Heterogeeous Talet a Optmal mgrato A Cotrbuto to te New coomcs of te Bra Dra Nguye Duc Ta Natoal Grauate Isttute for Polcy Stues GRIPS, Tokyo Vetam Developmet orum VD Tokyo mal: 3@stu.grps.ac.jp September 3, 24 Abstract Ts paper s a cotrbuto to a ew le of teory argug tat a certa outflow of uma captal bra ra s ot always a ba tg to te source coutry. rst, t erces te metoology by solvg te problem wt assumpto o workers eterogeeous talet a sows tat te strbuto of talet s mportat. Seco, cotrast to te prevous lterature, ts paper sows tat postve effect of te outflows may ever take place uer some certa cotos. Tr, f tere s a postve effect, tere exst cotos emgrato costrats tat maxmze te ga from bra ra - or te optmal bra ra cotos. Relevat polces o emgrato for te source coutry are te suggeste. Keywors: Mgrato, Bra ra, Huma captal formato, Small-ope ecoomy Prepare for te Coferece o Growt, Developmet, a Macroecoomc Polcy, December 7-8, 24, Ia Statstcal Isttute, Del, Ia. I am grateful to Professor Kec Oo GRIPS, Assocate Professor utos Yamauc IPRI a ASID, a semar partcpats at GRIPS Tokyo for ter valuable commets a elpful suggestos.
. Itroucto Te term bra ra was popularze after WW II we tere was a uge umber of leag scetsts mmgratg to Ute State from Wester urope, Caaa a Sovet Uo Rapoport 22. However, te causes a cosequeces of bra ra le to ebates a resolutos te Ute Natos oly as early as 967, cocerg te argumet tat te poor coutres lost ter most talete people to te rc coutres Lowell 22. Durg te 97s, may ecoomsts pa atteto to te ssue, creatg te frst wave of bra ra ecoomcs. Notably Jags Bagwat, amog oters, may be te most fluecg fgure te ebates. coomsts urg te pero sare, more or less, a cosesus tat bra ra s a zero-sum game, wc te rc atos ga o te loss of te poor atos. Bagwat a Hamaa 974, Bagwat 976, Bagwat a Partgto 976, Hamaa 977, Bagwat 979a, 979b, a later Kowk a Lela 982, Myagwa 99. Ts frst wave seeme to fae away wt te ecle of te frst geerato of evelopmet ecoomcs te late 97s 2. It must wat for almost two ecaes to see te seco wave to take place, followg te rase of ew growt paragm, wc uma captal was realze as a mportat ege of ecoomc growt. Moutfor 997 for te frst tme argues tat bra ra s ot always a curse to poor coutres, f ot a effectve way to escape from te poverty trap. Hs argumet s tat people a poor coutry may ave stroger motvato to get more sklls f tey see some probablty of emgratg to a rc coutry, were tey ca ear more wt te same level of uma captal. Ts le of tkg as bee evelope teoretcally Val 998, Stark et al. 997, Dos Satos a Postel-Vey 23, a emprcally Bee et al. 2, 23. As a result, a ew geerato of bra ra polcy s trouce Stark et al 998, Stark a Wag 22, Dos Satos a Postel-Vey 23, Stark 24 a graually cosere Drkwater et al. 22, Lowell 22, Lowell a ley 22. Ts paper s a cotrbuto to ts le of teory. It evelops a moel recofrmg tat bra ra s ot always a ba tg to te source coutry. But t ffers from prevous lterature some aspects. rst, te paper looses te omogeeous worker assumpto a sows tat eterogeeous talet s a mportat etermat of bra ga or bra ra. Seco, cotrast to te prevous stues, ts paper sows tat postve effect of bra ra s ot evtable: uer some cotos, ts effect ever occurs. Tr, f tere s postve effect, tere exst cotos of emgrato costrats to maxmze te ga from bra ra - te optmal bra ra cotos. 2 Barro a Sala--Mart 24: 6-2 prove a goo bref revew of pases evelopmet of growt teory.
