Albagli, Elías; A ias Gu ié ez, Agus ín H.; Ki chne , Ma kus
A icle
Collec i e sa ings pension policy in an economy wi h
he e ogenei y and in o mali y
Es udios de Economía
P o ided in Coope a ion wi h:
Depa men o Economics, Uni e si y o Chile
Sugges ed Ci a ion: Albagli, Elías; A ias Gu ié ez, Agus ín H.; Ki chne , Ma kus (2024) : Collec i e
sa ings pension policy in an economy wi h he e ogenei y and in o mali y, Es udios de Economía,
ISSN 0718-5286, Uni e sidad de Chile, Depa amen o de Economía, San iago de Chile, Vol. 51, Iss. 2,
pp. 325-381
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325
Es udios de Economía, Vol.51 - Nº 2, Diciemb e 2024. Págs 325-381
Collec i e Sa ings Pension Policy in an Economy wi h
He e ogenei y and In o mali y*
Sis ema de pensiones de aho o colec i o en una economía con
he e ogeneidad e in o malidad
ELÍAS ALBAGLI**
AGUSTÍN H. ARIAS***
MARKUS KIRCHNER****
Abs ac
We compa e he mac oeconomic e ec s o a ully unded indi idual de ined
con ibu ion (IDC) pension scheme, an un unded pay-as-you-go (PAYG) sys-
em, and a collec i e de ined con ibu ion (CDC) egime. Unde he la e , con-
ibu ions o wo ke s om a gi en coho a e in es ed in capi al ma ke s and
epaid o ha coho upon e i emen ; i s collec i e na u e a ises om an in a-
gene a ional p og essi e edis ibu i e ule. Ou esul s om an o e lapping
gene a ions model calib a ed o Chile show ha he CDC scheme has simila
mac oeconomic e ec s as an IDC plan, including a mode a e posi i e e ec on
he o mal labo ma ke , agg ega e sa ings, and ou pu . The PAYG sys em has
nega i e e ec s on all hese dimensions. C i ical o he success o he CDC
scheme is condi ioning bene i s on con ibu ions, o incen i ize o mal labo
s a us. We conclude ha a CDC policy s ands as a sus ainable al e na i e o
coun ies wi h signi ican labo in o mali y and income inequali y.
Key wo ds: O e lapping gene a ions models; Pension sys em; In o mal labo
ma ke ; He e ogeneous agen s.
JEL Classi ica ion: E26, E27, H55, J46.
*
**
***
****
The iews and conclusions exp essed in his pape a e exclusi ely hose o he
au ho s and do no necessa ily e lec he posi ion o he Cen al Bank o Chile o i s
Boa d membe s. The au ho s would like o hank Juan Gue a, Ma ías Tapia, Albe o
Naudon and Ignacio Rojas o hei commen s and suppo , as well as Alejand a Cox
and semina pa icipan s a he con e ence “50 yea s o Es udios de Economía” a
he Uni e si y o Chile, he 28 h Colloquium on Pensions and Re i emen Resea ch
and he Cen al Bank o Chile o use ul commen s and sugges ions.
Mone a y Policy Di ision, Cen al Bank o Chile, [email p o ec ed].
Economic S udies A ea, Cen al Bank o Chile, [email p o ec ed].
Mac oeconomic Analysis A ea, Cen al Bank o Chile, [email p o ec ed].
Recei ed: Augus , 2023 Accep ed: Augus , 2024
326 Es udios de Economía, Vol.51 - Nº 2
Resumen
Se compa an los e ec os mac oeconómicos de sis emas de pensiones de capi a-
lización indi idual, epa o y de capi alización colec i a. Bajo es e úl imo, las
con ibuciones de los abajado es de una coho e de e minada son in e idas
en los me cados de capi ales y u ilizados pa a paga las pensiones de esa
misma coho e una ez que es a se e i a. Su na u aleza colec i a su ge de una
egla de edis ibución in agene acional p og esi a. Nues os esul ados ba-
sados en un modelo de gene aciones solapadas calib ado pa a Chile mues an
que es e sis ema iene e ec os mac oeconómicos simila es al égimen de capi-
alización indi idual, incluyendo un e ec o posi i o mode ado sob e el me -
cado labo al o mal, el aho o ag egado y la ac i idad. El sis ema de epa o
iene e ec os nega i os en odas es as dimensiones. Cen al pa a el desempeño
exi oso del sis ema de capi alización colec i a es el condiciona pensiones
en las con ibuciones de los abajado es, de mane a de incen i a el abajo
o mal. Concluimos que la implemen ación de un sis ema de capi alización
colec i a es una polí ica sos enible pa a países con signi ica i a in o malidad
labo al y desigualdad de ing eso.
Palab as cla e: Modelos de gene aciones solapadas; Sis ema de pensiones;
Me cado labo al in o mal; Agen es he e ogéneos.
Clasi icación JEL: E26, E27, H55, J46.
1. INTRODUCTION
In many coun ies, demog aphic changes, economic ans o ma ions and
social demands o be e li ing condi ions pu inc easing p essu e on exis ing
pension egimes. Al hough hese challenges a e aced by bo h de eloped and
de eloping coun ies, he la e o en need o cope wi h hem in an en i onmen
o high-income inequali y, signi ican deg ees o labo in o mali y and iscal
sus ainabili y conce ns in he mids o ageing popula ions and changing in e -
na ional in e es a es. Gi en his con ex , his pape examines he e ec s o
al e na i e pension sys em e o ms in an economy wi h la ge labo he e ogene-
i y in e ms o income and employmen s a us. We se up a gene al equilib ium
o e lapping gene a ions model (OLG) ea u ing he e ogeneous agen s and a
dual labo ma ke calib a ed o he Chilean economy, which se es as ou case
s udy, and we use his model o quan i a i ely e alua e he long- e m mac o-
economic e ec s o al e na i e pension e o ms, compa ing indi idual capi al-
iza ion wi h collec i e pension schemes wi h a edis ibu i e componen .
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Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
Collec i e de ined con ibu ion (CDC) schemes a e one so o collec i e
pension policy. Unde such a scheme, con ibu ions o wo ke s om a gi -
en coho a e pooled in o one common und and in es ed in capi al ma ke s,
and he p oceeds a e epaid o he same coho upon e i emen . This ype o
scheme has been p oposed as an al e na i e o indi idual de ined con ibu ion
(IDC) p og ams, whe e con ibu ions a e deposi ed in o indi idual accoun s,
in o de o imp o e isk sha ing. CDC schemes a e o en conside ed a “ hi d
way” be ween IDC schemes, whe e all isks a e aken on indi idually, and
de ined bene i (DB) schemes such as many pay-as-you-go (PAYG) sys ems
ha p opose a secu e e i emen income bu a e mo e di icul o main ain,
especially in he ace o slowing popula ion g ow h. A simila i y o CDC
schemes wi h IDC schemes is ha hey a e ully unded, con a y o un und-
ed PAYG sys ems. Howe e , unlike IDC schemes, CDC schemes imply some
in agene a ional edis ibu ion. The p ope ies o collec i e pension schemes
ha e been analyzed om di e en pe spec i es in he li e a u e, including isk-
sha ing o s abili y o pa icipa ion i pa icipa ion is olun a y (see Go don
and Va ian, 1988; Shille , 1999; Ball and Mankiw, 2007; Gollie , 2008; Cui e
al., 2011; Chen e al., 2016, 2017; Ku begu, 2018). Howe e , o ou knowl-
edge no exis ing s udy has a emp ed o quan i y he mac oeconomic e ec s o
such schemes, which a e c i ical o e alua e hei o e all cos s and bene i s in
compa ison o al e na i e pension policies. In addi ion, no s udy has analyzed
hese issues, no he design o in agene a ional edis ibu i e ules o pension
bene i s, o economies wi h la ge in o mal labo ma ke s and income he e o-
genei y, such as in many de eloping coun ies.
Hence, in his pape we s udy he long- e m mac oeconomic e ec s o im-
plemen ing al e na i e pension sys em e o ms in an economy wi h a signi i-
can in o mal sec o and income he e ogenei y. To unde s and hese e ec s,
and o quan i a i ely e alua e hem, we cons uc an OLG model wi h speci ic
ea u es ele an o de eloping and eme ging economies. We calib a e he
model o Chile, a coun y whose indi idual e i emen pension accoun s sys-
em has been a model o many coun ies.1 We conside ou di e en pension
schemes, all inanced h ough an iden ical inc ease o pension con ibu ions
aking he o m o a pay oll ax, ha solely di e om each o he in he way
hey ea he addi ional unds and alloca e hem among e i ees. In pa icula ,
we conside wo e sions o a CDC scheme ha di e in whe he hey make
pension bene i s depend upon he deg ee o labo e o du ing wo king li e,
1 Chile has swi ched o a p i a e e i emen accoun s sys em in he ea ly 1980s. Many
o he La in Ame ican coun ies ha e ollowed he Chilean model. These include (wi h
yea s o adop ion in pa en heses): Pe u (1993), Colombia (1994), A gen ina (1994),
U uguay (1996), Boli ia (1997), Mexico (1997), El Sal ado (1998), Cos a Rica
(2001), he Dominican Republic (2003), Nica agua (2004), and Ecuado (2004) (see
K asnoku skaya e al., 2018).
328 Es udios de Economía, Vol.51 - Nº 2
i.e., a condi ional (C-CDC) scheme and an uncondi ional (U-CDC) scheme,
and compa e hei mac oeconomic pe o mance o an IDC scheme and a
PAYG al e na i e.2 The collec i e na u e o he CDC schemes a ises om a
p og essi e edis ibu i e ule ha alloca es p opo ionally mo e bene i s o
lowe -income wo ke s. These a e also he wo ke s wi h mo e pa icipa ion in
in o mali y. Condi ioning he eceip o bene i s on employmen s a us wi hin
a edis ibu i e design is he key aspec ha incen i izes a s ong o maliza ion
o labo supply a low-income le els, which a e hose who make o he la ges
sha e o in o mal wo k, su passing he e ec s ound unde he IDC plan.
