How harmful is a high share of public expenditure in GDP?

Menguy, Sé e ine
A icle
How ha m ul is a high sha e o public expendi u e in GDP?
Economies
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GDP?, Economies, ISSN 2227-7099, MDPI, Basel, Vol. 13, Iss. 3, pp. 1-22,
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Published: 13 Ma ch 2025
Ci a ion: Menguy, S. (2025). How
Ha m ul Is a High Sha e o Public
Expendi u e in GDP? Economies,13(3),
74. h ps://doi.o g/10.3390/
economies13030074
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A icle
How Ha m ul Is a High Sha e o Public Expendi u e in GDP?
Sé e ine Menguy
Facul é Socié és e Humani és, Uni e si é Pa is Ci é, 12 Rue de L’école de Médecine, 75270 Pa is CEDEX 06,
F ance; [email p o ec ed]
Abs ac : We aim o analyze he po en ial posi i e o nega i e e ec s o public expendi u e
on economic g ow h, as well as hei de e minan s. To his goal, we use a simple heo e ical
model, which has he speci ici y o dis inguish be ween public in es men and consump ion
expendi u e, and which could be applied o a wide ange o de eloped o de eloping
coun ies. Rega ding public spending, we ind ha public consump ion expendi u e usually
ha ms global economic g ow h, whe eas public in es men expendi u e bene i s economic
ac i i y: i can inc ease income pe head, p o ided eal e u ns on capi al a e no oo small.
We can also heo e ically unde line he exis ence o an in e ed U-shaped ela ion be ween
he a ia ion o public in es men o consump ion expendi u e and economic g ow h.
An inc ease in public spending would bene i economic g ow h only up o a maximal
a ia ion, which posi i ely depends on eal capi al e u ns, bu nega i ely depends on
he capi aliza ion o he economy. Rega ding iscal esou ces, we ind ha inc easing he
consump ion axa ion a e and he sha e o iscal esou ces collec ed h ough consump ion
axes could bene i global economic g ow h, e en i i is de imen al o p i a e consump ion.
Keywo ds: go e nmen size; public consump ion; public in es men ; axa ion a es; eco-
nomic g ow h; non-linea ela ion
1. In oduc ion
A wo ldwide endency was he huge inc ease o he sha e o public expendi u e in
GDP du ing he XXe cen u y, om a ound 10% o GDP on a e age in 1900 o 40% on
a e age and e en mo e han 50% o GDP in some O ganiza ion o Economic Coope a ion
and De elopmen (OECD) coun ies in 2000: highe p o ision o public goods and se ices,
s onge ans e s. This can be due o he inc easing demand o social se ices in mo e
de eloped coun ies (Wagne law), o he highe acili y o poli icians o inc ease public
expendi u e and budge de ici s a he han educe social bene i s. The e o e, economic
analysis has been conduc ed o de e mine whe he his public expendi u e g ow h bene i s
o ha ms economic g ow h. Howe e , he e ec o he global size o public spending can
hen be dis inguished om he e ec o he s uc u e o his public spending.
The link be ween go e nmen size and economic g ow h is no unambiguous. Fi s ,
public expendi u e can bene i economic g ow h, as public consump ion is one explici
elemen o inal consump ion in calcula ing he G ow h Domes ic P oduc Indica o . By
iscal mul iplie s, inc easing public expendi u e (in as uc u e, Resea ch and De elopmen ,
heal h, educa ion) can imply an inc ease mo e han p opo ional in inal p oduc ion and
global economic ac i i y. A e e se causali y explana ion o he posi i e link is ha highe
economic g ow h implies highe iscal e enues (VAT o pe sonal axes). Howe e , wi h
he e ic ion e ec , high public expendi u e can also inc ease in e es a es and ha m he
mo e p oduc i e p i a e in es men and p i a e consump ion. Highe axa ion a es o
inance public expendi u e can educe he e u ns o in es men , he p oduc i i y and
Economies 2025,13, 74 h ps://doi.o g/10.3390/economies13030074
Economies 2025,13, 74 2 o 22
he incen i e o in es and o p oduce. A e e se causali y explana ion can also jus i y
he nega i e link be ween he size o he go e nmen and economic g ow h. In case o
weake economic g ow h o ecession, au oma ic s abilize s and iscal ans e s such as
unemploymen p o isions, mus inc ease o compensa e o he weake pu chasing powe
o households.
Since 2005, F ance and Denma k ha e he highes sha e o public expendi u e in GDP.
Fo example, eaching 57.5% o GDP in 2024, o al public gene al expenses in F ance a e
he highes in he Eu opean Union and nea ly 10 pe cen age poin s abo e he Eu opean
a e age (49.2%). The high le el o public expendi u e in F ance is due o social spending,
and o he la ge deg ee o isk socializa ion: mainly e i emen expendi u e, bu also
housing subsidies, amily allowances, and heal h. A high le el o public spending is no
a p oblem in i sel , as i can mos ly e lec di e en choices in he socializa ion o ce ain
ypes o spending (pensions, heal hca e, educa ion, e c.). Fo example, in he Scandina ian
coun ies, a high le el o public expendi u e is no a p oblem as unde lying policies and
ins i u ions a e mainly e icien . Ne e heless, i can become a p oblem when high public
spending e lec s ine icien public policies, when e enues canno be inc eased o balance
he budge , he eby pu ing deb sus ainabili y a isk, o when he le el o public spending
limi s he space a ailable o espond o la ge mac oeconomic shocks. The p oblem is
when s uc u al public expendi u e is oo high, hinde ing p oduc i e in es men o he
u u e (ecological ansi ion, a i icial in elligence). Public expendi u e co e s collec i e
public spending (police, secu i y, public adminis a ion, e c.) and insu ance ans e s
(heal hca e, pensions sys em, unemploymen bene i s). In he Bisma kian adi ion, he
insu ance ans e sys em was ini ially based on indi idual con ibu ions quali ying o
bene i s. Ne e heless, because o a lack o iscal esou ces and high budge de ici s, in
he Be e idgian adi ion, e en he insu ance sys em was, o e ime, mo e and mo e o en
o ced o be inanced wi h axes and by collec i i y. To a iable deg ees, coun ies ha e
in oduced minimum pension paymen s and ee minimal heal h se ice p o ision. I is
mainly o hese social p o ec ion expendi u es (social wel a e, heal h, educa ion) ha he
gap is la ge be ween F ance, Denma k, Sweden, Finland, and o he de eloped coun ies.