Te ext sectos preset tree moels. Te frst s a geeral moel, wc setups basc assumptos a pots out a geeral approac to solve te problem. Te seco s a basele moel apple to te case of geeral emgrato. Te tr extes te applcatos to a more realstc case of bra ra. Te fal secto s te cocluso. 2. A Geeral Moel 2.. Assumptos Workers talet : Coser a small ecoomy clug N workers wt fferet egrees of talet. ollowg Lucas 988, assume tat a worker s talet follows a gve probablty esty fucto pf, f. Ts meas, te probablty of a worker wt egree of talet s f umber of people wt talet a lm f gure. f, or te s: Nf. Te followg cotos ol: lm f gure. Probablty esty fucto of a worker s talet As a cotuous pf, ts coto ols: f Cost of eucato c a vual uma captal formato : Workers work a at te same tme coose to vest o ter ow uma captal. Te uma captal vestmet expeture s c. If oe vests c, se wll accumulate a stock of uma captal : 2
c,. 2 c, may be calle te uma captal formato fucto. Ts fucto sows tat te vual uma captal accumulate epes o te worker s talet a er uma vestmet expeture. I prcple, se ca receve more eucato from scool or more sklls from leag-by-og or from ay source, but tese actvtes are costly terms of real resource, wc are coute c. I geeral, ols te followg propertes: > ; c 2 < 2 c ; > 3 rom tese propertes, oe ca obta te substtuto rate of vestmet expeture a talet: c < ; c 4 Ts mples tat to atta a same amout of uma captal, te more talete te worker s, te less real resource se as to sacrfce. Worker s total come TU: Te compesato for worker s labor s assume equal to er level of uma captal stock: U. Tat meas, er lfe come s: TU + [ c + c, ]. 5 Were s te worker s tal uma captal eowmet. Probablty of emgrato π : ac worker as cace to emgrate to aoter coutry were te margal uma captal prouct s ger. Terefore, at ay level of uma captal stock, te successfully emgratg worker wll receve a comeω tmes ger ta te same worker workg omestcally, or: U ωu mgrate > ω. 6 Suppose te probablty of success s π π. π ca be a creasg, ecreasg, or costat fucto of, epeg o mgrato polces or oter curret stuatos. 3
Objectve fucto: It s assume tat eac worker eces ow muc real resource to vest er uma captal to maxmze er expecte lfe come. Or: 2.2. Geeral Soluto Max TU c + U. 7 c Our cetral cocer s te omestc aggregate uma captal formato Η, wc as bee wely realze as a mportat source of prouctvty a ecoomc growt. Te moel s soluto clues some steps. rst, we calculate Η cases wt a wtout emgrato. Seco, we compare tem a exame ow te fferece betwee tem epes o te moel s parameters. Huma captal formato wtout emgrato Η : I te case of close ecoomy, avg solve te maxmzato problem of er ow, te worker at gve talet level of all wll coose to vest c er uma captal, terefore se possesses a of uma captal. Tus, total uma captal stock of te ecoomy Η s te sum. Sce s a cotuous varable: Η + ] N f [ Η Η, f 8 were f s parameter vector of te pf of talet. Huma captal formato wt emgrato Η : We emgrato s possble, workers wll coser weter to vest more ter ow uma captal to emgrate. However, ue to a set of emgrato costrats eote by Ψ, tere may be ot all workers wllg to emgrate. Oly te workers wose talet belogs to a certa rage Τ ω,, f, Ψ ave cetve to go abroa wt some probablty of success. Tese workers wll vest more eucato ope of emgrato. Tose wose talet Τ ω,, f, Ψ ece ot to emgrate because to tem emgrato s mpossble or ot optmal. 4