Ou esul s show ha he C-CDC scheme has simila mac oeconomic e -
ec s as an IDC plan, including a mode a e posi i e e ec on he o mal labo
ma ke , which oge he wi h he ise o compulso y sa ings and he capi al
s ock, gene a es and expansion o ou pu . Mo eo e , he C-CDC scheme p o-
duces a s onge educ ion o in o mali y han he IDC plan. While he C-CDC
scheme also induces opposi e incen i es a highe wages, he lowe o mal
sec o pa icipa ion a he low end o he skill dis ibu ion deli e s an o e all
inc ease o labo o mali y in he economy. The mac oeconomic pe o mance
o he C- CDC policy is signi ican ly be e han ha o he PAYG sys em,
which has a s ong nega i e e ec on all dimensions. The condi ionali y o
pension bene i s upon labo e o unde he C-CDC plan is c i ical o hese
posi i e esul s, as a compa ison wi h he U- CDC scheme shows. Fu he -
mo e, he C-CDC plan sha es wi h he IDC al e na i e he abili y o cope wi h
de e io a ion o he old-age dependency a io, con a y o he PAYG sys em.
S uc u al models, as he one we de elop he e, a e well sui ed o he
analysis o pension policies, as well as o o he public policies, o se e al
easons. Fi s , hey allow o consis en exe cises, in he sense ha i a gi -
en policy has, o ins ance, an impac on he labo supply o households bu
his e ec depends, in u n, on how he policy a ec s hei sa ing decision, a
s uc u al model will be able o cap u e he in e ac ion be ween he decisions
on bo h ma gins, some hing ha canno be add essed in models whe e hese
decisions a e aken sepa a ely. Among o he bene i s, his consis ency allows
unde s anding he gene al equilib ium e ec s o a gi en policy. Fo example,
changes in agen s’ p opensi ies o sa e implied by al e na i e pension e o ms
ha e di e en e ec s on capi al accumula ion in equilib ium, and h ough his
channel, on wages and labo ma ke ou comes. Wi hou a s uc u al model, i
would be impossible o de e mine how hese second- ound e ec s ope a e, o
whe he hey a e ele an . In addi ion, a s uc u al model pe mi s a quan i a i e
e alua ion o he e ec s o al e na i e policies, as long as i s pa ame e s a e ap-
p op ia ely calib a ed (es ima ed) o cap u e he main cha ac e is ics o agen s
and o he ele an ma ke s a play.
2 All o hese op ions — IDC, CDC and PAYG schemes — ha e been unde discussion
as pa o he p oposals o e o m he Chilean pension sys em o e he las yea s.
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Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
Since a ini e li e- ime ho izon o households is c i ical o cap u e he
labo , sa ing and consump ion decisions o agen s o e bo h a wo king and
a e i emen pe iod, we depa om a s anda d OLG model wi h h ee gen-
e a ions. In o ha amewo k we inco po a e an in o mal sec o , which can
be hough o as home p oduc ion, and whe e wo ke s a e exemp o paying
pension con ibu ions and axes. The in oduc ion o his dimension esponds
o he signi icance ha in o mali y has in many de eloping and eme ging econ-
omies; in Chile, o example, he in o mal economy accoun s o abou a hi d
o o al employmen . Faced wi h highe pension con ibu ions, agen s will ha e
incen i es o swi ch o in o mali y o a oid paying a pe cei ed ax — an in-
cen i e pa icula ly ele an o low-p oduc i i y households.3 We also allow
o di e en skill g oups and discoun ac o s, as well as di e en p oduc i -
i ies ac oss sec o s, pe mi ing us o be e ma ch he income dis ibu ion, he
obse ed sa ing a es, and labo ma ke pa icipa ion in he economy. These
sou ces o he e ogenei y in he model a e c ucial o unde s anding he dis i-
bu ional implica ions o he di e en pension e o ms ha we conside , as well
as o cap u ing he di e en ial e ec s ha a gi en policy can ha e on dis inc
agen s. Finally, he model ea u es ce ain small open economy cha ac e is ics
ha allow us, o example, o cap u e he weigh ed dependency o pension
ounds’ e u ns on bo h he domes ic and he in e na ional in e es a e; ollow-
ing he obse a ion ha abou 40% o he o al ounds’ esou ces a e in es ed
ab oad.4
Pension policies wo k mainly h ough manda o y con ibu ions and, he e-
o e, hei e ec s depend on he abili y o agen s o e e o ced sa ings
3 A anasio e al. (2011), when analyzing he 2008 pension e o m in Chile, ound ha
e en hough pensions inc eased, he p obabili y o paying in o he pension sys em
o wo ke s olde han 40 yea s dec eased by 4.1%. Camacho e al. (2014) ound ha
he componen co esponding o a 10% ax inc ease in Colombia in 1993 o inance a
heal h plan led o an inc ease o 4% in in o mali y.
4 Ou model ea u es a single in es men ins umen , p ecluding agen s om choosing
among di e en in es men po olios o hei pension unds. In Chile, unde he cu -
en IDC pension sys em, agen s can op be ween 5 al e na i es, which di e om each
o he in hei domes ic/ o eign, ixed/ a iable-income composi ion. The implemen a-
ion o an addi ional con ibu ion may a ec his composi ion and, hus, he unds’
e u ns. Fo simplici y and ac abili y, we abs ac om his dimension. Howe e , i is
wo h no icing ha a CDC scheme is no sui ed o an indi idual choosing he po o-
lio composi ion. I could be assumed ha he in es men s a egy ha such a scheme
would ollow would be simila o he de aul s a egy o he cu en sys em, i.e., when
agen s do no ac i ely choose a po olio, hei unds a e in es ed ollowing an age-de-
penden de aul s a egy. E idence sugges s ha allowing agen s o mo e hei unds
be ween po olio ypes — unde an IDC scheme — is de imen al o he unds’ pe -
o mance; see Con e as e al. (2022). As such, ou esul s may be biased in a o o he
IDC scheme. Ou main goal is o s udy he s uc u al di e ences be ween he di e en
pension schemes conside ed. The in es men s a egies hey would ollow lie beyond
he scope o his pape .
330 Es udios de Economía, Vol.51 - Nº 2
by educing olun a y ones. In his ega d, empi ical e idence sugges s ha
agen s do ace bo h inancial and in o ma ional ic ions ha limi hei access
o c edi o he deg ee in which hey associa e cu en con ibu ions wi h u-
u e pension paymen s and, consequen ly, hei abili y o willingness o educe
p i a e sa ings. To his end, we include a simple mechanism ha p oxies o
in o ma ional ic ions ha lead o an incomple e in e naliza ion o u u e pen-
sion bene i s. Akin o he ole o bo owing cons ain s, his mechanism p e-
en s manda o y sa ings om being comple ely o se by highe indeb edness
h oughou wo king li e. Indeed, he li e a u e sugges s ha , hough signi ican ,
subs i u ion o manda o y sa ings is no pe ec . This mechanism is c ucial o
ou esul s, as i makes agen s eac o pension con ibu ions as i hey we e,
pa ially, axes. A anasio and Rohwedde (2003), exploi ing he empo al and
c oss-sec ional a ia ion o h ee pension e o ms, ind ha his subs i u ion is
somewhe e be ween 0.65 and 0.75 o men olde han 45, weake o younge
men, and non-exis en o basic pensions, which sugges liquidi y cons ain s
o lack o in o ma ion. A anasio and B ugi iani (2003) ind simila esul s
o I aly. Bo azzi e al. (2006) u he ind ha subs i u ion is highe o mo e
in o med indi iduals. Fo Chile, Mo andé (1998) inds esul s in he same line,
wi h a deg ee o subs i u ion o abou 0.5.
Rega ding he deg ee o which agen s comp ehend pension sys ems and
hei own need o sa e, he li e a u e shows a la ge dispe sion among indi id-
uals, and ha many wo ke s igno e how hei pension plans ac ually wo k.
Lusa di (1999) shows how household sa ings in he Uni ed S a es a e ypically
insu icien when e i emen a i es. Gus man and S einmeie (2005), also o
he U.S., ind ha , on a e age, wo ke s’ expec a ions on hei u u e pensions
a e no s ongly aligned wi h eali y. O he empi ical e idence shows how in-
di iduals use cos ly c edi sou ces e en hough hey ha e access o less expen-
si e c edi s, o how hey simul aneously hold liquid asse s wi h low e u ns
and c edi ca d deb s. The explana ions o his kind o beha io ange om
lack o inancial educa ion o inconsis en in e empo al p e e ences; see, o
example, Laibson (1997), Angele os e al. (2001) and Campbell e al. (2011).
These easons a e consis en , in u n, wi h he so-called e i emen consump-
ion puzzle (o e i emen sa ings puzzle), which desc ibes he signi ican d op
in consump ion a e e i emen (A anasio and Webe , 2010). O he explana-
ions include a p oblem o p ocess in o ma ion, which is no he esul o lack
o in o ma ion and canno be o e come by public educa ion; see New (1999).
A salien poin o his pape is he s udy o he in e ac ion o he di e en
pension schemes wi h he in o mal sec o . As men ioned abo e, he in oduc-
ion o a new pension sys em c ea es incen i es o agen s o mo e in o in o -
mali y whe e hey can a oid he paymen o hose con ibu ions. Howe e , we
ind ha unde bo h he IDC and he C- CDC plans in o mali y ac ually de-
331
Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
c eases, and ha his e ec is s onge o he second scheme. This di e s om
he conclusions p esen ed in Joube (2015), who inds ha ising he pension
con ibu ion paid by wo ke s om 10 o 15% in Chile would inc ease in o -
mali y by abou 10%. The eason o he di e en esul s is he amewo k, in
pa icula , he p esence o wo ea u es. Fi s , in ou model agen s in e nalize
— hough impe ec ly — ha he newly in oduced pension con ibu ions will
ansla e in o u u e pensions. This has a lowe e ec on he o al ne e u n o
o mal labo han a plain ax ha is used by a go e nmen o, say, pay expenses,
a ec ing labo supply decisions less. Second, ou gene al equilib ium model
— Joube (2015) uses a pa ial equilib ium one — allows o second- ound e -
ec s ha ope a e expanding he economy and c ea ing incen i es o mo e in o
o mali y, whene e he in oduc ion o a pension plan inc eases o al sa ings.
