Improving the quality of life and longevity of the elderly: the role of private versus public health

ORIGINAL RESEARCH
Recei ed: 29 Ap il 2025 / Accep ed: 21 Oc obe 2025
© The Au ho (s) 2025
Mau o Ma ia Baldi
[email p o ec ed]
Ra aella Coppie
[email p o ec ed]
Elisabe a Miche i
[email p o ec ed]
1 Depa men o Economics and Law, Uni e si y o Mace a a, Mace a a, I aly
Imp o ing he quali y o li e and longe i y o he elde ly: he
ole o p i a e e sus public heal h
Mau o Ma iaBaldi1· Ra aellaCoppie 1· Elisabe aMiche i1
Annals o Ope a ions Resea ch
h ps://doi.o g/10.1007/s10479-025-06920-1
Abs ac
We de elop an o e lapping gene a ions model o s udy how he combina ion o public
and p i a e heal h expendi u es a ec s he heal h s a us and/o longe i y o he elde ly, as
well as i s impac on s eady-s a e economic g ow h. We ind ha wo dis inc scena ios
may a ise-one wi h and one wi hou eliance on p i a e heal h ca e-depending on he
ela i e alue o p i a e e sus public heal h spending. In bo h cases, a posi i e locally
asymp o ically s eady s a e eme ges in e ms o capi al pe wo ke , and a swi ch be ween
egimes may occu depending on he sha e o public balance spen on he heal h sys em.
Fu he mo e, inc easing such a sha e inc eases he equilib ium longe i y, while he e ec s
on heal h s a us a e ambiguous. Speci ically, when he e ec i eness o public expendi u e
is low, inc easing public esou ces alloca ed o heal hca e does no necessa ily lead o im-
p o emen s in heal h s a us. In con as , when public spending is highly e ec i e, g ea e
alloca ion o public esou ces becomes a powe ul ool o imp o e heal h s a us in old age.
Keywo ds Public and p i a e heal h expendi u es · Longe i y · Li e quali y o he
elde ly · Economic g ow h · O e lapping gene a ions model
1 In oduc ion
In ecen decades, nea ly all coun ies ha e expe ienced a subs an ial inc ease in human lon-
ge i y. Li e expec ancy es ima es he a e age numbe o yea s a pe son is expec ed o li e
based on a ious ac o s such as hei bi h yea , cu en age, and demog aphic cha ac e is ics.
I is a key indica o used o assess he gene al heal h and well-being o popula ions, o en
e lec ing he e ec i eness o heal hca e sys ems, socioeconomic condi ions, li es yle choices,
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and public heal h policies. Be ween 1990 and 2022, li e expec ancy imp o ed in all egions
o he wo ld, leading o a educ ion in dispa i ies be ween coun ies. In 2022, Chad (52.997
yea s), Leso ho (53.036 yea s), and Nige ia (53.633 yea s) had he lowes li e expec ancy a
bi h (LEB), while Japan (84.82 yea s), Liech ens ein (84.656 yea s) and Swi ze land (84.255
yea s) epo ed he highes LEB. In 1990, he di e ence be ween he highes and lowes LEB-
Japan and Sou h Sudan, espec i ely-was 49.049 yea s (Ay emiz e al., 2024).
By 2022, his gap had na owed o 31.823 yea s be ween Japan and Chad. Despi e
his p og ess, he emaining dispa i y is s ill subs an ial, highligh ing he need o ocus on
achie ing he Sus ainable De elopmen Goals (SDGs) se o h in he 2030 Agenda o
Sus ainable De elopmen . (Uni ed Na ions, 2024)1
Longe i y has been a co ne s one in public heal h and medical deba es and li e expec-
ancy is o en used o measu e he heal h and well-being o a popula ion (Annas & Galea,
2018). A leas in he de eloped wo ld, longe expec ed li espans ha e been accompanied by
a signi ican inc ease in heal hca e expendi u es. A g owing body o li e a u e shows a clea
link be ween heal h expendi u es and heal h ou comes (Glied & Smi h, 2013). In pa icula ,
da a om ad anced coun ies show ha public heal h spending plays a e y impo an ole
in enhancing longe i y, a much g ea e ole han p i a e heal h spending. This e idence
s ems om he ac ha : “public heal h expendi u e is de o ed, i s and o emos , o inance
ac ions ha a ec an impo an ac ion o he popula ion and in ol e signi ican posi i e
ex e nal e ec s (i.e., wha we could iden i y as basic heal h: accina ion campaigns, p e-
en ion o disease, basic amewo k o heal h cen e s, e c.). Bu once basic p og ams a e
me , addi ional public expendi u e is likely o be de o ed o ac i i ies ha he p i a e sec o
also o e s, so he p oduc i i ies o he wo sec o s con e ge." Aísa e al. (2014).
In ac , he e a e some coun ies, e.g. he Uni ed S a es, whe e heal hca e expendi u e is
high, bu he public componen o his expendi u e is low, wi h limi ed e ec s on imp o ing
longe i y. Wi h ega d o his, in 2023, he li e expec ancy in he Uni ed S a es was 78.4
yea s, which is 4.1 yea s lowe han he a e age o compa able coun ies a 82.5 yea s. As
Aísa e al. (2014) s a es: “In pa icula , he abo e esul s o e a plausible explana ion o
he appa en ly pa adoxical da a o he USA. While his is he coun y ha de o es he la g-
es amoun o esou ces o heal h (o e 13 % o GDP, close o wice he a e age alue in
he sample), li e expec ancy was 76.2 yea s in 2000, below he a e age. The key elemen
ha enables us o unde s and his puzzle is p ecisely he composi ion o heal h expendi u e.
The a e age alue in he sample shows a a io o public o p i a e expendi u es o 3.72. In
con as , his a io is only 0.82 o he USA. Tha is, he public heal h sys em in he USA is
esponsible o only 45 % o he esou ces de o ed o heal h, compa ed o he almos 80 %
a e age in he OECD coun ies. The abo e esul s show he need o a edesign o he heal h
sys em h ough an in ensi e p omo ion o he public heal h sys em."
Al hough li e expec ancy has gene ally inc eased, heal hy li e expec ancy (HALE) has
no p og essed a he same a e.2 Be ween 2000 and 2019, HALE inc eased by 5.3 yea s,
om 58.1 o 63.5 yea s, while o e all li e expec ancy g ew by 6.4 yea s du ing he same
pe iod. This indica es ha , al hough people a e li ing longe , hey a e also spending mo e
1 A ailable a : h ps://sdgs.un.o g/goals..
2 Heal hy li e expec ancy is a measu e ha combines bo h quan i y o li e and heal h s a us. This e e s o he
a e age numbe o yea s ha a pe son can expec o li e in ull heal h, ee om disease o disabili y. I is no
jus abou how long people li e, bu how long hey li e wi hou majo heal h p oblems.