Economic de elopmen is limi ed wi hou a go e nmen insu ing p ope y igh s and
he u ni u e o public goods such as in as uc u es, social se ices, heal h, o educa ion.
Bu economic g ow h is also limi ed wi h an o e whelming go e nmen p e en ing he
de elopmen o p i a e ini ia i e, as public goods ha e diminishing e u ns. An excessi e
inc ease in public expendi u e is dange ous o he sus ainabili y o public inances. The
budge de ici and he public deb mus be con ained a a ime when he e a e p essu es on
hese public inances because o demog aphic ends (spending on pensions and long- e m
ca e wi h a highe li e expec ancy). Public spending is highly necessa y bu mus be used
e icien ly and in he mos p oduc i e way: o imp o e long- e m g ow h pe spec i es,
conside ing equi y conside a ions. The goal is o achie e be e ou comes a a sus ainable
le el o public spending. A balance has o be d awn be ween educing public indeb edness,
cu ing axes, and inancing public in es men in key a eas.
Ceppa ulo and Mou e (2020) p o ide a su ey o he li e a u e on he e ec o
composi ion and size o public expendi u e. In e ms o size, public spending would be
ei he bene icial in he Keynesian li e a u e o sus ain global demand in he sho - e m, o
ha m ul in he classical adi ion because o he c owding-ou e ec on p i a e in es men
in he medium e m. The e is a la ge amoun o economic li e a u e on he impac o
go e nmen size on economic g ow h. Theo e ical as well as empi ical analysis would end
o conclude ha public expendi u e would be necessa y and bene icial up o a ce ain sha e
in a pe cen age o GDP. Mo e p ecisely, he ‘A mey Cu e’ men ions ha public spending
would be bene icial up o a maximal poin ; howe e , i would become ha m ul he ea e
Economies 2025,13, 74 3 o 22
(see he ollowing Sec ion 2). The majo i y o he empi ical li e a u e suppo s he exis ence
o such an in e ed U-shaped ela ion wi h a maximal le el o public expendi u e beyond
which i can become ha m ul. E en i public spending is bene icial, beyond a ce ain poin ,
he diminishing e u ns o public capi al, he e ic ion e ec o mo e p oduc i e p i a e
in es men because o high in e es a es, as well as bu eauc a ic ine iciencies end o
domina e. In e ms o composi ion, acco ding o endogenous g ow h heo y and empi ical
obse a ions, p oduc i e public in es men (physical capi al accumula ion, bu also human
capi al: educa ion o esea ch) could ha e a posi i e e ec on g ow h, con a y o ope a ing
expendi u es: cu en spending, go e nmen consump ion, o subsidies. The e icacy o
public spending could also depend on i s inancing and he soundness o public inances:
budge de ici and public deb .
Ou pape con i ms he bene i s o public in es men expendi u e o economic g ow h,
as well as he ha m ulness o public consump ion expendi u e. We also p o e heo e ically
he exis ence o an in e ed U-shaped ela ion be ween inc easing public expendi u e and
economic g ow h, as well as he co ela i e exis ence o an op imal size o he go e nmen .
These esul s ha e al eady been men ioned in he heo e ical o empi ical li e a u e. How-
e e , he main con ibu ion o ou modeling is o highligh he heo e ical de e minan s
o his op imal size o he go e nmen and o shed ligh on he p ecise de e minan s o
such a h eshold, which should be use ul o policymake s. The e icacy o highe public
spending seems o depend, in pa icula , on he eal in e es a e, bu could also be ela ed
o he capi al sha e in he p oduc ion unc ion, o o he p e e ence o households o public
se ices. To de i e such indica ions, Sec ion 2pa ially e iews he economic li e a u e.
Sec ion 3explains ou heo e ical modeling. Sec ion 4p o ides heo e ical esul s on he
e ec o public in es men and consump ion expendi u e on economic g ow h, on he
de e minan s o he e icacy o public spending, and on he adequa e epa i ion be ween
consump ion, labo , and capi al axa ion a es o collec hese iscal esou ces. Sec ion 5
concludes he pape .
2. A B ie Re iew o Economic Li e a u e
The go e nmen p o ides in e media e goods necessa y in he p oduc ion p ocess,
in es men goods imp o ing p oduc ion capaci ies, bu also inal public consump ion
goods. I p o ides public goods ha he ma ke canno p o ide e icien ly, and i emo es
dis o ions in he alloca ion o esou ces due o ex e nali ies. The e o e, he link be ween
public expendi u e and sho - un o long- un economic g ow h has long been s udied
in he economic li e a u e [see, o example, Fo e and Magazzino (2011), Chobano and
Mladeno a (2009)o Cenc (2022) o a li e a u e e iew].
A oo-low le el o public expendi u e would no allow he S a e o gua an ee he
p ope unc ioning o he economy, whe eas an excessi ely high le el o public expendi u e
would discou age a su icien in es men in he p i a e sec o because o he high iscal
bu den. When he public sec o is small, i can con ibu e o inc easing he p oduc i i y o
capi al. Howe e , when he size o he public sec o is e y high, i can c owd ou mo e
p oduc i e p i a e in es men . Mo eo e , go e nmen al ac i i ies would ha e diminishing
e u ns and hei p oduc i i y could become oo low beyond a ce ain le el. The go e n-
men could begin o be in ol ed in ac i i ies o which i is ill-sui ed in compa ison wi h
he ma ke . Public ac i i ies can also some imes be conduc ed by in e es g oups o he
de imen o global social wel a e. So, Be gh and Hen ekson (2011) unde line ha he e
is a nega i e ela ion be ween he size o he go e nmen and economic g ow h in many
economic s udies.
