Technological Change, Population Dynamics, and Natural Resource Depletion

Transcription

1 Technological Change, Poplaion Dynamic, and Naral Reorce Depleion Andrea Schäfer Univeriy of eipzig Inie of Theoreical Economic / Macroeconomic Grimmaiche Srae eipzig, Germany chaefer@wifa.ni-leipzig.de Preliminary Verion Thi Verion: Ocober 2010 Abrac How doe echnological change affec he depleion of non-renewable naral reorce? We arge ha o anwer hi qeion i i neceary o accon for he ineracion beween kill-biaed echnological change, poplaion dynamic and heir effec on naral reorce e. Skill-biaed echnolgical change lead in a non-malhian world o a decline in he long-rn growh rae of he poplaion. Frhermore, kill-biaed echnological change i aociaed wih a lower long-rn depleion rae of naral reorce, wherea he raniion o a long-rn eqilibrim i characerized by an invere relaionhip beween he growh rae of he poplaion and he depleion rae of naral reorce. In he conex of a hrinking or aionary poplaion, he emergence of ained economic growh depend on he exience of poiive ineremporal knowledge pillover wih repec o reearch and developmen. Keyword: OG-Model, Endogeno Feriliy, Direced Technological Change, Nonrenewable Naral Reorce JE: J13, O40, O41, Q32

2 1 Inrodcion How doe echnological change affec he depleion of non-renewable naral reorce? We arge ha o anwer hi qeion i i neceary o accon for he ineracion beween kill-biaed echnological change, poplaion dynamic and heir effec on naral reorce e. Recen hiory in he developed world ha been characerized by an increae in he pply of killed labor relaive o nkilled labor indcing boh kill-biaed echnological change and - among oher facor - a decline in poplaion growh rae. Addiionally hi proce coincide wih an increaing demand for naral reorce. In recen year everal economi clarified he economic link beween poplaion dynamic, edcaion and economic growh, where a formal analyi of he non-malhian ineracion beween poplaion, echnology and reorce depleion wihin he conex of dynamic general eqilibrim model i ill miing in he lierare. Or reearch gge ha he non-malhian ineracion beween poplaion and echnology in erm of kill biaed echnological change indce a decline in poplaion growh and a decline in he depleion of naral reorce in he long-rn. Dring he raniion o a long-rn eqilibrim, poplaion growh and naral reorce depleion are inverely relaed. One of he major ylized fac ha characerized he developmen proce of indrialized conrie i a decline in feriliy rae. In developed economie, he raniion from rapid poplaion growh o low ne feriliy rae began a he ar of he econd phae of indrializaion in he nineeenh cenry. Birh rae declined faer han moraliy rae, yielding a banial redcion in ne poplaion growh, indcing he o-called demographic raniion Galor, 2005). Dring he la cople of decade, ne feriliy rae have reached excepionally low level, and have fallen hor of he replacemen hrehold even in conrie ha have radiionally exhibied qie high feriliy rae - e.g. Spain and Ialy. In le developed conrie, he feriliy raniion ared in he mid-1960, and i wa pariclarly rapid in Ea Aia. 1 Since we do no oberve a demographic raniion wiho economic developmen or vice vera, i eem ha demographic raniion can be conidered o be an inheren facor of economic developmen. Several economic channel ha are reponible for he oberved feriliy decline have been iolaed. The mo imporan one wih repec o he ineracion beween echnology and poplaion are propoed by Galor and Weil 2000,1996) aing ha: a) declining feriliy rae may be de o echnological progre ha, via i impac on he demand for hman capial, revere he relaionhip beween income and poplaion growh wih repec o he regime of Malhian agnaion and b) increaing real wage raie he opporniy co of having children, where lower feriliy generae poiive feedback effec on economic growh by mean of capial accmlaion. 2 In he pa ixy year, he relaive pply of killed labor ha increaed harply in he U.S. a well 1 For a comprehenive overview over apec of he demographic raniion, ee ee 2003). For he emergence of lowe-low feriliy rae in Erope, ee Kohler e al. 2002). 2 For a more deailed dicion of oher channel ee for example Schäfer and Valene 2010). 2

3 a in oher indrialized conrie. Moreover, and conrary o he predicion of a neoclaical framework wih concave prodcion echnologie, here ha been a harp increae in he killed wage premim ince he The andard explanaion for hi paern i an acceleraion in he kill bia of capial-embodied echnological change Aor e al., 1998 and Hornein e al., 2005). A he ame ime, increaing demand for hman capial and increaing wage can be een a being reponible for he decline in average feriliy rae, ince beer edcaed familie end o have fewer children and provide more edcaion per child, de o a rade off beween he nmber of children paren wih o raie and he amon of reorce hey pend on edcion per child ee for example de la Croix and Doepke 2003). Given a high inergeneraional perience, feriliy deciion and invemen in edcaion per child are ranferred from one generaion o anoher and inerac wih macroeconomic aggregae. 3 Figre 1 abo here Since he world-wide oil crii of he mid 1970 a he very lae, weern ocieie have been increaingly concerned abo he ainable developmen of heir economie a large, and we oberve a rong poiive relaionhip beween income per capia and demand for naral reorce. In 2007, OECD perolem conmpion amoned o 57% of he world perolem conmpion U.S. Energy Informaion Adminiraion, 2008) and per capia energy e differ beween he riche and he poore grop of conrie by a facor en Weil, 2005). In addiion he world depleion rae of crde oil i increaing, ee Figre 1, where OECD perolem conmpion increaed by a facor greaer han wo beween 1960 and 2005 U.S. Energy Informaion Adminiraion, 2006). From he oe of economic heory, carciy of naral reorce and poplaion dynamic have been a i core. Wherea Malh and Ricardo primarily had land in mind a caing diminihing rern o oher poibly reprodcible) inp, he concern of he Clb of Rome and he repor imi o Growh by Meadow deal wih non-renewable reorce a eenial inp for prodcion, while neglecing any ineracion beween carciy of naral reorce, relaive price and echnology. A regard he long-rn developmen of he world, heir predicion have proven wore han Malhian agnaion, ince hey arrived a he conclion ha achieving even a age of a lea conan per capia income wold no be poible. In hi conex, Solow 1974), Sigliz 1974), and Dagpa and Heal 1974) provided he fondaion for he formal analye of non-renewable reorce wihin he conex of model of endogeno growh ha emerged in he la wo decade. 4 Thi paper inegrae he feare of kill-biaed echnological change, feriliy decline and naral reorce e ino a comprehenive framework. More in deail, we conider an overlapping 3 For empirical and heoreical evidence, ee Kremer and Chen 2000); Roenzweig and Wolpin 1980); de la Croix and Doepke 2003,2004). 4 For a comprehenive overview on he heory of endogeno economic growh and he role of non-renewable naral reorce ee Groh 2007) and Piel 2002). 3