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o hose addi ional yea s dealing wi h heal h p oblems.3 Bu , as Za ulli and Caswell (2024)
s ess, su i ing in good heal h is essen ial because heal hy yea s o li e a e also a c ucial
pa o he dynamics o he li e cycle.
These ends unde sco e he impo ance o add essing no only longe i y, bu also heal h
s a us du ing ex ended li espans. E o s o imp o e access o heal hca e, p e en i e mea-
su es, and heal h educa ion a e c ucial in ensu ing ha inc eased li e expec ancy ansla es
in o heal hie yea s li ed. Go e nmen s a ound he wo ld ecognize he impo ance o he
heal h sys em; he e o e, heal h expendi u es h oughou he wo ld ha e inc eased wi h
ime.4 Heal h expendi u es a e mos ly inanced h ough public axa ion and a e g owing
mo e han he global economy, accoun ing o 10% o he wo ld g oss domes ic p oduc
(GDP) (25). The a e age heal h expendi u es as a sha e o GDP ha e inc eased om 7.8%
in 2005 o 9.8% in 2020 in he OECD coun ies (Anwa e al., 2023).
In ou pape , we p opose an economic amewo k able o accoun o bo h elemen s
ele an o heal h: one ela ed o li e expec ancy (i.e., li e quan i y), and one exp essing
quali y om he poin o iew o he elde ly pe son’s heal h (i.e., heal h s a us). We he e o e
cons uc a 2-pe iod o e lapping-gene a ions model ha inco po a es an analysis o endog-
enous longe i y oge he wi h he endogenous elde ly heal h s a us. The e o e, we combine
wo s ands o esea ch by conside ing bo h he ‘quan i y’ o li e and heal h s a us.
Following (Chak abo y, 2004) and Cip iani and Fio oni (2019), we assume ha he
p obabili y o su i al om he i s pe iod (adul hood) o he nex (old age) is endog-
enously de e mined h ough public in es men in heal h.5
I is also impo an ha elde ly su i o s inc ease hei heal h s a us. Hence, we assume,
ollowing (Va a igos & Zaka ia, 2013), ha his s a e o heal h depends on bo h public
expendi u es on medical ca e, and also on p i a e spending by indi iduals du ing he inal
pe iod o hei li es. In ac , he elde ly can decide o alloca e pa o hei income o heal h
ca e expendi u es o imp o e hei u ili y-enhancing heal h s a us. Unde s anding how pub-
lic and p i a e heal h expendi u es can in e ac is essen ial o shaping o e o ming heal h
ca e policies.
Bha acha ya and Qiao (2007) and Va a igos and Zaka ia (2013) conside ha p i a e
and public heal h expendi u es a e complemen a y, i.e., an inc ease in one leads o an inc ease
in he o he , a he han subs i u ing o i . This occu s when p i a e and public in es men s
ein o ce each o he in imp o ing elde ly heal h.Unlike(Va a igos & Zaka ia, 2013), we
in e p e he public componen as a "compe i o " o he p i a e inpu in de e mining elde ly
heal h because public and p i a e expendi u es ha e a ce ain deg ee o subs i u abili y.
The choice o conside bo h p i a e and public expendi u e by he elde ly a ises om he
ac ha p i a e expendi u e in his con ex - such as medical ea men s and in e en ions
3 GHE: Li e expec ancy and heal hy li e expec ancy. The Global Heal h Obse a o y. h p s : / / w w w . w h o . i n / d
a a / g h o / d a a / h e m e s / m o a l i y - a n d - g l o b a l - h e a l h - e s i m a e s / g h e - l i e - e x p e c a n c y - a n d - h e a l h y - l i e - e x p e c a n c y
- Wo ld Heal h O ganiza ion (2022).
4 WHO. Coun ies a e Spending Mo e on Heal h, Bu People a e S ill Paying Too Much Ou o Thei Own
Pocke s. WHO (2019). A ailable online a : h p s : / / w w w . w h o . i n / n e w s / i e m / 2 0 - 0 2 - 2 0 1 9 - c o u n i e s - a e - s p e n d i
n g - m o e - o n - h e a l h - b u - p e o p l e - a e - s i l l - p a y i n g - o o - m u c h - o u - o - h e i - o w n - p o c k e s .
5 The decision no o conside p i a e spending as a de e minan o longe i y is based on he ac ha da a on
de eloped coun ies, such as he Uni ed S a es, as highligh ed in he in oduc ion, show ha public heal hca e
spending is he main de e minan o a coun y’s longe i y. Fu he mo e, in ou model, longe i y is he p ob-
abili y o su i ing in o he second pe iod, and he e o e p i a e spending on longe i y should be done by
young adul s, which is no e y common. Thus, his modelling makes he analy ical model mo e manageable
wi hou implying a loss o gene ali y o he model i sel .
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o exis ing diseases and condi ions - ends o imp o e he heal h s a us o he elde ly. These
expendi u es a e p ima ily ocused on ca e, as p e en ion is o en no an immedia e p io i y
o p i a e indi iduals who alloca e hei esou ces mainly owa d ea men s and he apies.
As a esul , his ype o heal hca e can be p o ided in e changeably by he public heal h
sys em o , in cases o de iciencies o excessi ely long wai ing imes, by he p i a e sys em.
In his ega d, (Li e al., 2016) conduc a quan i a i e exe cise o examine whe he he
obse ed di e ences in he public-p i a e mix o heal h expendi u e can be accoun ed o by
a ia ions in he elas ici y o subs i u ion be ween p i a e and public spending, as well as by
di e ences in he e ec i eness o public expendi u e in heal h p oduc ion ac oss a sample
o OECD coun ies. Following he quan i a i e analysis by Li e al. (2016), ou model con-
side s wo di e en scena ios ega ding he e ec i eness o public heal h expendi u e, each
ep esen ing wo ma kedly di e en models o p i a e-public heal h expendi u e mixes.
This pape p esen s a simple amewo k o s udy he dynamic subs i u abili y be ween
public heal h p og ams and p i a e e o s, aimed a imp o ing heal h s a us in old age,
gi en ha longe i y depends on he public heal h sys em.
