Specialized Human Capital Investment, Growth and Convergence. Robert Tamura Clemson University. Michael Sadler Kansas State University

Transcription

1 Specialized Hman Capial Invemen, Growh and Convergence Rober Tamra Clemon Univeriy Michael Sadler Kana Sae Univeriy

2 Specialized Hman Capial Invemen, Growh and Convergence Model Inrodcion Over he pa decade, hman capial ha aken cener age a a deerminan of economic growh. Model of endogeno or exogeno growh emphaize hman capial accmlaion a an imporan engine of growh, if no i primary deerminan. Given he dramaic rie in chooling ha accompany economic growh, i i hard o dipe he emphai of hee model. However, here have been few aemp o model he perinen family-level deciion ha inflence children chooling. For example, he level and diribion of kill wihin a family i deermined by feriliy rae a well a by paren deciion on invemen in hman capial and which children hold receive hee invemen. All of hee deciion reflec rae of rern o vario aciviy. Moreover, hee rae of rern are inflenced by facor ha can only be died in a general eqilibrim eing. Thi paper conrc ch a model and aemp o hed ligh on imporan feare of he developmen proce. Thi paper preen a model economy wih overlapping generaion of familie whoe offpring receive eiher killed or nkilled hman capial. Feriliy deciion are endogeno, and paren m decide no only how many children o raie, b alo he proporion ha will receive each ype of hman capial. Child-rearing reqire parenal ime, which involve acrificing ime a work. Boh killed and nkilled paren are capable of raiing boh ype of children, b killed paren have a comparaive advanage in raiing killed children. Prodcion ilize inp of boh ype of hman capial. The parameer of he model are calibraed o mach cerain crcial feare of he world economy nder differen cenario, and he model i imlaed nder hee differen cenario. For cerain parameer combinaion, he rel of he model are broadly conien wih he hiorie of developed economie and he world economy a a whole. Uing hi framework, we derive predicion from he model economy relaed o he following empirical reglariie of he developmen proce:. Hman capial ( kill ) expand a per capia income rie. 2. The diperion in edcaional aainmen ( kill level ) decline.

3 3. Feriliy rae decline. 4. Life expecancy, and he average age of new labor marke enran boh increae. The fir reflec he kill deepening and broadening ha accompanie he modernizaion proce. A Becker (993) noe, I i clear ha all conrie which have managed perien growh in income have had large increae in he edcaion and raining of heir labor force. Fir, elemenary chool edcaion become niveral, hen high chool edcaion pread rapidly, and finally, children from middle-income and poorer familie begin going o college. (p. 24). Eaerlin (998) docmen he banial increae in primary enrollmen rae for a broad ample of conrie from he end of he 9 h cenry o he 990. Baier, Dwyer, and Tamra (999) ilize he daa in Michell (993) o docmen he increaing proporion of he male poplaion ha have been expoed o primary, econdary, and eriary edcaion in an even more comprehenive e of conrie. For he U.S., Denion aribe one-forh of he rie in per capia income from 929 o 992 o increae in he average level of chooling. The decline in he diperion of enrollmen rae i eviden from Table 0, which again ilize daa from Michell (993). In order o mmarize he daa, individal conrie are aggregaed ino coninen or region, alhogh he conien paern exhibied i a feare of all of he nderlying obervaion, a well. Alhogh in earlier year he ample are limied by daa availabiliy and herefore eimae for hee year end o be dominaed by a few developed economie, by 960, he ample i largely complee. A Table 0 make clear, commenrae wih he rie in average year of male chooling, each region how a marked decline in he diperion of expore o primary, econdary, and eriary edcaion, a meared by he coefficien of variaion. Thi illrae he fac ha economic growh and indrializaion involve pward kill mobiliy, and pecializaion in kill decline. Declining feriliy rae and he demographic raniion are a feare of every developmen experience. All developed economie pa hrogh a period of declining moraliy rae, followed by declining feriliy. According o Eaerlin (998), he feriliy decline ar in he 950 and 960 in mo developing economie, he excepion being b-saharan Africa, where feriliy rae remain high. While many heoreical model are The excepion i France, where he wo rae declined concrrenly. See Chenai (985).

4 able o generae a feriliy raniion, he model preened here allow o inveigae he relaionhip beween family ize and he edcaion opporniie of children. The lierare on he effec of ibhip ize on he allocaion of ime and reorce wihin he family indicae ha larger familie behave differenly han maller familie, and ha birh-order effec may be imporan, a well (ee, for example Behrman and Tabman (986), Linder (979) for he modern U.S.). Thee die repor wo major rel ha are efl in he aemen of or model: fir, when familie are larger, each child receive le parenal invemen, and econd, paren eem o rea heir children yemaically differenly. In addiion, Behrman, e al. (989) find ha chooling i more neqally diribed in larger familie in he U.S. And, in a dy of Philippine rice village by Qimbing (994), beer-endowed paren (in erm of eiher land or edcaion) rea children more eqiably han do poorer paren. Thee rel gge he need o inveigae a general eqilibrim model ha allow ch behavior a he hoehold level. Among oher hing, i allow inveigaion change in inergeneraional mobiliy from killed o nkilled a an economy develop. Finally, i i well known ha a economie develop, hey rely le on he labor of children. For example, according o ILO daa (qoed in Doepke (999)), in Korea in 960,.% of children aged zero o fifeen were economically acive, compared wih 4.3% in Brazil. A Korea economy developed and hman capial increaed, he e of child labor wa nearly eliminaed: in 985,.3% of children beween en and foreen were acive. A i well know, Brazil developmen experience ha no been o forio and in 990, 24.3% of en-o-foreen-year-old were ill economically acive. Overall... NEED BROADER STATS (ILO daa on i way.) Almo all radiional, non-raegic economic analye of feriliy and he effec of alriic behavior by paren oward heir children imply ha idenical children will be reaed eqally. 2 The economic environmen faced by paren inflence feriliy and child-rearing deciion, b freqenly, here i no incenive o rea children differenly (e.g., Becker, e al. (990)). Even when hi incenive exi, he model are rericed o impoe eqal reamen of offpring. Hiorically peaking, hi may no be an innoco rericion. Ample 2 Excepion inclde Mlligan (997) who die endogeno parenal alrim, and Behrman, Pollak, and Tabman (995).

