Optimal education and pensions in an endogenous growth model

Op imal Educa ion and Pensions in an Endogenous G ow h ModelI Elena Del Reya, Miguel-Angel Lopez-Ga cia∗,b aUni e si y o Gi ona, Spain bApplied Economics Depa men , Au onomous Uni e si y o Ba celona, 08193 Bella e a (Ba celona), Spain Abs ac In OLG economies wi h li e-cycle sa ing and exogenous g ow h, compe i i e equilib ia in gene al ail o achie e op imali y because indi iduals accumula e amoun s o physical capi al ha di e om he one ha maximizes wel a e along a balanced g ow h pa h ( he Golden Rule). Wi h hu- man capi al, a second po en ial sou ce o depa u e om op imali y a ises, ela ed o educa ion decisions. We p opose o eco e he Golden Rule o physical and also human capi al accumu- la ion. We cha ac e ize he op imal policy o decen alize he Golden Rule balanced g ow h pa h when he e a e no cons ain s o indi iduals o inance hei educa ion in es men s, and show ha i in ol es educa ion axes. Also, when he go e nmen subsidizes he epaymen o educa ion loans, op imal pensions a e posi i e. Key wo ds: endogenous g ow h, human capi al, in e gene a ional ans e s, educa ion policy JEL classi ica ion: D90, H21, H52, H55 IWe g a e ully acknowledge he hospi ali y o CORE, Uni e si ´ e ca holique de Lou ain and he Uni e si y o Ex- e e Business School, as well as inancial suppo om Ins i u o de Es udios Fiscales (Spanish Minis y o Finances), he Spanish Minis y o Science and Inno a ion (mobili y g an s JC2008-00098 and PR2008-0069 and p ojec s ECO2010-16353 and ECO2012-37572) and he Gene ali a de Ca alunya (con ac s 2009SGR189 and 2009SGR600, he XREPP and he Ba celona GSE Resea ch Ne wo k). We a e indeb ed o Dolo s Be ga, Raou Boucekkine, Jo di Caball´ e, Da id de la C oix, Ch is os Koulo a ianos, Pie e Pes ieau, Jos´ e Ignacio Sil a and Xa ie Rau ich o in- sigh ul commen s and c i icism. We also hank he Edi o and an anonymous e e ee o hei aluable commen s on ou submi ed manusc ip . We e ain esponsibili y o any emaining e o . ∗Co esponding au ho . Tel +34 93 581 12 29. Fax +34 93 581 22 92. Email add esses: [email p o ec ed] (Elena Del Rey), [email p o ec ed] (Miguel-Angel Lopez-Ga cia) P ep in submi ed o Else ie Decembe 12, 2012 1. In oduc ion The o iginal app oach o endogenous g ow h heo y (Rome (1986), Lucas (1988), Rebelo (1991)) is based on he idea ha e u ns o some capi al goods, such as he s ock o human cap- i al esul ing om educa ion, do no diminish as economies de elop. Knowledge spillo e s and educa ion ex e nali ies a e some o he mechanisms ha help sus ain he e u ns o he accumu- la ion o capi al. In u n, he p esence o a posi i e ex e nali y implies ha a Pigou ian subsidy is equi ed o a ain op imali y. Wi hin g ow h heo y, he o e lapping gene a ions li e a u e has speci ically ocused on he exis ence o in e gene a ional ex e nali ies (see, among o he s, Aza i- adis and D azen (1990), Caball´ e (1995), Bold in and Mon es (2005), Docquie e al. (2007)). These a e a consequence o child en inhe i ing a po ion o he human capi al o hei pa en s, o o he spillo e s gene a ed by he agg ega e s ock o human capi al. Because indi iduals igno e he e ec o hei educa ional decision on u u e human capi al, hey unde in