hve nlyzed he negive exernliies due o FC nd o educed migrion nd we hve nlyzed heir effec on he redisribuion policies nd on he supply of educion s public good. 15 For simpliciy we normlize he populion in ech generion o uniy. 6

7 Le us ssume for simpliciy h he world is in sedy se equilibrium nd hus h he world ne re of reurn, r, is consn. Due o he perfec cpil mobiliy nd he nrrow dimension of he economy, his fixes he domesic ne re of reurn o cpil, r,equlor nd hus fixes he domesic cpil o efficiency lbor rio, k, s well. Thus k = k where k is consn. Le us ssume h he wge re per efficency uni of lbor is independen of lbor supply ( nd hus of migrion levels) nd is dependen of he level of echnology λ,hisgiven: w = λ w. The disribuion of biliy Individuls possess differen levels of len biliy, where e i denoes he len biliy level of individul i. These len biliies re ssumed o be disribued over he closed inervl [0,E]ccording o he densiy funcion g e i,where,bydefiniion, 0 g e i de i =1nd g e i > 0 e i [0,E]. Le us ssume h ll generions hve len biliies which re picked up from he sme disribuion nd h he biliies of children re independen from he biliies of heir prens. The growh exernliy Le us ssume h here is n economy wide growh exernliy reled o he proporion of educed workers in he economy in he previous period s 1.Thuswemodelλ o be posiive funcion of he proporion of educed workers in he previous period, h is λ = λ (s 1 )where s 1 = e 1 Le s lso ssume h λ (0) = 1 nd h λ (1) is finie. g e i de i. (1) The individul s decision o be educed Agens live in overlpping generions world nd live for hree periods, deriving uiliy only from he hird period consumpion 16.Inheirfirs period of life gens cn inves resources in educion. They hve no resources of heir own, so hey mus borrow from he cpil mrke he world s re of ineres, r. Le us ssume he prive cos of educion o be fixed c p unis of oupu. Le us furhermore ssume h, in bsence of governmen s subsidies, c p = c mx. 16 The inroducion of hree periods is necessry becuse gens borrow o finnce heir firs period of life nd hey cn evidenly no borrow from gens who will no be live o be repid in he nex period. 7

8 Agens h inves in educion obin e i efficiency unis of lbor in heir second period of life, where e i is he level of he len biliy of gen i. Furhermore le s ssume h he gens who do no inves in educion hve only one efficiency uni of lbor in heir second period of he life. Agens cn only work in heir second period of life nd in his period he genmusrepyhedebofhefirs period. In he hird period hey re reired nd use heir svings o consume. All gens hve he sme preferences nd ccess o he sme echnology, lhough hey do no hve he sme levels of len biliy. Le us now ssume h he governmen subsidizes pr of he educionl coss susined 17. The governmen influences he educion decision of he gens by xing he educed workers nd covering pr of heir educion coss 18. Le us ssume h if he governmen susin enirely or prilly he educion cos, hen scle economies rise nd we hve smller unirin cos. Le us define public cos of educion for ech gen o be fixed c unis of oupu. By ssumpion c<c mx. In he nex session we nlyse how he resuls chnge when we hve differen ssumpions on he educionl coss. In presence of governmen subsidies The prive cos becomes c p = c γ, where γ is he educion subsidy defined s he uni of oupu reimbursed o educed gens in generion nd γ [0, γ mx = c]. Le us define T o be he mrginl re of xion of educed workers in generion. Inroducing xion, he wge re per efficiency uni of lbor becomes w i = λ we i [1 T ]. The opiml decision for gen i will be o inves in educion if [1 T ] λ we i > λ w +[c γ ](1+r ). (2) Thus, ll gens wih len biliy greer hn e will inves in educion, were e is uniquely defined by he following equliy: e = λ s 1 w +(1+r ) c γ. (3) 1 T λ s 1 w Le us ssume h he model is such h e [0 + ε,e ε], where 0 < ε < E 2. Dynmics nd sedy se produciviy The only dynmics in he model derive from he growh exernliy. From equion (1) i is cler h he proporion of workers who re educed ime 17 These subsidies re given direcly o educed. The nlysis does no chnge if we consider n equivlen verge educion invesmen of he governmen (cdemic nd reserch infrsrucures, school plces, echers, ec.). 18 We ssume h only he educed workers re xed. Then we focus our nlysis on priculr quo of he xion reserved o py he educion s subsidies. 8