5 evidence exi ha children have no been reaed eqally by paren in many ocieie. 3 I herefore eem naral o inveigae he implicaion of allowing for hi phenomenon in a model of growh and feriliy choice. Wiho ome or of imperfecion in he economic environmen, i eem clear ha paren wold chooe o rea all children idenically. Of core, differen abiliy level may encorage differen level of parenal ranfer in order o maximize prodciviy. However, oher feare of he economic environmen are eqally likely o deliver hi ocome. In or model, differenial parenal ranfer can emerge even hogh children are ex ane idenical. The pariclar feare of he model ha generae hi propery are a prodcion nonconvexiy in he form of increaing rern o cale in killed hman capial accmlaion and cloed financial marke (o ha paren canno borrow again he earning of fre generaion). In ch an environmen, when paren care abo he average iliy of heir children, i may no longer deirable o eqalize ranfer or oher invemen acro children, epecially a lower income. However, a income grow, ranfer and invemen may become more eqal. The rel of he model alo add omehing o he ongoing debae abo he relaive meri of modeling growh a de o endogeno or exogeno force. Since we e a CES echnology, endogeno growh i poible when he elaiciy of biion beween he wo facor of prodcion killed and nkilled labor i high. When hi i no he cae, he model will approach a aionary ae in he abence of exogeno echnical progre. In he imlaion, we inveigae boh cenario, afer calibraing he model o mach cerain feare of he world economy, boh pa and preen. In order o mach hee feare, i i neceary ha he model economy exhibi growh in per capia income, a propery ha wold no be poible wih a low elaiciy of biion. Th, nder hi cenario, we ame ha nkilled hman capial grow exogenoly. Since hi i he only way ha he model can be made o fi he hiorical daa nder he low elaiciy of biion ampion, he differen rel ha emerge beween he endogeno and exogeno growh cenario allow o make claim abo he reaonablene of hee mechanim for generaing growh. The nex ecion of he paper formally develop he model economy. In ecion 3, he model i calibraed 3 A i mo exreme, hi endency i refleced in he iniion of primogenire, where only he fir born on receive inheriance, or, more generally, nigenire, where inheriance i paed down o a ingle heir. Primogenire wa codified a law in many Eropean ocieie and heir colonie (ee Sadler (2000) for example), and many grea ancien civilizaion alo adoped he pracice among he nobiliy (ee Bergrom (995)). A hee economie grew and reorce expanded, eqiy in inheriance eem o have become he rle.

6 and imlaed and we compare rel from differen parameer combinaion. Secion 4 conclde. The Model Economy In hi ecion we preen he nderlying model of he paper. Iniially. here are wo ype of agen, hoe wih killed hman capial and hoe wih nkilled hman capial. The diincion beween killed hman capial and nkilled hman capial i qie imilar o ha conained in Becker, Mrphy and Tamra, hereafer BMT, (990). However nlike BMT (990), nkilled hman capial can work wih killed hman capial. In fac he wage per ni of nkilled hman capial i riing he in he level of killed hman capial available in he economy. Skilled and nkilled agen have idenical preference. Among killed agen here can be many ype, where a ype i given by he level of killed hman capial. Frhermore here are a coninm of agen. The Lebege meare of nkilled agen we refer o a he poplaion of he nkilled agen. For each ype among he killed hman capial poplaion here are a poiive meare of agen. 4 An agen live for poibly wo period, yong and old. A in Jone (999) and Tamra (999b), we inrodce moraliy ino he problem. If a yong peron live hrogh he fir period, hen he or he live hrogh heir old age wih probabiliy one. However here exi he poibiliy ha he or he will pa away before hey complee heir fir period of life. Thi deah occr immediaely afer he ime a paren pend rearing and edcaing he yoh. 5 While yong an individal receive hman capial invemen, if any, from hi or her paren. If he or he rvive hi or her yoh, when an individal i old, he or he chooe own conmpion, feriliy and he invemen in hman capial of hi or her children. A paren doe no care abo he hman capial level of hi or her children, b raher, only abo he income of hi or her children. Unlike previo work on endogeno feriliy and endogeno hman capial invemen, Tamra (996,997,999ab), an agen can 4 In oher word, if here were a ype of killed hman capial ha had zero Lebege meare, ha ype wold have no effec on prodcion, or conmpion or poplaion. Since hi ype ha no effec on any mearable economic qaniy, we ignore hem. 5 Deah can occr a wo poible age, if a child receive no hman capial invemen, i can occr immediaely afer he rearing ime. If a child receive hman capial invemen, hen he deah occr immediaely afer he edcaion.