Mo e p ecisely, ega ding he composi ion o public spending, public expendi u e is
classi ied in o en p ima y expendi u e i ems
1
plus in e es paymen s. Then, he p oduc-
Economies 2025,13, 74 4 o 22
i i y o a ious public expendi u es and he s uc u e o public spending seem o impac
he consequences o he sha e o public expendi u e in GDP on economic g ow h. Indeed,
p oduc i e (unp oduc i e) public expendi u es a e mo e likely o ha e a posi i e (nega i e)
e ec on economic g ow h. Ne e heless, Ceppa ulo and Mou e (2020) show ha g ow h-
iendly spending (educa ion, R&D, heal h, anspo , and communica ion in as uc u es)
only ep esen s a hi d o o al expendi u e in he Eu opean Union. The e o e, he nega i e
e ec on economic g ow h o public spending beyond a ce ain le el would co espond o
he ac ha as public spending inc eases, he sha e o unp oduc i e public expendi u e is
p opo ionally highe . Ku asi and Ma on (2020) analyze he co ela ion be ween a ious
ypes o public expendi u es and GDP g ow h in 25 EU coun ies in he pe iod 1996–2017.
They ind ha social p o ec ion o gene al public se ice ha ms GDP g ow h, whe eas
lagged educa ion and heal h posi i ely impac GDP g ow h.
Fou nie and Johansson (2016) obse ed a la ge panel o coun ies be ween 1987 and
2014. They ind ha some public spending i ems (public in es men , heal h, esea ch
and de elopmen , educa ional le el) boos po en ial g ow h, while o he s (pensions and
public subsidies) lowe po en ial g ow h. The e is also e idence ha oo la ge go e nmen s
educe long- e m po en ial g ow h, unless he unc ioning o go e nmen is highly e ec i e
(as in No dic coun ies), wi h a high us o ci izens in he go e nmen , a low le el o
co up ion, and mo e decen aliza ion. In his con ex , he Swedish public expendi u e mix
could appea he mos g ow h- iendly among he OECD coun ies. Be gh and Hen ekson
(2011) show ha some coun ies wi h high axes can enjoy abo e-a e age g ow h, such as
Scandina ian coun ies, pa icula ly Sweden. Indeed, by implemen ing ma ke - iendly
policies, coun ies wi h highe social us le els can de elop la ge go e nmen sec o s
wi hou ha ming he economy.
Föls e and Hen ekson (2001), Fall and Fou nie (2015), and Cenc (2022) ind obus
nega i e econome ic co ela ions be ween g ow h and o al go e nmen expendi u e. In a
panel o 29 OECD equally ich coun ies, be ween 1970 and 2005, Be gh and Ka lsson (2010)
also show ha go e nmen size obus ly co ela es nega i ely wi h g ow h, e en when
con olling o economic eedom and globaliza ion. Howe e , hey also ind some e idence
ha coun ies wi h big go e nmen s can use economic openness and sound economic
policies o mi iga e he nega i e e ec s o big go e nmen s. In he same way, o 11 Cen al
and Eas e n Eu opean coun ies, Mi u and S anciu (2023) ind ha expendi u e pe o mance,
as well as e iciency sco es, a e highe in he g oup o medium-sized go e nmen s. Neg i
and Dinca (2023) also e alua e e iciency sco es o all he 27 Eu opean Union membe
S a es, in he 2005–2020 pe iod. They ind ha go e nmen quali y, e icacy, and public
pe o mance depend on ac o s o quali y o go e nance, such as human esou ces, eedom,
democ acy, co up ion, and digi aliza ion. Ne e heless, mos Eu opean coun ies could
imp o e hei e iciency sco e. The e o e, bo h pape s ein o ce he adequacy o a balanced
choice be ween go e nmen expendi u es and hei e iciency. Tha is why i was absolu ely
necessa y o dis inguish be ween public in es men and consump ion expendi u e in he
ollowing Sec ion 3o he model, as a p oxy o he quali y o public expendi u e.
Rome o-A ila and S auch (2008) s udy 15 EU coun ies o he 1960–2001 pe iod.
They es ima e a dis ibu ed lag model, con olling o bo h sides o he go e nmen budge .
They ind ha o al go e nmen e enue (because o he nega i e e ec o di ec axa ion on
physical capi al accumula ion), go e nmen consump ion, and ans e s nega i ely a ec
GDP pe capi a g ow h a es o e he business cycle, while public in es men has a posi i e
impac . They also p o ide obus e idence ha dis o iona y axa ion a ec s g ow h in he
medium e m h ough i s e ec on he accumula ion o p i a e physical capi al. Analyzing a
se o OECD and a subse o EU coun ies be ween 1970 and 2004, A onso and Fu ce i (2010)
ind ha bo h size as a pe cen age o GDP and business cycle ola ili y, o go e nmen

Economies 2025,13, 74 5 o 22
e enue (indi ec axes, social con ibu ions) and spending (go e nmen consump ion,
subsidies) ha e a sizeable and s a is ically nega i e e ec on g ow h. Fo a panel o 108
coun ies be ween 1970 and 2008, A onso and To a -Valles (2011) show ha he size o
he go e nmen o go e nmen consump ion hu s g ow h, whe eas ins i u ional quali y
has a posi i e impac on eal g ow h. Ne e heless, many s udies ha e shown ha hese
ela ions could be non-mono onic.