By combining analy ical me hods and nume ical expe imen s, ou model shows ha cap-
i al pe young wo ke e ol es o e ime acco ding o di e en dynamics. Speci ically, wo
scena ios may eme ge: in he i s he alue o p i a e heal h spending is highe han public
spending, leading he elde ly o alloca e pa o hei sa ings o heal h; in he second he
alue o p i a e heal h spending is lowe han public spending, and he elde ly choose no o
in es in heal h. In bo h cases, he model admi s a unique, posi i e, and locally asymp o i-
cally s able s eady s a e. Howe e , as he sha e o public esou ces alloca ed o heal h ca e
changes, a swi ch be ween he wo scena ios may occu . While his ansi ion does no a ec
he equilib ium le el o longe i y, i s impac on he heal h s a us may be ambiguous.
The pape p oceeds as ollows. Sec ion 2 illus a es he se up o he model. Sec ion 3
p esen s he dynamics o he sys em. Sec ion 4 concludes.
2 The economic se up
2.1 Heal h s a us and quan i y in he elde ly
We conside a p oduc ion economy popula ed by o e lapping gene a ions o agen s who li e
o wo pe iods: adul hood and old age. Following (Chak abo y, 2004), longe i y is endog-
enous, i.e. he p obabili y o su i ing om adul hood o old age depends on public expendi-
u es on heal h. Hence, we assume ha longe i y depends only on public heal h expendi u es
made by he go e nmen a ime
∈N
, and he mo e he go e nmen in es s in public heal h-
ca e, he g ea e he p obabili y o an adul su i ing as an elde ly pe son.
We de ine as
w
he wage a e o an adul o a uni o e ec i e labo , hen, being
τ∈(0,1)
he axa ion a e, he public heal h inanced by he s a e posi i ely depends on
iscal e enues i.e.,
τw
.
We deno e by
g
he amoun o public expendi u e on heal h a ime , hen, le
γ∈(0,1)
be he ac ion o he public balance de o ed o he heal h sys em, we ob ain
g =γτw .
(1)
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Al hough he leng h o each pe iod is no malized o one, an indi idual’s li e ime is unce -
ain. Mo e p ecisely, he indi idual p obabili y o su i ing om he i s pe iod (adul hood)
o he nex (old age) depends on public expendi u es on heal h when young. We e e o his
componen as quan i y o li e in he elde ly, and i is desc ibed by a a iable measu ing he
su i al p obabili y o an adul , hus becoming old in he second pe iod. We deno e his
p obabili y as
p ∈[¯p, 1)
, gi en a ixed exogenous h eshold o longe i y, which is associ-
a ed wi h no in es men in public heal h, deno ed by
¯p∈[0,1)
.
Then, ollowing (Chak abo y, 2004) and Cip iani and Fio oni (2019), we conside a
unc ion
P:R+→[¯p, 1)
which associa es he longe i y
p
wi h public heal h expendi u e
g
, such ha ,
P(0) = ¯p
. Speci ically, he e exis s an exogenous le el o longe i y whe e
limg →+∞P(g )→1
, meaning ha i public heal h expendi u e is su icien ly high, hen
longe i y is close o one. The unc ion P is con inuous, wice di e en iable and such ha
P′>0
, indica ing ha inc eased public expendi u e on heal h imp o es longe i y, and
P′′ <0
, e lec ing diminishing ma ginal bene i s.
A unc ion sa is ying hese assump ions is as ollows:
p
=¯p+
(1 −¯p)g
1+g
.
(2)
Rega ding he heal h s a us in he elde ly, ollowing (Va a igos & Zaka ia, 2013), we
assume i is ela ed o he expendi u es in he heal h sys em h ough wo componen s: he
public one, inanced by he s a e ia axa ion, and he p i a e one, unded by he elde ly
hemsel es, who di e a po ion o hei esou ces om consump ion in old age. Howe e ,
unlike (Va a igos & Zaka ia, 2013), we conside p i a e and public heal h expendi u es o
ac as a pe ec subs i u e o imp o e he heal h s a us o he elde ly.
The subs i u abili y be ween p i a e and public heal h expendi u es can be explained by
he es ima es o Li e al. (2016), which sugges ha o mos OECD coun ies, ei he he
Cobb-Douglas o m o a linea o m is a easonable ep esen a ion o heal h echnology.
In ac , in old age indi iduals p ima ily alloca es esou ces owa d ea men s and he apies
aimed a managing he heal h e ec s o exis ing condi ions. In his con ex , p i a e spend-
ing may be a solu ion o he excessi e wai ing imes in public heal hca e and conges ion
p oblems ela ed o public spending.
We deno e p i a e heal h expendi u e in old age as
x +1
, which ep esen s a ac ion o
he indi idual’s sa ings. Thus, hese sa ings a e used bo h o consump ion in old age and
o imp o ing heal h s a us.
Rega ding he heal h s a us o he elde ly, i is assumed ha he heal h echnology ha
de e mines he heal h s a us depends on bo h p i a e and public heal h expendi u es a ime
+1
, wi h a pe ec deg ee o subs i u abili y be ween hem.6
The heal h s a us in old age, deno ed by
h +1
, can be o malized as ollows:
h +1 =αg +1 + (1 −α)x +1.
(3)
6 Conside ing impe ec subs i u abili y be ween public and p i a e spending on heal h echnology (conside -
ing, o example, a Cobb Douglas unc ion) implies ha he echnical subs i u ion a e be ween he wo ypes
o spending is no cons an bu dec eases as he use o one o he wo ypes o spending inc eases. As we
al eady men ioned, based on he empi ical analysis o Li e al. (2016), bo h ypes o unc ion (linea o Cobb
Douglas) a e suppo ed by empi ical da a, bu he use o a Cobb Douglas unc ion would make he heo e ical
analysis much mo e complex.
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whe e
α∈(0,1)
ep esen s he e ec i eness o public expendi u e in imp o ing he heal h
s a us o he elde ly, while
(1 −α)
is he e ec i eness o p i a e expendi u e by he elde ly
hemsel es. The a io
α
(1−α)
measu es he a e a which p i a e and public heal h expendi-
u e can be exchanged, while main aining a cons an le el o heal h.
2.2 In e empo al cons ained u ili y maximiza ion
We conside an economy consis ing o an in ini e sequence o o e lapping gene a ions, each
po en ially li ing o wo pe iods, alongside an in ini ely-li ed go e nmen . Time is disc e e,
ha is,
=1,2, ...
. Following (Chak abo y, 2004), we assume ha each indi idual bo n in
gene a ion gi es bi h o one o sp ing a he end o pe iod , be o e expe iencing hei mo -
ali y shock. The new indi idual becomes economically ac i e only a he beginning o
+1
.
The e o e, in each pe iod a measu e-one coho o adul s is bo n, each endowed wi h one uni
o ime, which hey inelas ically supply o he labo ma ke , ea ning a wage income
w
. Fol-
lowing (Cip iani & Fio oni, 2019), o keep he model mo e ac able, we assume ha all adul s
e i e a he end o he i s pe iod.