4 generaion OG) economy poplaed by killed and nkilled hoehold. The edcaional choice of paren for heir offpring generae differenial feriliy beween killed and nkilled hoehold, in he piri of Galor and Monford 2006), and de la Croix and Doepke 2003). The prodcion ide of he model i characerized by a cale invarian wo-ecor growh model wih mliple prodcion age. Moreover, he direcion of echnological change i deermined by he relaive pply of killed and nkilled labor Acemogl 1998, 2002) reling from he edcaional choice of paren for heir offpring. In addiion, prodcion i bjec o a non-renewable naral reorce a an eenial inp. 5 The foc of or analyi will be he ineracion beween kill-biaed echnological change, poplaion dynamic and reorce depleion in he long-rn and dring he raniion o a long-rn eqilibrim. In order o addre he raniional dynamic, we perform hree nmerical experimen nder realiic parameer, dealing wih emporary and/or long-rn increae in he killed-nkilled poplaion raio. The remainder of he paper i organized a follow: Secion 2 inrodce he opimizaion problem of hoehold. Secion 3 decribe he prodcion ide of he model. Secion 4 preen he eqilibrim rcre, he dynamic yem and he properie of he balanced growh pah long-rn eqilibrim). In Secion 5, we calibrae he model and explore he dynamic behavior of he model. Finally, Secion 6 provide a mmary and a conclion. 2 Hoehold 2.1 Preliminary Remark We conider wo grop of hoehold, killed and nkilled. In accordance wih empirical obervaion reporing a high inergeneraional perience in edcaion, we ame ha he grop of killed hoehold raie only killed offpring. The grop of nkilled hoehold, in rn, raie boh ype of children, b on he hoehold level a ingle hoehold raie eiher killed or nkilled offpring. The fracion of nkilled hoehold raiing killed offpring will be deermined endogenoly in eqilibrim. Hence, here exi hree ype of agen, killed hoehold raiing killed offpring, nkilled hoehold raiing nkilled offpring and nkilled hoehold raiing killed offpring. The eing we preen here generae pward mobiliy or no mobiliy a all. In oher word, downward mobiliy and dicriminaion among offpring wih repec o edcaional choice are aben a we foc on developed economie wih low feriliy rae, which make he emergence of he former and laer nlikely. 5 Model of direced echnical change have been applied in environmenal economic by Andre and Smlder 2005), Di Maria and Smlder 2004), Di Maria and van der Werf 2005), and Di Maria and Valene 2006). Di Maria and Smlder 2004) analyze he effec of pollion and Andre and Smlder 2005) dy a laborreorce economy. Di Maria and Valene 2006) are probably he fir who provide a micro-fondaion of prely reorce-agmening echnical progre. 4

5 2.2 Preference and Bdge Conrain The economy nder conideraion i poplaed by a coninm of overlapping generaion. In hi eing, ime i dicree, indexed by and range from 0 o. Hoehold live for hree period: childhood, adlhood, and old age. All economically relevan deciion are made in he econd period of life, adlhood. Adl agen pply one ni of work ime each o firm, raie children and ave, wherea old agen conme heir aving. The economy i poplaed by wo grop of hoehold, killed and nkilled, denoed by i i, ), repecively, earning a wage income w i in heir econd period of life. Moreover, here are wo ype of children, ni,j,eiher rained o be killed worker or nkilled worker in + 1, denoed by j,. Preference of a member i, of generaion ha i born in 1 are defined over conmpion in and +1 a well a he poenial aggregae income of her children. They are pecified a i lnc i + ν lnw j +1 ni,j )+ρ ln c i +1, 1) where c and c +1 are preen and fre conmpion. w j +1 ni,j reflec oal poenial income of an individal offpring weighed by he alrim facor ν. Moreover, ρ repreen a al he individal dicon facor of fre conmpion. We denoe he fracion of nkilled hoehold inveing in edcaion for heir children by θ [0, 1]. Hence, he wo poplaion grop evolve according o and +1 θ n, + n, 2) +1 1 θ )n,. 3) The edcaion yem i privaely fnded. A eaching reqire killed labor, E, chooling fee depend on he wage rae for killed labor, w, and he exogenoly fixed eacher-den raio, φ. 6 Since he edcaion ecor i bjec o a non-defici condiion, iion fee have o cover he wagemofeacher w φn, + n, θ )w E. 4) Coneqenly he edcion co per child amon o w φ. Regardle of he ype of children paren wih o raie, feriliy i bjec o forgone wage earning, in erm of opporniy co and conmpion need of children. To hi end, paren have o relinqih he fracion z of heir wage income per child. Therefore, child-rearing co for nkilled children amon o zw ini, wih i becae only nkilled hoehold raie nkilled offpring. Toal child-rearing co 6 For imilar ampion regarding he chooling ecor ee Eicher 1996,1999), and Bhagwai and Srinivaan 1977). 5

6 for killed offpring on he oher hand amon o zw i + w φ)n i,,wihi,. In order o cover old age conmpion, member of generaion can by propery righ o naral reorce naral capial) and inve in he capial marke man-made capial). We denoe he ock of he exhaible naral reorce in period by M 7, and i exracion allocaed o prodcion by R. The economy i iniially endowed wih a reorce ock M 0 > 0. A he beginning of he crren period,, he ock of exhaible naral reorce i deermined by he pa reorce ock min exracion in he crren period, hence, M M 1 R.Eachmember of generaion by m i ni of naral reorce from he old age generaion a he compeiive price p R. When he reire in + 1, he ell a par of her naral reorce o firm for e in prodcion and he propery righ of he remaining par o he adl cohor born in period a price p R +1 per ni of naral reorce. The level of fre conmpion eqal revene from invemen in man-made capial on he capial marke 1 + r +1 ) i ), pl earning from elling naral reorce o firm exracion) and he elling of he propery righ o naral reorce o he adl cohor born in p R +1 mi ), hence c i +1 1+r +1 ) i + pr +1 mi, 5) wih i,. The bdge conrain of a killed hoehold raiing killed offpring i, j ) hreada w zw n, + w φn, for nkilled hoehold raiing nkilled offpring, i, j ) w zw n, + c + pr m +, 6) }{{} c +1 1+r +1 + c + p R m +, 7) }{{} c +1 1+r +1 and finally for nkilled hoehold raiing killed offpring, i and j ) w zw + φw )n, + c + pr m +. 8) }{{} c +1 1+r Opimizaion Amemberi, of generaion chooe {c i,ni,j c i +1,i,mi } o a o maximize he iliy fncion given by Eq.1), wih j, ifi and j,, ifi. Regardle of he ype of hoehold and which ype of children a hoehold wihe o raie, he maximizaion over he wo 7 We denoe aggregae level in capial leer and per capia level in lower cae leer. 6