These wo di e ences explain why he ul ima e esul we ind is a educ ion o
in o mali y unde he IDC and he C-CDC plans.
The s and o li e a u e ha add esses he e ec s ha social secu i y e-
o ms end o concen a e on pa icula aspec s, such as job mobili y, e i e-
men iming, sa ings, capi al ma ke de elopmen , and economic g ow h; see
Thomas and Spa a o (2016) o a e iew o econome ic wo ks ha conside
he e ec s o pension unds, and Kohl and O’B ien (1998) o a su ey on he
empi ical li e a u e on he e ec s o PAYG sys ems, mos ly on sa ings. In
his g oup o s udies, Holzmann (1997), using a Solow esidual speci ica ion
o TFP, inds ha he 1981 pension e o m in Chile had a posi i e e ec on
economic g ow h h ough he imp o emen o inancial ma ke s. In he same
line, Schmid -Hebbel (1998) concludes ha he same e o m boos ed p i a e
in es men , he a e age p oduc i i y o capi al and TFP. Encina (2013), using
he pension e o m o 2008 in Chile as a ea men and panel da a, inds ha
he imp o emen o he pension condi ions o he poo es indi iduals in Chile
induced a highe wi hd awal om he labo ma ke and less pe iods con ib-
u ing o he pension sys em by hese indi iduals. This is o be expec ed, as he
mechanism de e mining hose agen s’ pensions depends nega i ely on hei
pension le el (s emming om hei ac ual con ibu ions). We ea u e such a
mechanism in ou U-CDC scheme, and a simila o ce is a play, hough o he
complemen a y speci ica ions and ini ial condi ions deli e di e en esul s in
some dimensions. Da is and Hu (2008), in u n, using a panel o 38 OECD and
eme ging economies ind ha he pension-asse s- o-GDP a io has a signi i-
can posi i e e ec on g ow h. Ou s udy con ibu es o his li e a u e h ough
an analysis o he gene al equilib ium e ec s o al e na i e pension schemes in
a model wi h a dual labo ma ke and income he e ogenei y.
Pa allel and independen wo k by F assi e al. (2019) is closely ela ed o
ou pape . They s udy he e ec s on he labo ma ke and on capi al accumu-
la ion o h ee di e en pension plans: a PAYG sys em, and wo ully unded
332 Es udios de Economía, Vol.51 - Nº 2
ones, including an in agene a ional edis ibu i e componen . Howe e , we
di e om hei wo k in se e al dimensions: i s , we se up a iche cali-
b a ed model, p o iding an empi ically ealis ic quan i ica ion o he e ec s
o in oducing di e en pension plans. Second, we show he impo ance ha
inco po a ing in o ma ional and/o inancial ic ions has on he conclusions
eached; o example, con a y o hei indings, an IDC scheme is no longe
neu al on capi al, consump ion, GDP and he labo ma ke when agen s do no
ully in e nalize u u e pensions. Finally, and mos impo an ly, we conside
he e ec s ha pension plans ha e on in o mali y, a ma gin ha suppo s a
C-CDC- ype scheme.
O e all, ou esul s a e especially ele an o de eloping and eme ging
economies simila o Chile, which p esen signi ican labo in o mali y and
income inequali y. These esul s show ha p ope ly designed CDC schemes
may be an economically sus ainable and poli ically iable pension policy in
such a con ex .5
The es o he pape is o ganized as ollows. In he nex sec ion we p es-
en he de ails o he model, including he main equa ions ha desc ibe he
p oblems aced by he di e en agen s and hei op imal decision ules.6 In
sec ion 3, we discuss he calib a ion o he model. In sec ion 4, we p esen he
esul s o he quan i a i e analysis o he di e en e o m scena ios, oge he
wi h some obus ness analysis. In sec ion 5, we p esen an exe cise o analyze
how he di e en schemes pe o m when he popula ion g ow h a e dec eases.
Finally, sec ion 6 concludes.
2. THE MODEL
The model is based on he neoclassical g ow h models wi h o e lapping
gene a ions ollowing he wo ks o Samuelson (1958) and Diamond (1965). To
cap u e aspec s ele an o de eloping o eme ging economies and in pa icula
he Chilean case, we add se e al ea u es o an o he wise s anda d OLG mod-
el wi h h ee gene a ions. These ex ensions include, i s , endogenous labo
supply wi h he e ogeneous p oduc i i ies, as in B unne (1996) and Somma-
5 A limi a ion o ou model is ha i la gely abs ac s om iscal policy, since he only
iscal ins umen is a ax a e on o mal labo income o c ea e a wedge be ween ne
and g oss income. A iche ax s uc u e, including o he ins umen s such as axes on
i ms o consump ion, i s in e ac ion wi h in o mali y and i s ole in he inancing o
pension sys ems — pa ially o o ally, di ec ly o h ough deb — could be a ele an
ex ension. While his is an impo an opic ha dese es a en ion, i lies ou side he
scope o his pape . Addi ionally, o simplici y, we abs ac in he model om he
cu en solida i y pilla in Chile.
6 A de ailed desc ip ion o he model and he s eady s a e compu a ion is p o ided in he
appendix.
339
Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
while he indi idual pension paid ou o each indi idual when e i ed can
be w i en as
(11)
PRSRS
i
F
i
PF m
i
PF y
i i
,,
,
,
,
,,
2121 1
2242
PPP
F
BS
i
W
ii
23
22
01
3
1
max, ,
whe e 1
ii
< is an indica o unc ion aking he alue
1
when
ii<
and
0
o he wise, and whe e P
BS
+2 deno es a basic solida i y pension — exp essed in
pe capi a e ms — de ined as
(12) PP PN
N
BS
F
i
i
i
W
i
i
i
i
22 1
1
2
1
1
1
3
1
,,
,
The la e is ele an only o he hi d scheme (
31
). No ice P
BS
+2 is
de ined so as o exhaus P
F
+2 among all e i ees in
+2
.12 Fo he second
scheme (µ2 = 1), we de ine
(13)
i
i
p s
PTS
,
,
2
2
2
as he ac ion o he unds o he componen inanced by he i ms’
con ibu ions, P
F
+2, ha each e i ee o skill g oup
i
ecei es. He e
PTSp
s N
i
n
i
i
l
212
,,
a e he o al agg ega ed poin s assigned unde his
scheme and whe e he indi idual poin s, p si s,+, a e assigned acco ding o
he ollowing ule (ac ua ially ai ), which agen s know and in e nalize in hei
decisions:
(14)
p sRp sRp s
Rl
W
i s i
m
i
y
i
mi
m
,,,
,
,
211
20 1
1
WW
Rl
W
W
m i
yi
y
y
1
10
11
,
,
The pa ame e s
00
and
1
R con ol he edis ibu ion in ensi y o
he ule, and WNWN
y
i
n
i
y
i
l
11
/
,,
and
WNWN
m
i
n
i
m
i
l
111
1/
,,
12 Following he logic behind he solida i y pilla in place in Chile un il ecen ly, un-
de he hi d scheme (
31
), we assume ha hose agen s o whose sel - inanced
pension, Pi
W
,+2, is lowe han a ce ain uppe bound Pi
MAS
,+2 — om Pensión Máxima
con Apo e Solida io in Spanish — a e en i led o a pension acco ding o he o mula
PPPPP
i
F
MAS
i
W
BS
MAS
,,
max, /
2222 2
0, whe e P
BS
+2 is he highes pension ha any agen
could ecei e unde his pilla , i.e., when Pi
W
,
20. As a simpli ying assump ion, we
implemen his scheme assuming a cons an a io, PP
BS
MAS
22
13//
.
340 Es udios de Economía, Vol.51 - Nº 2
deno e he a e age sala ies o bo h age g oups.13 Fo he ou h scheme (
4
1
) we
de ine wo auxilia y a iables,
i ,2 and x +2, whe e he i s a iable de-
no es he ac ion o P
F
+2 co esponding o each wo ke and whe e he second
a iable deno es a eplacemen a e. The la e eme ges as he solu ion o he
ollowing sys em o equa ions:
(15)
i
F
i
m
i
m
P
RW l
xi
,
,,
22
211
2
i
i
05.�
(16) i
n
i i
lN
121
,,
Equa ion (15) ensu es ha P
F
+2
is dis ibu ed such ha all e i ees ge he
same eplacemen a e wi h espec o hei o mal middle-aged labo income,
while he second equa ion u he gua an ees ha all unds a e used. The solu-
ion o his sys em o equa ions is gi en by
(17)
i
i
m
i
m
m
RW l
WL
,
,,
2
211
1
(18) xP
WL
F
m
2
2
1
whe e WL
RN
m
W
i
i
nl
i
m
1211,
,.14
To al and indi idual pensions paid ou each pe iod a e he sum o bo h pen-
sion componen s and a e, espec i ely, gi en by:
(19) PP P
W
F
(20) PPP
i i
W
i
F
,,,
Public sa ings (in he o m o he pension ound) a e de ined as
(21)SS RS SSRS S
P
PW m
PW y
PW y
PF m
PF y
,,
,,
,
11231
PF y,
13 One can show ha in he special case whe e
g=0
, implying
WAWA
i
y
i
m
,,
//
1, he sec-
ond scheme (
2
1
) wi h pa ame e s
0
0
and
11
is equi alen o he i s scheme
o indi idual de ined con ibu ion (
11
) since, unde hese special condi ions, one
has ha
i
F
i
PF m
i
PF y
PRSRS
,,
,
,
,
22 21 1. Relaxing he condi ions on
e
and
g
, i.e.,
when
e<1
and/o
g>0
, bo h schemes a e only app oxima ely equi alen when
α
0
and
α
1 a e se o
0
and
1
espec i ely.