By aggregatg uma captal stocks of all workers stll te coutry ew equlbrum, oe ca f: Compare Η Η, f, Ψ, ω 9 Η a Η we wll see te uma captal stock wc case s ger. As, f, Ψ are gve, suc fferece Η Η Η epes o ω. Now, suppose tat polcy makers ca affect Ψ, for eac ω we exame fucto Η Ψ a efe as follows: If Η Ψ Ψ : te source ecoomy s a emgrato trap. If Ψ : max Η Η Ψ > : exstece of optmal emgrato costrats Ψ. Te followg sectos preset two moels of emgrato costrats. Secto 3 presets a smple case wc emgrato s possble to every worker, a te probablty of successful emgrato s te same to all geeral emgrato case. Secto 4 proves a moel wc oly tose wose bestowe uma captal are ger ta a certa level ca emgrate, a ter success probablty epes o ow muc uma captal tey possess bra ra case. 3. A Moel of Optmal mgrato 3.. Assumptos I ts moel, some assumptos are ae to smplfy te geeral moel. rst, te uma captal formato fucto s assume as: < < c Seco, tal uma captal eowmet s zero. Tat meas, a worker s lfe come s: TU c + c,. Tr, te probablty of emgrato π s assume exogeous a epeet from 3. 3.2. Soluto ollowg te above settgs, we ca solve for total uma captal stocks of te source coutry cases wt a wtout emgrato, a te compare te fferece betwee tem. Huma captal formato wtout emgrato Η : 3 Ts assumpto s smlar to te geeral emgrato assumpto Moutfor 997. 5
We tere s o cace to emgrate, te worker s objectve fucto s: Max TU c + c TU Solvg te problem: + c c c c 2 c 3 Tus, te aggregate uma captal formato of te ecoomy wtout cace for emgrato s: Η 4 or: Huma captal formato wt emgrato Η : Η N f 4. We t s possble to emgrate, te worker faces a probabltyπ of gog abroa a recevg te come c ω, a a probablty π of stayg to work te ome coutry a recevg a come c. Terefore, er expecte come wt cace of emgrato s: Now, er objectve fucto s: U ω c π + c π Max TU c Solvg te problem by takg OC: c + ω c π + c π TU + c π + ωc c π [ + γπ ] c 5 were γ w->. Tus, te uma captal to be accumulate by eac worker s: 6
[ + γπ ] c 6 Te aggregate uma captal formato of te ecoomy wt cace for emgrato s: Η were s te umber of worker at talet stayg te coutry. It s obvous tat: π π N. f Te, Η N π f [ + γπ ] After rearragg: Η + γπ π. N f Η or: Η + γπ π 7 quato 7 expresses te aggregate omestc stock of uma captal Η as a fucto of possblty of emgrato π : Η π Η. If π : Η Η. Ts s te case of o emgrato. If π : Η. Ts s te case of eftely free emgrato. Te ecoomy s aggregate uma captal s totally estroye or sappeare because all uma captal stock of te coutry wll flow abroa were uma captal come s ger. We ow coser te case π,. rom 7 Η > π, t s possble to take log bot two ses of 7 a takg ervatve wt respect to π, a after rearragg te formulato: Η π Η [ γ + ] γπ + γπ π Sce Η > π. + γπ π Η π sg sg{ [ γ + ] γπ } 7
Recall tat Η ω γ + ω sg π π ω sg 8 Proposto 3.: If ω, te source ecoomy always suffers from losg uma captal stock regarless probablty of emgrato π. Te ger te probablty s, te more te coutry loses ts uma captal stock. Ts stuato may be calle emgrato trap. Proof. ω ω ω π π > ω Η π > Η π s ecreasg π >, a Η Η Η π π >. I ts case, te relatosp betwee te omestc uma captal stock Η a te probablty of emgrato π s presete gure 2. H ω H gure 2. π ω : Te ecoomy a mgrato Trap Proposto 3.2: If ω >, tere exsts a crtcal value of emgrato probablty π * maxmzg omestc uma captal stock. π * s te optmal emgrato probablty. Proof. ω ω > ω > π > ω so tat: ω ω ω π > ω Η π > π f, π f π, π 8
ω π ω ω π < ω f Η π π π Η π < π f π, f π π f π π, Η s maxmze at Η M ω ω ω ω H we π π. Beavor of te omestc uma captal stock s epcte gure 3. Η ω * Η ω H M Huma Captal ga ω > H N Huma captal loss gure 3. π π ω > : xstece of Optmal mgrato π Proposto 3.2 sows tat we te coto ω > s satsfe, a small probablty of emgrato at frst wll ave postve effect o aggregate uma captal formato of te source coutry, because cetve effect omates flgt effect. It s sow tat tere exsts a ω ω crtcal value of emgrato probablty π tat maxmzes te et uma captal ga, or te aggregate omestc uma formato pot M gure 3. If te possblty of emgrato becomes ger, te et uma captal ga wll ecrease, a at a level π 4 g 4 π s te soluto for te problem + γπ π tersects te orzotal le Η Η., as sow gure 3 at pot N, were te curve Η π 9
