An OLG-CGE Modeling Framework for Bilateral Migration

Transcription

1 An OLG-CGE Modeling Framework for Bilaeral Migraion Marcel Méree Deparmen of Economics Universiy of Oawa Guohan Zhu Human Resources and Social Developmen Canada Governmen of Canada Absrac To accommodae possible negaive impacs of ageing and compensae for poenial labour shorages, many counries are adoping more open immigraion policy oward he developing counries. Meanwhile, he demand from many emerging developing economies for foreign inelligence is increasing wih a deeper degree of globalizaion and he ouspread of mulinaional enerprises and ousourcing. Based on a simple wocounry, hree-inpu and wo-generaion OLG model, his paper inroduces he concep of bilaeral migraion ino a general equilibrium framework. In he model, wo counries are assumed o be perfec symmeric in every aspec excep for heir populaion growh. Labour sock in each counry is differeniaed by boh heir skill levels and counries of origin. In his dynamic general equilibrium modeling, we inroduce a series of CET (consan elasiciy of ransformaion) equaions o capure he supply of each kinds of labour. Simulaion resuls from our model sugges some sylized facs which are in accord wih he new era of globalizaion. For example, i is shown ha more people migrae from he younger developing counries o he older developed counries. There will also be wage differenials beween naive and immigran skilled or unskilled workers. Finally, as he degrees of ageing diverge across wo counries, here will be significan endency for workers o move from he developed counries o he developing counries wih he booming of bilaeral rade beween hese hem.

2 1. INTRODUCTION Over he nex several decades, counries around he world will experience differen degrees of populaion ageing. Demographic projecions show ha all developed counries will be in he process of fas ageing afer 2010 when he baby-boom generaion (hose born beween 1946 and 1966) begins o reire (Méree 2002). Populaion ageing is widely viewed as a hrea o living sandard and economic growh, especially for he ageing developed counries, as shown by relaed researches by World Bank (1994) and OCED (1998). Pessimisic views urged hese counries o find soluions o accommodae possible negaive effecs of ageing. Besides some inernal reform scenarios, such as deb sabilizaion and delayed reiremen (OECD 1998), adoping more immigrans is considered by many counries as an opion o relax he pressure of ageing on labour marke by opening door o inernaional labour marke so as o improve he age srucure of an ageing economy. Inernaional migraion had already become a ho issue in almos all indusrial counries. Differen from he European mass emigraion o he new coninens before 1914, he pos-wwii wave of world migraion is accompanied wih a deeper inegraion of naional markes (globalizaion) and an acceleraing process of populaion ageing. On one hand, developed counries have incenives for aracing immigrans o deal wih he shorage of labour supply and alleviae possible fiscal impacs of ageing. Wih he developmen of informaion and elecommunicaion (ICT) indusry, many developed counries have also pu emphasis on aracing more high-skilled foreign labour from he developing world. This is a new characerisic of he pos-wwii wave of inernaional migraion. On he oher hand, deeper inegraion of naional markes provides incenives for a large scale of

3 indusry ou-sourcing, from he developed world o he developing world, and more and more mulinaional enerprises (MNEs) are operaing overseas. Consequenly, many emerging developing economies (he Eas-Asian dragons, for example) are demanding more and more foreign inelligence, especially hose from advanced counries, as a means of upgrading local indusries and filling he echnology gaps beween hem and he developed world. Overall, over he pas decade, we have observed a large scale of migraion of labour, a all kinds of skill levels, from he developing counries o he developed counries. In addiion, more and more people from he developed world have moved o emerging economies. Thus, i is highly relevan o sudy he so-called bilaeral migraion o capure he new rend of double-way inernaional migraion boh from he developing world o he developed world and from he developed world o he developing world. This model uses a simple 2*3*2*2 ype overlapping-generaion compuable general equilibrium (OLG-CGE) model o provide a basic modeling framework o sudy he issue of double-way migraion beween wo regions. Two counries in he model are assumed o be perfecly symmeric a he iniial seady saes. Working individuals in he wo counries are allowed o make choices on heir ime allocaion ino eiher working in his or her homeland or working abroad. In he dynamic, populaion growh raes of wo counries are assumed o be differen so ha one counry grows ino an older economy. Changes of age srucures differeniae hese wo counries labour markes, and he resuling wage differenials creae incenives for people o move. A series of CET funcions are used o model he supply side of labour so ha working individuals in each