14 Depending on he de ini ion o he eplacemen a e, R +2 could be emo ed om he
abo e equa ions and om he de ini ion o WL
m
+1
. The p esence o he e m R
+2
is o
exp ess income in e ms o i s alue in he pe iod in which pensions a e paid.
341
Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
and he budge cons ain o he agg ega e pension ound in pe iod
is
gi en by
(22) PS RS SS
SS
P
P
PW m
PW y
PF m
PF y
1
,,
,,
le ied axes
2.4 Wo ke s o Skill G oup I
We add ess now he p oblems aced by he households in his economy.
Each wo ke ’s li e- ime u ili y is a unc ion o hei consump ion when young,
Ci
y
,, middle-aged, Ci
m
,+1, and old,Ci
o
,+2, o he labo hey supply in he o mal
sec o when young and middle-aged, l
i
y
, and li
m
,+1, and o he labo hey dedi-
ca e o home p oduc ion, also when young and middle-aged, h
i
y
, and hi
m
,+1.
Fo
in
l
1, ,, i akes he ollowing o m:
(23)
U
CCC
i
i
y
i
i
m
i
i
o
i
,
,,,
1
1
1
22
1
11 1
,,
,, ,
,
,,
yi
y
i
y
i
y
ii
mi
m
i
m
lh hlh
1
11
hi
m
,
1
whe e A
0
is he in e se o he elas ici y o in e empo al subs i u ion,
1
de e mines he wage elas ici y o labo supply, which equals
11
/
,
i
01, is he subjec i e discoun ac o o skill g oup i, and
0
s ands
as a speci ic cos o wo king a home.15 The a iable Θ
i
j
,, wi h
jym=,
, is an
endogenous p e e ence shi e based on Galí e al. (2012) ha is aken as gi en
by he wo ke s and sa is ies
i
y
i
i
y
AC
A
,
,
1
i
m
i
i
m
AC
A
,
,
1
11
wi h
01, and
i
0. The pu pose o his p e e ence shi e is o al-
low o an incomple e weal h e ec on labo supply. When
0
, we ob ain
i
j
i,
, and, hus, he s anda d cons an ela i e isk a e sion (CRRA) u il-
i y unc ion implying a non-ze o weal h e ec ; ins ead, when
1
, he e is
no weal h e ec .16
15 This speci ic cos can be jus i ied, o ins ance, by he lack o heal h insu ance, o by
job insecu i y.
16 In any case, he disu ili y o wo k is assumed o g ow wi h he ac o A
1
so ha he
model has a balanced g ow h pa h when
1
.
342 Es udios de Economía, Vol.51 - Nº 2
Home p oduc ion is implemen ed h ough a dec easing e u ns o scale p o-
duc ion unc ion, wi h labo as he only inpu ; wi h Ab ha
i
i
1
as he e u n
o bo h he young and middle-aged wo ke s, and wi h bb b
nn
ll
11
0.
This speci ica ion allows o labo p oduc i i y di e ences be ween he o mal
sec o and he in o mal sec o , while p oduc i i y g ow h is equal ac oss sec-
o s.
Le ing S
i
y
, and S
i
m
, deno e olun a y sa ings when young and middle-
aged, espec i ely, a gi en gene a ion’s budge cons ain s o pe iods
,
+1
,
and
+2
sa is y:
(24) CWlAbhah S
i
yW
i
y
i
y
i
ii
y
i
y
,,
,,
,
1
1
(25) CWlAbhah
i
mW
i
m
i
m
i
ii,,
,,
1111
1
1
1
m
i
y
i
m
RS S
11 1
,,
(26) CR
SP
i
o
i
m
i
,,
,
22
12
whe e τ is an income ax used o he calib a ion o he in o mal labo sup-
ply.17 Following Joube (2015), we will se
ξ
so ha a 5 pe cen age poin s
income ax inc ease yields a 10% inc ease in in o mal wo k. The budge con-
s ain s can be combined in o he pe cei ed in e empo al budge cons ain
(IBC) ha agen s use o make hei decisions:
(27)
He e, he pa ame e
i
01, de e mines he ac ion o he p esen alue
o he u u e pension, P
RR
i
,+
++
2
12
, ha agen s in e nalize in hei consump ion,
17 No e ha he hi d equa ion inco po a es he simpli ying assump ion ha indi iduals
ha e ini e ho izons and, he e o e, choose o end up wi h ze o asse s when hey die
(i.e., hey make no beques s).
343
Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
sa ing and labo decisions. This is a simple, educed- o m s a egy o cap u e
wo di e en ypes o ic ions, inancial and in o ma ional. Fo ins ance,
ϕ
i
can accoun o bo owing cons ain s, meaning ha only a ac ion o u u e
pensions can be used as colla e al. Al e na i ely,
ϕ
i
e lec s he igno ance o
agen s abou how he pension sys em wo ks, cap u ing, o example, he pe -
cep ion ha agen s may ha e o pension con ibu ions as axes, wi hou any
bene icial coun e pa . I could also cap u e, o ins ance, he con idence agen s
ha e on he u u e p omised paymen s o ac ually ma e ialize, which is no
necessa ily a negligible ac o , a leas in eme ging economies. The p ac ical
pu pose o his speci ica ion is o p e en a pe ec subs i u ion be ween man-
da o y sa ings — in he o m o pension con ibu ions — and olun a y sa -
ings, which is achie ed o hose agen s wi h
i
1.18
Wo ke s, hen, ace he p oblem o maximizing hei li e- ime u ili y (23)
subjec o (27), (20), (9), (11), (13) ,(14), (17) and he de ini ions o Si
kj
,
,, o
kPWPF=,
and
jym=,
, ea ing Θ
i
y
,, i
m
,1, W
i
y
,, Wi
m
,+
1
, P
BS
+2
, W
y, W
m
+1
, PTS +2, WL
m
+1, P
F
+2, R +1, and R +2 as gi en.19 Nex , we p esen he i s -
o de condi ions associa ed wi h his p oblem. Fi s , he equa ions o labo
supply in he o mal sec o , l
i
y
, and l
i
m
, a e gi en by
(28) i
y
i
y
i
y
i
y
y
lh
CW
,,
,,
,
1
l
(29) i
m
i
m
i
m
i
m
m
lh
CW
,,
,,
,
11 1
1
11
l
whe e ne sala ies a e de ined as
(30) WW
W
W
P
y
i
y
i
W
i
Fi
y
y
l
l
,,
,
,
11 12
1
2
1
1
FF
W
PTS
2
3
1
3
(31)
WW
W
W
m
i
m
i
W
i
Fi
m
l
l
,,
,
,
11
12
1
1
1
11
1
mm
F
W
F
m
P
PTS
P
WL
1
2
2
34
2
1
1
3
The abo e equa ions show how, a he op imum, agen s wo k up o he
poin whe e he disu ili y o wo king — bo h in he o mal and in o mal sec o
— equals he ma ginal u ili y o consuming he pe cei ed ne e u ns on ha
18 See he in oduc ion o a discussion o ela ed li e a u e.
19 We assume he e, o simplici y, ha agen s do no in e nalize he e ec s o hei labo
e o s on P
F
+2
. A simila assump ion is made in Sommacal (2006).
344 Es udios de Economía, Vol.51 - Nº 2
wo k, which depends on he pension scheme in place. Second, he equa ions
associa ed wi h labo supply in he in o mal sec o , h
i
y
, and h
i
m
,, a e gi en by
(32) h
CAb haW
i
y
i
y
i
y
i
ii
y
,
,
,
,
§
©
¨
¨
·
¹
¸
¸
§
©
¨
¨
¨
¨
¨
·
¹
¸
T
[
K
N
4
1
¸¸
¸
¸
¸
ª
¬
«
«
«
«
«
«
º
¼
»
»
»
»
»
»
§
©
¨
¨
·
¹
¸
¸!
1
1
1
0
I
[
K
i Ahba W
ii
y
,l
o he wise
®
°
°
°
°°
¯
°
°
°
°
(33) h
CAbhaW
i
m
i
m
i
m
i
ii
m
,
,
,
,
§
©
¨
¨
·
¹
¸
¸
1
1
1
1
1
1
T
[
K
N
4
§§
©
¨
¨
¨
¨
¨
·
¹
¸
¸
¸
¸
¸
ª
¬
«
«
«
«
«
«
º
¼
»
»
»
»
»
»
§
©
¨
¨
·
¹
¸
1
1
1
1
1
I
[
K
i Ah
¸¸ !
ba W
ii
m
l
,1
0
o he wise
®
°
°
°
°
¯
°
°
°
°
Acco dingly, a he op imum, agen s spend ime wo king in he in o mal
sec o only i he income hey ob ain om wo king in ha sec o is la ge han
wha hey pe cei e hey could ea n i hey we e o wo k in he o mal sec o .
Finally, we can de i e he ollowing consump ion equa ions o C
i
y
, and
C
i
m
,:
(34) C
Wl Wl
RAb h
i
y
W
i
y
i
yi
m
i
m
i
,
,,
,,
111
1
1
1
1
12
12
1
1
ah
RRR
ii
y
i i
,
///
/
+
Abhah
R
P
RR
R
i
ii
m
i
i
i
1
1
1
1
2
12
1
1
,
,
/
i
RR
1
12
12
1
///
(35)
C
RS Wl Abh
i
m
i
yW
i
m
i
m
i
,
,,,
1
1111
1
1
ah P
R
R
ii
m
i
i
i
,
,
//
1
2
2
1
2
1
1
The consump ion when old, Ci
o
,+2, is de e mined by equa ion (26). The e-
o e, young and middle-aged wo ke s consume a ac ion o he p esen alue
o hei pe cei ed o al li e- ime income (which dec eases as
ϕ
i
dec eases,
implying a la ge ic ion), and sa e he es . When old, he wo ke s simply
345
Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
consume all hei sa ings plus, i applicable, he pension ecei ed om he
go e nmen .