9 is n incresing funcion of he proporion of workers who were educed ime 1, h is Since hus e = λ s 1 s = ψ (s 1 ). (4) s 1 (1 + r ) c γ λ 2, (5) s 1 w 1 T s = g e λ s 1 (1 + r ) c γ s 1 λ 2. (6) s 1 w 1 T Le us ssume h E is high enough so h he mos ble worker will lwys chooses o be educed even if no one ws educed in he previous period. Since we know h gen i wih e i = 0 will never chooses o be educed, hen his implies h here mus exis les one sedy se equilibrium for s,which we denoe s s. Wheher his is unique sedy se depends on he properies of he funcion λ = λ (s 1 ). If his funcion hs convex regions, represening criicl msses of educed people in he economy, hen here my be muliple sedy ses. The unique Sedy Se cse is depiced in figure (1). s +1 1 Ψ * ( s ) s g s Figure 1: Ψ (s ) indices he proporion of educed gens in urkic cse, when here is no migrion nd here re governmen s subsidies. 9

10 3 Aurchic cse In his secion we solve he model in n urchic conex where here is no migrion beween regions. Le us resume he iming of he model. Time The governmen decides T nd γ nd influence he prive cos of educion. Ech gen i decides wheher o inves in educion or no ccording o heir len biliy e i. Agens who inves in educion receive γ nd borrow c γ from he cpil mrke. Time +1 The educed gens py T o heir governmen nd repy he deb of he firs period of he life. Time + 2 All gens re reired nd use heir svings o consume. I is possible o solve he governmen mximizion problem hrough he Bckwrd Inducion mehod (BI herefer) 19. Firs we solve he mximizion problem of he gens ime +1 nd hen we solve he mximizion problem of governmen. In ech period we ssume h s 1 is given, hen we cn define λ s 1 w. (7) The gen s decision is given by equion (3) e g = +(1+r ) c γ. (8) 1 T The governmen. Le us define Ω mesure of he welfre of he region derived from he produciviy of he gens h in ime re residen in region. 20 When here is no migrion (Aurchic cse), we define Ω A, Ω A, s " e g # de i, c (1 + r ) +. (9) The firs erm on he righ hnd side of (9) denoes he ne gin in produciviy of region due o he presence of educed workers. The second 19 See he Appendix for ll he compuion of his urchic cse. 20 This is non-sndrd funcion of socil welfre. I is mesure of he region s gin derived from he produciviy of ech generion, ne of he educionl coss. I llows o compre he differen scenrios nlyzed in his model nd o cpure he educionl decisions of he governmen. In he nex chper he figure (2) gives grphic inuiion of Ω. I is possible exend his sic simplificion of he model by defining socil welfre funcion h ke in ccoun he exernliies linked o he educion. See in Appendix for furhermore deils. 10

11 erm corresponds o he ol produciviy of region independenly from he presence of educed workers. For ech ime he governmen mximizes he Ω subjec o blnce consrin for ech generion. Furhermore, le us ssume h he governmen decides independenly by he posiive exernliy of educion of generion for he fuure generions nd h he blnce consrin is binding. So we hve s g, (1 + r ) γ = " Ã Z! # E de i, T e The mximizion progrm for he governmen is Mx: s g, T " e g s g,. (10) # de i, c (1 + r ) +. (11) TheopimlvlueofhexionT (nd indirecly, by he equion 10, he opiml vlue of he subsidies o educed) is where =1+ h T h E e g E 2e g i i, (12) 0 <T < 1 if e g > 2 E. (13) 3 We cn resume his firs resul wih he following proposiion. ³ Proposiion 1 When he number educed i is no high e g > 2 3 E, hen exiss posiive opiml level of xion nd, consequenly, posiive level of educionl subsidies. This opiml level is T =1+ [E e g ] [E 2e g ]. 3.1 Role of governmen nd effecs on he region s growh To undersnd beer he role of he governmen on he educionl decisions of he gens, we hve o compre how he welfre chnges in presence of posiive subsidies o educion. In figure (2) we show grphiclly hese chnges. In bsence of governmen subsidies equion (3) becomes e = +(1+r ) c mx e 0. (14) 11

12 E e 0* =+c mx 0 s 0 1 E e 0* e * =+c 0 s o s g 1 Figure 2: Aurchic cse. Role of he educionl subsidies on he welfre nd on he number of educed workers. Compring expression (8) nd (14) we hve h e 0 e g = (1 + r )[c mx c] E 2e g T. (15) By ssumpion c mx >cnd, by he proposiion 1, we know h here is posiive xion when e g > E 2. Then we cn conclude h e 0 >e g when e g > E 2. (16) The equion (15) explins excly he effec of n cive role of he governmen. The firs erm shows he chnge due o he lower cos of he public educion hnks he scle economies. I depends from he difference (c mx c). When here is no difference beween he public nd prive cos, his erm disppered. The second erm shows he chnge due he presence of proporionl 12