es in educa ion. In o e lapping gene a ions economies (OLG he ea e ), his sou ce o depa u e om op imali y adds o he po en ial disc epancy be ween laissez- ai e and op imal physical capi al accumula ion well known since Diamond (1965). As a esul , wo policy ins umen s a e equi ed o a ain op imali y, ypically educa ion subsidies and in e gene a ional ans e s. The pe sis ence in he esul ha educa ion should be subsidized in a g ow h con ex masks he ac ha i hinges on he op imali y c i e ion adop ed, o wi , he maximiza ion o a discoun ed sum o u ili ies de ined o e consump ion le els. In he case o in ini ely li ed indi iduals, hey a e assumed o discoun u u e u ili y, so he e is no di e ence be ween indi idual and social objec i es. In an OLG economy wi h labou p oduc i i y g ow h ha ansla es in o inc easing consump ion le els, he planne canno maximize indi idual u ili y along a balanced g ow h pa h simply because i g ows wi hou limi . The social planne is hen assumed o maximize he sum o p esen and u u e u ili ies o consump ion pe uni o na u al labou , discoun ing he la e a an a bi a y discoun a e. In his pape we assume, on he con a y, ha he social planne pu posi ely wan s o ea all gene a ions alike, while being espec ul wi h indi idual p e e ences. One way o do his is o maximize a u ili y unc ion whose a gumen s a e consump ion le els pe uni o e icien labou . In a h ee o e lapping gene a ion economy whe e human capi al accumula ion is he engine o g ow h, we sea ch o he balanced g ow h pa h ha maximizes his no ion o li e ime wel a e subjec o he cons ain ha he wel a e o he ep esen a i e indi idual o e e y o he gene a ion is ixed a he same le el. This op imali y c i e ion would appea as he coun e pa in he cu en amewo k o he ( wo-pa ) Golden Rule (Diamond, 1965, Samuelson, 1968). As will be made clea , emb acing his c i e ion implies ha i is no longe w ong o indi id- uals o igno e he (posi i e) e ec o hei decisions on he human capi al o o he s. Wha is now w ong om he poin o iew o he social planne is o indi iduals o igno e he e ec o hei decisions on he g ow h a e o he economy. This e ec can gene ally be posi i e o nega i e, bu i is de ini ely nega i e when he indi idual aces he Golden Rule wage and in e es a es. Thus, an educa ion ax is equi ed o main ain he economy on i s op imal balanced g ow h pa h. Fu - he mo e, when he go e nmen subsidises he epaymen o educa ion loans, he op imal policy also in ol es posi i e pensions. In ou model, like in Bold in and Mon es (2005) and Docquie e al. (2007), agen s a e bo n 1 wi h an endowmen o knowledge equal o he human capi al o hei pa en s, and bo ow when young in o de o in es in educa ion. C edi ma ke s a e pe ec . Inhe i ed and acqui ed human capi al in e ac o p oduce human capi al o he middle-aged. As middle-aged, agen s wo k, pay back he loan, sa e and pay axes o ecei e subsidies. Finally, as old-aged, hey consume and pay axes o ecei e subsidies. We use his model o iden i y he op imum, compa e i o he laissez ai e and iden i y he op imal policy. The objec i e o he social planne is o maximize a wel a e index de ined in e ms o consump ion pe uni o e icien labou , and he decision a iables a e also de ined in e ms o ou pu pe