2.5 Agg ega ion
We now close he model by speci ying a numbe o de ini ions and ag-
g ega e condi ions, ha a e equi ed o hold in equilib ium. To al o mal labo
supply o each skill ype and coho needs o equal he co esponding labo
demand, which equi es
(36) lN L
i
y
i i
y
,, ,
=
(37) lN L
i
m
i i
m
,, ,
1
o
in
l
1, ,.20 Agg ega e olun a y p i a e sa ing is de ined as
(38) SSS
y
m
whe e
SS
N
y
i
n
i
y
i
l
1
,,
and
SS
N
m
i
n
i
m
i
l
11
,,
, while manda o y pub-
lic sa ing, S
P, is de ined in (21). Gi en ha only a ac ion
1
o o al sa -
ings, SSS
T
P
, is in es ed in o capi al, while he emaining ac ion
γ
is
in es ed ab oad, capi al each pe iod sa is ies
(39)
KS
T
11
*
No ice ha we allow, in addi ion, o o eigne s o own some exogenous
ac ion o he domes ic capi al s ock,
AN
**
, whe e
*0.21 This
las assump ion, oge he wi h he assump ion s a ing ha pa o domes ic
sa ings a e in es ed ab oad, amoun — as al eady men ioned — o a simple
and s ylized s a egy o model inancial openness in a small open economy as
he Chilean one.22
Finally, agg ega e consump ion in pe iod
is de ined as
(40) CC
CC
y
m
o
whe e
CC
N
y
i
n
i
y
i
l
1
,,
,
CC
N
m
i
n
i
m
i
l
11
,,
and
CC
N
o
i
n
i
o
i
l
12
,,
20 No e ha Ni , young indi iduals o ype
i
a e bo n each pe iod
, while, also in pe iod
, he e a e Ni ,−1 middle-aged indi iduals, and Ni ,−2 old indi iduals, bo h o ype
i
.
21 We assume
γ
*
g ows wi h AN
, in o de o ha e a balanced g ow h pa h.
22 I is possible o show ha his way o modelling inancial openness, implies ha he
in e es a e di e en ial is dec easing in he le el o domes ic sa ings.
346 Es udios de Economía, Vol.51 - Nº 2
a e he co esponding agg ega e consump ion le els o each coho . And, ag-
g ega e axes amoun o
TWLWL
i
n
i
y
i
y
i
m
i
m
l
1,,
,,
3. CALIBRATION
The model is calib a ed o ma ch a se ies o s a is ics o he Chilean econ-
omy. Table 2 summa izes he base calib a ion. The du a ion o a pe iod, gi en
by pa ame e
T
, is se o be
20
yea s. The e o e, he model implies ha agen s
a e e i ed o
20
yea s ollowing a wo king li e o
40
. We assume he e a e
5
di e en skill g oups in he economy, i.e., nl
=
5, ha a e ep esen a i e
o he income quan iles.23 The subjec i e discoun ac o s o he di e en skill
g oups,
β
i o
in
l
1, ,, in u n, a e calib a ed o eplica e he a e age sa -
ing a es o each income quan ile, which a e compu ed om he Family Bud-
ge Su ey (Encues a de P esupues os Familia es — EPF) om 2012, using
he me hodology p oposed by Madei a (2015, 2018).24 The ma ched sa ing
a es a e
9%
,
−01.%
,
42.%
,
82.%
and
17%
o he i s o i h quan ile,
espec i ely, compu ed as he a e age sa ings du ing ac i e li e. These a es
imply an agg ega ed sa ing a e, weigh ed by income, o
77.%
.25 The social
secu i y con ibu ion a e paid by he wo ke s,
τ
W, is se a
10%
, in line wi h
he de ined con ibu ion o he cu en pension sys em in Chile. The social
secu i y con ibu ion a e paid by he i ms on behal o wo ke s,
τ
F, is se o
0% in he ini ial s eady s a e o he model and i is inc eased o
5%
when he
al e na i e pension e o ms a e implemen ed in he di e en exe cises. The
pa ame e ha de e mines he disu ili y o wo k,
χ
i, is calib a ed so as o se
ela i e o al wo k — o mal and in o mal — ac oss abili y g oups o
048.
,
063.
,
074.
,
086.
, and
1
, i.e. implied by he
36%
,
47%
,
56%
,
65%
, and
75%
pa icipa ion a es o each quan ile ( he model has no unemploymen ).26
The annual popula ion g ow h a e,
n
, is se o
05.%
, and he echnolog-
ical g ow h a e co esponding o he g ow h a e o ou pu pe capi a, wages
and o he pe capi a a iables in he model,
g
, is se o
2%
, in line wi h he
23 Quan ile numbe 1, ep esen s he lowes income g oup, and, he e o e, he less skilled
g oup.
24 Sou ce: Na ional S a is ics Ins i u e (Ins i u o Nacional de Es adís icas — INE).
25 This numbe is close o he agg ega e sa ing a e o Chile ha was es ima ed o be
be ween 8.3% and 9.8% since 2010 by he OECD (h ps://da a.oecd.o g/hha/house-
hold-sa ings.h m).
26 Sou ce: he da a is p o ided by he Minis e io de Desa ollo Social y Familia, Subsec-
e a ía de E aluación Social, based on in o ma ion om he Encues a Casen and he
Encues a Casen en Pandemia 2020. In a obus ness exe cise o al labo pa icipa ion is
se equal acc oss skill g oups; esul s hold.
347
Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
a e age g ow h end es ima ed o Chile un il 2050 in he base p ojec ion sce-
na io o Albagli e al. (2015).27 Following he same s udy, he labo sha e in
p oduc ion,
1
, is se o
52%
, which co esponds o he 2008-2013 a e age
o he a io be ween sala ies paid by he co po a e sec o and he agg ega e al-
ue o ha sec o , ne o axes, acco ding o Na ional Accoun s da a o Chile.28
The capi al dep ecia ion a e,
δ
, is se a
4%
pe yea .
27 This las es ima ion is based on da a and es ima ions om he Cen al Bank o Chile,
Na ional Socioeconomic Cha ac e iza ion Su ey (Encues a de Ca ac e ización Socio-
económica Nacional — CASEN), he INE and he OECD. I s me hodology is based on
he p oduc ion unc ion.
28 Sou ce: Cen al Bank o Chile.
TABLE 2
BASELINE CALIBRATION, * QUANTILES
quan iles
QQQQQ
12345
,,,,
Pa ame e Desc ip ion Value/Ta ge Sou ce
T
Du a ion o a pe iod 20 yea s -
i
T
11
/Annual discoun a es Sa ing a es Madei a (2018)
τ
W Employee con ibu ion
a e 10% Exis ing a e in Chile
τ
F Employe con ibu ion
a e
05→%
Exe cise
τ
Income ax 30% -
χ
i
Labo disu ili y
li
048063 074086 1.,.,.,., CASEN and CASEN
en Pandemia 2020
n
Annual popula ion
g ow h a e 0.5% Albagli e al. (2015)
g
Annual echnological
g ow h a e 2% Albagli e al. (2015)
θ
In e emp. elas ici y o
subs., in 1 Li e a u e
φ
Labo supply elas ici y,
in e se 3 O si e al (2014)
α
Capi al sha e 0.48% Albagli e al. (2015)
11
1
/
T
Annual dep ecia ion 4% Li ea u e
κ
Wo k a home speci ic
cos . 1.3 in o mal e u n be ween
g oss-ne o mal e u n
ξ
In o mal sec o p od.
c . 0.53 10% in o mali y
inc ease
:%30 35
η
In o mal sec o p od. c
hin i s ss
1
1.
.No maliza ion
348 Es udios de Economía, Vol.51 - Nº 2
The pa ame e ha de e mines he speci ic u ili y cos o wo king a home,
κ
, is se o
13.
, so as o make he e u n in in o mali y lowe han he g oss
wage and la ge han he ne wage. The pa ame e
ξ
, in u n, is se so ha a
5
pe cen age poin s income ax inc ease, yields a
10%
inc ease in in o mal wo k
ollowing Joube (2015), and
η
is se so ha h
1
equals
1
in he i s
s eady s a e. Joube (2015) inds ha in o mali y inc eases
10%
when a
5
pe cen age poin s inc ease in he pension con ibu ion is in oduced in a model
o Chile, no as a consequence o an inc ease o he income ax. Howe e ,
he model o Joube (2015) is a pa ial equilib ium one whe e he pension
con ibu ion beha es mo e close o an income ax as he one in oduced he e,
han o a pension con ibu ion. The pa ame e
υ
, which de e mines he size
o he weal h e ec on labo supply is se o
1
. This speci ica ion a o s he
ϕ
iF ic ion
0505050505., ., ., ., .*
Li e a u e
υ
P e e ence shi e 1 G eenwood e al.
(1998)
ρ
Labo elas ici y o subs.
b/ skills 0.33 Ciccone and Pe i
(2005)
a5
P oduc i i y, F. sec o ,
skill g . 5 1 Calib a ion a ge s
a4
P oduc i i y, F. sec o ,
skill g . 4 0.48 Labo income quan iles
a
3
P oduc i i y, F. sec o ,
skill g . 3 0.21 (CASEN 2015)
a2
P oduc i i y, F. sec o ,
skill g . 2 0.19
a1
P oduc i i y, F. sec o ,
skill g . 1 0.05
429714 5211505.,., ., ., .*
ab
55
P oduc . In . sec o ,
skill g . 5 0.21 Calib a ion a ge s
agg ega e
ab
44
P oduc . In . sec o ,
skill g . 4 0.09 pa icipa ion in in o -
mal sec o
ab
33
P oduc . In . sec o ,
skill g . 3 0.06 and quan ile dis i-
bu ion
ab
22
P oduc . In . sec o ,
skill g . 2 0.04 (NESI 2012)
ab
11
P oduc . In . sec o ,
skill g . 1 0.02
57 18 18 17 14,,,, *
355
Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
— mainly due o he gene al equilib ium e ec o highe agg ega e sa ings
and capi al — and s ongly dec eases o he lowe skilled ones, which now
pe cei e a lowe ne income o wo king in he o mal sec o and he e o e,
decide o ela i ely in ensi y hei in o mal labo .