4 counry could make migraion decision based on boh heir preferences in working locaions and changes of heir expeced earnings. 2. MODEL DESCRIPTIONS In he 2*3*2*2 ype OLG-GE model, he world economy has been divided ino wo counries (J1 and J2). In each of he hem, wo homogenous i goods (S1, S2) are produced and wo major inpus (capial and labour, labelled as Kd and Ld) are used in he producion of each good. Labour inpu used in he producion of each good is a consan elasiciy of subsiuion (CES) aggregaion of wo kinds of labour differeniaed by heir skill levels (Skilled and unskilled). In he model, each counry is populaed wih wo generaions (he working generaion G1, or gj, and he reired generaion G2, or gm) of populaion, and G1 is assumed o be he only ferile generaion. The firs generaion, G1, is also he only working generaion in each counry as he labor inpu in he producion processes. A he iniial seady sae, we assume populaion of each of he wo generaions equals o 1/2 of oal populaion, and he shares of skilled and unskilled workers in each generaion are equal across counries. In he dynamic as he degrees of ageing diverge across counries, he resuling wage differenials beween he wo counries labor markes generaes incenives for boh he skilled and he unskilled workers o move in boh direcions (from he younger counry o he older counry and from he older counry o he younger counry). Consequenly, he shares of skilled and unskilled workers in each counry labor force vary overime, as a resul of bilaeral migraion.

5 To simplify our model and pu emphasis on he decision making of differen ypes of labor, governmen behaviour is no aken ino accoun. We also ignore he exisence of beques and heriance in he opimizaion process for a ypical individual. 2.1 Producer behavior In each counry, here are wo secors producing each one of he wo goods s (S1 or S2) using Cobb-Douglas (CD) producion echnologies. A represenaive firm s problem is o minimize he producion cos subjec o he embedded consrain ha characerizes he firm s echnology: Maximize Pq X R Kd + W Ld (1) Kd js,,, Ld js,, js,, js,, j, js,, j, js,, K K αs (1 αs ) js,, = s js,, js,, s.. X SPKd Ld (2) In equaion (1), Pq j,s, is he producion price of good s in counry j a ime. R j, and W j, represen he renal price and wage rae in counry j a ime respecively. Kd j,s, is he capial demanded by producion secor s in counry j a ime. Ld j,s, is he aggregae labour demanded. In equaion (2), SP 1 s is a scaling consan parameer and α K s is he expendiure share measuring he inensiy of use of capial in producion. S1 is capial-inensive good and S2 is labour-inensive good. The echnology is represened by a CD producion funcion in which he sum of expendiure shares for capial and labour inpus is equal o one. 1 In his paper, SP s, are assumed o be one in all secors.

6 In his model, we assume ha he aggregae labour supply, Ld j,s,, is a CES mixure of wo kinds of labour supply (qual) in he economy: he skilled labour and he unskilled labour (equaion 3). LD LD LD s 1 s 1 s LD LD LD Q s Q s s 1 j,, s = δs, skill j,, s skill, + δs, unskill j,, s unskill, Ld ( LdQ LdQ ) (3) Skilled and unskilled labour is differeniaed by he quaniy of labour supply hey could provide per uni of ime. In equaion (3), δ Q s,qual is he share of labour a skill level qual demanded by he producion of good s and he sum of δ Q s,skill and δ Q s,unskill is equal o one. Ld s is he elasiciy of subsiuion beween differen kinds of labour in secor s. The value of δ Q s,qual is calibraed o be differen boh across secors and across counries. Differeniaing equaion (1) wih respec o equaion (2), we ge he following equaions (4) and (5) as he firs order condiions for a producer: K js,, js,, R j, = αs (4) Kd js,, W j, = Pq X Pq X K js,, js,, (1 αs ) (5) Ld js,, Afer he renal price and he aggregae wage rae are deermined, a firm s opimizaion problem urns o be minimizing is expendiure on labour inpu. Tha is, i minimizes he following equaion (6) subjec o he above equaion (3): W Ld WdQ LdQ WdQ LdQ j, j, s, = j, skill, j, s, skill, + j, unskill, j, s, unskill, (6) where WdQ j,qual, is he wage rae paid for labour inpu a skill level qual in counry j a ime. The following equaion (7) and equaion (8) are he firs order condiions:

7 LdQ W DQ s DQ s Q W j, j, s, qual, = δ s, qual Ld j, s, WdQ j, qual, DQ DQ DQ 1 Q s 1 s j, δ s, qualwdq jqual,, s = (7) (8) Equaion (7) shows he demand for labour inpu a skill level qual, LdQ j,s,qual,, increases wih he aggregae labour demanded, he share of his kind of labour in he echnology and decreases wih he wage rae paid for his kind of labour. Equaion (8) shows he aggregae wage in region j is a weighed sum of wage rae paid for all kinds of labour. In his model, migraion is inroduced so ha he aggregae labour demand a skill level qual by secor s, LdQ j,s,qual,, could be supplied by workers from boh he naive labour marke and foreign labour marke. In oher words, his labour supply is a CES aggregaion of labour supply differeniaed by heir counries of origin (LdM j,j,s,qual, and LdM i,j,s,qual, ), as shown by he following equaion (9): DM DM DM s 1 s 1 s DM DM DM Ldm s Ldm s s 1 j, s, qual, = δ j, j, s, qual j, j, s, qual, + δi, j, s, qual i, j, s, qual, LdQ ( LdM LdM ) (9) Ldm where δ j,j,s,qual is he share of labour supplied by workers born in counry j o secor s in counry j and δ Ldm i,j,qual,s is he share of labour supplied by workers born in counry i o secor s in counry j. LdM j,j,qual, (LdM i,j,qual, ) is he quaniy of labour supplied by individuals born in counry j (i) and work in counry j. In oher words, LdM j,j,s,qual, and LdM i,j,s,qual, are labour supplied by naive and foreign workers respecively. DM s is he elasiciy beween naive labour demand and foreign labour demand in secor s.

8 A ime, he problem for a producer in secor s in counry j is o minimize is aggregae wage expendiure on naive and foreign workers subjec o he above equaion (9). The aggregae wage expendiure is deermined by he following equaion (10): Wdq LdQ WdM LdM WdM LdM j, qual, j, s, qual, = j, j, skill, j, j, s, qual, + i, j, qual, i, j, s, qual, (10) where WdM j,j,s,qual, and WdM i,j,s,qual, are demand wage raes paid by producers in counry j o naive and immigran workers working in secor s in counry j. The firs order condiions for his sage of opimizaion are he following equaions (11) and (12): LdM WdQ DM s DM s WdQ Ldm j, qual, i, j, s, qual, = δ i, j, s, qual LdQ j, s, qual, WdM i, j, qual, DM DM DM 1 s s Ldm 1 s j, qual, δ i, j, s, qualwdm i, j, qual, i s = (11) (12) Equaion (11) shows he demand for naive (immigran) workers increases wih he raio of aggregae wage rae for workers a skill level qual o he wage rae for naive (immigran) workers, he share parameer of naive (immigran) workers in oal labour force and he aggregae demand for workers a skill level qual in secor s. Equaion (12) shows he aggregae demand wage rae is a weighed sum of demand wage raes for naive and immigran workers working in counry j. 2.2 Household behavior An Allais-Samuelson overlapping generaion framework characerizes households in he economy, so ha his model is based on he life-cycle heory of savings. In period,