uni o e icien labou . The Golden Rule hus ob ained consis s o i e condi ions ha need o be sa is ied simul a- neously. One o he condi ions o he op imal alloca ion is ha he ma ginal p oduc o physical capi al (pe uni o e icien labou ) equals he (endogenous) g ow h a e o he economy. This is like in models wi h exogenous p oduc i i y g ow h (Bui e , 1979). As a as human capi al accumula ion is conce ned, a condi ion equa ing ma ginal bene i s and ma ginal cos s o in es - ing in educa ion is ob ained. Unlike he case o exogenous g ow h models, whe e laissez- ai e and socially op imal physical capi al accumula ion can coincide by chance, we show ha , in ou amewo k, he laissez ai e wi h pe ec c edi ma ke s canno possibly a ain he Golden Rule. Indi iduals will always o e o unde in es ei he in physical capi al, human capi al, o bo h. To be mo e p ecise, as illus a ed in Table 1 below, when he laissez- ai e accumula ion o physical capi al is smalle han he op imal one, he laissez- ai e expendi u e in educa ion can ei he exceed o all sho o he op imal one. Howe e , a laissez- ai e accumula ion o physical capi al g ea e han o equal o he op imal one can only coexis wi h a oo la ge expendi u e in educa ion. As a consequence, i he laissez- ai e amoun o physical capi al coincides wi h he socially op imal le el, he e will be o e -accumula ion o human capi al. The e o e, as s a ed abo e, in o de o a ain he Golden Rule, educa ion should be axed. The eason is ha indi iduals choose hei human capi al in es men s accoun ing only o he e ec s on hei ea nings and loan epaymen cos s, no on he g ow h a e o he economy. I aced wi h he op imal (i.e., Golden Rule) ac o p ices in he laissez ai e, hey would igno e he cos s associa ed wi h main aining hese ac o p ices a hei Golden Rule le el as human capi al inc eases due o hei in es men s. Unde hese ci cums ances, hey would o e -in es in educa ion. Fo his eason, a ax is equi ed o main ain he economy on he op imal balanced g ow h pa h. The es o he pape is o ganized as ollows. Sec ions 2 and 3 espec i ely p esen he model and he social op imum. Sec ion 4 de i es he decen alized ma ke equilib ium in he p esence o go e nmen . The ax ins umen s a ailable a e lump-sum axes on bo h he wo king and he e i ed popula ion and educa ion subsidies. The laissez- ai e can be easily cha ac e ized by se ing he ax pa ame e s equal o ze o. Sec ion 5 compa es he laissez- ai e balanced g ow h pa h wi h he Golden Rule. Finally, Sec ion 6 cha ac e izes he op imal policy and Sec ion 7 concludes. 2. The Model The basic amewo k o analysis is he o e lapping gene a ions model wi h bo h human and physical capi al de eloped in Bold in and Mon es (2005) and Docquie e al. (2007). A pe iod , L +1indi iduals a e bo n. They coexis wi h L middle-aged and L −1old-aged. Popula ion g ows a he exogenous a e nso ha L =(1 +n)L −1wi h n>−1. Agen s a e bo n wi h an endowmen 2 o basic ”knowledge”. This knowledge is assumed o be equal o he le el o human capi al o hei pa en s h −1and is measu ed in uni s o e icien labou pe uni o na u al labou . Human capi al in pe iod is p oduced ou o he amoun o ou pu in es ed in educa ion d −1and basic knowledge h −1acco ding o he p oduc ion unc ion h =E(d −1,h −1). Assuming cons an e u ns o scale, he p oduc ion o human capi al can be w i en in in ensi e e ms as h /h −1=e(˜ d −1), whe e e(.) sa is ies he Inada condi ions and ˜ d −1=d −1/h −1is he amoun o ou pu de o ed o educa ion pe uni o inhe i ed human capi al. A single good Y is p oduced by means o physical capi al K and human capi al H , acco ding o a cons an e u ns o scale p oduc ion unc ion Y =F(K ,H ). As explained below, only he middle-aged wo k and hey inelas ically supply one uni o na u al labou , so ha H =h L . Physical capi al is assumed o ully dep ecia e each pe iod. I we de ine k =K /L as he physical capi al pe uni o na u al labou a io and ˜ k =K /H =k /h as he physical capi al pe uni o e icien labou a io, he echnology can be desc ibed as Y /H = (˜ k ), whe e (.) also sa is ies he Inada condi ions. The li e ime u ili y unc ion o an indi idual bo n a pe iod −1 is U =U(cm ,co +1),(1) whe e cm and co +1deno e he consump ion le els as middle-aged and old-aged, espec i ely. This u ili y unc ion is assumed o be s ic ly quasi-conca e and homogeneous o deg ee j>0. The ea- son why we only impose s ic quasi-conca i y ins ead o s ic conca i y o he indi idual u ili y unc ion is ha , as i will be made clea e sho ly, we a e only in e es ed in o dinal p e e ences. To al ou pu p oduced in pe iod ,F(K ,H ),can be de o ed o consump ion, cm L +co L −1, in es men in educa ion (o human capi al), d L +1, and in es men in physical capi al, K +1. Thus, he agg ega e easibili y cons ain w i es F(K ,H )=cm L +co L −1+d L +1+K +1(2) o , exp essed in uni s o na u al labou : h (k /h )=cm +co 1+n+(1 +n)d +(1 +n)k +1(3) Al e na i ely, we can di ide (3) by h , which is gi en a ime , and ob ain he agg ega e easibili y cons ain in pe iod measu ed in e ms o ou pu pe uni o e icien labou : (˜ k )=˜cm +˜co e(˜ d −1)(1 +n)+(1 +n)˜ d +e(˜ d )(1 +n)˜ k +1(4) whe e ˜cm =cm /h and ˜co =co /h −1deno e espec i ely consump ion when middle-aged and con- sump ion when old-aged pe uni o e icien labou .1No e ha h +1/h =e(˜ d )=1+g +1, whe e g +1is he g ow h a e o p oduc i i y om pe iod o pe iod +1. Along a balanced g ow h pa h, all a iables exp essed in e ms o ou pu pe uni o na u al labou a e g owing a a e g. In consequence, all a iables exp essed in e ms o ou pu pe uni o e icien labou emain cons an : ˜cm =˜cm +1=˜cm, ˜co =˜co +1=˜co,˜ k =˜ k +1=˜ kand ˜ d =˜ d +1=˜ d. 1No e ha cm L and co L −1a e exp essed in uni s o ou pu . Since middle-aged indi iduals supply one uni o na u al labou , cm and co a e exp essed in uni s o ou pu pe uni o na u al labou . The in e p e a ion o ˜cm and ˜co in e ms o uni s o ou pu pe uni o e icien labou ollows na u ally. 3 3. The Social Op imum: he Golden Rule in he p esence o Endogenous G ow h In he p esence o p oduc i i y g ow h ha ansla es in o consump ion g ow h (as is, o cou se, he case along a balanced g ow h pa h), consump ion le els will g ow wi hou limi . I is o his eason ha , in o de o cha ac e ize he balanced g ow h pa h, consump ions (and all he o he a iables) ha e o be exp essed in e ms o ou pu pe uni o e icien labou . As a gued in he in oduc ion, an ob ious consequence o cm and co +1g owing o in ini y is ha a social planne will be unable o choose