In summa y, adul s bene i om he consump ions du ing adul hood, consump ions in
old age, and hei heal h s a us
h +1
when old, depending on he su i al p obabili y. The
li e ime u ili y o an indi idual o gene a ion is hen gi en by he ollowing unc ion7:
U = ln c +p {βln c +1 +θln h +1},
(4)
whe e
c
is consump ion du ing adul hood,
c +1
is consump ion du ing old age, and
h +1
is he heal h s a us du ing old age. The pa ame e s
β
,
θ
a e posi i e, and
p
ep esen s he
p obabili y o su i ing om you h o old age.
The budge cons ain in he i s young adul hood pe iod is gi en by:
(1 −τ)w =c +s
(5)
whe e
τ∈(0,1)
is he ax a e on labo income and
s
e e s o sa ings (unde s ood as he
pu chase o annui ies). All a iables a e assumed o be non-nega i e.
Gi en he adul s’ sala y and he in e es a e, he cos o heal h expendi u e educes he
esou ces a ailable o bo h u u e consump ion and sa ings.
The second-pe iod budge cons ain is gi en by:
s ˆ
R +1 =c +1 +x +1
(6)
whe e
x +1
is he p i a e expendi u e o heal h s a us, and
ˆ
R +1
is he g oss e u n on i s
sa ings. In ac , consump ion in old age is inanced by he e u ns on sa ings accumula ed
du ing adul hood. Fu he mo e, we assume ha all goods a e pe ishable and ha agen s can
7 The assump ion o addi i i y in he u ili y unc ion in o e lapping gene a ions models is o en used (see, o
example (De La C oix & Michel, 2002)). Addi i i y implies ha he well-being de i ed om consump ion
does no depend on he le el o heal h and ice e sa. In eali y, consump ion is o en mo e “use ul” i you
a e heal hy and ice e sa: in ac , g ea e consump ion can imp o e heal h s a us. Howe e , his hypo hesis
allows us o analyze choices ela ing o consump ion and heal h sepa a ely: in ac , he agen decides o in es
pa o hei sa ings in heal h only by looking a he di ec con ibu ion o u ili y, wi hou conside ing how
heal h a ec s he ma ginal u ili y o consump ion.
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only ans e alue o e ime h ough capi al ma ke s. Indi iduals a e assumed o ha e no
beques mo i es. Following (Chak abo y, 2004), in o de o elimina e he isks associa ed
wi h unce ain li espans, we assume he exis ence o a pe ec annui y ma ke , whe e all sa -
ings a e managed h ough mu ual unds. A he end o hei you h, indi iduals deposi hei
sa ings in o a mu ual und. These unds a e exclusi ely in es ed in capi al, and he mu ual
und gua an ees a g oss e u n o hose who su i e in o old age. I hese unds yield a g oss
e u n o
ˆ
R +1
on i s in es men , hen unde pe ec compe i ion, equilib ium in he annui y
ma ke is main ained:
ˆ
R
+1 =
R
+1
p
,
(7)
whe e
R +1
is he g oss in e es a e.
To summa ize, we can collec all he equa ions p esen ed so a o o mula e he ollow-
ing cons ained op imiza ion p oblem:
max U = ln c +βp ln c +1 +θp ln h +1
(8)
s. .: g +1 =γτw +1
(9)
c +s =(1 −τ)w
(10)
c +1 +x +1 =s ˆ
R +1
(11)
ˆ
R
+1 =
R
+1
p
(12)
h +1 =αg +1 + (1 −α)x +1
(13)
x +1 ≥0.
(14)
The solu ion o he maximum cons ained op imiza ion p oblem esul s in he ollowing
P oposi ion.
P oposi ion 2.1 The i s -o de condi ions o p oblem (8–14) a e as ollows.
Case A: I
(1 −α)θR +1(1 −τ)w ≥(1+βp )αγτw +1
(15)
hen
c
=
1
1+(
β
+
θ
)
p
[
p
R
+1
α
1−
αγτw +1 + (1
−
τ)w
],
(16)
1 3
Annals o Ope a ions Resea ch
c
+1 =βR +1
1+(β+θ)p
[
p
R
+1
α
1
−
αγτw +1 + (1 −τ)w
],
(17)
x
+1 =
1
1+(β+θ)p [
θR +1(1
−
τ)w
−
(1+βp )
α
1−α
γτw +1
],
(18)
s
=p
1+(β+θ)p
[
(β+θ)(1 −τ)w −
1
R
+1
α
1
−
αγτw +1
].
(19)
Case B: I
(1 −α)θR +1(1 −τ)w <(1+βp )αγτw +1
(20)
hen
c
=
(1 −τ)w
1+βp
,
(21)
c
+1 =
βR
+1
(1 −τ)w
1+βp
,
(22)
x +1 =0,
(23)
s
=βp
(1 −
τ
)
w
1+βp
.
(24)
Mo eo e , in bo h cases, hese condi ions a e also su icien .
P oo See Appendix A.
□
The p e ious condi ions e e ed o Case A and Case B become easie o in e p e eco-
nomically. In ac , he condi ion ela ing o Case A is:
(1 −α)θR +1(1 −τ)w ≥(1+βp )αg +1.
(25)
The le -hand side o he inequali y ep esen s he alue (
θ
) o he (ne , discoun ed) income
spen on p i a e expendi u e when elde ly (also aking in o accoun he e ec i eness
1−α
),
while he igh -hand side, on he o he hand, ep esen s he alue o public expendi u e
g +1
o which he e ec i eness
α
is also aken in o accoun . The e o e, when he inequali y o
Case A is e i ied, i means ha he alue o p i a e heal h expendi u e is g ea e han public
expendi u e. As a consequence, p i a e spending on heal h by he elde ly is posi i e.
Con e sely, in Case B he alue o p i a e spending on heal h is lowe han public spend-
ing, and he e o e he elde ly will no in es hei sa ings in heal h spending, i. e.
x +1 =0
.
Thanks o he i s -o de condi ions, we can also de i e
h +1
by subs i u ing he exp es-
sion o
x +1
in bo h cases A and B ( espec i ely gi en by (18) and by (23)) in o he o -
mula (3) and conside ing ha
g +1 =γτw +1
. A e some algeb aic manipula ions, he
exp ession o
h +1
in Case A is as ollows:
1 3
Annals o Ope a ions Resea ch
h
A
+1 =
[
αγτw +1p
+ (1 −
α
)(1 −
τ
)
R +1w
]
θ
1+(β+θ)p
,
(26)
while o Case B, we ob ain:
hB
+1 =
αγτw
+1.