Mo e p ecisely, Coayla (2021) e iews empi ical es ima es con i ming ha he e would
be an op imal le el o public expendi u e beyond which i could become de imen al. Using
a Gene alized Me hod o Momen s app oach o a la ge da ase o 129 coun ies be ween
1980 and 2009, Asimakopoulos and Ka a ias (2016) assess he exis ence o a s a is ically
signi ican ela ion be ween go e nmen size and economic g ow h, o bo h de eloped and
de eloping coun ies. Howe e , hey also show empi ically he exis ence o a h eshold, an
op imal le el o go e nmen size, which would be a ound 18% o GDP. Indeed, economic
esul s would sugges a U-shaped ela ionship o he link be ween public expendi u e in
GDP and economic g ow h. Indeed, in unde –de eloped coun ies, inc easing he size o
he public sec o , in as uc u es, heal h, and public se ices would a o highe economic
g ow h. Howe e , beyond a gi en le el, o de eloped coun ies, excessi ely inc easing
he size o he public sec o , axa ion a es, and compulso y con ibu ions, would ha m
economic g ow h.
By inco po a ing h eshold analysis in a c oss-coun y g ow h eg ession, Ch is ie
(2014) also shows a non-linea ela ion (exis ence o h esholds) be ween go e nmen size
and long- un economic g ow h o a b oad panel o 136 coun ies be ween 1971 and 2005.
The nega i e e ec o public spending on g ow h would be signi ican only abo e 33%
o GDP, whe eas he e ec would be posi i e below his le el, especially in he case o
p oduc i e public spending. Using he sys em-GMM es ima o o linea dynamic panel
da a models, o 156 coun ies be ween 1980 and 2010, Ma ins and Veiga (2014) also ind
ha go e nmen size has a quad a ic (in e ed U-shaped) e ec on he g ow h a e o he
Human De elopmen Index (HDI), pa icula ly in de eloped and high-income coun ies.
The e o e, he e ec is no only on economic g ow h bu also on o he quali a i e and
social aspec s o economic de elopmen (po e y, inequali y, educa ion). Go e nmen
expendi u e would ha e a posi i e e ec on de elopmen only un il 17% o GDP, bu om
hen on, he e ec could become nega i e.
Fo e and Magazzino (2011) s udy 27 Eu opean Union coun ies om 1970 o 2009.
Panel da a analysis shows ha an excessi e a io o public expendi u e o GDP educes
he g ow h a e. The e o e, public spending would be bene icial un il i ep esen s a ound
40% o GDP, bu i would become ha m ul beyond his maximal le el. Fo mos coun ies,
gi en he high public expendi u e- o-GDP a ios in de eloped coun ies, cu ing public
spending would be bene icial in inc easing economic g ow h and long- e m social wel a e.
Chobano and Mladeno a (2009) con i m he exis ence o an in e ed U cu e whe e he
size o he go e nmen is on he ho izon al axis and economic g ow h is on he e ical
axis. In 28 OECD coun ies, om 1970 o 2007, go e nmen spending maximizing g ow h
should no exceed 25% o GDP. In 81 coun ies om 1961 o 2005, go e nmen consump ion
o inal goods and se ices ( ans e s o housing, heal h ca e, schooling) should no exceed
10.4% o GDP.
In he same way, ega ding iscal esou ces, Myles (2009) shows heo e ically ha he e
would be a maximal axa ion a e beyond which iscal esou ces would dec ease (La e
cu e). Empi ically, o example, Scully (1994) inds ha o maximize economic g ow h,
axa ion a es should emain mode a e: he a e age axa ion a e should be be ween 21.5%
and 22.9% o GNP. Mo eo e , Johansson e al. (2008) s udy he design o ax s uc u es
o p omo e economic g ow h. They ind ha consump ion axes a e he mos g ow h-
Economies 2025,13, 74 6 o 22
iendly, ollowed by pe sonal income axes, whe eas co po a e axes a e he mos ha m ul
o g ow h. In he ollowing sec ions, ou goal is o p o ide a simple model, o shed ligh
on he heo e ical de e minan s o a po en ial op imal size o he public sec o , in e ms o
public spending o in e ms o axa ion a es o collec iscal esou ces.
3. The Model
As men ioned in he p e ious Sec ion 2, nume ous econome ical s udies y o ind a
link be ween iscal policy and economic g ow h. Howe e , he cu en s udy aims a illing
a heo e ical gap: explaining he ole o he a ious de e minan s o his link: capi al sha e
in he p oduc ion unc ion, eal in e es a e, households’ p e e ences o public se ices, e c.
Ou model aims o shed ligh on he ansmission channels o iscal policy, by unde lying
he consequences o he le el o public expendi u e and a ious axa ion a es on p i a e
consump ion and economic g ow h. We conside a s anda d Dynamic S ochas ic Gene al
Equilib ium (DSGE) model, wi h a ep esen a i e household, a ep esen a i e i m, and a
go e nmen . Rega ding budge a y au ho i y, he con ibu ion o he cu en pape is o
conside a de eloped iscal block, whe e public consump ion expendi u e is di e en ia ed
om public in es men expendi u e, o s udy he consequences o he p oduc i i y o
public spending on economic ac i i y. Mo eo e , we also dis inguish be ween he use o
a ious iscal esou ces and dis o iona y axes on consump ion, capi al, o labo .
3.1. Households
In a gi en pe iod ( ), he ep esen a i e household maximizes an in e empo al u il-
i y unc ion:
max
∞
∑
m=0
βmE (U +m)(1)
whe e E
is he a ional expec a ion ope a o condi ional on in o ma ion a ailable a da e
( ), and (β) is he ime discoun ac o .
We suppose ha he u ili y unc ion o a ep esen a i e household is as ollows:
U =θ
(θ−1)αc(C )(θ−1)
θ+αg(Gc
)(θ−1)
θ−αl
1
(1+φ)L(1+φ)
(2)
wi h: (
C
): eal p i a e consump ion; (
GI
): eal public in es men ; (
Gc
): eal public con-
sump ion; (L ): hou s wo ked by he labo o ce.
The indices (0 <
αc
< 1), (0 <
αg
< 1), and (0 <
αl
< 1) a e he espec i e weigh s
gi en o consump ion o p i a e goods, consump ion o public goods, and leisu e in he
u ili y unc ion.