7 pecialize hi or her hman capial invemen on a fracion of hi or her children. Le percrip refer o killed earning, and percrip refer o nkilled earning. We ame ha paren care abo he average nmber of adl rvivor hey have. Finally we ame log preference for racabiliy; ignoring individal bcrip for impliciy: α c + γ [ b d ] + β{ y + + ( ) y + } ln ln ln ln, () where ", $, ( > 0, 0 # #, and d i he average nmber of deah prior o reaching adlhood and b i he nmber of birh a paren ha. We foc on one pariclar eqilibrim. We ame ha all paren wih he ame level of kill chooe he ame acion. In pariclar we ame ha all idenical paren chooe he ame fracion of children o inve in killed hman capial. 6 Each individal ha he following bdge conrain, for he momen we ppre wheher earning are for killed or nkilled hman capial individal: [ ( )] c = y b θ + τ, (2) where 2 >0 i he fracion of ime each child ake o rear, i he fracion of children receiving killed hman capial invemen, and J i he eaching ime pen per child receiving killed hman capial. We ame ha if a child receive no killed hman capial invemen, he or he i endowed wih h ni of nkilled hman capial. Noice ha a paren m pend rearing ime and edcaion ime on all children, regardle of wheher hey rvive o adlhood. The killed hman capial accmlaion echnology ha wo branche. The fir branch i for paren wiho any killed hman capial. The econd branch i for paren wih killed hman capial. We ame ha he fncional form for each i imilar o Tamra (99,996): [I CHANGED LAMBDA TO KAPPA HERE] h h h + ( κ ) ( ) Ah h if nkilled = ϕ ϕ µ τ, Ah π π µ h τ if killed, = max{ }, 0 < n < B <, : > 0, > 6 > 0. The fir branch how ha an nkilled paren can (3) iliy. 6 An alernaive eqilibrim wold be one where all idenical hman capial individal have he ame

8 prodce killed hman capial. Since, n < B, relaive o a killed paren, an nkilled paren i le able o ake advanage of he exiing body of knowledge, conained in exiing ock of nkilled hman capial, hman capial, h h. Frhermore even he nkilled paren h, i le prodcive, 6 <, han an eqivalen amon of killed. Thi indicae ha he nkilled paren ha a comparaive advanage in prodcing nkilled children, and h, ceeri parib, he or he will have more children han a killed paren. Over ime if he body of knowledge in he poplaion rie, hen perhap all nkilled paren will have killed children. We ame ha nkilled hman capial grow exogenoly (if a all) over generaion a rae, F $. The final ep in eing p he model i o preen prodcion. There i a ingle conmpion good in he economy. I i prodced nder conan rern o cale in he diribion of hman capial. If all hman capial level, killed and nkilled, were mliplied by., oal op of he conmpion good wold alo be mliplied by.. There i diminihing rern in each individal ype of hman capial, ha will be elaboraed in more deail below. Ame ha he wo ype of hman capial, killed and nkilled are combined ing he following CES prodcion echnology: ρ { } ρ ( ) ρ ( ) ( ) Y = ε ε N h + ε K M K = m jh j ω j= ω > > ρ, ε > 0 ω, ρ, (4) where here are N nkilled individal; M ype of killed individal, here are m j killed individal of ype M m j j= j, each wih h ni of killed hman capial, and killed individal. 7 The echnology can be hogh of a a andard CES prodcion echnology combining labor wih a capial inp. The imporan difference are ha he capial inp come from he diribion of killed hman capial, and ha he capial inp demonrae increaing rern o cale in he nmber of killed agen, T>. 8 To ee hi, ame ha all killed agen are idenical wih h ni of killed hman capial, and he poplaion of killed agen i M, hen he capial inp i given by: j i m j. 7 Th he Lebege meare of nkilled agen i N and he ebege meare of killed agen of ype 8 The capia inp i a verion of Tamra (992).

9 K = ω M h Thi increaing rern in killed hman capial paricipaion will play an imporan role in deermining if nkilled paren have only killed children. Since each individal i a e of meare 0, each agen ha no conrol on hi or her wage per ni of hman capial. Each agen, killed or nkilled, earn an income ha i a prodc of he wage per ni of hman capial and he amon of hman capial hey have. Earning for an nkilled worker and a killed worker of ype i are: (5) y { } ( ) ρ ( ) ρ ( ) ( ) w = ε ε N h + ε K ε N h y i i i { } ρ ( ) ρ ( ) ( ) w = ε ε N h + ε K K h i h = h ineqilibrim i = w h, = w h, i ρ ρ ρ ρ ρ ρ ω ω i,, (6) Since all nkilled agen have he ame level of nkilled hman capial, all nkilled agen earn he ame income. Of he killed agen, he wage per ni of killed hman capial depend on he amon of killed hman capial an individal ha. Since each agen i a e of meare 0, no agen ha any conrol on he wage per ni of hman capial. We ame ha agen do no form coaliion of mearable ize o ry and affec he wage per ni hman capial of heir ype. Th we ame ha all agen ake all wage in he nex period o be independen of heir acion. Unkilled Problem In hi becion we analyze he nkilled agen problem. Wriing o he problem for he nkilled agen: ( ) = max { n,,τ } U h + [ ( + )] + [ ] [ ln + ln + ] ( ) ln + α ln y α ln b θ τ γ ln b d + β{ w + h + y } The fir order condiion for opimal choice of feriliy, b, killed hman capial invemen ime, J, and hare of children receiving killed hman capial invemen,, are: (7)

10 ( + τ ) ( θ τ ) α θ γ = b + b d αn βµ = b ( θ + τ ) τ αnτ = β b + ( θ τ ) { ln y + ln y + } (8) Noice ha he effec of expeced deah cae an increae in he marginal benefi of an addiional birh. I ha no effec on marginal benefi of hman capial invemen. However becae of he iming of yoh deah, i raie he co of hman capial invemen. Th he effec of yoh moraliy i o increae feriliy and decreae invemen. Wheher he redcion in invemen occr via lowering he hare of children receiving invemen, and or he lowering of he amon of invemen per child receiving invemen i nclear. The fir wo eqaion can be olved in erm of he hare of children receiving killed hman capial invemen,, and he expeced nmber of deah, d. The reling fncion are: 9 b τ Α Β C = ( α + γ )( θ + τ ) = Β + Β 4Α 2Α ( α βµ ) = + d ( d θ ) C 2 = γ + αd θ βµ + 2βµ d θ = βµθ γ (9) Sbiing in for n J ino he final eqaion in (8) and implifying prodce: [ ln y + ln y + ] µ = Th he income gap beween a killed child from an nkilled paren and hi or her nkilled ibling from he ame nkilled paren i a conan percenage, approximaely eqal o :. Below we will olve for he pecializaion rae,, nder perfec foreigh, b before we do hi we m examine he killed agen (0) (999b). 9 Excep for he addiion of invemen fracion hi i he ame rel a conained in Tamra