7 ae, naral reorce m i ), and invemen in he capial marke i ) imply a non-arbirage condiion beween he wo ae ha i known a Hoelling rle, ch ha marginal rern on boh ae are eqalized 1+r +1 pr +1 p R. 9) Hence, he marginal rern of invemen on he exhaible reorce ock, pr +1,meqalhe p R marginal rern of invemen on he capial marke ed o finance reearch and developmen R&D) Skilled Hoehold Skilled hoehold raie killed offpring and maximize 1) bjec o 6), implying ha c n, 1 1+ν + ρ w, 10) ν 1 + ν + ρ)z + φ), 11) ρ 1+ν + ρ w pr m. 12) Since killed hoehold finance edcaion wih heir labor income, w, heerogeneiy doe no inflence hem. However, i i worh noing ha heir feriliy deciion alo depend negaively on he fracion of wage income ha i ranferred o heir offpring, z, and he den-eacher raio, φ Unkilled Hoehold Raiing Unkilled Offpring Unkilled hoehold raiing nkilled offpring, maximize 1) bjec o 7), implying ha c n, 1 1+ν + ρ w, 13) ν 1 + ν + ρ)z, 14) ρ 1+ν + ρ w p R m. 15) Since hi ype of hoehold doe no inve in edcaion, he nmber of nkilled children raied in nkilled hoehold i no inflenced by he parameer φ. Comparing 11) and 14) how ha killed paren raie fewer children han nkilled paren, de o he exience of edcaion co. Moreover, when w <w, nkilled hoehold have le reorce available for preen and fre conmpion. 7

8 2.3.3 Unkilled Hoehold Raiing Skilled Offpring Maximizing lifeime iliy 1) bjec o 8) implie c n, 1 1+ν + ρ w, 16) ν w 1 + ν + ρ) w z + w φ) ν ν + ρ) z + ω φ), 17) ρ 1+ν + ρ w pr m, 18) wih ω w w. Apparenly, he wage differenial beween killed and nkilled labor ω play a crcial role in he deciion of nkilled hoehold ha wih o edcae heir offpring o killed worker. The higher he killed-wage premim, he higher he edcaional co per child compared o he wage income of an nkilled hoehold, ch ha he nmber of killed children born in nkilled hoehold i inverely relaed o ω The Share of Unkilled Hoehold Raiing Skilled Offpring Unkilled hoehold raie eiher nkilled or poibly killed children. Given ha dicriminaion among offpring i aben, he fracion of nkilled hoehold raiing killed offpring i denoed by θ [0, 1]. For θ > 0, he lifeime iliy of nkilled paren raiing killed offpring m a lea eqal he lifeime iliy of nkilled paren raiing nkilled offpring. In ligh of he olion o he opimizaion problem of nkilled hoehold 13)-15) and 16)-18), we can eablih he following propoiion Propoiion 1 The grop of nkilled hoehold raie boh ype of children, i.e. n,,,, ch ha 8 w+1 w+1 > 0 and n, > 0, if ω +1 z + ω φ. 19) z Relaion 19) i an indifference condiion, which will deermine he fracion 0 θ 1 of nkilled hoehold in eqilibrim ha edcae heir offpring o killed worker. A paren are alriic oward heir offpring wih repec o he offpring aggregae poenial labor income, he fre wage differenial beween killed and nkilled labor, ω +1, m a lea compenae for he co raio beween killed and nkilled offpring. Moreover, a he righ-hand ide of 19) i greaer han one, i follow ha 19) hold only if w+1 >w +1.9 Comparing he repecive feriliy 8 The proof follow direcly from 13)-15) and 16)-18) given ha,,. 9 Similar o de la Croix and Doepke 2003) feriliy differenial are generaed by wage differenial. The difference i ha de la Croix and Doepke conider a conino wage diribion. 8

9 deciion yield n, >n, >n,, 20) given ha 19) hold. The feriliy of nkilled hoehold raiing nkilled offpring i he highe ince hey do no allocae reorce o edcaion, wherea he oppoie i re for nkilled paren raiing killed offpring. Since he laer have o pay iion fee in erm of w,wherew <w, hey rade a lower nmber of children again a higher income for heir offpring. Hence, he feriliy of nkilled paren raiing killed offpring i he lowe. 3 Prodcion 3.1 Preliminary Remark The prodcion ide of he model bild on Acemogl 1998,2002). Aggregae op, Y,i compoed of wo inermediae, Y and Y, emming from wo differen prodcion procee, one ing killed labor, and he oher one ing nkilled labor and a e of machine, x and x, which are complemenary o each kind of labor, repecively. The prodcion of machine reqire he exience of echnological knowledge a bleprin or deign), labor and naral reorce. Bleprin are he ocome of prpoefl invemen in reearch and developmen R&D). abor marke for killed and nkilled labor are compeiive and each kind of labor i amed o move verically wihin he prodcion procee. Thi ampion ake differen kill-ineniie and a low iner-ecoral mobiliy of killed and nkilled labor ino accon. Moreover, killed labor i employed in prodcion a well a in edcaion. 3.2 Final Good Prodcion Final op i compoed of inermediae good, Y and Y, boh emming from wo differen prodcion procee. The elaiciy of biion beween Y and Y i deermined by ε 0, ) ch ha he prodcion of final op i bjec o he following need CES-prodcion fncion [ ] Y γy ) ε 1 ε +1 γ)y ) ε 1 ε ε 1 ε. 21) The parameer γ 0, 1) i a diribion parameer deermining how imporan he wo good are for aggregae op. In each period he price of final op, i normalized o 1 - ha i p Y 1, where he price of Y 10 For deail ee Appendix B.1. and Y are denoed by p Y and p Y, ch ha 10 [ γ ε p Y ) 1 ε +1 γ) ε p Y ) 1 ε] 1 1 ε p Y 1. 22) 9

10 3.3 Prodcion of Inermediae and Machine The prodcion of Y and Y reqire killed and nkilled labor, a well a a range of labor complemenary machine, x and x, repecively. In each period of ime,, herearen i i, ) differen ype of machine available. Prodcion fncion of boh inermediae read a Y 1 N x 1 j)1 dj Y ), 23) 0 Y 1 N 1 0 x j) 1 dj Y ), 24) wih 0 <<1. Machine, x i, are manfacred wih killed or nkilled labor, xi, repecively, and naral reorce, R xi i, ), where labor a well a naral reorce are conidered o be eenial for he prodcion of machine. Moreover, we ame ha crren echnological knowledge increae facor prodciviy in he machine prodcing ecor. 11 Therefore, prodcion fncion for a machine of ype j in ecor or read a x j) AN x j)) 1 α R x j)) α, 25) wih A, B > 0. x j) BN x j)) 1 α R x j)) α, 26) 3.4 Reearch and Developmen R&D conie he earch for new deign bleprin) of machine. To hi end, reearch firm ren labor ervice, capial inp and naral reorce. I implifie he analyi coniderably hogh, if we ame ha all he hree of hee facor combine o prodce bleprin in exacly he ame way ha hey combine o prodce final op, i.e. we apply he o called lab-eqipmen approach. By doing o, we alo ake ino accon he criicim reed by reorce economi wih repec o he o-called knowledge driven pecificaion, in which killed) labor i he only inp o R&D and i herefore eemingly overly opimiic ampion ha R&D cold ake place wiho naral reorce. In addiion we ame ha bleprin are depreciaed enirely afer one period. We conider hi o be qie realiic, ince one period encompae approximaely 30 year and firm prodcing a eam engine m become obolee a ome poin in ime I i debaeable weher hi ampion indce an over-opimiic perpecive wih repec o non-reprodcible capial ino he model, b i i neceary o generae eady-ae growh. 12 Wiho complee depreciaion, he old generaion wold ell i ae on he capial marke o he adl cohor. The laer wold pli p i aving beween exiing bleprin and invemen in R&D. Under fll depreciaion he amon of aving i enirely allocaed o R&D. 10