eoug, te cetve effect s omate by te flgt effect, makg te total effect equal to zero. ally, f emgrato becomes certa π, te ecoomy wll lose all of ts uma captal stock. 3.3. ffects of Wage Gaps ω. rom Proposto 3.. I te case ω, te moel mples tat gve omestc coto of te source coutry, te wage gap betwee te source a recevg coutres s ot g eoug to create suffcet motvato for accumulatg ew uma captal te source coutry, terefore te out-flowg uma captal s always larger ta te ewly create uma captal. Now, gve te value ofω, f s too small, makg ω, te source coutry s also euce to a emgrato trap. I our moel, ca be uerstoo as te source coutry s egree of tecology of kowlege trasfer or uma captal formato. Te ger s, te more prouctve te formato s. A ger meas a more effcet eucato c system or ks of socal orgazato, wc allow more effectve learg-by-og. Ts coto sows te avatage ecoomc tegrato of tose coutres wose uma captal formato capablty s g. Cosequetly, a polcy mplcato s erve: mprovg omestc eucato qualty a oter ways of trasferrg kowlege, suc as learg-by-og workplace, s a goo reacto to a tegrate worl. We tegratg to te worl, wage a opportuty fferetal abroa wll create a ema for eucato, but f te omestc eucato fals to meet te creasg ema, te coutry wll lose ts uma captal. 2. rom Proposto 3.2. We te coto ω > s satsfe, te moel suggests tat a postve probablty of emgrato s ot always as ba as tougt. It s ot a zero-sum game betwee te source a te recevg coutres. mgrato possblty motvates people te source coutry to accumulate more uma captal, a at a approprate probablty of emgrato, te source coutry ca ga uma captal from ts process e. bra ga from bra ra. Moreover, tere exsts a value of emgrato probablty tat maxmzes te uma captal ga of te source coutry. It s te pot of optmal emgrato. Te source coutry s govermet ca use emgrato polces to cotrol ts probablty to lea te ecoomy to te optmalty.
Te moel also cofrms tat, ay case, a suffcet g value of emgrato probablty wll amage te source coutry s uma captal stock et loss of uma captal. Ts meas tat a cotrol emgrato s always ecessary. 4. A Moel of Optmal Bra Dra 4.. Assumptos I ts moel, te frst two atoal assumptos te prevous moel are stll kept, but te tr assumpto s aapte to te fact of bra ra tat oly workers wose uma captal s greater ta a certa level ca emgrate, a worker s emgrato probablty π s epeet o er uma captal stock eogeous emgrato possblty. Itutvely, te followg propertes of π soul ol: > π f >, a π π π oterwse; > f > ger sklle perso s easer to emgrate; lm ts coto guaratees te cotuty of te fucto; v lm π te perso bestowe wt extremely g sklls ca certaly emgrate. or a cocrete soluto, we may assume: π f >, a π oterwse. may be cosere as a tresol emgrato costrat polcy. It s easy to see tat π satsfes te propertes above. Te relatosp betwee π a s presete gure 4. π gure 4. mgrato probablty as a fucto of uma captal
4.2. Soluto ollowg te same proceure te case of geeral emgrato, we frst solve for total uma captal stocks of te source coutry cases wt a wtout emgrato, a te compare te fferece betwee tem. Huma captal formato wtout emgrato Η : As sow te geeral emgrato case, equato 4, te aggregate uma captal formato of te ecoomy wtout cace for emgrato s: Η * Huma captal formato wt emgrato Η : I case of gog abroa, a worker s expecte come s: U ω π + π 9 were c a π. Now, er objectve fucto s: Max TU c Solvg te problem by takg OC: c ω + + c + ω + TU c + ω ω c Tus, te uma captal to be accumulate by eac worker s: c 2 ω c 2 However, oly tose wose uma captal s greater ta ca ave cace gog abroa, terefore oly tose wose talet s greater ta a crtcal level tat ece to vest more eucato ope of emgratg. Tose wose talet s lower ta o ot cage ter beavor, because ay case, tey ave o cace to emgrate. Te sape of te uma captal fucto, terefore, s a curve wt a jump at See gure 5. 2
3 Te relatosp betwee a s: ω 22 As a result, te aggregate uma captal formato of te ecoomy ts case s: + Η * 23 were s te umber of worker at talet stayg te coutry. It s obvous tat: π Te, + Η * + Η * 24 Te fferece betwee H a H s + Η Η Η * * Η * gure 5. Huma captal accumulato wt bra ra v