Le us now y o unde s and why he inc ease in ac i i y as a consequence
o implemen ing he IDC scheme is la ge han when implemen ing he C-
CDC one unde he baseline calib a ion, whe eas his e e ses unde he he -
e ogeneous one. In he C-CDC case, he mos skilled agen s educe hei ol-
un a y sa ings less han unde he IDC one because hei pe cei ed addi ional
pension, gi en he newly imposed con ibu ion, ends up being lowe unde his
scheme han unde he IDC one. In o he wo ds, hey canno allow hemsel es
o educe hei p i a e sa ings as much, since, as a consequence o he edis-
ibu i e ea u es o he C-CDC sys em, o each uni con ibu ed hey ge paid
a lowe u u e pension han unde he IDC one. The lowe skilled g oups, on
he o he hand, ecei e highe pensions pe uni con ibu ed unde he C-CDC
scheme han unde he IDC one, so hey end o undo a highe ac ion o he
new manda o y sa ings unde he o me scheme han unde he la e in an
a emp o smoo h consump ion.
356 Es udios de Economía, Vol.51 - Nº 2
TABLE 4
LONG-TERM MACROECONOMIC EFFECTS, CONDITIONAL COLLECTIVE DEFINED CONTRIBUTION SCHEME.
*: EXPRESSED IN PERCENTAGE POINT CHANGES. CONSUMPTION ENCOMPASSES TWO ROWS, THE UPPER ONE SHOWS THE PERCENTAGE CHANGE WHILE ACTIVE,
WHEREAS THE LOWER ONE WHEN RETIRED.
Exp essed in
pe cen age changes
Baseline calib a ion F ic ion sensibili y Openness
obus ness
Agg ega e
Skill g oups
ϕ
γ
5 4 3 2 1 1 0 He e oge.
0
++
Wages -2.2 -1.1 -2.5 -3.4 -4.3 -6.6 -4.5 0.1 -1.8 -1.1
Fo mal wo k 0.8 -0.9 1.3 2.6 4.1 8.0 0.5 0.1 0.9 1.5
In o mal wo k -2.7 8.3 -1.9 -6.7 -12.3 -2.6 -3.9 -0.1 -2.2 -4.8
Fo mal ne income 2.4 1.3 2.3 2.9 3.6 6.1 3.6 0.1 1.9 3.5
0.8 0.5 0.7 0.9 1.0 2.0 0.7 0.1 0.6 1.9
Volun a y sa ings/GGP* -2.5 -0.8 -0.6 -0.5 -0.5 -0.2 -4.4 -0.8 -2.2 -2.7
Public sa ings/GDP* 4.3 4.7 3.9 4.3 4.4
To al sa ings/GDP* 1.8 0.3 3.1 2.0 1.7
Annual in e es a e* -0.1 -0.0 -0.2 -0.1 -0.2
Capi al 6.6 1.0 10.9 7.5 9.8
Consump ion 6.3 1.2 2.2 2.9 3.6 3.6 2.9 7.9 6.9 7.4
4.5 16.7 26.7 41.3 34.3
Fo mal GDP 3.5 0.7 5.2 4.1 5.4
357
Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
Now, when mo ing om he baseline o he he e ogeneous calib a ion, he
lowes skilled agen s unde he C-CDC scheme espond by inc easing hei
sa ings ela i ely mo e han he same skilled agen s unde he IDC scheme.
The eason is ha when he deg ee o in e naliza ion dec eases, unde he C-
CDC scheme, agen s also s op in e nalizing he edis ibu i e bene i s o he
scheme. The e o e, sa ings unde he he e ogeneous calib a ion inc ease ela-
i ely mo e o he C-CDC scheme han o he IDC one, pushing he ac i i y
change unde he C-CDC sys em abo e he change unde he IDC one. O he
gene al equilib ium e ec s a e also a play, such as he change in public sa -
ings which depends on he changes in wages and in o mal employmen .
4.3 Uncondi ional Collec i e De ined Con ibu ion
Table 5 p esen s he esul s ob ained om implemen ing he U-CDC e-
o m. Unde his scheme he wo mos skilled g oups o agen s a e no en i led
o an addi ional pension, e en hough i ms do pay a pay oll con ibu ion on
hei behal .35 The emaining skill g oups, in u n, ecei e an addi ional pen-
sion, inanced h ough he i ms’ con ibu ions, ha is dec easing in he i s
componen o hei pension, i.e., he one inanced h ough he employees’ con-
ibu ions. The model p edic s a much lowe ou pu g ow h be ween he ini ial
and inal s eady s a es, o only 1%. This is pa ially explained by he lowe cap-
i al s ock g ow h, abou 4.6% in he long un, and, mainly, by he signi ican ly
la ge educ ion in o mal labo , -2.3%. This lowe agg ega e pe o mance ul-
ima ely inc eases in o mali y in he economy, showing how a edis ibu i e
mechanism in a collec i e de ined con ibu ion scheme can be a poo design. In
ac , in o mali y unde his scheme inc eases conside ably, 4.1% in agg ega e
e ms. This pension plan does, no wi hs anding, imp o e he consump ion o
he h ee less quali ied skill g oups in he economy. No ice ha o hose skill
g oups ha bene i om his scheme, 1 o 3, he e is an implici ax because o
each addi ional uni hey ecei e om hei sel - inanced pension, he ac ion
o hei pension ha is inanced h ough i ms’ con ibu ions dec eases by 1/3
uni s (simila ly o he solida i y pilla cu en ly in place in Chile). This is also
esponsible o he dynamics ha end in a la ge in o mali y and in a educ ion
o o mal wo k.
35 This assump ion ollows he edis ibu ion cu en ly used in he pension scheme in
place in Chile, whe e only wo hi ds o he e i ees, he leas skilled ones, a e en i led
o edis ibu i e bene i s.
358 Es udios de Economía, Vol.51 - Nº 2
TABLE 5
LONG-TERM MACROECONOMIC EFFECTS, UNCONDITIONAL COLLECTIVE DEFINED CONTRIBUTION SCHEME.
*: EXPRESSED IN PERCENTAGE POINT CHANGES. CONSUMPTION ENCOMPASSES TWO ROWS, THE UPPER ONE SHOWS THE PERCENTAGE CHANGE WHILE ACTIVE,
WHEREAS THE LOWER ONE WHEN RETIRED.
Exp essed in
pe cen age changes
Baseline calib a ion F ic ion sensibili y Openness
obus ness
Agg ega e
Skill g oups
ϕ
γ
5 4 3 2 1 1 0 He e oge.
0
++
Wages -1.6 -2.0 -2.1 -0.3 -0.3 -0.4 -3.3 0.1 -0.8 -1.1
Fo mal wo k -2.3 -1.6 -1.6 -4.1 -4.1 -4.0 -4.6 0.1 -1.3 -1.5
In o mal wo k 4.1 2.5 1.5 10.5 10.3 0.4 8.3 -0.1 2.3 3.4
Fo mal ne income -2.4 -2.0 -2.1 -2.9 -2.9 -3.0 -4.7 0.1 -1.4 -1.8
Volun a y sa ings/GGP* -2.6 -0.3 -0.1 -0.7 -0.8 -0.8 -4.5 -0.8 -1.8 -2.8
Public sa ings/GDP* 4.1 4.3 4.0 4.1 4.2
To al sa ings/GDP* 1.5 -0.2 3.1 2.2 1.4
Annual in e es a e* -0.1 -0.1 -0.2 -0.2 -0.2
Capi al 4.6 -1.6 10.9 7.5 6.8
Consump ion 3.5 -3.4 -3.3 4.0 8.0 17.9 -1.5 7.9 5.7 4.1
-6.9 -7.0 34.4 69.7 172.1
Fo mal GDP 1.0 -3.2 5.2 2.9 2.5
359
Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
4.4 Pay-As-You-Go
Finally, Table 6, shows he e ec s p edic ed by he model i a PAYG sys em
would be implemen ed. This scena io is signi ican ly wo se han he p e ious
pension e o ms in all dimensions conside ed. In pa icula , i is he only sys-
em ha p oduces a nega i e impac on capi al, consump ion, and ou pu ; all o
hem signi ican . The dec ease in sa ings, caused by he ini ial ans e o he
old gene a ion leads o a con ac ion o o al sa ings and, consequen ly, o he
capi al s ock. This lowe capi al s ock le el, in u n, educes he demand o la-
bo o he i ms beyond he di ec e ec on he labo supply implied by he ax
on wages. This educes o mal labo a ound 4.1%, while inc easing in o mal-
i y o all skill g oups. As a consequence, GDP in he long un dec eases abou
4.8%. I is wo hwhile men ioning ha he ule ha de e mines he dis ibu ion
o unds is p opo ional o he con ibu ions made by each agen . This educes
he dis o ions in he labo ma ke , by s ongly linking con ibu ions and u-
u e bene i s. O he al e na i e pay-as-you-go schemes should ha e addi ional
nega i e e ec s o he ones p edic ed he e.