11 Boh R&D ecor generae new bleprin according o he following cale invarian prodcion fncion N +1 η N )δ D ) λ and N+1 η N )δ D ) λ λ 0, 1], 27) where D and D are pending on R&D in ni of he final good) for killed- and nkilled-labor complemenary ype of machine, repecively. The parameer η and η are prodciviy parameer ha allow he co of innovaion o differ. A a non-criical b implifying ampion wih repec o he qaliaive rel of or analyi, we e λ 1, ha i lab-eqipmen linearly ener he prodcion fncion of new bleprin. Finally, we will diingih he cae δ>0 and δ<0. In he former, echnological advance are parially hedged o by diminihing echnological opporniie Evenon, 1984; Korm, 1993; Jone and William, 2000), when i may become more and more complicaed o achieve prodciviy gain. In he cae of δ<0, here are ineremporal knowledge pillover; a cae which i labeled a anding on he holder of gian in he lierare. In eiher cae, aomiic R&D firm ake he level of echnological knowledge in he crren period a given and neglec he impac of heir own R&D effor on he fre level of echnological knowledge. For λ 1andδ 0, or R&D pecificaion read a N+1 η D N and N )δ +1 η D. 28) )δ N 4 Eqilibrim 4.1 Preliminary Remark In hi ecion, we pecify he eqilibria on he marke for inermediae good and machine, he labor marke and he marke for naral reorce. The properie of hee eqilibria deermine marginal prodcion co in he machine prodcing ecor and he diribion of aggregae aving beween he wo R&D-ecor. Wih he informaion a hand, he rae of echnological progre for killed and nkilled-labor complemenary innovaion are known, which allow o deermine he dynamic of he depleion rae of naral reorce. 4.2 Prodcion, abor, and Naral Reorce Inermediae good marke are compeiive, ch ha marke clearing deermine facor price, p Y and p Y, according o marginal prodciviie of Y and Y in he prodcion of final op, Y, ch ha p py p Y 1 γ γ ) Y 1 ε. 29) Y 11

12 The al characeriic of a ymmeric eqilibrim imply eqal price and qaniie prodced and demanded) for any ype of machine wihin he and he -ecor, ch ha p x j) p x, p x j) p x and x x j) x x, x x j) x x which implicae ha he demand for machine i eqally pread over all ype of machine ) 1 p x Y Y p x Iniively, a higher price p Yi p and x Y p x ) 1 Y. 30) increae he vale marginal prodc of all facor employed in he prodcion of Y i, encoraging firm o employ more machine. A higher level of labor employmen imilarly increae he demand for machine a more worker are ing hem marke ize effec). Converely, demand for machine i inverely relaed o he renal price of machine, p x and p x. Since machine prodcer have o cover R&D expene for repecive bleprin, machine are old nder monopoliic compeiion, facing demand a decribed in 30). Hence, he profi of machine prodcer are obained a π x [p x c x ]x and π x [p x c x ]x 31) wih c x ) α w,p R ) w )1 α p R AN 1 α)1 α α α and c x w,p R ) w )1 α p R BN 1 α)1 α α α, 32) ) α repreening he marginal co of machine prodcing firm 13, opening o ino a markp over marginal prodcion co, which i deermined by he elaiciy of biion beween differen ype of machine p x cx w,p R ) 1 and p x cx w,p R ). 33) 1 In conclion, he profi of machine prodcer 31) wrie a and π x 1 ) 1 π x 1 ) 1 p Y p Y ) 1 Y ) 1 Y c x ) 1, 34) c x ) 1. 35) Becae of he marke ize effec, he profi of echnology owner increae wih he employmen of he ype of labor ha i complemenary o he repecive ype of machine, he vale marginal prodc of inermediae in final good prodcion, p Yi, i,, ince hi increae machine demand ee 30)) and decreaing in marginal prodcion co of machine, c xi,i,, a <1, where prodcion co are increaing fncion in he facor price for labor and naral 13 For deail ee Appendix B.2. 12

13 reorce. Given he rcre of hi ymmeric eqilibrim, he level of Y and Y are obained from 30) ogeher wih 23) and 24) a Y 1 ) 1 2 N p Y c x ) 1 Y, 36) and ) Y 1 ) 1 2 p N Y 1 c x Y. 37) Coneqenly, he relaive price for he wo inermediae i idenified by 29) ogeher wih 36) and 37), ch ha ) ε σ p 1 γ N Y ) σ c x ) 1 σ γ N Y c x 38) wih σ ε ε 1)1 ). Given ha labor marke for killed and nkilled labor are compeiive, he killed wage premim ω w w i given by p Y / Y Y ). Hence, we can wrie he wage differenial in ligh of 38) Y ogeher wih 36) and 37) a follow ) ε 1 γ σξ ω N ) σ 1 σξ Y ) ) σξ σ 1)1 ) A σξ γ N Y, B 39) wih ξ +1 α)1 ). If σ>1, hen he killed wage premim of he crren period i increaing in he preen bleprin raio N N becae he wo inermediae are gro bie in final good prodcion. The relaive facor reward of killed labor decline, however, in he crren killed-nkilled employmen raio, Y Y for a given bleprin raio, N N, a labor i bjec o diminihing marginal rern. In he nex ecion, we will ee ha he evolion of he bleprin raio depend likewie on σ 1. More pecifically, an expeced increae in he employmen raio of killed and nkilled labor Y +1 will Y +1 increae he profiabiliy of killed-labor complemenary innovaion by mean of he marke ize effec and will bia echnological progre oward he ecor, whenever σ>1. We now rn o he employmen rcre in eqilibrim. Skilled labor i employed in prodcion and edcaion. Each kind of labor i allowed o move verically wihin he and he -ecor. By doing o, we ake ino accon differing kill ineniie and a low mobiliy of labor beween he killed and nkilled labor inenive ecor. Obvioly, fll employmen of labor reqire Y + x + E and Y + x. 40) Therewih, compeiive labor marke generae an employmen rcre a follow For deail ee Appendix B.3. Y ξ E Y ξ, ), x x ξx ξ E ), 41) ξx ξ, 42) 13