13 xion o he educed. In his erms here is he redisribuive role of he governmen h increses he educionl coss for he gens wih greer len biliy nd decreses he coss for he gens wih lower biliy. In figure (3) re shown he effecs of xion on he individul income of he gens. In Appendix we exend his sic simplificion of he model by defining socil welfre funcion h ke in ccoun he exernliies linked o he educion. In his exension, he opiml vlue of he xion T e becomes T e = T ³ Z 0 (s g, ) E 2e g (17) In ccording o he economic inuiion, when he governmen inernlizes hese posiive exernliies, hen here is greer level of xion, greer level of subsidies n so n increse in he number of educed workers. The equion (15) becomes in his cse e 0 e g = (1 + r )[c mx c] E 2e g T + Z0 (s g, ), (18) where he second erm shows he chnge due he presence of he fc h he governmen ke in ccoun he posiive exernliies due o he educion. E (1-T)E e 0* (1-T)e * 0 s s 0 s * 1 Figure 3: Effecs of xion on he individul income of he gens In figure (2) is shown he effec of xion on he welfre funcion. The governmen increses he number of educed workers by decresing he educion cos of he gens wih lower len biliy nd finncing hese subsidies by 13

14 xing more he gen wih higher len biliy. When he number of educed workers increses, welfre nd growh effecs rise. Furhermore, if here re muliple sedy se equilibri hen he economy cn move from low o high educion sedy se. The sedy se of he wo cses is shown in figure (4). We cn resume hese resuls wih he following proposiion. Proposiion 2 In presence of opiml xion (proposiion 1), he number of educed workers increses respec he cse wih zero educionl subsidies. This increse is e 0 e g = (1+r )[c mx c] E 2e g T Corollry 3 Theincreseinhenumberofeduced,giveninheproposiion 2, idenifies wo differen effecs of direc role of he governmen in he educionl decisions. Firs, decrese in he individul educionl coss hnks he scle s economies of public educion. Second, redisribuive role of he governmen h decreses he educionl coss for he worker wih lower len biliies. Corollry 4 The increse in he number of educed workers implies n increse in he welfre nd in he growh of he economy respec he cse wih zero educionl subsidies. s +1 1 * Ψ ( s ) 0 Ψ ( s ) s0 s g s Figure 4: Ψ (s ) indices he proporion of educed gens in urkic cse, when here is no migrion nd here re governmen s subsidies. Ψ 0 (s ) indices he proporion of educed gens in urkic cse,when here is no migrion nd here re no governmen s subsidies o educed. 14

15 4 Mobiliy cse (full mobiliy - wo regions) Le us inroduce in he model workers mobiliy. We exmine he cse in which only he educed gens cn migre (BD) 21. We ssume h here re only wo regions A nd B 22. The iming of he model is he sme of he Aurchic cse. The only difference is h in ime + 1 educed gens decide wheher o migre or no. They py T o he governmen of he region in which hey work nd hey repy hedebofhefirs period of he life. I is possible o solve he governmen mximizion problem hrough he BI mehod 23. Firs, we solve he mximizion problem of he gens ime + 1nd hen we solve he mximizion problem of governmen. In period, he gen i chooses wheher if educe himself or no given he governmen decisions bou T nd γ wih = A, B. The opiml decision for gen i, borninregion, is o inves in educion if where rg mx 1 T ª ; 1 T I b e i >+ c γ (1 + r ) (19) ³ λ s g, 1 w ³ λ s g,i 1 w b. Thus, ll gens wih len biliy greer hn e g e is uniquely defined by he following equliy: inves in educion, were e g = + c γ (1 + r ) rg mx ª. 1 T (20) ; 1 T I b The sme resul follows for gen i, born in region I, e gi = b + c γ I (1 + r ) rg mx ª. 1 T (21) ; 1 T I b 21 This hypohesis is compible wih he ssumpion h here re no mobiliy coss. The resuls do no chnge if we ssume h he coss of mobiliy (rnsfers coss, socil coss, inegrion s coss, ec...) re very smll for educed workers (closed o zero) nd very high for non educed. I is furhermore possible exend he nlysis o he cse in which here re no educionl requiremen for emigrion bu becomes hrd disinguish he BD specs of he workers migrion. 22 I is possible, wihou chnging he resuls, ssume h he region B represens he res of he Union nd so he ssumpion h he region i is smll open economy is verified. 23 See he Appendix for ll he compuion. 15