Η [ ] 25 Recall from 3, 22 tat * ω a Η N f ω 26 Ts meas tat oe ca rewrte: Ω Η N 27 were : Ω ω f. 28 Now we vestgate ow Η epes o recall tat epes o te value of, as sow equato 22. Note tat f, or tere s o restrcto of emgrato N f Η Η Ts s cosstet wt te extreme case, we all uma captal fles out of te source coutry. I aoter extreme, f, leag to Η, or te case of close ecoomy. Betwee te two extremes, Η wll cage ts rage of -H, ue to te cage of ts oma of,, or as, fuctoal relatosp betwee Η a or. respectvely. Our purpose ow s to exame suc Because >, N>, we ca terefore suffcetly vestgate te cage of te value of Ω as cages. Rewrte equato 28 as Ω ω f f 29 a set u a f v so tat v, cf of workers at talet. f uv uv vu 4
5 Plug to 29: [ ] + Ω ω Now we take te frst ervatve of Ω wt respect to. Because te upper bou oes ot epe o, ts ervatve must be zero a elmate from te equato: + Ω ω Remember tat f : + + + + Ω ω f f Rearrage te rgt a se: Ω ω ω f Deote:, + K K ω ω, 3 a f g, 3 [ ] g K Ω 32 Now, plug to 27 to get:
Η Ω N, a eote : A N, 33 Η We ca te erve from 32: A [ K g ] Η or: A [ K g ] 34 As > from equato 22, > A >, we fally come to te followg equato: Η sg sg[ K g ] 35 quato 35 s mportat because by examg te sg of [ K g ], we ca uersta ow Η cages as vares. Because f s gve a so far we o ot ave more assumptos cocerg ts propertes, geeral we ave o clear-cut cocluso of te sape of fucto g. However, 3 reveals some basc features of g : g ; a g > >. We ow cocer te upper lmt of g. I geeral, g ca coverge or verge. But, for our curret purpose, we coser te case of vergece s merely a specal case of covergece to +. We ca terefore stgus two cases: A B lm g > K lm g < K clug lm g + Proposto 4.: If lm g > K Case A, tere exsts a crtcal level of tat above t te source ecoomy wll ave bra ga from bra ra Η becomes postve. Tere also exsts a level of maxmzg te et bra ga Η s maxmze a postve, or te optmal bra ra value of. 6
g g 2 K G g v gure 6. Case A: g coverges to a value greater ta K Proof. xstece of : As lm g > K Η lm [ K g ] < sg lm sg lm A [ K g ] <. We also kow tat lm Η, terefore, as approaces +, Η must approac zero from above. Ts meas tat at a great eoug value of, Η must be greater ta zero, a, cosequetly, tere must exst at least oe pot were Η excees zero. Suc a pot s llustrate as pot N gure 7. xstece of : As g s cotuous, a g < K a lm g > K, g must somewere excee te orzotal le k K, or were [ K g ]. Suppose t s pot G gure 6 take g for example. Tat g excees k K at G ecessarly meas tat g s creasg at G, or, matematcally, g >. Tese facts mply tat at least oe value of so tat: [ K g ] a [ K g ] g <. By recallg 35 a efg as 2, oe comes to a equvalet cocluso tat Η 2 [ K g ] a Η [ g ] sg sg < * 2 K Η maxmzes at see gure 7 5. 5 Sce a exact from of g s ot kow, tere may be multple roots as g as a sape lke g 2 gure 6. To keep te llustrato smple but wtout loss of te geeral mplcatos, we wll coser te case of uque root oly curve g gure 6 a Η gure 7. Te cases of multple roots are presete te fgures as te ase curves. 7
Η M Η N Η 2 Η gure 7. xstece of Optmal Bra Dra Case A Proposto 4.2: If lm g < K Case B, te exstece of optmal bra ra s coclusve. Depeg o te partcular propertes of te talet probablty esty fucto, te source ecoomy may always loss ts uma captal at ay level of - te case of bra ra trap, or ga uma captal oly lmte rages of. g g g 4 3 K P g 5 v gure 8. Case B: g coverges to a value smaller ta K 8
Proof: As lm g < K see gure 8 for fferet possble sapes of g lm [ K g ] > Η sg lm Η sg lm[ K g ] > lm >. Terefore, as approaces +, Η must approac zero from below. I ato, tat g may sometmes excee te orzotal le k K or always below suc le suggests tat Η may ave maxma or may ot ave at all, respectvely. Tese facts mply tat tere are 3 possble sub-cases: B xstece of postve maxma of Η Η a 3 Η gure 9 4 B2 No-xstece of postve maxma of Η, a cosequetly, Η s a o-postve fucto Η gure 5 B3 No-exstece of maxma of Η, a, cosequetly, Η s a strctly egatve fucto Η gure 6 Tus, sub-case B tere exsts optmal bra ra, but te rage were et bra ga s postve s lmte some tervals of gure 9, see te parts of te curves above orzotal axs. Take curve Η for example: Η > oly we, 3 2 I sub-cases B2, B3, Η s o-egatve. gure. Ts mples tat te source ecoomy suffers from losg uma captal at all level of. It s by our efto te case of bra ra trap. Η Q Η 3 Η 4 Η 2 gure 9. xstece of Optmal Bra Dra Case B 9