So, u ili y is an inc easing and conca e unc ion o (
C
), an index o households’
consump ion o all goods ha a e supplied; (
θ
) is he elas ici y o in e empo al subs i u ion
(how p i a e consump ion eac s o he eal in e es a e in a dynamic amewo k). U ili y
is also an inc easing and conca e unc ion o eal public goods and se ices p o ided in he
home coun y, i.e., public consump ion expendi u e (
Gc
). Indeed, wi hou conside ing he
well-being p o ided by public consump ion expendi u e, i would no be easy o jus i y he
use ulness o his public spending. Finally, u ili y is also a dec easing and con ex unc ion
o he hou s wo ked (
L
), whe e (
φ≥
0) is he in e se o he F isch elas ici y o labo supply,
he in e se o he elas ici y o he wo k e o o he eal wage.
This maximiza ion is subjec o a li e ime and in e empo al budge cons ain , o
wha e e da e ( ) conside ed a which he ac ualiza ion is ealized. Households consume
goods (including axes), ealize in es men s in physical capi al, and pu chase go e nmen
bonds. Capi al is en ed by households o i ms, o which hey ecei e a en al a e.
Rega ding hei esou ces, households ecei e labo and capi al e enues, and gains om
Economies 2025,13, 74 7 o 22
go e nmen bonds held om he p e ious pe iod. P ices o goods, in e es a es, axa ion
a es, and wages a e aken as gi en by he ep esen a i e household. A he equilib ium,
he nominal in e es a e o capi al se ices en ed ou o i ms mus be he same as he
nominal in e es a e on public bonds, o a oid any non-exploi ed a bi age oppo uni y.
The e o e, households’ budge cons ain o each pe iod ( ) akes he o m:
(1+τc
)P C +P INV +B =1−τl
W L +1−τk
R K +(1+R )B −1(3)
Wi h, in pe iod ( ): (
INV
): eal in es men in new physical capi al; (
K
): s ock o
physical capi al; (
B
): nominal alue o go e nmen bonds and public deb ; (
P
): le el
o consume p ices; (
W
): nominal hou ly wage; (
τl
): axa ion a e on labo income; (
τc
):
axa ion a e on consump ion; (τk
): axa ion a e on capi al; (R ): nominal in e es a e.
Fu he mo e, he capi al s ock a ies acco ding o he ollowing equa ion:
K +1=(1−δ)K +INV (4)
whe e (δ) is he dep ecia ion a e o capi al.
I we suppose ha (n) is he g ow h a e o he popula ion, we ha e
L +1=(1+n)L (5)
Using Equa ions (3)–(5), we ob ain he ollowing in e empo al budge cons ain pe
popula ion uni . In he pape , all lowe case le e s indica e a iables pe head, i.e., pe
popula ion uni (di ided by L).
Maximiza ion o he u ili y o he ep esen a i e consume in Equa ion (2) unde
he budge cons ain (3) implies ha he p ima y de ici is null [
B =(1+R )B −1]
,
and he e o e:
c +
j
∑
m=1
(1+n)m(1+τc
+m)
(1+τc
)h(1−τk
+1)R +1
P +1+(1−δ)i...h(1−τk
+m)R +m
P +m+(1−δ)ic +m
=1
(1+τc
)h1−τk
R
P +(1−δ)ik +(1−τl
)
(1+τc
)
W
P
+
j
∑
m=1
(1+n)m(1−τl
+m)
(1+τc
)h(1−τk
+1)R +1
P +1+(1−δ)i...h(1−τk
+m)R +m
P +m+(1−δ)i
W +m
P +m
−(1+n)j+1k +j+1
(1+τc
)h(1−τk
+1)R +1
P +1+(1−δ)i...1−τk
+jR +j
P +j+(1−δ)
(6)
3.2. The Rep esen a i e Fi m
We suppose ha he coun y is popula ed by a con inuum o i ms. The ep esen a i e
i m p oduces a di e en ia ed good in a monopolis ic compe i ion se ing, wi h he help
o wo p oduc ion ac o s: capi al and labo , whose espec i e sha es in he p oduc ion
unc ion a e (0 <
α
< 1) and (1 –
α
). We assume ha ma ginal p oduc s a e posi i e and
diminishing and ha all ac o s a e complemen a y in he p oduc ion unc ion. Mo eo e ,
whe eas public consump ion expendi u e (
Gc
) is unp oduc i e, public in es men expendi-
u e (
GI
) is also a ac o aising public inpu ; p oduc ion inc eases wi h public in es men
acco ding o he pa ame e (
γ
). Ne e heless, public in es men emains limi ed and
ep esen s only 2–3% o GDP in he Eu opean Union. We assume all co po a e axes a e
included in he household capi al income ax le ied. So, he p oduc ion unc ion has he
ollowing o m:
Y =A K )α(L )1−αGI
γ(7)
Economies 2025,13, 74 8 o 22
wi h, in pe iod ( ): (A ): echnology o p oduc i i y shock; (Y ): eal economic ac i i y.