(27)
I is s aigh o wa d o obse e ha
hA
+1 ≥
h
B
+1
. In ac , om (13), (9), and (14) we ha e:
h +1 =
αg
+1 + (1 −
α
)
x
+1 ≥
αg
+1 =
αγτw
+1 =
h
B
+1.
(28)
In pa icula , (28) holds i
hA
+1
is subs i u ed on he le -hand side. Al e na i ely, one a i es
a a simila conclusion by bounding (26) using (15). In ac , om (15) we ha e:
h
A
+1 =
αγτw
+1
p
θ+ (1 −α)(1 −τ)R
+1
w
θ
1+(β+θ)p
≥
αγτw +1p θ+(1+βp )αγτw +1
1+(β+θ)p
=αγτw +1 =hB
+1
.
As o he exp ession o
p
, in bo h cases we ind ha by subs i u ing (1) in o (2), we ob ain:
p
=¯p+
(1 −¯p)g
1+g
=¯p+
(1 −¯p)γτw
1+γτw
.
(29)
Thus, i can be seen ha he quan i y o li e, i.e. longe i y, is equal in he wo scena ios as i
depends only on public spending; con e sely, heal h s a us is highe in scena io A whe e he
elde ly in es pa o hei sa ings in imp o ing hei heal h.
2.3 P oduc ion, in es men and sa ing
As p e iously men ioned, a each pe iod, a new gene a ion o adul s, each wi h measu e one,
en e s he economy. Each agen is endowed wi h one uni o labo du ing hei you h and is
compulso ily e i ed in old age.
The agg ega e p oduc ion echnology o he economy is assumed o ollow a cons an -
e u ns- o-scale p oduc ion unc ion, u ilizing bo h labo and capi al. Capi al s ock is
assumed o ully dep ecia e a e one pe iod o use, meaning ha he capi al s ock in any
gi en pe iod is equal o he sa ings in he p e ious pe iod.
P oduc ion in ime employs physical capi al
K
and labo L. We deno e
Y
he agg ega e
ou pu and ep esen he agg ega e echnology o he economy by he ollowing p oduc ion
unc ion:
Y =
F
(
K
,L
)=
AK
δ
L
1−δ.
(30)
In ha equa ion,
Y
,
K
, and L espec i ely s and o agg ega e ou pu , physical capi al s ock
and e ec i e labo in he economy in pe iod , while A is he o al ac o p oduc i i y, and
δ∈(0,1)
is he p oduc i i y o physical capi al.
1 3
Annals o Ope a ions Resea ch
A key ques ion, howe e , conce ns how he sys em ansi ions be ween hese scena -
ios as a pa ame e -speci ically, he sha e o he public budge alloca ed o he heal hca e
sys em- changes.
We i s conside he case wi h low
α
- alue, i.e.
α=0.5
so ha , as eme ges om Fig.
1b, he ixed poin always belong o
RA
, i.e., a posi i e ac ion o sa ings is alloca ed o
p i a e heal hca e o all le els o public budge de o ed o heal h sys em
γ
. The esul ing
equilib ium alue o capi al pe wo ke ,
k∗=k∗
A
as mo ing
γ∈(0,1)
, is shown in Fig. 4a.
The i ual ixed poin associa ed wi h he no-p i a e-spending scena io ypically esul s in
a lowe le el o
k∗
, assuming he easible equilib ium solu ion is maximizing. Howe e he
co esponding cu e (in cyan) exhibi s a hump-shaped pa e n: as
γ
inc eases, he equilib-
ium alue o capi al pe wo ke ini ially ises and hen declines. This sugges s he exis ence
o an op imal public budge sha e o heal hca e ha maximizes he equilib ium ou come.
We now u n o he case o a high
α
- alue, speci ically
α=0.8
. In con as o he p e i-
ous case, he esul ing scena io he e depends on he
γ
- alue, he sha e o he public budge
alloca ed o he heal h sys em. Fo low alues o
γ
, he model again p oduces scena io A,
cha ac e ized by posi i e heal hca e spending. In his case, he maximum capi al pe young
wo ke le el is eached a
γ≃0.11
. Howe e , as he sha e o he public spending on heal h-
ca e inc eases and c osses he h eshold alue
γ≃0.6
, a ansi ion occu s: he ixed poin
k∗
A
becomes i ual, while he ixed poin
k∗
B
becomes easible. This ma ks a shi o a new
si ua ion in which he e is no p i a e heal hca e spending.
In his case, since public heal hca e spending is highly e ec i e, hen he e exis s a
h eshold le el o ax e enue alloca ed o heal hca e ha shi s he equilib ium om Case A
o Case B. This implies ha , i public spending is highly p oduc i e in gene a ing “heal h",
he e is no need o he elde ly o alloca e pa o hei sa ings o heal h expendi u e. In hese
coun ies, an e ec i e and ex ensi e public heal hca e sys em alone is su icien o ensu e a
good heal h s a us. Con e sely, i public and p i a e heal hca e spending a e equally e ec-
i e, only he scena io in ol ing p i a e heal h expendi u e can eme ge. In his case, no
le el o public spending is su icien o p e en he elde ly om in es ing in “heal h".
Finally, we aim o asses whe he , and o wha ex en , he op imal choices eme ging om
bo h scena ios a ec bo h li e quan i y and heal h s a us.
Fig. 4 The equilib ium poin
k∗
o egion A (in cyan) and egion B (in magen a) being
γ∈(0,1)
1 3

Annals o Ope a ions Resea ch
Longe i y is gi en by
p (k )
as de ined in equa ion (35), while heal h s a us in old age
-depending on he scena io- is desc ibed by he ollowing equa ions:
h
A
+1 =
{
αγτkδ
+1
[
¯p+
(1
−
¯
p
)
γτA
(1
−δ
)
k
δ
1+γτA(1−δ)kδ
]
+ (1 −α)(1 −τ)
(
1+Aδkδ−
1
+1
)
kδ
}
A(1 −δ)θ
1+(β+θ)
[
¯p+(1−¯p)γτA(1−δ)kδ
1+γτA(1
−
δ)kδ
],
(42)
in he case o posi i e p i a e heal hca e spending, and
hB
+1 =
αγτA
(1 −
δ
)
k
δ
+1,
(43)
in he case o no p i a e heal hca e spending.
Wi h ega d o longe i y, e en i i s unc ional o m emains he same ac oss bo h sce-
na ios, i a ies wi h he equilib ium le el o capi al pe wo ke , which in u n depends on
he sha e o he public budge
γ
alloca ed o he heal hca e sys em.