14 wih ξ x 1 α)1 ) andξ +1 α)1 ). Frher, demand for labor in he edcaion ecor i deermined by he oal nmber of children en o chool and he rcral parameer φ - he den-eacher raio, ch ha E φn, + θ n, ). 43) Coneqenly, employmen raio in prodcion are pecified a follow Y Y x x 1 φn, ) φθ n,. 44) Naral reorce are mobile wihin he machine prodcing ecor and an efficien e of naral reorce reqire ha reorce demand of he machine prodcing ecor eqal he overall amon of naral reorce exraced. Hence, R x + R x R. 45) Moreover, perfec compeiion on he marke for naral reorce rel in 15 R x R x ϕ x {}}{ 1 1+ ) ε ) 1 γ σ N σ 1 σ γ N ) ε ) 1 γ σ N σ 1 σ γ N 1 γ Y Y ) σ 1 σ ) Y σ 1 σ Y ) c x σ 1)1 ) σ c x ) c x σ 1)1 ) σ c x ) ε ) σ N 1+ σ 1 ) σ Y σ 1 ) σ γ N c x σ 1)1 ) Y σ c x }{{} 1 ϕ x ϕ x R 46) R. 47) If σ 1, which necearily implie ha ε 1, hen he hare of exraced naral reorce, ϕ x and ϕ x, depend only on he relaive imporance of inermediae Y and Y for final good prodcion Y, i.e. ϕ x γ and ϕ x 1 γ. Ifσ>1he marke ize for machine play a crcial role, i.e. relaive profiabiliy and demand for machine increae in accordance wih ha ype of labor which i complemenary o he repecive ype of machine. In conclion, demand hare for naral reorce depend no only on γ, b alo on he bleprin raio, N N, he employmen raio in inermediae prodcion, Y, and he marginal prodcion co raio, Y c x c x A B N N ω α 1. 48) Eqaion 48) ae a poiive link beween he killed wage premim and relaive marginal prodcion co of killed labor complemenary innovaion. In ligh of 48), 47), 46) and 39), we obain for deail ee Appendix A.1.): 15 For deail ee Appendix B.4. 14

15 Propoiion 2 An increae in he killed-nkilled employmen raio in inermediae prodcion, Y, and/or an increae in he bleprin raio, N N, raie machine demand and prodcion in he -ecor relaive o he -ecor. Therefore, he hare of exraced naral reorce allocaed o he -ecor, ϕ x, m decline and ϕ x m increae, given ha σ>1. Having eablihed he eqilibria on he marke for inermediae good and machine, he labor marke and he marke for naral reorce, we now rn o he capial marke and he dynamic of reorce exracion, i.e. he depleion rae of naral reorce. Y 4.3 Man-made Capial and Depleion of Naral Reorce Free enry in R&D drive profi down o zero in boh R&D-ecor. The vale of each bleprin eqal he diconed profi ream i.e. π+1 xi +1)) generaed by paen owner, ha i, machine prodcer. On he oher hand, Eq. 28) implie ha he marginal prodciviy of one ni of final op allocaed o R&D eqal η i N i ) δ. A p Y ha been normalized o 1, in eqilibrim hi yield π+1 xi η i N i 1+r ) δ 1. 49) +1 Coneqenly, in eqilibrim he following non-arbirage condiion m hold and herefore π x +1η N ) δ π x +1η N ) δ, 50) π x +1 π x +1 ) η N δ η N. 51) Thi echnology marke clearing condiion can be olved for he bleprin raio in he beqen period by making e of 34), 35) and 38) 16 N+1 N+1 [ ) η N η 1 γ γ N ) δ ] +σ 1)ξ ) ξ R σ 1) ) ε Y +1 Y +1 ) σ 1)ξ A B ) σ 1)1 ) 1 ξ R σ 1), 52) wih ξ R α1 ) and ξ R σ 1) > 0 wihin he range of plaible parameer vale. 17 Since inermediae Y and Y are gro bie, meaning ha σ>1, he bleprin raio in 16 For deail ee Appendix B Noe ha: 0.65 and ξ R arond ee Secion 5.2 for more deail, ch ha σ > wold be neceary, in order o violae ξ R σ 1) > 0. 15

16 + 1 will be biaed oward killed labor complemenary innovaion if i i expeced ha he killed-nkilled labor raio, Y +1, increae in he inermediae ecor. An increae in A Y B enhance +1 he efficiency of labor and naral reorce in he machine prodcion of ecor relaive o ecor, ch ha marginal prodcion co, cx c increae. If δ>0, he nmber of exiing bleprin x dampen he peed of innovaion in he fre becae i i comparaively difficl o innovae kill-complemenary machine for he nex period. For δ<0, in conra, he exiing level of bleprin peed p innovaion de o poiive ineremporal knowledge pillover. Olay for R&D are financed by aggregae aving S whicharenoinveedinownerhipof naral reorce implying ha he amon of aggregae aving can be wrien a 18 S p Y Y [ ρ ] 1+γ + ρ ξ 1+ω ξr 1 τ ϕ x. 53) τ }{{} Clearly, a prereqiie no only for ained economic growh, b [ alo for ] a non-rivial inerior ρ olion of he model i, S > 0impliedbyZ 1 > 0, i.e. 1+γ+ρ ξ 1+ω > ξr 1 τ ϕ x τ. Therefore, i i no fficien ha he labor income hare exceed he naral reorce income hare, i.e. ξ >ξ R. A neceary condiion for S > 0 and ained economic growh herefore i a fficienly high expendire hare for old-age conmpion ρ 1+γ+ρ. Everyhing ele being conan, he ize of he wage differenial and he killed-nkilled poplaion raio can alo have a poiive impac on aggregae aving. A we will ee frher below, however, he ize of Z 1 govern no only crren aving, b alo he dynamic of reorce exracion. Th, he ineracion beween he wo erm conained in Z 1 i alo crcial for he depleion of naral reorce dring he raniion and in he long-rn. We come back o hi poin frher below. Free enry in R&D and perfec capial marke imply eqaliy beween fre aggregae profi and revene of aggregae aving allocaed o he R&D-ecor or, ch ha N+1 i πi r +1 )D. i Th, we obain from he echnology marke clearing condiion 51), given ha D + D S : D Z Ω S and D Ω 1+Ω S, 54) ) wih Ω N +1 η N δ N N+1 η N and +1 N+1 given by 52). Bearing in mind ha he price per ni of naral reorce eqal i vale marginal prodc in inermediae prodcion, he evolion of he naral reorce price, g pr +1, i given by he change in he vale marginal prodc of naral reorce. 19 On he oher hand, in eqilibrim, he change of hi facor price i ied o he inere facor by Hoelling rle 1 + r +1 g pr +1 ), where he echnology marke clearing condiion 1 + r +1 π+1 i ηi N i ) δ,wihi, ) link he 18 For deail ee Appendix B We denoe he gro growh rae of x, x +1 x,byg+1. x 16