9 each economy is populaed by wo generaions g (G1 and G2) of populaion. A represenaive cohor born in period works in he firs period of is life (from 16 o 50 years of age), reires in period +2 and lives anoher period of ime (from 51 o 85 years of age). Thus each period approximaely represens 30 years in our model. The lifeime uiliy for a represenaive cohor is represened by a ime-separable nesed CES funcion wih he elasiciy differen from one. The opimizaion problem for a cohor a skill level qual can be divided ino hree seps in an open economy wih inernaional rade. Firs, immediaely afer birh, a cohor decides on he allocaion of is labour income on eiher curren or fuure consumpions. On he second sep of opimizaion, he aggregae consumpion expendiure is allocaed among differen kinds of consumpion goods. Finally, on he hird sep of opimizaion, an individual decides on he composiion of each consumpion good s in erms of he counry of origin. In oher words, consumpion of good s is an aggregaion of boh locally- produced producs and impored producs following he Armingon assumpion. The ineremporal opimizaion problem for a represenaive cohor a he skill level qual in counry j is o maximize is lifeime uiliy subjec o he budge consrain: C jg,, IT IT g IT j, qual, g, j, qual, = IT C j, qual, g, + g 1 1 g= 1 1+ ρ Maximize U( C ) ( ) (9) 7 C j, qual, = j, g, + g 1 + τ j, qual, g. + g 1 g= 1 s.. LW PCon (1 )( C ) (10) The CES-form uiliy funcion is shown by equaion (9), and he lifeime uiliy is an aggregaion of he presen values of curren consumpion and fuure consumpions. In each counry j, U(C j,qual,g, ) j,qual, is he lifeime uiliy for a represenaive cohor a skill

10 level qual and C j,qual,g, is he quaniy of aggregae consumpion. IT is he ineremporal elasiciy of subsiuion, and ρ is he pure rae of ime preference. The larger ρ, he more of is lifeime resources a cohor would spend in he firs sage of life and he less i saves. As shown in equaion (10), he lifeime wealh for an individual a he skill level qual, LW j,qual,, is allocaed ino curren consumpion and fuure consumpions. In his equaion, PCon j,g, is he aggregae consumpion price for an individual of age g a ime. LW j,qual, is represened by he following equaion (11): LW j, qual, 7 1 = g= 1 1+ Rin j, + g 1 [ LInc (1 CR ) + Pens ] (11) j, qual, g, + g 1 j, qual, g, + g 1 where LInc j, qual,g, is he labour income for an individual a skill level qual of age g a ime. Rin j, is he ineres rae in region j a ime. CR represens he conribuion rae of pension ou of oal labour income; Pens j,qual,g, is he pension income received by he las wo generaions. In boh counries, an individual s labour income depends boh on he individual s age-dependen produciviy and he skill level. Thus, he labour income for an individual in he working generaion gj a ime can be represened by he following equaions (12) and (13): LInc = wage EP j, qual, gj, j, qual, j, g, qual EP = α γ + λg ψg γ λ ψ j, g, qual (12) Q 2 qual ( ),,,, 0 (13) The ineremporal maximizaion problem is as follows. Households in all counries are forward looking and have perfec foresigh. A represenaive cohor is born in he firs

