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