11 problem. Skilled Problem In hi ecion we olve he problem facing he ypical killed paren. The appendix how ha if he nkilled paren are inveing in a poiive fracion of heir children, hen all killed paren are inveing in all of heir children. Frhermore he appendix how he condiion neceary for all killed paren o inve in heir children, even when nkilled paren are prodcing no killed children. Sppoe he condiion in he appendix hold o ha all killed paren inve in all of heir children. The problem facing a killed paren become: { i [ i i ] [ i ] } α α θ τ γ β i ( ) ( ) { } U h = max ln y + ln b + + ln b d + ln y + i n,τ () The fir order condiion deermining he opimal feriliy and hman capial invemen ime for a killed paren are: ( + τ i ) ( θ τ ) α θ γ = b + b d ατ i βµ = b + τ i i i ( θ τ ) i i i (2) Oberve ha he expeced nmber of deah facing a killed paren can differ from he expeced nmber of deah facing an nkilled paren. 0 A before, he effec of yoh moraliy i o raie he nmber of birh and lower he amon of invemen per child. Since we are focing on he cae where all killed paren inve in all of heir children, we can nambigoly predic ha moraliy will lower he amon of hman capial invemen per child. Solving for feriliy and hman capial invemen ime reveal: 0 Urban worker had differenial moraliy compared o rral worker hrogho mch of hman hiory, ee Diamond (999).

12 b τ i Α Β i C = ( α + γ )( θ + τ ) = Β + Β 4Α 2Α C ( α βµ ) ( 2 ) ( d θ ) = d + γ 2 = γ βµ + α + βµ d θ = βµθ i αd + α + γ (3) Since each killed paren face he ame expeced nmber of deah, oberve ha feriliy and hman capial invemen ime i independen of he level of kill an individal ha. Frhermore noice ha if an nkilled paren inve in all hi of her children in (9), hen he or he will have idenical feriliy and idenical invemen per child a a killed paren only if he moraliy i he ame. A a coneqence of (3) and (9), i i clear ha nkilled paren will have higher feriliy han killed paren. Frhermore nkilled paren will inve le per child han a killed paren inve per child. Finally ince all killed paren inve he ame amon of ime per child, all killed individal converge o he ame hman capial level, a in Tamra (99). Th he rnpike heorem hold in hi model. The raio of killed hman capial of children of killed hman capial paren i and j i given by: h h i + j + ϕ ( h ) µ Ah i τ i h = = Ah ϕ ϕ µ τ h ( h j ) ϕ If all nkilled hman capial individal evenally prodce only killed hman capial individal, hen all individal will become idenical in he long rn. The long rn behavior of he model i given by he olion o (3) when d converge o 0. Solving for hi cae prodce he balanced pah feriliy and hman capial invemen for killed paren: j i j ϕ (4) b τ = = γ βµ ( + γ ) θ α βµθ γ βµ (5) I i qie poible for heir o exi branche of dynaie ha remain nkilled forever. In pariclar, given he prodcion fncion, if D # 0, hen boh killed and nkilled individal are eenial facor of

13 prodcion. If boh ype are eenial hen here cold be a aionary ae of conan flow of new killed worker, i.e., ome children of nkilled paren will forever become killed, while all killed paren prodce killed worker. We preen boh phenomena below in he nmerical olion. The Appendix preen he fficien condiion for all killed paren o raie only killed children. Frhermore Appendix B preen he fficien condiion for endogeno growh. A in Barro and Sala-i- Marin (995), endogeno growh i only poible if D > 0. However D > 0 i a neceary condiion, b i i no fficien. Endogeno growh reqire ha he killed capial aggregae m grow a a fficienly fa rae o pll he nkilled worker ino he killed poplaion. Eenially he killed capial aggregae m grow faer han he poplaion growh rae of nkilled worker. In he erminology of he neoclaical growh model, capial accmlaion m be more rapid han he poplaion growh rae, i.e., capial deepening m occr. Nmerical Solion In hi ecion we deail he nmerical olion of he model for vario parameer vale. We examine everal cae. In pariclar we choe hree differen vale of D, D = -2.5, -.25,.525. A we varied D, we varied he rae of exogeno echnological progre for nkilled worker. We did hi in order o mainain a conan average growh rae of income per capia. The parameer of he imlaion were adjed o mach world poplaion in he year 800 of arond billion and in he year 2000 of 6 billion, and per capia income of killed worker a roghly $30,000. Cerain parameer are held conan in he for imlaion. The vale of he hee parameer are " =.395, $ = 5., ( =.4775, 2 =.25,: =.05, A = 2, and B =.5. The vale of T, F, n, and 6 are adjed in order o mainain he coniency of he daa wih hiorical realiy nder each cenario. The vale of hee parameer are lied a he op of each panel of Figre. The nmber of ae variable in hi problem varie over ime. The economy i characerized by he hman capial diribion. I i neceary o know he level of nkilled hman capial a ime, h, and he poplaion of nkilled individal, N and he diribion of killed hman capial a ime, { h j, m j } M j= where h j i he level of killed hman capial ha all individal in grop j ha, and m j i he