The e o e, pe popula ion uni , Equa ion (7) implies
y =Y
L
=A (k )αgI
γ(L )γ(8)
The ep esen a i e i m maximizes i s nominal p o i :
Π =P Y −R K −W L (9)
P oduc ion ac o s a e paid a hei ma ginal p oduc s, and we suppose cons an
e u ns. So, he maximiza ion o he p o i in Equa ion (9) implies
P ∂Y
∂K
=αP Y
K
=R >0 (10)
P ∂Y
∂L
=(1−α)P Y
L
=W >0 (11)
The e o e, by combining Equa ions (10) and (11), we ob ain he ollowing ela ion
be ween he nominal wage and he nominal in e es a e:
W
P
=(1−α)K
αL R
P =(1−α)
αR
P k (12)
3.3. Fiscal Policy and Global Equilib ium
The iscal au ho i y ixes public expendi u e and ax a es. The go e nmen is sup-
posed o c edibly commi o epaying he public deb . Con a y o many economic pape s,
as Rome o-A ila and S auch (2008) o example, we do no suppose a p io i ha he budge
cons ain is balanced e e y pe iod. In a pe iod ( ), he budge cons ain is as ollows:
B =(1+R )B −1+P GC
+GI
−τc
P C −τl
W L −τk
R K (13)
The public deb in pe iod ( ) equals he public deb in he o me pe iod ( – 1) in-
c eased by he o me public deb ’s in e es a e, plus he cu en pe iod’s public spending
o be inanced, dec eased by he cu en pe iod’s iscal esou ces. The la e includes con-
sump ion, labo , and capi al axa ion. Howe e , we ha e men ioned ha he p ima y de ici
should be null acco ding o he op imiza ion cons ain o he ep esen a i e consume . So,
using Equa ions (5) and (13), we ob ain he ollowing in e empo al budge cons ain :
gC
+gI
=b
P
+τc
c +τl
W
P
+τk
R
P
k (14)
We can now de i e he equilib ium on he goods ma ke ega ding global demand.
Neglec ing he comme cial balance, clea ing he goods ma ke , and equali y be ween he
supply and demand o goods and se ices du ing he pe iod ( ), equi es he ollowing:
Y =C +GC
+GI
+INV (15)
So, pe popula ion uni , Equa ions (4), (5) and (15) imply he ollowing in e empo al
global equilib ium:
j
∑
m=0
(1+n)m
(1−δ)my +m=
j
∑
m=0
(1+n)m
(1−δ)mc +m+gc
+m+gI
+m
−(1−δ)k +(1+n)j+1
(1−δ)jk +j+1
(16)
Economies 2025,13, 74 15 o 22
In his con ex , economic au ho i ies could be emp ed o a bi a e be ween inc easing
p i a e consump ion (
αc
) and global economic g ow h (
αy
). The e o e, using Equa ions (26)
and (29), we ob ain he ollowing a e age well-being in Equa ion (30):
αc
.
c +αy
.
y = (αc+αy)

"β(1+τc
−1)
1+τc
−11+.
τc
#θ
−1


+αyα(1+n)β2θ
R −1
P −1

"(1+τc
−1)
1+τc
−11+.
τc
1+.
τc
+1#θ
−"(1+τc
−1)
1+τc
−11+.
τc
#2θ


(30)
The e o e, as men ioned abo e, inc easing he consump ion axa ion a e could bene i
global economic g ow h, whe eas mode a ing he consump ion axa ion a e could bene i
p i a e consump ion. So, i is only in he in e media e case o an a bi age sligh ly in
a o o p i a e consump ion (3
<αc
αy<
3.5) ha he e could be an in e ed U-shaped
ela ion be ween he inc ease o he consump ion axa ion a e and an indica o o economic
well-being, wi h a maximal axa ion a e o collec he op imal le el o iscal esou ces [see
Figu e 4].
Economies 2025, 13, x FOR PEER REVIEW 15 o 23
𝛼𝑐󰇗+𝛼𝑦󰇗=(𝛼+𝛼)󰇯𝛽(1+𝜏
)
󰇡1+𝜏
1+𝜏󰇗󰇢󰇰−1
+𝛼𝛼(1+𝑛)𝛽
𝑅
𝑃 󰇯(1+𝜏
)
󰇡1+𝜏
1+𝜏󰇗1+𝜏
󰇗󰇢󰇰
−󰇯 (1+𝜏
)
󰇡1+𝜏
1+𝜏󰇗󰇢󰇰
(30)
The e o e, as men ioned abo e, inc easing he consump ion axa ion a e could ben-
e i global economic g ow h, whe eas mode a ing he consump ion axa ion a e could
bene i p i a e consump ion. So, i is only in he in e media e case o an a bi age sligh ly
in a o o p i a e consump ion (3<
<3.5) ha he e could be an in e ed U-shaped
ela ion be ween he inc ease o he consump ion axa ion a e and an indica o o eco-
nomic well-being, wi h a maximal axa ion a e o collec he op imal le el o iscal e-
sou ces [see Figu e 4].
Calib a ion: 𝜃=0.5; 𝛽=0.99; n = 0.01; 
 =0.08; . 𝛼=0.34
Figu e 4. Va ia ion o he consump ion axa ion a e and economic wel a e.
Fu he mo e, in hese condi ions, i he go e nmen 's goal is o a ge an equilib ium
budge , a highe consump ion axa ion a e should be compensa ed wi h a dec ease in he
labo o capi al axa ion a e. The e o e, acco ding o Equa ion (A22) in Appendix A, hese
a ia ions should be such as
-0.100
-0.080
-0.060
-0.040
-0.020
0.000
0.020
0.040
0.060
-0.50
-0.44
-0.38
-0.32
-0.26
-0.20
-0.14
-0.08
-0.02
0.04
0.10
0.16
0.22
0.28
0.34
0.40
0.46
economic wel a e
c °
ac/ay=1
ac/ay=3.2
ac/ay=4
Figu e 4. Va ia ion o he consump ion axa ion a e and economic wel a e.
Fu he mo e, in hese condi ions, i he go e nmen ’s goal is o a ge an equilib ium
budge , a highe consump ion axa ion a e should be compensa ed wi h a dec ease in he
labo o capi al axa ion a e. The e o e, acco ding o Equa ion (A22) in Appendix A, hese
a ia ions should be such as
ατk
−1
.
τk
+(1−α)τl
−1
.
τl

=τc
−1
.
τc
(1+τc
−1)


α(1+n)βθ1+τc
−11+.
τc
θ
1+τc
−11+.
τc
1+.
τc
+1θ−α(1+n)βθ(1+τc
−1)θ
1+τc
−11+.
τc
θ+R −1
P −1


{α(1+n)βθ"(1+τc
−1)
1+τc
−11+.