As shown clea ly in bo h Figs. 5a and 6a, li e expec ancy inc eases as
γ
inc eases, e en
in he absence o p i a e heal hca e spending. This e idence holds ega dless o whe he
public spending is ela i ely e ec i e o no and i pe sis s e en when ansi ions be ween
scena ios occu . This is because longe i y is de e mined solely by public heal hca e spend-
ing, no p i a e expendi u e, which makes he ou come simila in bo h scena ios.
Howe e , a di e en beha io eme ges when conside ing heal h s a us. Speci ically,
when he e ec i eness o public heal hca e spending is low, inc easing he sha e o he
public budge dedica ed o heal h does no necessa ily imp o e heal h s a us- his is e i-
den in Fig. 5b. In con as , when public spending is highly e ec i e, e en a shi be ween
scena ios, does no hinde imp o emen s: inc easing he public budge sha e dedica ed o
heal h becomes an e ec i e ool o enhancing he equilib ium le el o heal h s a us (see
Fig. 6b). Indeed, when public heal hca e spending is signi ican ly mo e p oduc i e han
p i a e spending, a scena io wi h only public heal hca e -wi hou any p i a e heal h expen-
di u e-may esul in a highe heal h s a us. This p o ides impo an policy insigh s: i public
heal hca e spending is signi ican ly mo e e ec i e han p i a e al e na i es, and he S a e
Fig. 5 Equilib ium alues o
p∗
(panel a) and
h∗
(panel b) wi h
α=0.5
o egion A (in cyan) and egion
B (in magen a) being
γ∈(0,1)
1 3
Annals o Ope a ions Resea ch
chooses o alloca e a la ge sha e o i s ax e enue o heal hca e, a ully public sys em can
ensu e a highe heal h s a us han a mixed public-p i a e model.
4 Conclusion
We conside ed an o e lapping gene a ions model o in es iga e he ole o public e sus p i-
a e heal hca e in de e mining bo h he quan i y o li e and heal h s a us in old age, as well as
i s impac on s eady-s a e economic g ow h.
Depending on he compa ison be ween he alue o p i a e and public heal h expendi-
u es, wo scena ios can eme ge: ei he he elde ly alloca e a posi i e amoun o hei sa -
ings o p i a e heal hca e, o hey choose no o in es in p i a e heal h a all.
Wi h ega d o he exis ence and s abili y o s eady-s a e equilib ia in capi al pe wo ke
- and he co esponding dynamics o li e quan i y and heal h s a us - we combine analy i-
cal ools wi h nume ical me hods o demons a e ha , in each scena io, a posi i e, locally
asymp o ically s able s eady-s a e eme ges. This s eady s a e can be ei he easible o i -
ual, depending on he model’s pa ame e alues.
Acco dingly, we ixed he alues o he main pa ame e s and a y bo h he sha e o he
public budge alloca ed o heal hca e and he e iciency o public spending in he heal h
p oduc ion unc ion.
On he one hand, ou wo k is able o show ha i public and p i a e spending a e equally
e ec i e in he heal h p oduc ion unc ion, hen he easible s eady-s a e is cha ac e ized
by posi i e p i a e heal h expendi u e by he elde ly, ega dless o he sha e o he public
budge alloca ed o he heal h sys em. Di e en ly, i public spending is mo e e icien han
p i a e expendi u e, hen he e is a ansi ion om a si ua ion wi h posi i e p i a e heal h-
ca e spending o a one wi h no p i a e heal hca e spending, as long as he sha e o he public
budge de o ed o he heal hca e sys em is inc eased.
Fig. 6 Equilib ium alues o
p∗
(panel a) and
h∗
(panel b) wi h
α=0.8
o egion A (in cyan) and egion
B (in magen a) being
γ∈(0,1)
1 3
Annals o Ope a ions Resea ch
On he o he hand, i shows ha an inc ease in he sha e o he public budge alloca ed
o he heal hca e sys em has a posi i e e ec on longe i y in bo h scena ios, whe he o
no he elde ly engage in p i a e heal hca e spending. Howe e , he impac o his inc ease
on heal h s a us may be ambiguous, as i depends c i ically on he e ec i eness o public
spending wi hin he heal h p oduc ion unc ion. Speci ically, when public expendi u e is
ela i ely ine ec i e, inc easing he alloca ion o public esou ces o heal hca e does no
necessa ily esul in imp o emen s in heal h s a us. In con as , when public spending is
highly e ec i e, a la ge alloca ion o public esou ces becomes a powe ul ool o imp o e
heal h s a us in old age.
The abo emen ioned esul s esul s sugges ha policymake s could imp o e he e ec-
i eness o public heal h expendi u es by p io i izing p e en i e ca e p og ams, imp o ing
access o p ima y heal hca e se ices, and in es ing in accina ion and disease con ol ini-
ia i es. In addi ion, a ge ed subsidies o essen ial ea men s and he expansion o public
hospi al capaci y could help ensu e ha inc eased public spending is ansla ed in o angible
imp o emen s in bo h longe i y and heal h ou comes.
4.1 Fu he de elopmen s o he model can be conside ed in u u e esea ch
Fi s , di e en heal h echnologies could be conside ed ha ake in o accoun he impe -
ec subs i u abili y be ween public and p i a e heal hca e expendi u es. In ac , conside ing
impe ec subs i u abili y, such as by adop ing a Cobb-Douglas speci ica ion, he a e o
echnological subs i u ion be ween public and p i a e expendi u e would no longe emain
cons an bu would diminish. This means, o example, ha as public heal hca e expendi u e
inc eases, i s ma ginal ’p oduc i i y’ would all ela i e o p i a e expendi u e. This adjus -
men could al e he op imal alloca ion be ween public and p i a e spending; howe e , ana-
lyzing his case would equi e a new model, which may no be analy ically ac able gi en
he added complexi y in oduced by non-linea i ies.
Fu he mo e, he assump ion o addi i i y in he u ili y unc ion implies ha he u ili y
de i ed om consump ion is independen o he indi idual’s heal h s a us, and ice e sa.
The addi i e speci ica ion allows consump ion and heal h- ela ed decisions o be analyzed
sepa a ely. Unde his assump ion, agen s alloca e sa ings o heal h solely based on i s di ec
con ibu ion o u ili y, wi hou accoun ing o he way heal h migh in luence he ma ginal
u ili y o consump ion. In con as , a non-addi i e (e.g., mul iplica i e) u ili y speci ica ion
in oduces complemen a i y be ween heal h and consump ion. In such a amewo k, be e
heal h inc eases he ma ginal u ili y o consump ion, he eby inc easing he likelihood o
highe in es men in heal hca e. Simila ly, g ea e consump ion can ein o ce heal h and
imp o e i s ma ginal u ili y. While his app oach is mo e ealis ic, i signi ican ly inc eases
he model’s complexi y and elimina es he possibili y o ea ing he wo choices indepen-
den ly. Howe e , i ep esen s a p omising and na u al ex ension, which we plan o pu sue
in u u e esea ch.