16 In his model we ssume h for he educed gens here is no mobiliy coss so h educed workers decide wheher migre or no in response o differen ne wge h hey receive. Their fuure wge is reled o he xion/subsidies policies of he governmens nd o he differences of echnology beween regions. I is srighforwrd o verify h he educed workers will prefer o sy in region if where T < η+(1 η) T I (22) η b. Le us ssume h b. Wecnhereforedisinguishhreedifferen ses of he world Cse (1) T < η +(1 η) T I ll educed migre in region. Cse (2) T > η +(1 η) T I ll educed migre in region I. Cse (3) T = η +(1 η) T I here is no migrion. The hree cses depiced in figure (5) ** T T =η cse 2 cse 1 cse 1 * T I * = 0 I T Figure 5: The hree cses when >b The governmen. For ech ime he governmen mximizes he Ω M, subjec o blnce consrin for ech generion. We define Ω M, 16

17 Ω M, s, " e # " de i, + s I, b e I de i, # s [c (1 + r )] + (1 s ) (23) where s, is he number of gens which re educed in region ime nd work in region +1;s I, is he number of gens which re educed in region I ime ndworkinregion ime +1 nd b = λ s 1 I w The firs nd he second erms on he righ hnd side of (23) denoe he ol producion of region due o he presence of educed workers. The hird erm corresponds o he educion coss in region. The fourh erm corresponds o he produciviy of non educed gens. Furhermore, le us ssume h he governmen decides independenly by he posiive exernliy of educion of generion for he fuure generions nd h he blnce consrin is binding. So we hve, where [γ (1 + r )] s g, =[AT ] s g, (24) A s g, Ã de i, e g (T )! s g,, + s g, Ã de i, e gi (T I)! s g,i,. Le us now nlyze he governmen s decision by using he BI for ech differen ses of he world (See Appendix). Cse (1): Compring he welfre funcions, we hve h Ω > Ω > Ω,0 Ω I > Ω I,0 > Ω I (25) where Ω is he opiml vlue of he welfre funcion in he mobiliy cse nd where T > 0, Ω isheopimlvlueinheurchiccsendω,0 is he opiml vlue in he mobiliy cse wih zero xion. Cse (2): Compring he welfre funcions we hve Ω I > Ω I > Ω I,0 Ω > Ω,0 > Ω. (26) 17

18 Cse (3): We hve he sme resuls of he urchic cse. Ω = Ω > Ω 0. (27) Compring he (25), (26), nd (27) i is srighforwrd see h, when, hen he only Nsh Equilibrium in his gme is T = η ; T I =0 when >b nd T =0;T I =0 when = b. whereη = η ε, ε > 0ndε 0. The following proposiion summrizes he resuls obined in his secion. Proposiion 5 In presence of full mobiliy of educed gens, he only NE is zero xion (nd, consequenly, zero subsidies) when he regions hve he sme iniil echnology ( = b). When he counries re symmeric ( > b η > 0), hen he only NE is T = η ; T I =0. Where η is he higher level of xion sufficien o rc ll he educed workers of he oher region. Posiive effecs of he coordinion Le us ssume h he wo regions re member of n economic union like he EU so h he educed workers cn migre inside he union wihou impedimens. Furhermore, le us ssume h inside he Union here is Cenrl Auhoriy (CA) nd h my impose coordinion beween xion/subsidies policies of he regions. Le us nlyse differen coordinion policies h cn rise. Firs, he CA cn impose o ech region he opiml level of subsidies nd xes h is chosen in he Aurchic cse. This policy mximizes he welfre of ll regions if he regions re symmeric, oherwise one of he hree cse described bove rises nd so here is region h looses ll is educed workers. Second, he CA cn impose minimum level of xion/subsidies (T min > η). This policy chnges he ply off of he gme described bove. The hree cses re depiced in figure (6). The only NE becomes T = T min ; T I = T min. Also in his cse when he regions re symmeric his policy imply lower welfre for he region lower producive h looses ll educed workers. Third, he CA cn impose h he xion/subsidies of he region more producive re proporionl o he xion of he lower producive region. If T = η +(1 η) T I, hen he only NE becomes T = η (1 η) T I ; T I = T I when >bnd T = T ; T I = T I when = b. I is srighforwrd see h his NE is Preo Improvemen respec he NE obined in bsence of CA. The new cses despised in figure (6). This nlysis cn be exended wihou chnging he resuls o Union wih mos regions.. The resuls obined in his secion cn be summrized by he following proposiions. 18