τc
#θ
+1+τc
−11+.
τc

τc
−1
.
τc
hατk
−1+(1−α)τl
−1iR −1
P −1
−α(1−δ)−R −1
P −1o−hατk
−1+(1−α)τl
−1i
(31)

Economies 2025,13, 74 16 o 22
The e o e, o compensa e o a highe (smalle ) consump ion axa ion a e, bene icial
o economic g ow h, and o equilib a e he budge balance, he dec ease (inc ease) in he
labo axa ion a e could be mo e accen ua ed han he dec ease (inc ease) in he capi al
axa ion a e [see Figu e 5]. Indeed, acco ding o he pa ame e (
α
), he labo sha e in he
p oduc ion unc ion is abou wice highe han he capi al sha e in he p oduc ion unc ion.
The absolu e le els o he capi al o labo axa ion a es ha e no e ec in he long un, as
Equa ion (30) is always e i ied. Howe e , i he consump ion axa ion a e (
τc
−1
) is highe ,
compensa o y a ia ions in he labo o capi al axa ion a es a e bo h mo e accen ua ed.
Mo eo e , i he labo axa ion a e (
τl
−1
) is highe , a ia ions in he labo axa ion a e
a e s ongly accen ua ed, whe eas a ia ions in he capi al axa ion a e a e much mo e
mode a e. On he con a y, i he capi al axa ion a e (
τk
−1
) is highe , a ia ions in he
capi al axa ion a e a e only e y sligh ly accen ua ed, whe eas a ia ions in he labo
axa ion a e a e signi ican ly mo e mode a e.
Economies 2025, 13, x FOR PEER REVIEW 16 o 23
󰇣𝛼𝜏
𝜏󰇗+(1−𝛼)𝜏
𝜏󰇗󰇤
=𝜏
𝜏󰇗
(1+𝜏
)󰇯𝛼(1+𝑛)𝛽󰇡1+𝜏
1+𝜏󰇗󰇢
󰇡1+𝜏
1+𝜏󰇗1+𝜏
󰇗󰇢−𝛼(1+𝑛)𝛽(1+𝜏
)
󰇡1+𝜏
1+𝜏󰇗󰇢+𝑅
𝑃󰇰
{𝛼(1+𝑛)𝛽󰇯(1+𝜏
)
󰇡1+𝜏
1+𝜏󰇗󰇢󰇰+󰇡1+𝜏
1+𝜏󰇗󰇢
𝜏
𝜏󰇗𝛼𝜏
+(1−𝛼)𝜏
𝑅
𝑃
−𝛼(1−𝛿)− 𝑅
𝑃} − 󰇟𝛼𝜏
+(1−𝛼)𝜏
]
(31)
The e o e, o compensa e o a highe (smalle ) consump ion axa ion a e, bene icial
o economic g ow h, and o equilib a e he budge balance, he dec ease (inc ease) in he
labo axa ion a e could be mo e accen ua ed han he dec ease (inc ease) in he capi al
axa ion a e [see Figu e 5]. Indeed, acco ding o he pa ame e (α), he labo sha e in he
p oduc ion unc ion is abou wice highe han he capi al sha e in he p oduc ion unc-
ion. The absolu e le els o he capi al o labo axa ion a es ha e no e ec in he long
un, as Equa ion (30) is always e i ied. Howe e , i he consump ion axa ion a e (𝜏
)
is highe , compensa o y a ia ions in he labo o capi al axa ion a es a e bo h mo e ac-
cen ua ed. Mo eo e , i he labo axa ion a e (𝜏
) is highe , a ia ions in he labo ax-
a ion a e a e s ongly accen ua ed, whe eas a ia ions in he capi al axa ion a e a e
much mo e mode a e. On he con a y, i he capi al axa ion a e (𝜏
) is highe , a ia-
ions in he capi al axa ion a e a e only e y sligh ly accen ua ed, whe eas a ia ions in
he labo axa ion a e a e signi ican ly mo e mode a e.
Calib a ion: (α = 0.34), (𝜃=0.5), (𝑛=0.01), (𝛽=0.99), (𝑅/𝑝=0.08), (𝛿=0.02),
(𝜏
=0.17), (𝜏
=0.17), (𝜏
=0.22)
Figu e 5. Op imal labo and capi al axa ion a es.
Fu he mo e, acco ding o Equa ion (31), op imal a ia ions in he labo o capi al
axa ion a es also depend on a ia ions o o he a iables. Indeed, i he capi al sha e in
he p oduc ion unc ion (α) inc eases, a ia ions in he labo axa ion a es should be e-
duced, whe eas a ia ions in he capi al axa ion a es should be accen ua ed. Mo eo e ,
i he eal capi al e u ns (R/p) o i he in e empo al elas ici y o subs i u ion (θ) inc ease,
a ia ions in he labo and capi al axa ion a es should be accen ua ed. On he con a y,
-3.00
-2.00
-1.00
0.00
1.00
2.00
3.00
-0.5
-0.46
-0.42
-0.38
-0.34
-0.3
-0.26
-0.22
-0.18
-0.14
-0.1
-0.06
-0.02
0.02
0.06
0.1
0.14
0.18
0.22
0.26
0.3
0.34
0.38
0.42
0.46
0.5
l ° k °
Figu e 5. Op imal labo and capi al axa ion a es.
Fu he mo e, acco ding o Equa ion (31), op imal a ia ions in he labo o capi al
axa ion a es also depend on a ia ions o o he a iables. Indeed, i he capi al sha e
in he p oduc ion unc ion (
α
) inc eases, a ia ions in he labo axa ion a es should be
educed, whe eas a ia ions in he capi al axa ion a es should be accen ua ed. Mo eo e ,
i he eal capi al e u ns (R/p) o i he in e empo al elas ici y o subs i u ion (
θ
) inc ease,
a ia ions in he labo and capi al axa ion a es should be accen ua ed. On he con a y,
i he capi al dep ecia ion a e (
δ
), he popula ion g ow h a e (n), o he ime discoun
ac o (
β
) inc eases, a ia ions in he labo o capi al axa ion a es should be mi iga ed.