Clea ly, all modelling and calib a ion exe cises a e subjec o a subs an-
ial deg ee o unce ain y. In pa icula , i is di icul o calib a e he inancial/
in o ma ional ic ions ha make agen s in e nalize only pa ially he u u e
bene i s unde he di e en schemes. No wi hs anding, as i can be obse ed,
he anking o he di e en al e na i es, in e ms o hei agg ega e e ec s is
ela i ely obus : he indi idual de ined con ibu ion and he condi ional col-
lec i e de ined con ibu ion schemes, ha e he mos posi i e (o leas nega i e)
e ec s in e ms o capi al, GDP, and labo , while he uncondi ional collec i e
de ined con ibu ion scheme and, in pa icula , he pay-as-you-go sys em, ha e
he (less posi i e) mos nega i e e ec s on hose a iables. In addi ion, only he
i s wo schemes a e able o educe in o mali y.36
36 The collec i e de ined con ibu ion scheme beha es exac ly he same as he o he i s
wo schemes when he in o ma ional and/o inancial ic ion is se o 0 ac oss skill
g oups, as expec ed.
360 Es udios de Economía, Vol.51 - Nº 2
TABLE 6
LONG-TERM MACROECONOMIC EFFECTS, UNCONDITIONAL COLLECTIVE DEFINED CONTRIBUTION SCHEME.
*: EXPRESSED IN PERCENTAGE POINT CHANGES. CONSUMPTION ENCOMPASSES TWO ROWS, THE UPPER ONE SHOWS THE PERCENTAGE CHANGE WHILE ACTIVE,
WHEREAS THE LOWER ONE WHEN RETIRED.
Exp essed in
pe cen age changes
Baseline calib a ion F ic ion sensibili y Openness
obus ness
Agg ega e
Skill g oups
ϕ
γ
5 4 3 2 1 1 0 He e oge. 0
++
Wages -5.4 -5.4 -5.5 -5.5 -5.5 -5.5 -6.2 -4.2 -5.4 -6.3
Fo mal wo k -4.1 -4.1 -4.1 -4.1 -4.1 -4.1 3.3 -4.9 -4.1 4.7
In o mal wo k 6.0 11.8 8.3 8.0 7.9 -0.2 4.6 7.1 6.2 8.3
Fo mal ne income -5.4 -5.4 -5.5 -5.5 -5.5 -5.5 -6.2 -4.2 -5.3 -6.3
-2.0 -2.0 -2.1 -2.1 -2.1 -2.1 -0.5 -4.2 -2.3 -3.2
Volun a y sa ings/GGP* -0.6 -0.3 -0.1 -0.1 -0.1 -0.0 -0.8 -0.3 -0.6 -0.4
Public sa ings/GDP* -0.7 -0.9 -0.5 -0.7 -0.8
To al sa ings/GDP* -1.3 -1.7 -0.7 -1.3 -1.2
Annual in e es a e* 0.0 0.1 -0.0 0.0 0.1
Capi al -5.5 -6.3 -3.8 -5.4 -7.8
Consump ion -4.0 -5.7 -5.7 -5.7 -5.7 -5.7 -4.7 -2.9 -3.9 5.0
-1.9 -1.2 -0.6 0.2 -1.7
Fo mal GDP -4.8 -4.8 -4.4 -4.7 -6.2
361
Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
5. DEMOGRAPHIC TRANSITION
One pa icula ly ele an aspec o pension schemes is hei abili y o han-
dle popula ion dynamics, such as a dec ease in he e ili y a e o an inc ease
o li e expec ancy. In his sec ion we use ou model o analyze he i s o hose
si ua ions, simula ed by a d op in he popula ion g ow h a e, and how he di -
e en pension schemes can accommoda e such a change. We conside a simple
s eady s a e ansi ion exe cise in which he economy depa s o m he s eady
s a e associa ed o each pension scheme analyzed, and simula e a d op o
n
om he calib a ed annual
05.%
o
025.%
. In o he wo ds, we conside wha
happens unde each o he ou al e na i e pension e o ms a e hey ha e
been ully implemen ed and he popula ion g ow h a e d ops by hal .
The uppe -le g aph shows he basic popula ion dynamics igge ed by
he d op in he g ow h a e
n
. Remembe ha each pe iod in he model co e-
sponds o
20
yea s. In pa icula , besides he p opaga ion o he shock h ough
he ac i e and o al popula ions, one can obse e how as he popula ion g ow h
alls, he amoun o passi e agen s in he economy ela i e o ac i e wo ke s
inc eases. When he economy s abilizes a i s new s eady s a e he a io o
passi e indi iduals ela i e o he ac i e ones is abou
3
pe cen poin s highe .
Mo eo e , as popula ion g ow h slows down, capi al becomes ela i ely mo e
abundan in he economy inducing a dec ease in he in e es a e. By he same
mechanism he ela i ely mo e sca ci y o wo ke s induce an inc ease in wag-
es. This inc ease in wages, in u n, gene a es a mo emen o wo ke s om he
in o mal o he o mal sec o (see uppe -le g aph in Figu e 1). The highe
wages oge he wi h he inc ease in o mal labo make pension con ibu ions
inc ease. Unde he case o he i s h ee schemes, IDC, C-CDC and U-CDC,
he d op in
n
ul ima ely ansla es in o highe pensions. Howe e , unde he
PAYG pension al e na i e, a e age pensions inc ease less, as now ela i ely
less wo ke s inance he passi e indi iduals in he economy. I is in e es ing
o no e ha o all ou cases, du ing he ansi ion, pensions unequi ocally
dec ease, hough mo e s ongly unde he PAYG scheme. The eason is he de-
c ease in he in e es a e and he ac ha du ing he i s wo pe iods, he unds
used o pay ou pensions ha e been o ally o pa ially in pe iod
2
cons i u ed
unde he o iginal s eady s a e (when sala ies and o mal wo k we e lowe ).
The lowe - igh g aph shows he dynamics o he second componen o
o al pensions, o h ee pension sys ems, o he condi ional collec i e de-
ined con ibu ion, o he indi idual de ined con ibu ion, and o he PAYG
schemes. We can see how in he case o he PAYG al e na i e he d op o he
second componen , he one inanced by i ms’ con ibu ions, is he one push-
ing o al pensions downwa ds unde ha scheme. The i s componen — no
shown — beha es in he same way unde all ou schemes, as is o be expec -
362 Es udios de Economía, Vol.51 - Nº 2
ed. Hence a pension sys em ha is ully a icula ed as a PAYG sys em would
endu e mo e se e e p oblems, in e ms o pensions, in he ace o a popula ion
g ow h decele a ion.
FIGURE 1
DEMOGRAPHIC TRANSITION
As shown in Figu e 2, he slow down in popula ion g ow h yields la ge
GDP, consump ion and capi al s ock, o all pension schemes. Howe e , i is
wo hwhile no icing ha he pe o mance o he economy wi h a PAYG sys em
is clea ly domina ed by he o he h ee schemes he e conside ed, in pa icula ,
by he C-CDC and IDC al e na i es.
FIGURE 2
DEMOGRAPHIC TRANSITION
363
Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
6. CONCLUDING REMARKS
In his pape we p esen a quan i a i e analysis o he long- e m mac o-
economic e ec s o implemen ing al e na i e pension sys em e o ms in an
economy wi h a signi ican in o mal sec o . Fo his, we cons uc a h ee-pe i-
od OLG model wi h i e skill g oups, in o ma ional and/o inancial ic ions
and labo in o mali y ha we calib a e o he Chilean economy. We conside
ou di e en pension schemes — all inanced h ough an iden ical inc ease
o pension con ibu ions aking he o m o a pay oll ax — ha only di e
om each o he in he way hey ea he addi ional unds and alloca e hem
among e i ees. In pa icula , we conside wo e sions o a CDC scheme ha
di e in whe he hey make pension bene i s depend upon he deg ee o labo
e o du ing wo king li e o no , and compa e hei mac oeconomic pe o -
mance o an IDC scheme and a PAYG al e na i e. Condi ioning he eceip o
bene i s on employmen s a us wi hin a edis ibu i e design is he key aspec
ha incen i izes a s ong o maliza ion o labo supply a low income le els,
which a e hose who make he la ges sha e o in o mal wo k, su passing he
e ec s ound unde he IDC plan. In pa icula , he in o ma ional and/o inan-
cial ic ions play he main ole in he dynamics o ou model as hey es ic
he deg ee in which agen s in e nalize how pension con ibu ions ansla e in o
u u e pension paymen s.
The quan i a i e esul s sugges ha he C-CDC scheme has simila mac-
oeconomic impac s as an IDC plan unde he baseline calib a ion, including
a mode a e posi i e e ec on he o mal labo ma ke , which oge he wi h an
inc ease in capi al due o he ise o compulso y sa ings, gene a es an expan-
sion o ou pu and consump ion. Wi h espec o he e ec hese schemes ha e
on he in o mal sec o , he C-CDC c ea es s onge incen i es o mo e away
om in o mali y among lowe -income wo ke s, and since hese a e p ecisely
he ones mo e p esen in in o mali y, we obse e a s onge educ ion o ag-
g ega e in o mali y unde his pension plan. This, despi e he opposi e e ec
p oduced among highe -income wo ke s. The U-CDC al e na i e, in u n, has
a nega i e e ec on he labo ma ke , since by making u u e bene i s indepen-
den o con ibu ions, low-income agen s no longe ha e an incen i e o mo e
in o o mali y. The addi ional capi al is, he e o e, no longe complemen ed by
mo e wo k, and o al ou pu only inc eases ma ginally. This educes employ-
men and o mali y. The PAYG sys em cons i u es he mos ad e se scheme o
all he ones conside ed. The impo an educ ion o he capi al s ock, o a ound
5.5%, educes he demand o labo beyond he nega i e e ec o he imposed
con ibu ion on i ms. Fo mal employmen alls a ound 4.1% and in o mali y
inc eases abou 6%, while GDP and consump ion all 4.8 and 4% espec i ely.
364 Es udios de Economía, Vol.51 - Nº 2
Finally, we ind ha like he IDC plan, he C-CDC scheme’s sol ency is
obus o popula ion ageing, he main sho coming o un unded pay-as-you-go
(PAYG) sys ems. Hence, ou esul s sugges ha a C-CDC scheme may be an
economically sus ainable and poli ically iable al e na i e o coun ies wi h
signi ican labo in o mali y and income inequali y.