14 poplaion killed grop j, and M i he nmber of grop of killed hman capial a ime. Informaion on all of hee variable compleely characerize he economy a ime. The eqilibrim we conidered wa one in which killed paren only raied killed children. Alhogh clearly an approximaion, here i ample empricial jificaion for hi ampion. Mobiliy die verify ha pward mobiliy i mch more likely ha downward mobiliy in indrialized and indrializing conrie. When combined wih he obervaion ha he proce of indrializaion enail pgrading he kill level of all occpaion, we reach he conclion ha downward mobiliy i a rare occrrence, a conclion we hold noe ha i conien wih or view ha he developmen proce can be reaonable characerized a a proce of overcoming nonconvexiie in prodcion opporniie while faced wih imperfec (or cloed) financial marke. Under hi eqilibrim, he acion of he killed paren are imple o characerize. Their feriliy and hman capial invemen ime are idenical, no maer he level of kill any killed paren ha. Thee policy fncion are given by (3). Having olved for he policy fncion for all killed paren, he only hing lef o olve are he policy fncion of he nkilled paren. Recall ha all nkilled paren in period have h ni of nkilled hman capial. The growh rae of hee ni i amed exogeno and eqal o 8 $. The policy fncion for feriliy and hman capial invemen ime, a fncion of he hare of children receiving invemen, are given by (9). Therefore he only nmerical problem o be olved i he hare of children from nkilled paren ha receive invemen. The eqaion deermining hi i given by (0). Uing he rel from (3) and (9) and biing ino (0) yield he following implici fncion for : µ = ( ) ω M + ρ ε = ln K m j h j Ah h j ρ ε ( N ( )( b d )) σh ω ( ) ( σ ) ( τ ) ϕ ϕ µ ω (6) See, for example, he cro-conry dy of occpaional mobiliy by Erikon and Goldhorpe (992). Evidence on earning and occpaion mobiliy ha ppor or ampion can be fond in Zimmerman (992) for he U.S. and Chechi, Ichino, and Richini (999) for Ialy and he U.S.

15 Le ( b,τ ) be he opimal choice of feriliy and invemen ime of he killed paren, given by (3), and ( b,τ ) be he choice of feriliy and hman capial invemen of nkilled paren, a a fncion of he hare of children receiving invemen (given by (9)), hen [( ) ] [ ( ) ( ) ] ( )( ) ε ϕ ϕ µ ρ µ = σ τ σ ε + ln ln ω ln Ah h N b d h M ( ωρ ) ( ) [ ] ( τ ) ln mj b d Ah hj + N ( b d )( Ah ( σh ) ( τ ) ) + j= π π µ ω ϕ ϕ µ ω (7) can be olved nmerically o deermine he opimal hare of nkilled children receiving invemen. Once i deermined i i a imple maer o e (9) o olve for he choice of feriliy and invemen, ( b,τ ) and ( b,τ ). When hee policy fncion are deermined i i a imple maer o pdae he ae variable in he economy, h + ( )( ) N = N b d M π π µ { h j +, m j + } = Ah ( h j ) ( τ ), m j ( b d ) j= ϕ ϕ h j Ah ( h j ) ( µ + + = τ ) ( ) m = N b d M + j+ + + = σh = M M { } if = 0 + if > 0 M j= (8) The fir line of (8) pdae he nkilled hman capial level. If here i exogeno echnological progre,

16 hen he nkilled hman capial level rie. The econd line of (8) pdae he nmber of nkilled individal in he poplaion. The nkilled poplaion rie or fall if he gro growh rae of poplaion, n rie enogh o offe hoe ha become killed. The hird line of (8) i he law of moion of all killed children from killed paren. Since all killed paren chooe he ame feriliy and he ame amon of ime o inve in heir children, he law of moion for hi grop i qie imple. Oberve ha he exience of he hman capial pillover in he accmlaion echnology indce hman capial convergence. The forh line of (8) give he hman capial of he fir generaion of killed children from nkilled paren. The nmber of new killed worker i given by he fifh line of (8). Finally he evolion of he nmber of killed hman capial ype i given by he la line of (8). The for panel of Table conain he rel from hee olion. Each panel cover one of he for differen parameer e. The fir colmn give he imlaed year. The econd colmn li he average income in he world. The hird colmn conain he average income of he killed poplaion, where we average over all individal who received killed hman capial from heir paren. The forh colmn conain he average income of he nkilled poplaion. The final hree colmn are he poplaion of killed individal, he poplaion of nkilled individal and world poplaion. The for differen panel of Table prodce markedly differen rel. In he fir panel, D =.55, income growh occr for boh ype of individal. Income grow for he killed becae poplaion i riing and becae killed hman capial i riing. For nkilled individal, heir income rie becae of he riing capial conribion in he economy provided by he killed worker. Noice ha he poplaion of nkilled worker peak in 960 wih a poplaion of 3580 million. By 2000 he nkilled poplaion i redced o 250 million, and a minoriy of he world 6030 million poplaion. World poplaion growh low dramaically a he world become more and more kill dominaed. The dip in per capia income of he killed ha occr in 2000 arie becae of he iniial creaion of killed individal by nkilled paren. The old killed poplaion in 960 i 48 million. In 2000 he old killed poplaion i 5 million original killed decendan and 820 million newly killed. They are le killed han heir killed conerpar; he original killed worker have.2 ni of killed capial and he newly killed have only ni of killed capial. However by he nex generaion he kill level of heir decendan are 2. and.046 and he kill level of heir grandchildren are 3.95 and.59. So from.05 percen of income, he nex generaion rie o 2.2 percen and