19 T ** min T = T η cse 2 cse 1 cse 1 T min I * * T η = 1 η I T Figure 6: Differen echnology scenrio wih minimum x specific for ech region. Proposiion 6 The presence of Cenrl Auhoriy h is ble o impose h T = η +(1 η) T I implies new NE T = η (1 η) T I ; T I = T I. This NE is Preo Opimum respec he NE obined in bsence of CA. Corollry 7 When he region re symmeric, hen he new NE obined implies he opiml level of xion nd subsidies chosen in he Aurchic Cse. 5 Mobiliy cse (pril mobiliy-wo regions) Le us ssume h here re only wo regions U nd N, whereu is represenive Se of he Union 24 nd N is new Se h eners inside he Union 25. Le us ssume h he wge per efficiency uni of lbor of his economy is lwys lower hn he wge of region U independenly of is fiscl policies nd echnology chrcerisics. LeusssumehinheregionN he mobiliy of is educed workers is no perfec bu here is probbiliy of successful emigrion in he region U, π U, h is independen of he number of workers who re eligible o migre 26. Furhermore, we ssume h emigrion policy is fully niciped. We ssume h π < 1 nd h i is very smll for workers of region N. This ssumpion cn be jusified by he presence of srong mobiliy coss (pecuniry nd socil). We ssume h U cn influence π U by migrion policies h remove his coss. 24 Le us ssume h here is no differences beween he formes ses of he Union. 25 This nlysis cn be exended wihou chnging he resuls o he new enry of mos regions. 26 This ssumpion follows from he smll counry hypohesis. 19

20 When here is probbiliy of emigring nd erning higher wge, he gen s educionl decision becomes n expeced uiliy problem. For simpliciy, we ssume h gens re risk neurl, h only he educed workers cn migre nd h ll he oher ssumpions of he previous secion re verified 27. Le us define λ s N 1 w N w N (28) s he wge re per efficiency uni of lbor for he educed of region N. By ssumpion, in ech period we hve h where w U >w N (29) w U λ s U 1 w U 1 T U. (30) The opiml decision for gen i born in N will be o inves in educion if π U w U +(1 π U ) 1 T N w N e i >w N + c (1 + r ). (31) Thus, ll gens wih len biliy greer hn e N will inves in educion, were e is uniquely defined by he following equliy: e N w N +(1+r ) c γ N = πu w U +(1 π U ). 1 T N (32) w N As in he previous nlysis i is possible o idenify he opiml level of xion for he new enry: T N = h ³ E e N π U w U +(1 π U )w N w N (1 π U ) E 2e N i E 2e N + (1 π U ) E 2e N for π 6= 0 T N =0 forπ =0. (33) When π 1 (full mobiliy cse), hen T N 0. When π 0 (urchic cse), hen T N T N, where 27 Sme disribuion of biliy, sme educion coss, ec. 20

21 E e T N N =1+ (34) E 2e N is he opiml vlue of he xion for he region N in he urchic cse. The verge proporion of educed people in he economy N is given by he following ideniy s N = (1 πu ) R E g e i de i e N 1 π R U E g (e e i ) de. (35) N If π = 1 hen he source economy loses ll his educed workers nd s N =0. If π = 0 hen here is no migrion inside he union. Thus, sufficien condiion for he exisence of posiive level of BD such h he source economy benefis in erms of produciviy is h dsn dπ will be given where dsn dπ where s N π s N e N e N π > 0whenπ = 0. The opiml level of π =0. Differeniing equion (35) we obin ds N dπ = sn π + sn e N e N π, (36) R E g e i h de i 1 R E g e i i de e = N e N h 1 π R i U E 2 < 0 (37) g (e e i ) de N 1 π U g e N = h 1 π R i U E 2 (38) g (e e i ) de N w N = +(1+r ) ª c γ N w U 1 T N w N πu w U +(1 π U ) 1 T N w N 2 < 0. (39) Seing π u = 0 nd noing h R E g e i h de i 1 R E g e i i de is mos e N e N qurer, we obin he resuls summrized by he following proposiion. Proposiion 8 If here re srong differences on he wge per efficiency uni of lbor nd here re imperfec mobiliy of educed workers, hen posiive opiml level of BD emigrion rise if g e N {w N +(1+r )[c γ N ]}[w U (1 T N )w N ] > [(1 T N )w N ] nd 0 <TN T Mx. 21