17 inere rae o he profiabiliy of fre innovaion. Th, he dynamic of he depleion rae of naral reorce τ i redced o he following difference eqaion 20 τ +1 Z 1 τ ) 1 g ϕrx ) Z 2 1 τ ) wih Z 1 ρ 1+ν+ρ ξ 1+ω ξr 1 τ ϕ x τ, Z 2 1 )1 + Ω ). A we remarked before, a neceary condiion for aggregae aving being poiive i Z 1 > 0. Oherwie, aving, demand for machine and he depleion rae wold jmp o zero implying zero op in he machine and he inermediae good prodcing ecor. I i preciely he behavior of Z 1 Z 2 which eer naral reorce depleion dring he raniion and in he long-rn. Dring he raniion, an increae in aggregae aving caed by an increae in Z 1 a compared o Z 2 - which conain he aggregae relaive profi of fre innovaion ee 54)) - increae he depleion rae of naral reorce. In hi ene, higher aggregae aving in he crren period caed for example by an increae in Z 1, increae fre reorce depleion via a higher demand for machine. We will now clarify he mechanim behind difference eqaion 55) by aring from a differen poin of origin. In eqilibrim, demand for machine in ecor i, ha o mach i pply 21 X i N i x i N i p Yi) 1 Yi c xi N i G xi ) 1 α ) ϕ xi α R, 56) wih R τ M 1 and G A B, ifi. Accordingly, depleion of naral reorce m increae whenever demand for machine in one or boh ecor agmen in ch a way ha facor reallocaion change in ϕ xi ) alone i no fficien, and a higher exracion of naral reorce become neceary. In ligh of 56), change in demand for and he op of machine in ecor are given by g p Y c x +1 ch ha, afer ome maniplaion, ) 1 g +1 ) 1 α α g+1 g ϕx +1 +1) gr, 57) Z ) 1 1 g ϕx +1 g R Z ) 2 A g+1 R τ +1 τ 1 τ ), he la eqaion eqal exacly 55). In m, he evolion of he economy i governed by a for-dimenional yem of difference eqaion conaining he law of moion for he poplaion raio,, he bleprin raio, N N,he 20 For deail ee Appendix B.7a). 21 For deail ee Appendix B.7b). 17

18

19 iii) The bleprin raio and he depleion rae aify: ) N N [ η η ) +σ 1)ξ 1 γ γ ) ε Y Y ) σ 1)ξ ) ] 1 A σ 1)1 ) ψ, 65) B τ 1 Z 1 Z 2, 66) where ψ ξ R σ 1) + δ +σ 1)ξ ) > 0 in he range of plaible parameer. iv) Skilled and nkilled-labor complemenary innovaion evolve along he balanced growh pah in compliance wih [ g g N i n ξ 1 τ ) ξr] 1 δ, i,, 67) where n 1 θ )n, θ eady ae. ) n, + n, repreen he average nmber of children in In ligh of Iem iv) of Propoiion 3, i follow ha he long-rn growh rae of he poplaion ha a poiive impac on he growh rae of innovaion a long a δ>0. Whenever, long-rn poplaion growh abilize below or a he reprodcion level, n 1, he long-rn growh rae of prodciviy g 1) become negaive, given δ>0. In hi ene, kill-biaed echnological change wold dig i own grave via he negaive feedback effec indced by he decline in feriliy becae n θ < 0. If one conider poiive poplaion growh a no feaible de o pace rericion and/or making allowance for Unied Naion long-rn poplaion projecion ha predic a aionary world poplaion afer he year 2150, he eady ae exhibi no poplaion growh, ch ha n 1 hold be a reaonable predicion for he fre. Under hee circmance he maximm prodciviy growh rae wold be 0 ee 67)) for τ 0andδ>0. Hence for any inerior olion 0 <τ < 1, he economy wold exhibi negaive prodciviy growh and a negaive growh rae of he wage rae in boh ecor ee Corollary 1 below). In oher word, a aionary poplaion confroned wih non-renewable naral reorce i no able o generae ained economic growh in or eing, when he prodciviy of R&D i negaively relaed o he level of echnology δ >0). A prereqiie for long-rn growh in he face of aionary or hrinking poplaion herefore inclde poiive exernal rern in R&D wih repec o he exiing ock of bleprin, i.e. δ<0 in or pecificaion. The following propoiion characerize he behavior of he hare of nkilled hoehold raiing killed offpring in he long-rn, θ, and he depleion rae of naral reorce, τ, wih repec o variaion in he eacher-den raio φ and he prodciviy raio in R&D, η η for deail ee Appendix A.3): 19

20 Propoiion 4 Along he balanced growh pah: i) The killed wage premim i pecified a ω z θ z φ, ch ha φ > 0 and θ η η < 0. ii) The reacion of he long-rn depleion rae τ w.r.. change in φ and η τ φ, τ η η 0. η i ambigo, i.e. Iem i) of Propoiion 4 follow from Iem ii) of Propoiion 3. An increae in φ raie he righhand ide of 19), ch ha he killed-nkilled poplaion raio m increae which reqire an increae in he hare of nkilled hoehold raiing killed offpring θ, hence θ φ > 0. Converely, an increae in η η raie he lef-hand ide of 19) where he righ-hand ide remain conan. Th, he killed-nkilled poplaion raio m decline in he long-rn, ch ha θ decline and θ η η < 0. The ign of τ φ and τ η η poiive or negaive depend eenially on Z 1 φ Z 2 φ whenever Z 2 in rn i ambigo Iem ii) of Propoiion 4). Wheher τ φ i for deail ee Appendix A.3). Iniively, φ > Z 1 φ, he reacion of relaive profiabiliy of kill-complemenary innovaion wih repec o variaion in φ i larger han he reacion of aggregae aving, ch ha he depleion rae of naral reorce m increae, i.e. > 0 and vice vera. An analogo argmen τ φ applie for τ. Since an increae in φ or a redcion in η η η η cae an increae in θ, he average growh rae of he poplaion n m decline. The laer in rn cae a negaive impac on he long-rn growh rae of prodciviy, if δ>0, and i condcive o economic growh whenever poiive pillover effec exi wih repec o exiing echnological knowledge, i.e. δ<0. 22 The level of per-capia conmpion c i and ci +1 depend on wage income wi, i,, ch ha he evolion of per-capia conmpion level i ied o he evolion of wage. Corollary 1 In ligh of Propoiion 3 we obain for he evolion of wage, g wi,wihi,, in eady ae for deail ee Appendix A.4): 1 g wi gn i ξ [ ) ξ R n ξ 1 τ ) ξr] [ ] 1 δ 1 ξ R τ. 68) g pr n The economy move along a ainable growh pah, whenever g wi 1, which implie a lea non-declining per-capia conmpion level in he long-rn. From 68), ainable developmen in i weake form reqire ha prodciviy growh m a lea compenae for he increae in he naral reorce price caed by an increaing horage of naral reorce. 22 The impac of he change in τ on g wih repec o variaion in φ or η may work in he oppoie direcion η however. Since reaonable parameer range imply ξ >> ξ R, he change in n alway dominae he change in τ, ch ha we can expec nder realiic parameer conellaion g g 0and 0forδ 0. φ η η 20