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371
Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
(60) wawl
l
i
y
i
ii
,
,
1
(61) wgaw l
l
i
m
i
ii
,
,
1
1
(62) ll l
n
i i
yi
m
,,
,
1
o
in
l
1, ..., _ . And, also,
(63)
ww
y
i
n
i
y
i
l
1,
(64)
ww
m
i
n
i
m
i
l
1,
(65) wl
l
m
w
i
m
i
ii
nl
i
m
11,
,
(66) yk
gnl
11
1
(67) lal
i
n
iii
l
1
1
,
/
(68) wy
l
F
1
1
(69) wlwl
gn
i
n
i
y
i
yi
m
i
m
l
111
,,
,,
(70) y
k
gn
k
11
(71)
k
11
*
(72) ss s
gn
y
m
11
(73)
ss
y
i
n
i
y
i
l
1,
(74)
ss
m
i
n
i
m
i
l
1,
(75) ss s
gnss
P
PW m
PW y
PW y
PF m
,,
,
,
1
123
11
PF y
PF y
s
gn
s
1
11
,
,
372 Es udios de Economía, Vol.51 - Nº 2
(76) pp p
W
F
(77) p s s
gns
F
PF m
PF y
PF
123112
411
,,
,mm
PF y
gnns
111
2
1
,
(78) p s s
W
PW m
PW y
11
2
,,
(79)
sw
l
PW y
i
nW
ii
y
i
y
l
,
,,
1
(80)
sw
l
PW m
i
nW
ii
m
i
m
l
,
,,
1
(81)
sw
l
PF y
i
nF
ii
y
i
y
l
,
,,
1
(82)
sw
l
PF m
i
nF
ii
m
i
m
l
,
,,
1
(83)
ks
s
P
11
*
STEADY STATE
Le a iables wi hou ime subsc ip deno e s eady s a e alues. We
sol e o he s eady s a e by means o nume ical me hods, using as s a -
ing alues o he nume ical sol e he analy ical s eady s a e solu ion o
he special case whe e
WF
0
(no e ha imposing
F
0
is equi a-
len o
1
234
0
,
i
y
i
m
o all
i
,
ij
o all
i
and j,
1
and
0. F om (46) o (48)
c s wl gb hah s c
i
o
i
m
i
m
i
m
iii
m
i
y
i
m
1
1
wlgbhah wlebh
i
m
i
m
iii
m
i
y
i
y
i
1
11
ah
cc
ii
y
i
y
i
m
and we also ha e om (49) h ough (53) and (75) h ough (82) ha
(84) ss ppp
i
PW y
i
PW m
i
W
i
F
i
,,
= = = = =0
o
in
l
1, ,, and ha
373
Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
(85) ss ssspppp
PPWy PW mPFy PF mWFBS
= = = = = = = = =
,,,, 0
F om (41) h ough (43) and he p e ious esul s,
c s wl gb hah s c
i
o
i
m
i
m
i
m
iii
m
i
y
i
==
43 42 1
1
mm
=
41
11
1
wlgbhah w l bh
i
m
i
m
iii
m
i
y
i
y
i
ah c c
ii
y
i
y
i
m
⇔
c c c
wlgbhah w l
i
o
i
m
i
y
i
m
i
m
iii
m
i
y
i
y
2
1
1
bb hah
ii
i
y
1
We also ha e ha ,
(86) c c
i
m
ii
y
and
c c
i
o
ii
m
he e o e,
c
c
c
i
o
ii
m
ii
y
2
Combining bo h esul s, we ob ain ha
(87) c
wl gb hah w l bh
i
y
i
m
i
m
iii
m
i
y
i
y
i
1
11
ah
ii
y
ii
21
F om (60), and (61), and since
1
,
wa
w
i
y
i
=
wgaw
i
m
i
1
Thus,
(88) ww
g
i
yi
m
1
374 Es udios de Economía, Vol.51 - Nº 2
F om (58) and (59), and as long as
i
y
i
m
,
1
,
(89) ha
bhw
i
y
i
i
i
y
max
1
1
1
,, 0
(90) hga
bhw
i
m
i
i
i
m
§
©
¨
¨
·
¹
¸
¸
§
©
¨
¨
·
¹
¸
¸
ª
¬
«
«
«
«
«
«
º
¼
»
»
»
»
»
max1
1
[
K
FN
»»
®
°
°
°
¯
°
°
°
½
¾
°
°
°
¿
°
°
°
1
1
0
I
,
i b
i<
0 we ha e ha ,
hh
i
y
i
m
= =0
else,
ha
bhw
i
y
i
i
i
y
1
1
1
1
11
1
1
1
gga
bhw
i
i
i
m
1
1
1
1
1
1
ghi
m
which holds also o bi
≥
0. We u he choose bi such ha
(91) hlw
bh
i
yi
y
i
1
375
Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
and
(92) hlw
bh
i
mi
m
i
1
In addi ion, assuming ha such a bi is cons an o e i, we ha e ha ,
(93) cgl l
aw
i
yi
m
i
y
ii
i
1
1
2
2
and
(94) cgl l
aw
i
m
i
i
m
i
y
ii
i
1
12
2
F om (56) and (57),
law
w
bh
i
yi
i
y
i
1
1
1
1
(95)
i
m
i
m
i
i
lgaw w
bh
1
1
11
1
lw
bh
aw
i
y
i
i
i
y
1
11
1
1
and
(96) lw
bh
gaw
i
m
i
i
i
m
1
1
1
1
1
1
Then, i
1
, we ha e
(97) ll g
i
y
i
m
1
1
1
1
376 Es udios de Economía, Vol.51 - Nº 2
The e o e, young and middle-aged o mal wo k sa is y he same p opo ion
as young and middle-aged in o mal wo k. Then, om (62), (97) and (96),
(98)
ll l
nlg
l
n
iii
yii
m
ii
mii
m
1
1
11
1
1
ii
m
i
lgn
g
n
1
1
1
111
1
1
1
1
1
1
11
1
1
w
bh
aw
i
i
i
m
To no malize li o
1
- in his s eady s a e - one needs o se
11
11
1
1
1
1
g
nw
bh
i
i
1
aw
ii
m
i
mi
i
g
nw
bh
11
11
1
1
1
1
1
aw
i
F om (67), and since li
=
1,
(99) la
la
i
n
ii ii
n
ii
ll
11
Then, using (97) in (93), we can w i e
c
g g
awl
i
y
ii
ii
m
11
1
1
2
1
1
2
ll g
i
y
i
m
1
1
1
1
377
Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
(100) c
g g
awl
i
y
ii
ii
m
11
1
1
2
1
1
2
F om (41), we ha e ha
(101)
swlb
hah c
awlga
i
y
i
y
i
y
iii
y
i
y
ii
m
1
1
1
1
1iii
m
ii
i
wl g
g g
awl
1
1
11
1
1
2
1
1
1
1
2
ii
m
ii
g
g g
1
1
11
1
1
1
1
1
1
2
2awl
ii
m
F om (42), we ha e ha
swlgbhah s c
gawl
i
m
i
m
i
m
iii
m
i
y
i
m
ii
m
1
1
1
11 1
1
11
1
1
1
1
1
1
2
galw g
g g
ii
m
ii
aw
lc
ii
m
i
m
2
(102)
378 Es udios de Economía, Vol.51 - Nº 2
S a ing om (83),
ks
1
=
72
11
11
ss
gn
y
m
=
73 74
1
111
i
n
ii
yi
m
lss
gn
=
101 102
1
i
n
iii
lg
g g
1
1
1
1
1
2
1
1
11
1
1
1
1
gn g g
i
ii
111
111
1
1
1
2
2awl
ii
m
=
ij
w
12
1
1
11
1
1
1
11
1
1
1
1
2
g
g g
gn
ii
1111
1
1
1
2
g ee
g
i
ii
i
n
iii
m
lal
1
=
98
1
1
1
1
12
1
1
1
1
1
1
11
1
w
gn
g
g g
1
1
2
1
1
1
1
11 11
1
ii
gn g g
11
1
2
1
i
ii
i
n
i
lal
ii
379
Collec i e Sa ings Pension Policy in an Economy... / E. Albagli, A. H. A ias, M. Ki chne
=
99
1
1
1
1
12
1
1
1
1
1
1
11
1
gn
g
g g
1
1
2
1
1
1
1
11 11
1
ii
gn g g
11
1
2
i
ii
wl
=
66 68
1
1
1
121
1
1
1
gn
1
1
11
1
1
1
11
1
1
1
1
2
g
g g
gn
ii
11
111
1
1
1
2
g g
i
ii
1
11
gn
k
ll
Then, he capi al labo a io is hen a non linea unc ion o
, and he
model pa ame e s,
k
lgn
gn
112
11
1
1
1
1
1
1
1
(120) 1
1
11
1
1
1
11
1
1
1
1
2
g
g g
gn
ii
11
111
1
1
1
2
g g
i
ii
and hus, using (99), we can ob ain he s eady s a e alue o
k
,
380 Es udios de Economía, Vol.51 - Nº 2
(104) kk
ll=
F om (66) and (68), hen,
(105) wk
gnl
111
Nex , we de i e he bi ha ensu es ha (91) and (92) a e sa is ied and show
ha his bi is common o all
i
. Fo his, we depa om equa ions (91) and
(92) and use (58), (59), (95), (96) as well as (86), (97) and (100),
blw
hh
ww
bh
aw
i
i
y
i
y
i
i
i
y
1
1
1
1
11
1
1
1
1
a
bhw
h
i
i
i
y
1
bh
w
bh
bhw
bh
i
i
i
i
1
1
1
1
11
1
1
w
(106) bhwbhww
ii
11
1
o all
i
. The same esul is ound i depa ing om
(107) blw
hh
i
i
m
i
m
1