22 This proposiion ses h he source economy cn benefi fromhebdin he exen h here re sufficien number of people who would be eniled o inves in educion. The inroducion of xes nd subsidies implies wo differen resuls. The subsidies increse he number of educed workers nd so decrese he probbiliy for he new enry o be in he opiml BD condiions. Furhermore, he xes increse he wge differenils beween he enry region nd he ohers nd so increse he probbiliy o gin from he BD. The ssumpion h ll he regions inside he Union re similr, i is equivlen o ssume h here is Cenrl Auhoriy h collecs he migrion policies of he regions inside he union nd decides he opiml vlue of π U for he region N, hπ = 1 for he former regions nd h η =0. Le us consider he cse of uniformly disribued biliies e N g e i = 1 E g e i de i = 1 e N E (40) (41) ds N dπ > 0iff 1 π U wu 1 T N w N π U w U +(1 π U ) 1 T N w N > µ1 e N E (42) Thus, BD will increse he proporion of educed people in he economy if π is low, if w U is very high relive o 1 T N w N nd if he proporion of educed people in he economy ws previously low. Equion (42) implies h when biliies re disribued uniformly, if w U is lrge enough here is posiive level of π U such h nex period produciviy increses in he source economy. As in Mounford (1997), in presence of n opiml migrion policy under BD, he reurn funcion s = ψ (s 1 ) is everywhere bove he reurn funcion compred wih he cse of no emigrion. Thus clerly n opiml emigrion policy will increse he shor nd long run produciviy in he source economy. Finlly, if here re muliple sedy se equilibri hen emporry emigrion policy migh lif source economy from low o high educion sedy se. The figure (7) despics hese resuls 22

23 s ψ ( s ) wih 0 ψ ( s ) ψ ψ m m ( s ) wih ( s ) opiml opiml migrion migrion o 45 s m 0 s s Figure 7: despics he dynmics of he economy when here is unique sedy se equilibrium for he cse where here is no migrion nd when here is opiml emigrion (Ψ 0 is he cse wih opiml xion nd Ψ m is he cse wihou xion). 6 Mobiliy cse (pril mobiliy-hree regions) Le us ssume h here re hree regions A, B nd N. Where A nd B re former member of he Union nd N is new region h eners inside he Union. According he previous cse, le us ssume h when he new region (N herefer) is dmied inside he union he mobiliy of is educed workers is no perfec bu here is probbiliy of successful emigrion in he region, π wih = A, B. Le us lso ssume h π = 1 for educed workers of he Union. Le us ssume h he wge per efficiency uni of lbor of his economy is lwys lower hn he wge inside he union independenly of he specific fiscl policy nd echnology chrcerisics of ech region inside he union. Then, by ssumpion, in ech period we hve h where w U >w N (43) w U rg mx λ s 1 w 1 T ; λ s I 1 w I 1 T I ª (44) he bes wge re per efficiency uni of lbor vilble inside he union. The opiml decision for gen i born in N will be o inves in educion if πw U +(1 π) 1 T N w N e i >w N + c γ N (1 + r ) (45) 23

24 Le us ssume h here is no coordinion beween regions nd h ech region of he union decided independenly he vlue of π mximizing his own welfre nd do no ke in ccoun he welfre of he region N. Leuslso ssume h his vlue cn no be higher hn π mx < 1. Then,wessumeh 0 < π < π mx. Le us define Ψ mesure of he welfre of region. Ψ Ω + Γ(π ) wih, I = A, B (46) were Γ(π ) is he brin drin gin for he region deriving by he rcion of educed workers of region N. We define Γ(π ) π s N, " λ Z # s E 1 w de i, e N (47) Where π s N, is he number of gens which re educed in region N ime nd work in region +1. I is srighforwrd o show h If π 1 T > π I b 1 T I hen Γ(π )=π s N " e N de i, # (48) nd Γ(π I ) = 0 (49) we lso ssume h If π 1 T < π I b 1 T I hen Γ(π ) = 0 (50) " Z # E nd Γ(π I )=π I s N b de i, (51) e N If π 1 T = π I b 1 T I hen Γ(π )= 1 2 π s N nd Γ(π I )= 1 2 πi s N " b e N de i, # " e N de i, # (52) Le us ssume he sme iming defined before. Then, i is possible o solve he mximizion problem hrough he BI mehod. The opiml decision for gen i born in N is uniquely defined by e N (eq. 32) 24