17 he grandchildren are a 4.9 percen. The grea-grandchildren have 7.4 and 2.9 ni of killed capial and he relaive income i 39 percen. Thi parameerizaion prodce compee exod of nkilled worker ino he killed worker caegory. When hi occr he model become like Tamra (999ab). Long rn balanced growh per capia income growh i given by: y + y βµθ = A γ βµ µ ω γ βµ θ α [ + γ ] Thi i he long rn growh rae in hi economy, b hi rae will no be reached nil he year Oberve ha he world poplaion growh low remendoly afer By 2200, he world poplaion grew by only 2.6 percen over he previo 40 year. Th while poplaion growh i poiive, he model indicae ha world poplaion growh rae never rern o he level aained over he 800 o 2000 period. Eenially he enire world ener ino a demographic raniion, moving from nkilled o killed. Since he enire world become killed, i i obvio ha he hare of he economic pie prodced by he killed poplaion i 00 percen. The nex panel, D =.25 illrae a cenario wih conino rapid world poplaion growh. Poplaion grow a a.8 percen annal rae from 260 o 2200, compared wih he.07 percen annal rae in he previo olion. Noice ha he annal world poplaion growh i above he rae of poplaion growh from 800 o 2000 of.84 percen. [REWORD] Th he world never ener ino a demographic raniion and he poplaion of nkilled worker rie perpeally. In he long rn he model prodce he inereing rel ha half he world poplaion i killed and he oher half remain nkilled. Thi aionary diribion i inereing becae i reqire ha in every generaion each nkilled paren inve in a conan proporion of hi or her children. Since poplaion growh i faer in hi model, average per capia income growh i lower han per capia income growh of he killed worker. Th he relaive earning of killed worker rie compared o nkilled worker. Unkilled worker income growh i compleely driven by he riing level of capial inp provided by he killed worker. The hare of world op prodced by killed worker rie from abo 36 percen o abo 00 percen by he end of he olion in year Thi i obvio ince hey are half he world poplaion, b heir average income grow a a faer rae han he (9)

18 poor. In boh Table a and Table b, he olion are able o mach he world poplaion in 800 and 2000 a well a he average income in he world in However hey have differen predicion concerning he long rn poplaion growh rae in he world. By year 7800, he end of he olion, D =.55 prodce an annalized poplaion growh rae of.05 percen. For D =.25, he annalized poplaion growh rae i.96 percen. The final wo panel of Table provide he rel when D < 0. In each of hee cae he vale of D implie ha boh facor of prodcion are eenial. Hence i i no poible for all nkilled worker o prodce only killed progeny. However he cae prodce wo differen poplaion cenario. For D = -.5, he world iniially engage in a demographic raniion, and almo all children of nkilled paren become killed. Inereingly he world in 800 i abo 50 percen nkilled. In 840, only 3.7 percen of he poplaion i nkilled. However hi nderhoo he long erm fracion of he world poplaion ha will remain nkilled, which aympoe o 00 percen! Th he model predic ha he Malhian porion of he world, where Malhian implie low kill and high poplaion growh, become he enire world. Table c ha a differen predicion in erm of income. Per capia income growh occr for boh he killed and he nkilled. However, no rpriingly, he growh rae of income of he killed exceed he growh rae of he nkilled. Unkilled worker income rie becae he capial componen provided by he killed worker i riing and becae here i exogeno growh among he nkilled worker. The hare of he world op prodced by he killed fall from 92 percen in 800 o 0 by he end of he olion in year Table d preen he rel for D = -.7. Noice ha world poplaion i growing wiho bond a in he cae for D =.25. Here however world poplaion grow a a faer rae han for D =.25, becae he nkilled hare of he poplaion i riing and no conan a half. In fac he nkilled poplaion eenially become he enire world poplaion. The hare of he world op prodced by he killed poplaion rie iniially from 54 percen in 800 o 65 percen in 920. Afer 2400 he hare of op prodced by he killed poplaion fall below 80 percen and rend downward forever. There are occaional blip pward when nkilled paren prodce killed children. The poplaion hiorie prodced by D = -.5 and D = -.7 vary grealy from he cae D =.55. For D =

19 -.5 predic poplaion growh a remendo rae. From a vale of 630 million in 2000, poplaion reache 8000 million by Poplaion growh average almo 2 percen annally a he end of he olion in year Wih D = -.7, he long rn poplaion growh rae i imilar o ha obained nder D = -.5. By he end of he ample, poplaion growh average almo 2 percen annally. Table 2 prodce he ime erie for he relaive imporance of he killed poplaion boh in erm of relaive economic prodcion and relaive poplaion. The killed hare of he world poplaion aain hree poible vale for hee 4 cae. In he fir cae, D =.55, he model how ha all nkilled evenally chooe o raie only killed children by year For D =.25, he model prodce omehing like a aionary killed hare of he world poplaion. Thi aionary vale appear o be 50 percen. For boh negaive vale of D, he killed hare of he world poplaion goe o 0 by he end of he olion. For D > 0, he killed hare of GDP goe o 00 percen by he end of he olion. For D =.55, 2600 he enire poplaion i killed. For D < 0, he killed hare of GDP goe o 0 by he end of he ample (acally o 2.2 percen in he cae of D = -.7). Table 3 provide more informaion from hee olion. The for panel conain informaion concerning he average year of chooling in he poplaion, he average oal feriliy rae in he world, a well a he oal feriliy rae for killed and nkilled paren, and life expecancy of killed, nkilled and he world a a whole. Recall ha each period i 40 year, o ha he maximm life expecaion i 80. To calclae life expecaion for he progeny of paren, we e he following vale for killed paren and nkilled paren: lifeexp ecaion = lifeexp ecaion = ( ) + ( + ) 80 b d 40 θ τ d ( ) + ( + ) + ( ) b if killed 80 b d 40 θ τ d 40θd b if nkilled There are inereing difference beween he for e of olion. In he cae where D < 0, he long rn aionary world oal feriliy rae, TFR, i arond 4.33 children. Thi accon for he rapid poplaion growh. While life expecancy hi i heoreical maximm in all for of he cae, when D < 0, age a enry ino he labor force, eenially edcaion + 6 or 40(2 + J), i 0. Th by he meare of primary chooling, he ypical worker ha only 4 year of chooling. In conra, for D > 0, he world TFR fall well below 4.33, a 2.04 for D =.55 and 3.09 for D =.25. In boh of hee cae he world TFR i an average of he oal