Finally, an impo an ex ension o he model in u u e esea ch could be o conside ha
longe i y, as well as heal h in old age, depends no only on public spending bu also on p i-
a e spending. This inclusion could pe haps make he model di icul o s udy analy ically,
bu i would ce ainly add in e es ing elemen s o u he e lec ion.
1 3
Annals o Ope a ions Resea ch
Appendix A
We begin by making p elimina y subs i u ions o simpli y he maximiza ion p oblem,
educing he numbe o cons ain s. To do his, we subs i u e he exp ession o
g +1
gi en
by (9) in o (13), and he exp ession o
ˆ
R +1
gi en by (12) in o (11). Then, om (11), we
calcula e
s
as
s
=p
R +1
(
c +1
+
x +1
)
(A.1)
and subs i u e his exp ession in o (10). Finally, we subs i u e (13) in o (8) and ob ain he
ollowing equi alen model:
max U = ln c +βp ln c +1 +θp ln [αγτw +1 + (1 −α)x +1]
(A.2)
s. .: c +
p
R +1
(c +1 +x +1) = (1
−
τ)w (A.3)
−x +1 ≤0.
(A.4)
We sol e model (A.2–A.4) using he me hod o Lag ange mul iplie s. In his ega d, we
de ine
η
as he Lag ange mul iplie associa ed wi h cons ain (A.3), and
ξ
as he Lag ange
mul iplie associa ed wi h cons ain (A.4). We obse e ha model (A.2–A.4) is a mixed
cons ained maximiza ion p oblem, as i includes bo h an equali y cons ain and an inequal-
i y cons ain . Acco ding o, he heo em on Lag ange mul iplie s (see, o ins ance, Simon
and Blume 1994, pp. 434 and 435), he cons ain quali ica ion equi es ha he ank o he
Jacobian ma ix -e alua ed a a local maximize and co esponding o he se o equali y
cons ain s and binding inequali y cons ain s- be maximal. In ou case, his ma ix, ega d-
less o he poin a which i is e alua ed, is gi en by:
[1p
R +1
p
R +1
00
−
1
].
Since i s ank is always maximal, he cons ain quali ica ion condi ion is sa is ied.
We can he e o e cons uc he Lag angian unc ion, whose exp ession is:
L
(c
,c
+1
,η
,ξ
) = ln c
+βp
ln c
+1
+θp
ln [αγτw
+1
+ (1 −α)x
+1
]
−
η
[
c +p
R +1
(c +1 +x +1)
−
(1
−
τ)w
]
+ξ x +1
By di e en ia ing he Lag angian wi h espec o he decision a iables
c
,
c +1
, and
x +1
we espec i ely ob ain:
∂L
∂c
=0:
1
c
−
η
=0
(A.5)
1 3
Annals o Ope a ions Resea ch
∂L
∂c +1
=0:
βp
c +1
−
η
p
R +1
=0
(A.6)
∂
L
∂x +1
=0:
(1 −
α
)
θp
αγτw +1 + (1 −α)x +1
−
η
p
R +1
+ξ
=0.
(A.7)
I is also necessa y o sa is y he ini ial cons ain s:
c
+p
R +1
(
c +1
+
x +1
) = (1 −
τ
)
w (A.8)
and
x +1 ≥0.
(A.9)
Mo eo e , he complemen a y slackness condi ion equi es ha :
ξ x +1 =0.
(A.10)
Finally, he non-nega i i y condi ion o he mul iplie associa ed wi h he inequali y con-
s ain is gi en by:
ξ ≥0.
(A.11)
F om (A.5) we ge
c
=
1
η
.
(A.12)
Likewise, om (A.6) we ge
c
+1 =
βR
+1
η
.
(A.13)
We i s conside he case when
x +1 >0
. We e e o his case as Case A. In ligh o (A.10),
his implies ha
ξ =0
. By eplacing
ξ =0
in o (A.7), a e some algeb a, we ind he ol-
lowing exp ession o
x +1
:
x
+1 =
θ
η
R +1
−α
1−α
γτw +1
.
(A.14)
By subs i u ing (A.12), (A.13), and (A.14) in o (A.3), we ind
η
=
1
1
1+(β+θ)p
[
p
R +1
α
1
−
αγτw +1 + (1
−
τ)w
].
(A.15)
1 3

Annals o Ope a ions Resea ch
By subs i u ing his alue in o (A.12), (A.13), and (A.14), we ob ain he alue o
c
,
c +1
,
and
x +1
espec i ely. Finally, by imposing
x +1 >0
, we ge he ollowing condi ion:
(1 −α)θR +1(1 −τ)w >(1+βp )αγτw +1.
(A.16)
Case B a ises when
ξ >0
, which implies
x +1 =0
. F om (A.7), we ob ain
ξ
=
p
R +1
η
−(1 −α)θp
αγτw +1 + (1 −α)x +1
.
(A.17)
By subs i u ing (A.12), (A.13), and
x +1 =0
in o (A.3), we ind
η
=
1+βp
(1 −τ)w
.
(A.18)
By subs i u ing his alue in o (A.12) and (A.13), we ob ain he alue o
c
and
c +1
,
espec i ely.
Mo eo e , by subs i u ing (A.18) in o (A.17), we ind he condi ion cha ac e izing
Case B. The case when
ξ =0
and
x +1 =0
ep esen s he bounda y be ween he wo
egions co esponding o Cases A and B. This jus i ies he use o
≥
(ins ead o >) in he
inequali y desc ibing Case A.
The exp essions (19) and (24) o
s
a e ob ained by subs i u ing he alues o
c +1
and
x +1
in o (A.1), espec i ely o Cases A and B.
Finally, he Hessian ma ix o he Lag angian unc ion wi h espec o he decision a i-
ables
c
,
c +1
, and
x +1
is gi en by:
H
L =



−
1
c2
0 0
0−βp
c2
+1
0
00
−
θ(1−α)2p
[αγτw +1+(1−α)x +1]
2


.
(A.19)
This ma ix is clea ly nega i e de ini e. Consequen ly, he i s -o de condi ions a e also
su icien .
Appendix B
F om (31), i ollows ha
w +1 =
A
(1 −
δ
)
k
δ
+1.
(B.1)
Likewise, (32) implies
+1 =
Aδk
δ
+1.