25 For simpliciy, we nlyze only he opiml decision of governmen nd I bou he vlue of π. Hence we focus our enion on he migrion compeiion inside he union. I is srighforwrd see h ll hese nlysis cn be exended o he fiscl compeiion beween he wo regions wihou chnging he resuls. Ech governmen of he union mximizes he vlue of Γ(π) nd i is srighforwrd see h, wihou coordinion he only NE of his migrion compeiion is π = π I = π mx > π U (53) The governmen N decides T nd γ ccording o he opiml decision of he gens (eq. 32) nd of he region inside he union (eq. 53). The presence of Cenrl Auhoriy h collecs migrion policies of he region inside he union nd decides he opiml vlue of π U for he region N implies posiive Brin Drin when we re in he condiion delineed in he proposiion (7). Differenly, when here is no coordinion, hen here is migrion compeiion beween he governmens inside he union which involves in vlue of π > π U,hevlueofπ is oo high o hve posiive exernliies from BD lso for he region N. In his cse he BD hd negive effec on he growh of he region N nd heir opiml decision is o hve zero subsidies. This resul is mos imporn when he new enry region is required o hve sndrd in he growh o remin in he Union. The CA cn help he New region o do no decrese is growh, indeed he compeiion beween he former region implies posiive effecs for hese ones in he shor period bu in he long period hve negive effecs for he Union becuse he poores counries cn no sisfy he sndrd required. 25

26 7 Conclusion In his pper we inroduce role for he governmen in he educionl decisions of he gens hrough he inroducion of educionl subsidies nd xion. This mke i possible o sudy BD nd FC in unified frmework nd nlyze heimpcofhebsenceofcoordinioninsideheeu. In Secion 3, we solve he model in n urchic conex nd we obin he opiml level of xion nd subsidies [Proposiion 1]. This opiml level implies n increse in he number of educed workers [Proposiion 2]. Furhermore, his increse implies lower educionl coss [Corollry 3] nd n increse in he growh of he region [Corollry 4]. In Secion 4, we solve he model in full mobiliy conex where here is perfec migrion of he educed worker inside Union. The FC mong he regions desroyed he posiive exernliies due o he subsidies. According o he lierure, he FC cuses fll in he provision of public goods. Lower xion nd lower educionl subsidies rise [Proposiion 5]. The presence of Cenrl Auhoriy which coordines he fiscl policies, i is necessry o obin new NE h is Preo Opimum respec he NE obined wihou coordinion.[proposiion 6]. Finlly, when he regions re symmeric, hen he new NE is he sme obined in he urchic cse.[corollry 7]. In Secion 5, we solve he model in pril mobiliy cse where here is new enry inside he Union. If he mobiliy of he educed workers of he new enry is no perfec nd cn be influenced by he migrion policies of he former members of he Union. Then, I is possible idenify sme circumsnces in which he migrion of educed worker does no imply negive exernliies for he sending region. [Proposiion 8]. These circumsnces implies he so clled Brin Gin. In Secion 6, we nlyse he bsence of coordinion beween he former regions inside he union. This bsence implies migrion compeiion. The former region ries o rc educed workers of he new enry. This compeiion implies posiive effecs for hese ones in he shor period bu in he long period hve negive effecs for he Union becuse he poores counries mus qui he union. Exensions of he model The model presened in his pper cn be exended in order o nlyze differen economic nd poliicl nlyses. 1. We cn inroduce mobiliy cos for he educed workers. This cos cn be no only referred o he pecuniry coss direcly linked o he migrion (rnspor, new house, ec.) bu lso i cn be referred o he non pecuniry cos indirecly linked o he migrion (live in new nion, differen lnguge, ec.). The inroducion of his coss do no chnge he min resuls obined in his pper bu here re some imporn resuls: The more re he mobiliy cos, he lower is he role of FC. 26

27 While he pecuniry cos re normlly similr beween he differen regions, on he conrry he non pecuniry coss cn be very differen nd hey cn be direcly influenced by he policies of he governmen. These differences my increse or decrese evenully echnology s differences nd so he FC nd BD exernliies. Furhermore, by decresing hese coss, he governmen of he former region inside he EU cn ry o rc he educed workers of he new enry (migrion compeiion). 2. We cn inroduce enlrged role of he governmen. In his pper we hve nlyzed governmen which do no ke in ccoun redisribuion income policies. If we consider new version of he socil welfre funcion h he governmen wn mximize hen wehveoherimpornresuls: The FC implies no only lower provision of public good bu lso lower income redisribuion. This resuls, in ccording o he FC lierure, is due o he fc h ech governmen decreses he x in order o rc he educed worker nd so i mus decrese he income redisribuion. If we nlyze he redisribuion policies, hen we mus ke in ccoun lso he non educed migrions. The risk o rc mny non educed workers implies lower income redisribuion nd so increse he negive exernliies of he FC. 3. We hve nlyzed he impc of he FC nd BD when he new enry region hs jus decided o be in he EU. I is lso possible enlrge his nlysis by sudying new sep in which he new enry decides even if i is convenien be member of he EU. 4. We cn subsiue he ssumpion h he biliies of children re independen from he biliies of heir prens wih he ssumpion of exernliies of educion of he preceden educion. In his cse we obin more relisic model wih more rich dynmics nd we increse he negive exernliies due o he FC. Oherwise, he min resuls showed before do no chnge. 27