11 period and works in his period as he labour force in he economy. Par of is labour income is saved as invesmen in capial sock and he res is consumed. In he las wo periods, his cohor spends all is wealh, including invesmen in he firs period and ineres income, on consumpion. Immediaely afer birh, an individual decides on he allocaion of his or her expeced labour income ino curren consumpion and fuure consumpions. Differeniaing he above equaion (9) wih respec o he budge consrain (10) yields he following firs order condiion for he aggregae consumpion, C j,qual,g,, consumed by an individual a skill level qual of age g (Equaion 14): C IT K C (1 + Rin+ g(1 τ ))(1 + τ ) PCon g, + g 1 jqualg,, + 1, + g = C C jqualg,,, + g 1 (1 + ρ)(1 + τ ) PCong+ 1, + g (14) Equaion (14) shows he raio of aggregae consumpion by individual of age g a skill level qual a ime +1 o i is a ime increases wih he raio of aggregae consumpion price a ime o he aggregae consumpion price a ime +1. In oher words, higher are fuure consumpion prices, higher is curren consumpion and lower is fuure consumpion. On he second sep of opimizaion, an individual of age g allocaes he expendiure on he opimal aggregae consumpion, C j,qual,g,, across secors. The problem is o maximize he following CES uiliy funcion subjec o budge consrain: CS j, S1, qual, g,, CS j, S 2, qual, g, CS CS CS 1 1 CS CS CS CS CS 1 j, qual, g, = δs1 j, S1, qual, g, + δs 2 j, S 2, qual, g, Maximize C ( CS CS ) (15) s.. PCon C = PCons CS + PCons CS (16) j, g, j, qual, g, j, S1, j, S1, qual, g, j, S 2, j, S 2, qual, g,

12 In equaion (15), δ s CS as he parameer represening he preference of a consumer of age g on consumpion goods across secors. CS is he elasiciy of subsiuion for consumpions across secors. CS j,s,qual,g, is he quaniy of consumpion of good s by an individual of age g a skill level qual living in counry j a ime. In equaion (16), PCons j,s, is consumpion price of good s. The following equaions (17) and (18) are he firs order condiions for he second sep of opimizaion: CS CS CS CS PCon jg,, j, qual, s, g, = δ S C j, qual, g, PCons js,, PCon CS CS CS 1 CS 1 jg,, = δ S PCons js,, s (17) (18) Equaion (17) shows he consumpion of good s by individual of age g a skill level qual a ime increases wih he aggregae consumpion bu decreases wih he consumpion price of good s on he marke. Equaion (18) shows he aggregae consumpion price is a weighed sum of he consumpion prices for differen good s. In our Armingon-ype rade model, an individual in counry j allocaes he consumpion expendiure on good s by counry of origin, which is he hird sep of opimizaion. In oher words, a decision is made on how much o spend on eiher locallyproduced good s or impored good s. The following equaions show he hird sep of consumer s opimizaion: CSJ j, j, s, qual, g,, CSJi, j, s, qual, g, CE CE CE 1 1 CE CE CE CE CE 1 j, s, qual, g, = δ j, j, s j, j, s, qual, g, + δi, j, s i, j, s, qual, g, Maximize CS ( CSJ CSJ ) (19) s.. PCons CS = Pq CSJ + Pq,,,,, (20) j, s, j, s, qual, g, j, s, j, j, s, qual, g, i, s, CSJ i j s qual g

13 In equaion (19), δ i,j,s CE as he parameer represening he preference of a consumer on goods s produced in counry i, and CE is he elasiciy of subsiuion for locallyproduced and impored consumpion good s. CSJ i,j,qual,s,g, is he quaniy of consumpion of good s produced in counry i bu consumed by counry j s consumer. In oher words, se i shows he counry of origin of a consumed good and se j shows where his good is consumed. Accordingly, CSJ j,j,qual,s,g, represens locally-produced consumpion good s consumed by generaion g in counry j a ime. The following equaion (21) and equaion (22) are he firs order condiions for he hird sep of opimizaion: CSJ PCons CE CE CE PCons js,, i, j, s, qual, g, = δ i, j, s CS j, s, qual, g, Pqis,, CE CE CE 1 CE 1 js,, δ i, js, Pqis,, i = (21) (22) 2.3 Labour supply, migraion and demographics In his sub-secion of model descripion, we pu emphasis on our conribuion as he modeling of he supply side of labor marke in a dynamic CGE model. Since he evoluion of each counry s populaion is no longer exogenously deermined and becomes more dependen on he resuls of bilaeral migraion, we will also inroduce how he posmigraion populaion of each counry evolues. In each counry, he aggregae labour supply is provided by workers eiher born in his counry or born abroad. Thus, here are four ypes of labour on he marke: naive skilled workers, immigran skilled worker, naive unskilled workers and immigran unskilled workers. The decision making of labour supply is closely correlaed o household behavior. A ime, each individual in counry j is endowed wih a cerain