20 feriliy rae of each ype. 2 Life expecancy rie o i heoreical maximm, b he econd difference i in he age a enry ino he labor force. Oberve ha for D > 0, he age a enry i ignificanly above 0. I rie o 2.5 for D =.55 and 5 for D =.25. Th in he cae of D =.55, he average individal i a college gradae, wherea for D =.25, he average individal complee primary chool or abo 9 year of chooling. Thi difference in oal feriliy rae and he age a enry ino he labor force appear o be he mo efl way in which o compare hee olion wih he world hiory. Table 4 conain he informaion from Baier, Dwyer and Tamra (999). The daa how he fracion of he world labor force ha ha had no edcaion, he fracion of worker wih ome expore o primary edcaion (b no econdary chooling), he fracion of worker wih ome expore o econdary edcaion (b no higher edcaion) and he fracion of worker wih ome expore o higher edcaion. The nderlying daa come from B. R. Michell (993). While he daa i no compleely mooh, hi i moly de o he incorporaion of more conrie over ime. The ample of conrie i complee by 960. Over ha horer period, here ha been a monoonic decline in he fracion of worker wih no edcaion a all. There ha been a very ligh decline in he fracion of worker wih only expore o ome primary edcaion, from 42 percen o 40 percen. There ha been pracically a dobling of he worker ype wih ome econdary chooling, from 8.6 percen o 36.2 percen. Finally here ha been an exploion in he hare of worker wih ome higher edcaion expore, 2.2 percen o 4 percen. The calclaed average age a finihing edcaion if one calclae he average year of chooling pl 6 o hi ample, i given in he final colmn of Table 4. We aigned 0 year of chooling o hoe wih no edcaion expore, 3 year of chooling for hoe wih ome primary chooling, 0 year of chooling for hoe wih ome econdary chooling and 5 year for hoe wih ome higher edcaion. Since he 850 daa only conain he Unied Sae and he Unied Kingdom, and hee conrie lead he way in niveral edcaion, ee Goldin (9xx), i i clear ha he average age of enry ino he labor force ha rien by a lea 00 percen over he pa 50 year. Thi gge ha a model of riing nmber of nkilled worker raiing killed children beer fi he daa. Figre In hi ecion we preen ome rel of he nmerical olion in graphic form. We preen he hare 2 Where in he cae of D =.55, here i only one ype, killed worker.

21 of he poplaion ha are killed for he for parameer e. We alo preen he hare of GDP ha i prodced by he killed poplaion. Alo inclded are graph conaining log of average per capia income, log of average killed per capia income, log of nkilled income. Figre how he evolion of he hare of oal world poplaion ha i killed nder he for differen cenario. Thi i he ame inormaion ha wa mmarized for fory-year inerval in Table 2. When D =.55, hi hare iniially decreae, b hen rie rapidly, he economy converging o a iiaion where all worker are killed. Recall ha nder hi cenario, nkilled hman capial i no growing over ime. When combined wih he fac ha nkilled labor i non-eenial o prodcion, we arrive a he conclion ha one day in he fre, all worker will be killed. Under he cenario D =.25, however, he killed hare of he world poplaion approache a vale near.5. Under he oher wo cenerio, he killed hare of world poplaion approache zero. When D = -.5, hi variable fir increae, hen decreae monoonically. When D = -.70, here i ome flcaion dring cerain ime period, b a monoonic decline for approximaely 500 year. A can been een in Figre 6, he period of flcaion and monooniciy in he graph in Figre coincide wih flcaion in he rae of raniion from killed o nkilled (e.g., he fracion of children of nkilled paren who receive killed hman capial invemen. Th, hee are period where labor-force flcaion brogh on by parenal deciion cae flcaion in oher per-capia magnie. Figre 2 plo he killed hare of world GDP, exending he informaion in Table 3. Here, he diincion beween he cae of D > 0 and D < 0 are riking. In he former cae, he hare of op prodced by killed worker monoonically approache one, alhogh a a lighly lower rae wih he lower vale of D. When D < 0, on he oher hand, killed worker evenally accon for all of he op prodced. When D =.05, he decline in he killed hare of op i monoonic, wherea when D = -.70, hi hare increae for wo generaion, fall monoonically for almo 3000 year, hen ocillae a i converge o zero. A we will noice below, hee ocillaion are de o hoe in he hare of nkilled children who receive killed hman capial. Figre 3 plo he naral log of average per capia income. The anomolo rel ha log per capia income are higher when D =.25 compared wih log per capia income when D =.55 arie becae he poplaion nder he fir cae i mch larger han he poplaion in he econd cae. The average level of killed hman capial i lower in he former cae, b hi i compenaed for by a more rapidly growing killed