28 8 Appendix Aurchic Cse (fixed coss) The mximizions for he governmen is Mx T " : s sub o γ = The Firs Order Condiion is Foc(T ): sg, T " e g # de i, c (1 + r ) + (1 + r ) Ã Z! E de i, T. e de i, c g (1 + r ) e g (T ) # + s g, Ã! e g, T =0. By sg, T FOC becomes ( ³ e g, T g ³ = e g, g T e g, "Z E e g, nd by he opiml vlue of e g equion he de i, 1 T 1 T e g e g (T ) # ) + s g, =0 The opiml vlue of he xion T is =1+ h T h E e g E 2e g Where 0 <T < 1 if E 2 <e g <E 1. i i Aurchic cse (vrible coss) In he iniil semen of he model we hve ssumed h when he governmen subsidizes he educion hn lower educionl coss rise hnks scle s economies. Le us now inroduce new specificion of he educionl coss h ke in ccoun his effec. Le us define g prmeer which h cpures he i scle economies of public educion. Where g>0ndγ 0, γ mx = c 1 g. ½ c p = c (1 + g)γ c g = c gγ. (54) 28

29 Then e is uniquely defined by he following equliy: e = +(1+r ) c (1 + g)γ. 1 T The governmen s objecive funcion is Ω A, s " λ Z s E 1 w e g # de i, (c (1 g)γ )(1+r ) +. The mximizions for he governmen is Mx T " : s e g sub o γ = The Firs Order Condiion is Foc(T ): sg, T ( e g + s g, By e = +(1+r )[c (1+g)γ ] (1 T ) becomes 1 T T =1+ de i, 1 T e g (T ) sg, g R E e g, g T # de i, (c gγ )(1+r ) + Ã Z! E (1 + r de i, T. ) e de i, ) c gγ (T ) (1 + r ) + " Ã Z! E e g, T + g de i, e g (T ) de i, ³ e g, nd by sg, T e e g + sg, ³ E 2e g ³ = e g, g T 1+gT ³ + g e g, # g e g, T T =0.,heFOC g R E sg, e g e g, g T e g, ³ (T ) de i, e g, + sg, g ³ g e g, =0, 1. 29

30 The opiml vlue of he xion T is where =1+ T ³ g E e g e g, T 2 h 1 E (1 + g) e g (2 + g)i (55) T Compring e 0 nd e g we hve h < 1 if e g E (1 + g) > (2 + g). (56) e 0 e g =(1+g) E 2e g T. By ssumpion g>0, hen e 0 >e g when e g > E 2. (57) According wih he economic inuiion, higher g higher is he increse in he number of educed workers respec he cse wihou subsidies. Furhermore, when g increse hen he opiml level of xion nd of subsidies increse. Mobiliy Cse ( full mobiliy - wo regions) Le us solve he Mobiliy Cse by he BI mehod. The governmen mximizion problem is for he region Mx T : s g,, " e g # " de i, + s g,i, b +(1 s g, ) e gi # de i, s g, [c (1 + r )] + where sub o : (1 + r ) γ = AT A B b ³ R E ³ R E e g (T ) de i, e gi (T I) de i, ³ R s g,, E + s g, s g,i,i + s g,i ³ R E e gi (T I) de i, e g (T ) de i, s g,i, s g,,i Le us now nlyze he governmen s decision by using he BI for ech differen ses of he world when >b.. 30

31 Cse (1): T < η +(1 η) T I We hve h s g,, nd s g,i,i = s g, = s g,,i =0. The mximizion problem of he governmen becomes ; s g,i, = s g, Mx T : s g, " e g The (20) nd he (21) become # " de i, c (1 + r ) + s g,i sub o : (1 + r ) γ = AT. e gi de i, # + 1 T e g e g = +(1+r ) c AT <e 0 The Firs Order Condiion is e gi = b +(1+r ) c e 0I 1 T b >e gi. Foc(T ): sg, T " de i, c (1 + r ) e +g (T ) # +s g, Ã! e g, T =0. ³ g Compuing he FOC we obin e g, 1 T Ã de i, e g e g (T ) +s g, =0, h ³h i g s g,i 1 T E e g e h E 2e g h i s g,i + E e gi s g,! s g,i E e gi s g, i h T = i ³ R E E 2e g e gi (T I) de i, s g, T + sg, ³ g i h + e g, T =0, E e g i. + The opiml vlue of he xion T is T =1 + s g, s g, h h E e g E 2e g i h i s g,i E e gi i h + s g,i E e gi i <T, 31