14 quaniy of labour. Since here is no consrain on he movemen of labour across borders, so his individual has he freedom o allocae his or her ime ino eiher working in he home counry or working abroad. In he model, he aggregae labour supply a skill level qual in counry j a ime, LsQ j,qual,, is provided by a number of people a skill level qual (Pop j,qual, ), who are working and living in his counry: LsQ Pop EP j, qual, = j, qual, j, gj, qual (27) where LsQ j,qual, is a CET aggregaion of labour supplied by naive workers and foreign workers (LsM j,j,qual, and LsM j,i,qual, ), as shown by he following equaion (28): SM SM SM DM SM SM Lsm Lsm + 1 j, qual, = δ j, j, qual j, j, qual, + δ j, i, qual j, i, qual, LsQ ( LsM LsM ) (28) Lsm In equaion (28), δ j,j,qual,,s is he share of ime allocaed by an individual born in Lsm counry j ino working in counry j, and δ j,i,qual,s is he share of ime allocaed by an individual, born in counry j, ino working in counry i. LsM j,j,qual, (LsM j,i,qual, ) is he quaniy of labour supplied by an individual born in counry j and work in counry j(i). SM is he elasiciy of ransformaion beween labour supplied o producion in counry j by workers born in counry j and labour supplied o producion in counry i by workers born in counry i. This elasiciy value shows he responsiveness of naive and foreignborn labor o he wage raes paid for hem by producers in counry j. In he numerical model, boh counries are assumed o be perfecly symmeric a he iniial seady sae, so all share parameers and elasiciy values are assumed o be equal across counries. The values of relaed parameers are presened on he following Table 1.

15 As menioned above, an individual born in counry i allocaes he ime ino eiher working in his or her home counry or abroad. The opimizaion problem for his individual is o maximize is labour income gained from boh working in his or her home counry and abroad. In he model, he problem is o maximize he following equaion (29) subjec o equaion (28): WsQ LsQ WsM LsM WsM LsM j, qual, j, qual, = j, j, qual, j, j, qual, + j, i, qual, j, i, qual, (29) where WsQ j,qual, is he aggregae wage rae paid for an individual, a skill level qual, born in counry j a ime. WsM j,j,qual, (WsM j,i,qual, ) is he wage rae paid by producers in counry j (i) for naive (immigran) workers born in counry j. The following wo equaions (30) and (31) are he firs order condiions: LsM WsQ SM SM Lsm WsM j,, i qual, j,, i qual, = δ j,, i qual LsQ j, qual, WsQ j, qual, SM SM SM 1+ Lsm 1+ j, qual, δ j,, i qual WsM j,, i qual i = (30) (31) Equaion (30) suggess he quaniy of labour provided by naive (immigran) workers born in counry j increases wih he raio of wage rae for naive (immigran) workers in home (desinaion) counry o he aggregae supply wage rae for workers, a skill level qual, born in counry j. I also augmens wih he share of labour provided by naive or immigran workers in he aggregae labour supply and he aggregae quaniy of labour supply provided by workers a skill level qual and born in counry j. Equaion (31) suggesed he aggregae supply wage rae for an individual born in counry j is he