he consump ion le els ha maximize U =U(cm ,co +1) along a balanced g ow h pa h, because U ends o in ini y. The s anda d way o sides ep his p oblem is o assume ha he planne maximizes a discoun ed sum o u ili ies and o sea ch o he sequence o consump ions leading o he op imal balanced g ow h pa h. Assume, on he con a y, ha he social planne wan s o ea all gene a ions equally while being espec ul wi h indi idual p e e ences. One possibili y is o he o conside a alua ion unc ion de ined o e ˜cm and ˜co +1. Since he u ili y unc ion (1) is homogeneous, we can w i e: ˜ U =U(˜cm ,˜co +1)=Ucm /h ,co +1/h =(1/hj )U(cm ,co +1)=(1/hj )U (5) Thus, a ”new” u ili y unc ion is ob ained by means o a mono onic ans o ma ion o he i s one, his ensu ing ha o dinal p e e ences a e espec ed. The a gumen s o his u ili y unc ion a e measu ed in e ms o consump ion pe uni o e icien a he han na u al labou . No ice ha (5) and (1) ha e he same unc ional o m, and a e bo h homogeneous o deg ee j. Also, he slope, cu a u e and highe de i a i es o indi e ence cu es in (˜cm ,˜co +1) space a e he same as hose o he co esponding indi e ence cu es in (cm ,co +1) space. We can now posi ha he social planne ’s objec i e is o choose he balanced g ow h pa h ha maximizes he wel a e o a ep esen a i e indi idual, as measu ed by (5), subjec o he cons ain ha e e yone a ains he same u ili y le el, ˜ U=U(˜cm,˜co). We adop his app oach, which is eminiscen o Diamond (1965)’s o iginal ea men o he Golden Rule in an OLG amewo k wi h p oduc i e capi al.2No e ha , as shown by (5), al hough bo h U and hj end o in ini y, he a io ˜ U=(1/hj )U is well de ined along a balanced g ow h pa h. Then, he social planne will choose (˜cm,˜co,˜ k,˜ d) ha maximize U(˜cm,˜co) subjec o he balanced g ow h pa h e sion o (4) and he echnological ela ionship 1 +g=e(˜ d). We ob ain, om he i s o de condi ions: ∂U(˜cm ∗,˜co ∗)/∂˜cm ∂U(˜cm ∗,˜co ∗)/∂˜co=(1+g∗)(1 +n) (6) 0(˜ k∗)=(1+g∗)(1 +n) (7) e0(˜ d∗)      ˜co ∗ (1+g∗)(1 +n)2−˜ k∗     =1 (8) 2Phelps’ Golden Rule iden i ies he amoun o physical capi al ha maximizes consump ion pe capi a (Phelps, 1961). A wide iew o he Golden Rule concep , sugges ed by Diamond (1965) and Samuelson (1968, 1975a and 1975b) in an OLG model wi hou p oduc i i y g ow h, ocuses on he esou ce alloca ion ha maximizes he li e ime wel a e o a ep esen a i e indi idual subjec o he cons ain he e e yone else’s wel a e is ixed a he same le el. The esul ing wo-pa Golden Rule encompasses bo h Phelps’ Golden Rule and Samuelson(1958)’s biological in e es a e. 4 ˜cm ∗+˜co ∗ (1+g∗)(1 +n)= (˜ k∗)−(1+g∗)(1 +n)˜ k∗−(1 +n)˜ d∗(9) 1+g∗=e(˜ d∗) (10) De ini ion 1. The Golden Rule balanced g ow h pa h (˜cm ∗,˜co ∗,˜ k∗,˜ d∗)p o ides he maximum le el o wel a e ˜ U=U(˜cm,˜co) ha can be achie ed by a ep esen a i e indi idual subjec o he ea- sibili y cons ain and he addi ional cons ain ha e e yone else a ains he same le el. I is cha ac e ized by exp essions (6)-(10). The in e p e a ion o hese equa ions is simple i we s a om he exogenous p oduc i i y g ow h se ing, ha we ob ain o a gi en gand (wi hou loss o gene ali y) ˜ d=0. Then, he Golden Rule is cha ac e ized by (6), (7) and (9) wi h ggi en and ˜ d=0 (see Bui e , 1979). Equa ion (6) is