21 5 Nmeric 5.1 Preliminary Remark We now rn o hree nmerical experimen in order o illrae he behavior of he economy nder conideraion wih repec o: an exogeno increae in he killed-nkilled poplaion raio, a decline in relaive reearch co η η andanincreaeinchoolingqaliyφ. The fir experimen generae only raniory effec ince he eady ae remain naffeced by hi hock. The econd experimen hif he long-rn vale of all endogeno variable excep he killed wage premim which correpond o z z φ in he long-rn. Therefore a non-monoone raniion of ω in he fir and he econd experimen i he rel. The hird experimen obvioly alo aler he long-rn level of he killed wage premim. 5.2 Calibraion Since one period encompae approximaely hiry year, we chooe for he dicon facor of fre conmpion, ρ, a vale ha i andard in real-bine-cycle lierare: 0.99 per qarer, i.e. ρ in or conex. The parameer repreen he labor hare in inermediae good prodcion and i e o In he U.S., energy expendire a a hare of GDP amoned o 8.8% in 2006 wih a maximm cloe o 14% a he beginning of he eighie ee Energy Informaion Adminiraion, 2009). Hence 8.8% conie an pper limi for he reorce income hare of non-renewable in or model. We herefore e α 0.08, which implie ξ R α1 ) in each inermediae ecor. The parameer φ reflec he eacher-den raio and i e o 1/20. Moreover, child-rearing i bjec o forgone conmpion poibiliie and loe in poenial lifeime earning which amon o 13% for highly edcaed women and higher if women drop o of he labor marke compleely Dankmeyer, 1996). The direc ime co for paren raiing a child o adlhood amon o 50% of paren ime endowmen ee de la Croix and Doepke, 2003), which wold imply z a a lower limi. Taking loe in lifeime earning ino accon, we e z 0.15, which implie a killed wage premim ω 1.5 which mache U.S. daa Acemogl, 2002). The weigh ν of children in he iliy fncion drive he growh rae of he poplaion. We chooe a vale of ν 0.26, which generae approximaely zero poplaion growh along he balanced growh pah per year). Now here are five parameer lef: he elaiciy of biion beween inermediae good in final good prodcion, ε, he weigh of Y in final good prodcion, γ, he raio of he prodciviy parameer in R&D, η η η,, and he exernaliy parameer in he raio of prodciviy parameer in machine prodcion A B R&D, δ. We calibrae hee parameer o ha hey mach a long-rn prodciviy growh rae of 2.4%, an invemen hare I Y in he viciniy of 14.43%, fiing he 10 year average of US 21

22 privae fixed capial formaion a a hare of GDP 23 OECD Economic Olook Daabae), an employmen hare in edcaion of arond 2%, and a long-rn decline in he naral reorce ock of 2.4% per year, which cae, via Hoelling rle a long-rn inere rae cloe o 4%. The remaining parameer are herefore fixed a follow: ε 2.4,γ 0.655, η η 2.2,δ 0.045, and A B 8.24 Figre 2 abo here 5.3 Comparaive Saic and Eqilibrim Dynamic The behavior of he depleion rae wih repec o change in φ and η η i preened in Figre 2. Indeed, τ follow a -haped relaion wih repec o change in φ, where he increaing arm i no relevan ince he implied θ i cloe o 1 for φ>0.14 and a realiic range of φ fall wihin and 0.055, ch ha τ φ beween 2 and 2.5, ch ha τ η η > 0. η < 0. Similarly, in order o mach crren daa η Experimen 1: Increae in he killed-nkilled poplaion raio hold fall In or fir experimen, we analyze raniory effec of an exogeno increae in he killednkilled poplaion raio wih repec o he ineracion beween kill-biaed innovaion, poplaion dynamic and he depleion of naral reorce. 25 In or experimen, he nanicipaed hock occr in period 0, while or poin of reference i he eady ae deermined by he parameer conellaion menioned above. The rel are preened in Figre 3. Given he bleprin raio N N in he crren period, an increae in he killed-nkilled poplaion raio,, lead o a decline in prodciviy of killed labor relaive o nkilled labor. Therefore, he crren wage raio ω i redced ch ha i i more affordable for nkilled hoehold o raie killed offpring, i.e. n, increae ee Eq. 17), provided ha θ > 0. Recall now, ha he killed-nkilled poplaion raio alo adj by mean of change in he fracion of nkilled hoehold raiing killed offpring, θ. 23 Noe ha: I Y [ [γ ε +1 γ) ε p ) 1 ε ] 1 ε 1 γ+1 γ) Y Y ) ε 1 ] Z 1, for deail ee Appendix B.9. ε ε 1 ε 24 The nmeric mehod i decribed in Appendix B.10. Given he parameer conellaion he dynamic yem exhibi wo poiive eigenvale wihin he ni circle and wo oide he ni circle, ch ha he economy i bjec o addle-poin abiliy. Thi conellaion i rob for a very large range of parameer, excep for ε<2. Noe alo ha he raniory behavior i independen from δ Evoked by he Vienam war in he 1960, he US experienced an exogeno increae in he killed-nkilled labor raio, ee Acemogl 1998,2002). 22

23 Figre 3 abo here A he killed-nkilled poplaion raio i above i long-rn vale, θ m adj in compliance wih 19) from below o i long-rn vale which enre a decline of. The growh rae of he nkilled labor force, g+1, i now above i long-rn vale and he growh rae of he killed labor force, g+1, i below i. Since we know, moreover, ha: n, >n, >n,, he growh rae of he poplaion, n, m be above i long-rn vale a well and poplaion growh i eenially driven by an expanion of he nkilled labor force. The drop in he nmber of children en o chool caed by he decline of θ, lower he fracion of killed labor, E, allocaed o he edcaion ecor. The laer and he iniial increae in he killed-nkilled poplaion raio indce an increae in killed-nkilled employmen raio in boh prodcion ecor which again i compaible wih he oberved decline in ω. 26 Since he marke ize for killed-labor complemenary machine ha increaed, given he crren bleprin raio N N, relaive aggregae demand for killed-labor complemenary machine X X increae, ch ha he hare of exraced naral reorce allocaed o he -ecor, ϕ x, m increae. The increaed relaive marke ize for killed labor complemenary machine reflec alo higher employmen of labor in he ecor ch ha he depleion rae of naral reorce τ i redced. Hence, he crren working cohor inve more in naral reorce. A he ame ime, he invemen hare of GDP, I Y, decline. In he beqen period, he killed-nkilled poplaion raio hrink according o he feriliy deciion of he previo period θ declined in he previo period). The new level of he killed-nkilled poplaion raio i ill above i long-rn vale and ha been anicipaed by he allocaion of aving o boh R&D-ecor in he previo period, i.e. increaed in he previo period. Therefore, echnological progre i biaed oward killed-labor complemenary innovaion, ch ha N N i now above i long-rn vale a well ee Eq. 52)). Accordingly, he killed-wage premim increae and overhoo i long-rn level, which make edcaion for nkilled hoehold le affordable, ch ha n, decreae o below i long-rn vale. A he ame ime, more nkilled hoehold wih o rain heir offpring o killed worker. Coneqenly, θ rie in compliance wih indifference condiion 19), which rel in a higher demand for killed labor in he edcaion ecor. A he killed-nkilled poplaion raio hrink and poplaion growh i mainly creaed by an expanion of he nkilled poplaion grop, he increae in θ lead o an increae in demand for killed labor in he edcaion ecor relaive o he E overall working force, i.e. increae. Again, wih regard o adjmen in naral reorce e in prodcion, we have o conider wo effec: Fir, he adjmen in he hare of exraced naral reorce allocaed o he - or-ecor, ϕ i,i,, and econd, he adjmen in he exracion rae of naral reorce, i.e. τ. The aforemenioned decline in he killed-nkilled poplaion raio a compared o he previo period accompanied by an increae in θ and E 26 Noe ha Y Y x x de o ymmery ampion ee alo Eq. 41) and 42)). D D 23