16 weighed sum of supply wage rae paid by producers in counry j o naive workers and supply wage rae paid by producers in counry i o immigran workers born in counry j. In he firs chaper of his disseraion, we inroduce he demographic shock as given he projeced feriliy raes, we calculae each counry s populaion and effecive labor supply for each period. The changes of all economic variables in he dynamic are based on his demographic shock. In chaper 2, we inroduced a one-period exogenous shock of unilaeral migraion so ha he populaion of he source counry and he desinaion counry will be recalculaed for all periods afer he iming of migraion, as shown by he following equaions (32) and (33): POP MigCoe Pop i, qual, + 1 i, j, + 1, g i, qual, gi, + 1 (32) POP MigCoe Pop j, qual, i, j, + 1, g i, qual, gi, + 1 (33) where MigCoe i,j,qual,,g is he migraion coefficien calculaed based on wage differenials across counries. The economic implicaion of his migraion coefficien is he percenage of oumigraing people ou of he populaion of he source counries, so we only have o adjus he populaion for all counries for one period and allow heir populaion o regenerae based on he adjused populaion in he following periods. However, wih his seing, we are unable o model he bilaeral migraion as his raio is exogenously given. In his paper, modeling of he supply side of he labor marke enables us o differeniae naive and immigran workers. In addiion, we no longer rely on he migraion coefficien o exogenously deermine he number of migraing people. In oher words, for he fis ime in he dynamic CGE modeling, we rea labor sock in each

17 counry as commodiies on he good markes based on he raionale of Armingon assumpion. How much ime an individual allocaes ino working in he home counry and working abroad are deermined by boh his or her preferences a he iniial seady sae and he endogenously deermined changes of wage raes for naive and immigran workers. Consequenly, he populaion of boh counries is also exogenously deermined in each period. This raises a problem of he populaion of boh counries a each ime period could be eiher he number of people before migraion or he number of people afer migraion. Anoher obsacle is he iming for us o impose he exogenous feriliy raes. To solve hese problems, we firs assume ha boh naive and immigran workers follow he feriliy raes in he desinaion counries. Then, we inroduce wo ypes of populaion (he naive populaion PopM j,j,qual,,g, and he immigran populaion PopM i,j,qual,,g ) ino he model. The following four equaions are used o relae he wo variables of Pop j,qual,,g and PopM i,j,qual,,g and o capure he evoluion of populaion for counry j: PopM = LsM /( LuM EPM ) (34) j,, i qual,, G1 j,, i qual, j,, i qual, G1 j,, i qual, G1 Pop = PopM FR PopM Pop j, qual, + 1, G1 i, j, qual,, G1 j, i = PopM j,, i qual, + 1, G2 j,, i qual,, G1 = Pop j, qual, + 1, g+ 1 j, qual,, g (35) (36) (37) where PopM i,j,qual,,g is he populaion of people, a skill level qual, born in counry j bu working in counry i a ime. LuM j,i,qual,g is he uni labor supplied by his individual and EPM j,i,qual,g is he efficiency coefficien.

18 Equaion (34) is used o calculae he number of naive and immigran populaions based on he equilibrium levels of labor supply. Equaion (35) inroduces he exogenous feriliy rae and suggess ha he oal populaion in counry j a ime +1 equals o he sum of naive and immigran workers a ime augmens wih he feriliy rae in counry j. Equaions (36) and (37) show ha he populaion of he nex generaions of naive, immigran workers and he oal populaion of counry j a ime +1 equals o heir populaion of previous generaions a ime. 2.4 Deerminans of invesmen demand In each period, savings of he younger generaions finance invesmen equipped by firms o increase and mainain he sock of physical capial in period +1. The usual law of moion of capial sock can be represened by he following equaion (38): KS = j, 1 I + j, (1 + DR) KS j, (38) In equaion (38), I j, is he aggregae invesmen in counry j a ime. The capial sock in he economy a ime +1, KS j,+1, augmens wih invesmen in he previous period and declines wih depreciaion rae (DR). This equaion shows he aggregae invesmen level is deermined by he gap beween demand of capial sock in he nex period and curren capial sock ne of depreciaion. The capial sock is a composie good composed of he available goods in he economy. Consequenly, he capial sock in each economy is buil using an invesmen echnology wih boh he capial-inensive good and he labour-inensive good. Afer he