he equali y o he ma ginal a e o subs i u ion be ween second and hi d pe iod consump ions and he coun e pa in he cu en model o he so-called ”biological” in e es a e, i.e., he economy’s g ow h a e. Equa ion (7) is he equali y o he ma ginal p oduc o physical capi al (pe uni o e icien labou ) and he g ow h a e o labou measu ed in e iciency uni s (i.e., he sum o he a es a which he e iciency o labou and he na u al uni s o labou espec i ely g ow). O cou se, i g=0 we a e back o Diamond’s amewo k and we ob ain he o iginal wo-pa Golden Rule. Tu ning now o he endogenous g ow h amewo k, (6) and (7) con inue o hold wi h he same in e p e a ion, wi h g∗being ob ained om (10). Equa ion (8), howe e , equi es a ca e ul ex- plana ion. I poin s ou ha , along he op imal balanced g ow h pa h, he ma ginal bene i o an inc ease in he amoun o ou pu de o ed o educa ion (again pe uni o e icien labou ) mus equal i s ma ginal cos . This can be seen using he agg ega e easibili y cons ain (9). A ise in ˜ dhas a di ec cos in e ms o he hi d e m in he igh hand side o (9), as i educes consump- ion possibili ies by (1 +n). I also has an indi ec cos , gi en by e0(.)(1 +n)˜ kas a consequence o he e ec o a ise in ˜ don he a e o g ow h g: indeed, he g ea e he p oduc i i y g ow h a e g, he g ea e he amoun o ou pu ha mus be de o ed o in es men in physical capi al in o de o keep ˜ kcons an . Howe e , a ise in ˜ d(and hus in g) also has a bene i , since a g ea e gimplies a g ea e ma ginal a e o ans o ma ion be ween second and hi d pe iod consump ion in he LHS o (8). This amoun s o an expansion o consump ion possibili ies ha is cap u ed by e0(.)(1+n)˜co ∗/(1+g∗)(1 +n)2. A he op imum, he ma ginal bene i and he ma ginal cos s mus be equal: (1 +n)e0(˜ d∗)˜co ∗ (1+g∗)(1 +n)2=(1 +n)e0(˜ d∗)˜ k∗+(1 +n) (11) The e ms in ol ing (1 +n) in (11) cancel ou and (8) eme ges. Fo la e use, (8) can be ew i en, using (7) and (9), as e0(˜ d∗) (˜ k∗)−˜ k∗ 0(˜ k∗)−Λ∗(˜ k∗,˜ d∗)= 0(˜ k∗) (12) wi h Λ∗(˜ k∗,˜ d∗)=(1+g∗)(1 +n)˜ k∗+(1 +n)˜ d∗+˜cm ∗>0. No ice ha , in he exogenous g ow h amewo k, ˜ k∗is uni ocally de e mined by (7) and he op imiza ion p oblem can be sol ed sequen ially. In con as , wi h endogenous g ow h, (7) and (8) 5 a e no enough o de e mine ˜ k∗and ˜ d∗because (8) inco po a es also ˜co ∗, which can only be ob ained om (6) and (9). A sequen ial solu ion o he op imiza ion p oblem is now impossible: all op imal a iables a e de e mined simul aneously. To conclude his sec ion, no e ha he de ini ion o he social op imum ha is posi ed in his pape di e s in se e al ways om he s anda d one. Fi s , he op imal alloca ion esul ing om maximizing P∞ =0γ U(cm ,co +1) depends upon he choice o a pa icula social discoun ac o γ. Second, he in e - empo al ajec o y leading o he op imal balanced g ow h pa h depends on he ini ial condi ions. Thi d, he op imum co esponding o he s anda d app oach is con ingen upon he pa icula deg ee o homogenei y o he u ili y unc ion, and hus he p ecise ca dinaliza ion o p e e ences, (a poin s essed in Del Rey and Lopez-Ga cia, 2012). In con as , when one adhe es o he social planne ’s objec i e unde lying De ini ion 1, he social op imum is independen o he choice o an a bi a y social discoun a e and he e is no need o choose a speci ic ca dinaliza ion o u ili y. Finally, he ocus is on he choice o he bes op imal balanced g ow h pa h, his implying ha he ole o he ini ial condi ions is no an issue. 