(B.2)
Consequen ly, om (33) we ob ain:
1 3
Annals o Ope a ions Resea ch
R +1 =1+
+1 =1+
Aδk
δ
+1.
(B.3)
F om (29) and (31), we can ob ain a new exp ession o
p
as ollows:
¯
p
+
(1
−
¯
p
)
γτw
1+γτw
=¯p+
(1
−
¯
p
)
γτA
(1
−δ
)
k
δ
1+γτA(1 −δ)k
δ
=: p (k )
.
(B.4)
As s a ed in P oposi ion 2.1, he bounda y sepa a ing egions A and B is gi en by he ol-
lowing equa ion:
θR
+1
(1 −
τ
)
w
= (1 +
βp
)α
1−α
γτw +1
.
(B.5)
B inging all e ms o he le -hand side, applying he p e iously men ioned subs i u ions,
and making he dependence o
p
on
k
explici as s a ed in (B.4), we de ine he le -hand
side o his new equa ion as
C(k ,k
+1)
. This allow us o exp ess he bounda y condi ion
in e ms o he equa ion
[
θ
(
1+Aδkδ−
1
+1
)
(1
−
τ)kδ
−
(1+βp (k ))
α
1−α
γτkδ
+1
]
A(1
−
δ
)=0.
Nex , o bo h Cases A and Case B, we subs i u e he exp ession o
s
om (34), aking in o
accoun he p e ious subs i u ions bo h in he exp ession o
s
and in he condi ions cha ac-
e izing each case. F om his, he hesis ollows. As wi h he bounda y condi ion, all e ms
mus be b ough o he le -hand side. We hen de ine
A
o egion A and
B
o egion B.
Appendix C
(a) I is i ial o e i y ha
ˆ
B(0) = 0
.
(b) By applying he chain ule o compu e
ˆ
′
B(
k
)
, we ob ain:
ˆ
′
B(k )=β
(1 −
τ
)
A
(1 −
δ
)
(1+βp
(k ))
2
[
p′
(k )kδ
+δp
(k )kδ−1
(1+βp
(k ))
].
(C.1)
Since all he pa ame e s a e posi i e,
k ≥0
,
p (k )>0
o all
k ≥0
, and
p′(k )>0
o
all
k ≥0
, i ollows ha
ˆ
′
B(k )≥0
o all
k ≥0
. (c) Exploi ing he ac ha
o all
k ≥0
i holds ha
¯p≤p (k )<1
, and we ob ain:
ˆ
B(k )>
β¯p(1 −τ)A(1 −δ)
1+β
kδ
.
(C.2)
Applying he squeeze heo em, he claim ollows. (d) We compu e he i s de i a i e a
0 by aking he limi o he inc emen al a io as
h→0+
. Speci ically, also in ligh
o (C.2), we ha e:
1 3
Annals o Ope a ions Resea ch
lim
k →
0+
ˆ
′
B(k ) = lim
h
→
0+
ˆ
B
(
h
)
−
ˆ
B
(0)
h
= lim
h
→
0+
ˆ
B
(
h
)
h
>lim
h
→
0+
β
¯
p
(1
−τ
)
A
(1
−δ
)
1+β
1
h
1
−
δ=+
∞.
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Re e ences
Aísa, R., Clemen e, J., & Pueyo, F. (2014). The in luence o (public) heal h expendi u e on longe i y. In e -
na ional Jou nal o Public Heal h, 59, 867–875.
Annas, Geo ge J., & Galea, Sand o. (2018). Dying heal hy: Public heal h p io i ies o ixed popula ion li e
expec ancies. Annals o In e nal Medicine, 169(8), 568–569.
Anwa , Asim., Hyde , Shabi .,Mohamed No , No ashidah ., & Younis, Mus a a.(2023). Go e nmen heal h
expendi u es and heal h ou come nexus: A s udy on oecd coun ies. F on ie s in Public Heal h (11),
1123759 .
Ay emiz, Le en ., Sa ., Gamze, Baya ., Yilmaz, Danilina, Ma ina., & Sezgin Funda H. (2024). The long-
e m e ec o social, educa ional, and heal h expendi u es on indica o s o li e expec ancy: An empi ical
analysis o he oecd coun ies. F on ie s in Public Heal h, 12, 1497794.
Bha acha ya, J., & Qiao, X. (2007). Public and p i a e expendi u es on heal h in a g ow h model. Jou nal o
Economic Dynamics and Con ol, 31(8), 2519–2535.
Chak abo y, S. (2004). Endogenous li e ime and economic g ow h. Jou nal o Economic Theo y, 116(1),
119–137.
Cip iani, Giam Pie o., & Fio oni, Tama a. (2019). Heal h spending, educa ion and endogenous demog aph-
ics in an olg model. Human Capi al and Economic G ow h: The Impac o Heal h, Educa ion and Demo-
g aphic Change, pp. 209–249.
De La C oix, Da id., & Michel, Philippe. (2002). A heo y o economic g ow h: dynamics and policy in
o e lapping gene a ions. Camb idge Uni e si y P ess.
Glied, She y., & Smi h, Pe e C.(2013). The Ox o d handbook o heal h economics. Ox o d Uni e si y
P ess.
Li, Shuyun May, Moslehi, Solmaz, & Yew, Siew Ling. (2016). Public-p i a e mix o heal h expendi u e:
A poli ical economy and quan i a i e analysis. Canadian Jou nal o Economics Re ue canadienne
d’économique, 49(2), 834–866.
Wo ld Heal h O ganiza ion e al.(2022). Ghe: Li e expec ancy and heal hy li e expec ancy. The Global
Heal h Obse a o y [In e ne ].[ci ed 26 Aug 2022]. h ps://www who.in /da a/gho/da a/ hemes/
mo ali y-and-global-heal h-es ima es/ghe-li e-expec ancy-and-heal hy-li e-expec ancy.
Simon, C. P., & Blume, L.(1994). Ma hema ics o economis s. W. W. No on & Company, ISBN
978-0-393-11752-3.
Va a igos, Dimi ios., & Zaka ia, In an Zana iah. (2013). Endogenous e ili y in a g ow h model wi h pub-
lic and p i a e heal h expendi u es. Jou nal o Popula ion Economics, 26, 67–85.
1 3
Annals o Ope a ions Resea ch
Za ulli, Vi ginia., Caswell, Hal.(2024) Longe heal hy li e, bu o how many? A s ochas ic analysis o
heal hy li espan inequali y. Annals o Ope a ions Resea ch, pages 1–18.
Publishe 's No e Sp inge Na u e emains neu al wi h ega d o ju isdic ional claims in published maps and
ins i u ional a ilia ions.
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