Η Η 5 Η 6 Η 4.3. ffects of Wage Gaps ω I ts secto, we exame ow cages wage fferece betwee te recevg a te source coutres ω affect te et bra ga fucto, Η, a optmal bra ra level of tresol uma captal.. ffects of ω o Η. rom equato 26, we kow tat: Η ω 2 N ω 36 Terefore, at ay gve rate of tresol uma captal level, we ω creases ecreases, te et bra ga wll crease ecrease. 2. ffects of ω o. gure. Bra Dra Trap Cases B2, B3 As g >, > see Proof of Proposto 4., a K g, K K > g 37 2 K ω rom 3: < ω 2 [ ] ω 38 2
Combe 37 a 38, oe gets: K < ω K ω 39 3. Sft of te curve Η as ω cages. rom propertes of 34 a 37, oe ca erve te beavor of te et bra ra fucto Η we ω cages, as sow gure. or example, we ω creases, te curve Η wll sft upwars from Η to ω Η, a at te same tme, te maxmum pot ω 2 sfts backwars from M to M 2. Η M 2 Η ω 2 M Η ω Η ω 3 ω M 3 ω Η 2 3 gure. Sfts of Η by cages ω gure escrbes ow Η moves as ω cages. We ω ecreases, for example, te curve becomes flatter a lower, a tes to move outwars as te maxmum value creases. As a result, et bra ga becomes smaller a approaces zero. Ts mples a fact tat we come gap s bg, te cetve to accumulate more uma captal s ger a strogly omates te real loss of uma captal bestowe actual emgrats. Coser te optmal level of emgrato tresol, we f tat te coutry wt lower wage rate may be suggeste to coose a lower level ta te coutry wt ger wage rate or smaller come gap. Ts fg s terestg, because t mples tat te poorer coutres soul be more ope ts market of te gly sklle a omally volves more bra ra to te worl orer to accumulate more uma captal. 2
5. Cocluso I ts paper I ave costructe moels of optmal emgrato wt workers eterogeeous talets. Te moels prove tat uer some cotos te source coutry may get stuck a bra ra trap, were emgrato costrats always leas to et bra ra, or te bra ra effect at all tmes omates te bra ga effect. However, f te source coutry s ot a bra ra trap, t possbly accumulates uma captal by allowg a certa possblty of emgrato. I te case of geeral emgrato, a emgrato possblty small eoug may create motvato for uma captal accumulato te source coutry. Cosequetly, te moel ts case sows two crtcal pots ts process: a optmal emgrato probablty, were te source coutry s able to get maxmum uma captal stock; a a et bra ra probablty were te source coutry begs to lose ts uma captal. I te case of bra ra, e, oly workers wt uma captal bestowe ger ta a certa tresol may emgrate, a te ger sklle workers are easer to go, we also sow tat polcy makers ca affect suc tresol to maxmze te omestc aggregate uma captal formato. Cotos for exstece of bra ga from bra ra, owever, epe o te strbuto of workers talet. A cocluso of te exstece s oly aceve by examg more etals te propertes of suc strbuto fucto. Te moels propose ts paper are smple a statc. Altoug tey are smplfe may ways, ter message s basc a stragtforwar. or furter stues, we may exame te progress of uma captal stock a yamc framework, usg overlappg geerato approac wt uma captal bequest from workers parets. A more compreesve stuy may coser exteralty of uma captal as suggeste by Lucas 988. Moreover, to vestgate te yamcs of a eucato sector facg creasg ema may brg terestg results. ally, cocerg eucato polcy ope ecoomes, we ca aalyze te coutry s welfare wt fferet patters of eucato.e. publc versus prvate vestmet eucato a trag, cret wt a wtout costrats, etc. presece of emgrato. 22
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