4. Decen alized Ma ke Equilib ium wi h Go e nmen In his sec ion we analyse he beha iou o he economy in he p esence o educa ion subsidies and in e gene a ional ans e s and cha ac e ize he equilib ium balanced g ow h pa h o gi en alues o he policy pa ame e s. The p ope ies o he laissez- ai e balanced g ow h pa h a e discussed in sec ion 5. Since indi iduals no only decide abou he alloca ion o hei esou ces along hei li e cy- cle bu also how much o in es in educa ion, he e a e now wo po en ial sou ces o di e gence be ween socially op imal and indi idual choices. This is he eason why we posi ha he go e n- men has wo policy ins umen s a i s disposal: educa ion subsidies and in e gene a ional ans e s om he middle-aged o he elde ly. Among he di e en ways o ackling educa ion subsidies, we choose o model hem as subsidies o he epaymen , in he second pe iod o li e, o he loans aken in he i s one o pay o educa ion. Le zm >0 [ esp. <0] be he lump-sum ax [ ans e ] he middle aged pay [ ecei e], zo >0 [<0] he lump-sum ax he old pay [ he pension hey ecei e] and le θ be he subsidy a e, all o hem in pe iod . The laissez- ai e equilib ium can hen be e ie ed by se ing zm =zo =θ =0.3 Fac o p ices a e de e mined unde pe ec compe i ion by hei ma ginal p oduc s, so ha , i 1+ and w a e espec i ely he in e es ac o and he wage a e pe uni o e icien labou , 1+ = 0(˜ k ) (13) w = (˜ k )−˜ k 0(˜ k ) (14) Indi iduals choose, in hei i s pe iod, he amoun o educa ion ha maximizes hei li e ime esou ces. They do so by bo owing any amoun hey wish in pe ec c edi ma ke s. Conce ning 3I is wo h emphasizing ha , in his pape , he go e nmen subsidises he epaymen o educa ion loans. Al e na- i ely he go e nmen can di ec ly subsidise expendi u es in educa ion in he i s pe iod. Main esul s conce ning he op imal educa ion subsidy a e howe e obus o al e na i e designs o his policy. See below. 6 sa ings, hey beha e as pu e li e-cycle s, i.e., hey sa e o ans e pu chasing powe om he second o he hi d pe iod. Then, o an indi idual bo n a −1, consump ion when middle-aged and consump ion when old can be w i en espec i ely cm =w h −(1 + )d −1(1 −θ )−zm −s (15) co +1=(1 + +1)s −zo +1(16) whe e s a e he sa ings o a middle-aged. Thus, he li e ime budge cons ain o an indi idual bo n a pe iod −1 is: cm +co +1 1+ +1 =w h −(1 + )d −1(1 −θ )−zm − zo +1 1+ +1 (17) The i s o de condi ions associa ed wi h he indi idual decision a iables, d −1,cm and co +1, a e: w e0(d −1/h −1)=(1 + )(1 −θ ) (18) ∂U(cm ,co +1)/∂cm ∂U(cm ,co +1)/∂co +1 =(1 + +1) (19) whe e use has been made o he homogenei y o deg ee one o he E unc ion, i.e., h =e(d −1/h −1)h −1. Equa ion (18) shows ha he indi idual will in es in educa ion up o he poin whe e he ma ginal bene i in e ms o second pe iod ea nings equals he ma ginal cos o in es ing in human capi al allowing o subsidies. Rew i ing (18) as e0(˜ d −1)=(1 −θ ) (1 + (˜ k ))/w(˜ k ), his exp ession im- plici ly cha ac e izes he op imal a io ˜ d −1as a unc ion o ˜ k and θ , i.e., ˜ d −1=φ(˜ k , θ ).Since e00 <0 i can eadily be shown ha he g ea e ˜ k and θ a e, he g ea e ˜ d −1is. The go e nmen inances educa ion subsidies wi h e enues ob ained om axing he middle- aged and/o he old-aged: zm L +zo L −1=θ (1 + )d −1L (20) which plugged in o (17) yields cm +co +1 1+ +1 =ω (21) whe e ω is he p esen alue o he ne li e ime income o an indi idual bo n a −1: ω =w h −(1 + )d −1(1 −θ )−zm −1+n 1+ +1 [θ +1(1 + +1)d −zm +1] (22) The homogenei y assump ion on p e e ences implies ha he co +1/cm a io is a unc ion o +1only. Subs i u ed in o he budge cons ain (22) his allows us o w i e consump ion in he second pe iod as a ac ion o li e ime income, cm =π( +1)ω .Equilib ium in he ma ke o physical capi al is achie ed when he (physical) capi al s ock a ailable in +1,K +1, equals g oss sa ings made by he middle-aged in ,s L , minus he amoun o ou pu de o ed o human capi al in es men by he young in ,(1 +n)d L ,i.e., when K +1=s L −(1 +n)d L (23) 7 Using (23), he budge cons ain s (15) and (16), he go e nmen budge cons ain (20) and he equilib ium ac o p ices (13) and (14), one can ob ain he easibili y cons ain (3). Since s =w h −cm −(1 + )d −1(1 −θ )−zm , equilib ium condi ion (23) can be w i en in e ms o ou pu pe uni o e icien labou : ˜ k +1=(1−π( +1))˜ω e(φ(˜ k +1, θ +1))(1 +n) − ˜zm +1 1+ +1 − (1−θ +1)φ(˜ k +1, θ +1) e(φ(˜ k +1, θ +1)) (24) whe e ˜ω =ω /h and use has been made o he ac ha d =φ(˜ k +1, θ +1)h . This exp ession implici ly p o ides ˜ k +1as a unc ion Ψ(˜ k ; ˜zm ,˜zm +1, θ , θ +1). Along a balanced g ow h pa h, we can dele e he ime subsc ip s and w i e ˜ k= Ψ(˜ k; ˜zm, θ). An equilib ium a io o physical capi al o labou in e iciency uni s along a balanced g ow h pa h in he p esence o go e nmen in e en ion, ˜ kG, will hen be a ixed poin o he Ψ unc ion, i.e., ˜ kG= Ψ(˜ kG; ˜zm, θ). Such an equilib ium will be locally s able p o ided ha 0 < ∂Ψ(˜ kG; ˜zm, θ)/∂˜ k<1.In wha ollows, we will ocus on si ua ions whe e he equilib ium is unique and s able so ha he ela ionship be ween ˜ kand he ax pa ame e s can be w i en, wi h an ob ious no a ion, as ˜ k=˜ k(˜zm, θ) (25) We can now u n o he de e mina ion o ˜ do , wha is he same, he g ow h a e g. The amoun o ou pu de o ed o educa ion pe uni o inhe i ed human capi al along a balanced g ow h pa h will be go e ned by he ela ionship a ising om he educa ion decision (18), i.e., ˜ d=φ(˜ k, θ). Using (25) we can w i e ˜ d=φ˜ k(˜zm, θ), θo , o sho , ˜ d=˜ d(˜zm, θ) (26) Le ing gGbe he g ow h a e o any a iable exp essed in e ms o ou pu pe uni o na u al labou , since 1 +g=e(˜ d), we ha e 1 +gG=eφ(˜ kG, θ). Finally, he g ow h a e o all a iables exp essed in absolu e e ms (physical capi al, human capi al and ou pu ) is (1 +gG)(1 +n). This ollows om w i ing H[K] as hL [kL] and obse ing ha h[k] g ows a a e gGwhile Lg ows a a e n. Summa izing, he balanced g ow h pa h in he p esence o go e nmen in e en ion is cha ac- e ized by (25) and (26) sa is ying ∂U(˜cm,˜co)/∂˜cm ∂U(˜cm,˜co)/∂˜co=(1 + ) (27) we0(˜ d)=(1 + )(1 −θ) (28) ˜cm+˜co 1+ =˜ω(29) whe e ˜ω≡ω/his gi en by ˜ω=w−(1 + )˜ d (1 +g)+"(1 + )−(1 +g)(1 +n) 1+g#θ˜ d−"(1 + )−(1 +g)(1 +n) 1+ #˜zm(30) and ep esen s he p esen alue o li e ime esou ces exp essed in e ms o ou pu pe uni o e icien labou . 8