19 aggregae invesmen level is deermined, here are also wo seps of opimizaion for he invesors in he economy, which are depiced by he following four equaions: ISJi, js,,, ISJ j, js,, IS IS IS IS IS IS 1 j, = δs1 j, S1, + δs2 j, S2, Maximize I ( IS IS ) (39) IE IE IE IE IE IE 1 js,, = δ j, js, j, js,, + δi, js, i, js,, s.. IS ( ISJ ISJ ) (40) and PI I = PIS IS + PIS, IS, 2, j, j, js, 1, js, 1, js, 2 j S and PIS IS = Pq ISJ + Pq ISJ js,, js,, js,, j, js,, is,, i, js,, (41) (42) In equaion (39), δ IS s is he given share of good s in aggregae invesmen, and IS is he elasiciy of subsiuion for invesmen goods. IS j,s, is he invesmen good s used in counry j a ime. In equaion (40), δ IE i,j,s is he given share of invesmen good s produced in counry i bu used in counry j. ISJ i,j,s, is he invesmen good s used in counry j bu produced in counry i a ime. Accordingly, ISJ j,j,s, is he locally-produced invesmen good s used in counry j a ime. IE represens he elasiciy of subsiuion for locally-produced and impored invesmen good s. In equaions (41) and (42), PI j, represens he aggregae price of all invesmen goods in counry j a ime and PIS j,s, is he price of invesmen good s in counry j a ime. On he second sep of opimizaion, an invesor s problem is o maximize equaion (39) subjec o equaion (41). The following equaions (43) and (44) are he firs order condiions: IS PI IS IS IS PI j, js,, = δ s I j, PIS js,, = IS IS IS 1 IS 1 j, δ s PIS js,, s (43) (44)

20 Afer he IS j,s, is deermined, on he hird sep of opimizaion, an invesor allocaes is invesmen expendiure on invesmen good s across counries. The opimizaion problem is o maximize equaion (40) subjec o equaions (42). The following equaions (45) and equaion (46) are he firs order condiions: ISJ PIS IE IE IE PI js,, i, j, s, = δi, j, s IS j, s, Pqis,, IE IE IE 1 IE 1 js,, δi,, js Pq is,, i = (45) (46) 2.5 Foreign rade wih he res of he world As explained in he above secions, we allocae demand in counry j for goods produced in counry i based on he Armingon assumpion. In oher words, even hough individual producers are microscopic price akers, good s are assumed o be differeniaed in demand by counry of origin. A he iniial seady sae where all counries are assumed o be perfecly symmeric, each counry impors and expors he same quaniy of S1 and S2. Trade exiss when counries are perfecly symmeric because each of hem demands boh locally-made and foreign-made goods. During he processes of populaion ageing, changes in relaive facor abundances bring changes o wo counries comparaive advanages. Accordingly, demands for he four goods change overime. The aggregae level of one counry s impor or expor of good s (denoed as IMP j,s, and EXP j,s, ) is deermined by he above opimizaion seps for agens (consumer, invesor and governmen), which can be shown by he following equaion (47) and equaion (48):

21 IMP = POP CSJ + ISJ for i j j,, s j, qual, g, i,, j qual,, s g, i,,, j s iqualg,, EXP = POP CSJ + ISJ for i j j,, s i, qual, g, j,, i qual,, s g, j,,, i s iqualg,, (47) (48) 2.6 Equilibrium condiions A general equilibrium soluion is one in which all economic behavior are consisen wih boh curren prices and fuure prices and all markes clear. In his model, we impos he following four equilibrium condiions for good marke, labour marke, capial marke and he marke for financial asse respecively: X = POP CSJ + ISJ + GovSJ gj KS js,, ig,, jisg,,,, jis,,, jis,,, ig, g, qual POP LS = Lsq = j, qual, gj, j, qual, gj, j, qual, s, s Kd j, js,, s POP LD = PI KS + PGov Bond j, qual, gm, + 1 j, qual, gm, + 1 j, j, + 1 j, j, + 1 (49) (50) (51) (52) In equaion (49), LS qual,g, is he uni labour supply provided by an individual a skill level qual in he working generaion gj in counry j. In equaion (50), LD j,qual,g, is he financial asse owned by he old generaions gm living in counry j a ime. i Homogenous in his paper means ha he echnologies employed in he producion of each good are idenical across counries.