24 dring he raniion lead o a drop in killed-labor employmen raio in prodcion. Th, i appear ha he kill bia of innovaion i declining and he hare of naral reorce allocaed o he - -) ecor i diminihing growing). The ecor in rn, i bjec o declining poplaion growh a θ increae oward i long-rn vale. Coneqenly, he depleion rae τ increae in order o mach aggregae machine demand and follow he movemen of θ. A τ increae, hoehold hif heir invemen from naral capial o he capial marke, ch ha he invemen hare of GDP i increaing dring he raniion. The increae in θ,in rn, i reponible for a decline in poplaion growh n. Hence, he model gge an invere relaionhip beween τ and n afer an increae in. Experimen 2: Redcion in he relaive reearch prodciviy η η In or econd experimen we redce he relaive reearch prodciviy/reearch co, η η,by1%, where he nanicipaed hock occr again in period 0. A a poin of reference, we e again he eady ae menioned above. The rel are preened in Figre 4. a) ong-rn Indifference condiion 19) implie ha he long-rn wage differenial i deermined by ω which remain naffeced by a change in η η. Hence, we only oberve raniory change in ω caed by adjmen of and N N o heir new long-rn vale. Since relaive reearch co η η are redced, he long-rn killed-nkilled poplaion raio, ) mach 19), ppored by a rie in kill-biaed echnological change, N N ) wih an increae in relaive R&D expendire D D ) z z φ, m increae in order o, ha come along, relaive aggregae machine demand X X ) and he hare of exraced naral reorce ϕ x, allocaed o he -ecor. An increae in ) evoked by a higher fracion of nkilled hoehold raiing killed offpring repreened by θ, indce a decline in he long-rn growh rae of he poplaion n. In ligh of Propoiion 4, he increae in θ i aociaed wih a decline in he depleion rae of naral reorce τ, mirrored by a higher invemen hare of GDP, I Y ), and prodciviy growh increaed o 2.64%. b) Traniion A redcion in η η indce, via echnology-marke clearing condiion 51), an increae in he relaive profiabiliy of killed-labor complemenary innovaion, πx +1 π+1 x. Thi effec will be ppored by an increae in he killed-nkilled poplaion raio which cae an increae in nex period bleprin raio. 24

25 Figre 4 abo here A he killed nkilled poplaion raio i below i new long-rn vale afer he hock, he hare of nkilled hoehold raiing killed offpring, θ, m adj - in compliance wih 19) - from above o i new long-rn vale ch ha a mooh convergence of he killed-nkilled poplaion raio i enred. Accordingly, he growh rae of he killed poplaion grop m be above i long-rn vale and he growh rae of he nkilled poplaion grop below i. Moreover, given differenial feriliy of he kind ha: n, >n, >n,, overall poplaion growh n m be below i new and old long-rn vale. The iniial increae in θ afer he hock advance he demand for killed labor in he edcaion ecor relaive o he oal working force wih he coneqence of declining employmen raio of killed labor in prodcion. The killed wage premim ω i herefore above i long-rn vale. Moreover, relaive demand for killed labor complemenary machine i redced caing a lower fracion of naral reorce o be allocaed o he -ecor. Given he amon of nkilled labor in he crren period, he increae in aggregae demand for machine prodced in he ecor creae an increae in demand for naral reorce, ch ha τ rie. In he beqen period, he killed-nkilled poplaion raio and he bleprin raio increae. 27 Inhefollowingperiod,hemonoonedeclineofθ generae a mooh convergence of he poplaion raio o i long-rn vale, which i reponible for kill-biaed echnical change mirrored by an increae in he bleprin raio. Coneqenly, he wage raio converge from below i long-rn vale o i acal long-rn vale. The decline in θ lower he employmen raio of killed labor, E, in edcaion. Since echnological change and naral reorce are direced oward ha ecor, which benefi from higher employmen raio, he decline in θ allow for a redcion of he depleion rae where he growh rae of he poplaion i increaing. Experimen 3: Increae in he eacher-den raio φ In or hird nmerical exercie ee Figre 5), we increae he eacher-den raio φ by 1% which may reflec an increae in he qaliy of chool. a) ong-rn Indifference condiion 19) implie ha he long-rn wage differenial adj o ω z z φ.afer an increae in φ, i i more beneficial for he nkilled poplaion grop o raie killed offpring. η η 27 Iniially, he bleprin raio of he beqen period may fall a in he cae preened here, i.e. he decline in overcompenae he increae in Y +1, ee alo Eq.52). Y +1 25

26 Figre 5 abo here Coneqenly, θ increae. Wih a higher fracion of nkilled hoehold raiing killed offpring, he long-rn killed-nkilled poplaion raio m increae a well. Coneqenly, ) relaive demand for killed-labor complemenary machine X X increae which generae an ) increae in he raio of R&D expendire ) D D and he bleprin raio N N. Therefore, he ) hare of exraced naral reorce allocaed o he -ecor, ϕ x, m increae a well. Wih a higher fracion of nkilled hoehold wihing o edcae heir offpring o killed worker, he long-rn growh rae of he poplaion, n, m decline. Again, wih a higher hare of nkilled hoehold raiing killed offpring, he depleion rae of naral reorce m decline, wherea he invemen hare of GDP, I Y ), increae in he long-rn and long-rn prodciviy growh amon o 2.62%. b) Traniion An increae in φ implie ha he killed-wage premim ω m increae o i new long-rn vale eqal o w z z φ. Hence, he crren wage differenial i below i long-rn vale, a well a he killed-nkilled poplaion raio. A mooh convergence of he laer o i new long-rn vale reqire he hare of nkilled hoehold raiing killed offpring o adj from above i long-rn vale o i acal long-rn vale. Hence, he growh rae of he killed poplaion grop m be above i long-rn vale, oo. The oppoie i re for he nkilled poplaion grop. A n, >n, >n,, he growh rae of he poplaion i converging from below i long-rn vale o i acal long-rn vale. The increae in θ afer he hock raie he demand for killed labor in he edcaion ecor ch ha E increae a he expene of employmen raio of killed labor in prodcion. Coneqenly, he wage differenial beween killed and nkilled labor increae. Since he relaive) marke ize for killed-labor complemenary innovaion i crrenly redced, he hare of naral reorce allocaed o he -ecor m increae, i.e. ϕ x decline. A he ame ime, he depleion rae increae in order o aify aggregae demand for machine. The decline in he hare of nkilled hoehold raiing killed offpring dring he raniion redce he demand for killed labor in he edcaion ecor. A hi proce correpond o an increae in he killed-nkilled poplaion raio, he employmen raio of killed labor in prodcion m rie. Skill-biaed echnological change mee an increae in he killed poplaion grop. The decline in θ and he increae in relaive wage ha i reponible for a decline in n, cae a decline in he growh rae of he killed poplaion grop and an increae in he growh rae of he nkilled poplaion grop. Wih a lower demand for labor in he edcaion ecor and increaing employmen raio of killed labor in prodcion he depleion rae of naral reorce decline dring he raniion 26