22 poplaion. In boh of hee cae, growh i prely endogeno, and in he limi, he growh of nkilled and average earning i de o growh in he ock of killed hman capial. When D < 0, in cona, growh i de o a combinaion of hi effec and exogeno growh in nkilled hman capial. In fac, wiho he laer, he economy wold approach a aionary ae. We can ee from he figre ha he vale of he parameer ha were choen in order o mach cerain feare of he world economy a cerain dae imply ha he cae wih endogeno growh rel in higher per capia income, if only from 2040 onward when D =.25 (ee Table ). Figre 4 how he naral log of average killed per capia income. Since he op hare of killed worker approache niy o rapidly when D > 0, i hold come a no rprie ha he erie for hee cae hold mimic he erie for overall average income (Figre 3). When D < 0, he erie evenally ocillae a hey increae. Thee ocillaion do no appear in Figre 3 becae he killed hare of op i negligable (Figre ). Figre 4 plo hi erie for nkilled worker. Noe ha when D =.55, he erie diappear arond Thi i de o he fac ha here are no more killed worker in he economy, heir meare having converged o zero. Here, oo, nkilled worker eem o make o beer when D > 0 han when D < 0 and nkilled hman capial grow exogenoly. Figre 6 how he fracion of children from nkilled paren who receive killed hman capial invemen. In effec, i i a child probabiliy of pward mobiliy. When D > 0, hi probabliy remain poiive, converging o one when D =.55 and eemingly converging o approximaely.23 when D =.25. When D < 0, he probabliy converge o zero. The cae of D =.55 i perhap mo inereing. Here, in he iniial generaion, here i no mobiliy. Then, he vale jmp o almo.75 wihin wo generaion, decline eadily and converge o approximaely.55. Then, i jmp diconinoly o one and remain here for he draion. In hi cae, he indrial revolion i alo a mobiliy revolion, and a he economy mare, mobiliy level off. The preence of he econd mobiliy revolion prediced by he model i inriging. Once again, he erie ocillae for cerain ime period when D = Thee ocillaion are de o flcaion in he rern o killed hman capial, a can be een from Figre 4. Figre 7 how a meare of ineqaliy of income, he coefficien of variaion of per capia income. In wo of he cae, a Kzne-crve paern emerge, wih incqaliy iniially increaing, hen converging o zero. Thi occr boh when D =.55 and when D = When D = -.70, a imilar paern emerge, albei wih

23 higher ineqaliy embroidered wih he flcaion ha have become familiar for hi cae by now. For he cae wih D < 0, ineqaliy converge o zero becae all he world worker are evenally nkilled (ee Figre), and nkilled agen hare eqal earning. When D > 0, killed income ha a endency o converge, (ee eqaion (8)). The effec of hi endency are mo apparen when D =.55. In hi cae, we have already een ha he hare of he poplaion ha i killed rapidly approache one. Since he enire poplaion i killed and killed income converge, ineqaliy approache zero. When D =.25, here conine o be mobiliy from killed o nkilled. Even hogh killed earning converge, he differen kill level and earning of hee new enran ino he killed labor force keep imply perien (and even increaing) income ineqaliy. Figre 8 plo he coefficien of variaion of killed per capia income. Again, ince in he limi, he enire poplaion i killed when D > 0, hee erie merely mimic hoe in Figre 7. Wih D < 0, ineqaliy in killed per capia income increae and evenally begin o flcae, hee flcaion become progreively more evere when D = -.50.

24 Table 0: Average Year of Schooling and Diperion of Schooling Expore By Coninen Erope Norh America Soh Aia & Norh Africa & Africa & Caribbean America Oceana Middle Ea (exc Norh) H CV H CV H CV H CV H CV H CV H average year of chooling CV coefficien of variaion in male expore o primary, econdary, and eriary edcaion. Sorce: Baier, Dwyer, and Tamra (999) and

25 Table a: Average Income, Average Skilled Income and Unkilled Income (in 990 dollar) Skilled, Unkilled and Toal Poplaion (in million) D =.55, T = , F =, n =.35, 6 =.25e-4 Year Average Income Average Skilled Income Unkilled Income Skilled Poplaion Unkilled Poplaion Toal Poplaion e e e+08.24e Year Average Income Table b: Average Skilled and Unkilled Earning and Poplaion D =.25, T = , F =, n =.35, 6 = 5.85e-0 Avg Skilled Income Unkilled Income Skilled Poplaion Unkilled Poplaion Toal Poplaion

26 Year Table c: Average Income, Average Skilled Income and Unkilled Income (in 990 dollar) Skilled, Unkilled and Toal Poplaion (in million) D = -.5, T =.9, F = , n =.59, 6 = 3.5e-6 Average Income Avg Skilled Income Unkilled Income Skilled Poplaion Unkilled Poplaion Toal Poplaion Year Table d: Average Income, Average Skilled Income and Unkilled Income (in 990 dollar) Skilled, Unkilled and Toal Poplaion (in million) D = -.7, T =., F = , n =.25, 6 = 2.5e-3 Average Income Avg Skilled Income Unkilled Income Skilled Poplaion Unkilled Poplaion Toal Poplaion

27 Year Table 2: Relaive Imporance of Skilled and Unkilled D =.55 D =.25 D = -.50 D = -.7 hare of pop. hare of GDP hare of pop. hare of GDP hare of pop. hare of GDP hare of pop hare of GDP

28 Year Table 3a: Average Edcaion, Life Expecancy and Toal Feriliy Rae D =.55, T = , F =, n =.35, 6 =.25e-4 edcaion + 6 killed life exp. nkilled life exp. life exp. killed TFR nkilled TFR average TFR

29 Year Table 3b: Average Edcaion, Life Expecancy and Toal Feriliy Rae D =.25, T = , F =, n =.35, 6 = 5.85e-0 edcaion + 6 killed life exp. nkilled life exp. life exp. killed TFR nkilled TFR average TFR

30 Year Table 3c: Average Edcaion, Life Expecancy and Toal Feriliy Rae D = -.5, T =.9, F = , n =.59, 6 = 3.5e-6 edcaion + 6 killed life exp. nkilled life exp. life exp. killed TFR nkilled TFR average TFR

31 Year Table 3d: Average Edcaion, Life Expecancy and Toal Feriliy Rae D = -.7, T =., F = , n =.25, 6 = 2.5e-3 edcaion + 6 killed life exp. nkilled life exp. life exp. killed TFR nkilled TFR average TFR