The e o e, a highe consump ion axa ion a e is likely o a o a highe global economic
g ow h a e. The capi al and mainly he labo axa ion a e could hen simul aneously be
educed wi hou consequences on global iscal esou ces and he budge . Ne e heless,
he in ensi y o he dec ease allowed in capi al o labo axa ion a es depends on o he
economic a iables.
Economies 2025,13, 74 17 o 22
4.3. Empi ical Obse a ions
The cu en pape is heo e ical, and we lea e o u u e esea ch he empi ical e i i-
ca ion by econome ical me hods o ou hypo heses. Howe e , a i s look a empi ical da a
show ha hese heo e ical hypo heses could be e i ied. Fo he Eu opean Union, hese
da a come om he AMECO da abase o global economic g ow h a es, inal consump ion
expendi u e, and g oss ixed capi al o ma ion o he go e nmen s in 2024. Da a come
om he websi e o he Eu opean Commission o he yea 2022 (las a ailable da a) o he
implici ax a e on consump ion.
In 2024, he a e age annual GDP g ow h a e was 1.12% in he Eu opean Union,
whe eas o he go e nmen s, on a e age, inal consump ion expendi u e ep esen ed
21.5% o GDP and g oss ixed capi al o ma ion 3.7% o GDP a cu en ma ke p ices.
Howe e , in con o mi y wi h ou hypo heses, we can no e ha economic g ow h was
highe in a g oup o coun ies whe e he sha e o public in es men expendi u e was high.
Indeed, public g oss ixed capi al o ma ion and economic g ow h we e bo h highe han
he Eu opean a e age in C oa ia ( espec i ely, 5.7% and 3.4%), Romania (5.7% and 1.91%),
Slo enia (5.2% and 1.5%), Poland (5% and 2.96%) o in G eece (4.9% and 2.28%). On he
con a y, public g oss ixed capi al o ma ion and economic g ow h we e bo h weake han
he Eu opean a e age in he Ne he lands ( espec i ely, 3.1% and 0.63%), Ge many (3%
and 0.01%), o I eland (2.5% and
−
0.18%). Mo eo e , economic g ow h was also highe
in a g oup o coun ies whe e public consump ion expendi u e was weak. Indeed, he
sha e o public inal consump ion expendi u e in GDP was weak in a g oup o coun ies
whe e economic g ow h was highe han he Eu opean a e age: Po ugal ( espec i ely
16.8% and 1.93%), Mal a (17.2% and 5%), Li huania (18.1% and 2.43%), Cyp us (18.6% and
3.29%), Romania (18.8% and 1.91%), G eece (19.2% and 2.28%), Spain (19.4% and 2.91%),
o in Bulga ia (19.6% and 2.32%). On he con a y, he sha e o public inal consump ion
expendi u e in GDP was high in a g oup o coun ies whe e economic g ow h was weake
han he Eu opean a e age: in Sweden ( espec i ely, 26.6% and 0.9%), Finland (26.4% and
−0.1%), o in he Ne he lands (25.1% and 0.63%).
Fu he mo e, in 2024, he adjus ed wage sha e in labo cos s ep esen ed 62.9% o he
GDP on a e age in he Eu opean Union. Howe e , in con o mi y wi h ou hypo hesis, we
can no e ha inc easing public spending could mainly be bene icial in coun ies whe e
capi aliza ion (sha e o capi al in he p oduc ion unc ion) is weak. Indeed, he wage sha e
in labo cos s was highe in a g oup o coun ies whe e public g oss ixed capi al o ma ion
o inal consump ion expendi u e was highe han he Eu opean a e age: in Slo enia
( espec i ely, 74.3%, 5.2%, and 20.9%), La ia (73.8%, 5.3%, and 22.5%), F ance (66.8%,
4.3%, and 24.3%), C oa ia (66.1%, 5.7%, and 22.6%), o in Es onia (64.7%, 7.3%, and 21.1%).
On he con a y, he wage sha e in labo cos s and public g oss ixed capi al o ma ion
o inal consump ion expendi u e we e all weake han he Eu opean a e age in I eland
( espec i ely, 36%, 2.5%, and 13%), Mal a (50.3%, 4.1%, and 17.2%), Slo akia (54%, 3.7%,
and 20.8%), o in Cyp us (56.5%, 2.9%, and 18.6%).
Mo eo e , in 2024, he a e age GDP g ow h a e was 1.12% in he Eu opean Union,
whe eas he implici ax a e on consump ion was 17.2%. Howe e , in con o mi y wi h ou
hypo hesis, we can no e ha economic g ow h was highe in a g oup o coun ies whe e
consump ion axes we e high. Indeed, implici consump ion axes and economic g ow h
we e bo h highe han he Eu opean a e age in Denma k ( espec i ely, 23.4% and 1.94%),
Hunga y (22.9% and 1.48%), C oa ia (21.3% and 3.4%), G eece (20.2% and 2.28%), Slo enia
(19.6% and 1.5%), Bulga ia (19% and 2.32%), Poland (18% and 2.96%), Li huania (17.6% and
2.43%), Cyp us (17.4% and 3.29%) o in Mal a (17.3% and 5%). On he con a y, implici
consump ion axes and economic g ow h we e bo h weake han he Eu opean a e age in
Ge many ( espec i ely, 16.0% and 0.01%) o in I aly (16% and 0.67%).
Economies 2025,13, 74 18 o 22
5. Conclusions
Ou pape aims o shed ligh on he po en ial e ec o iscal policy on economic
g ow h: he de e minan s o a posi i e o nega i e impac o public expendi u e, and he
a ious ways o collec he necessa y iscal esou ces. Rega ding public spending, we show
ha public consump ion expendi u e usually ha ms global economic g ow h, and all he
mo e as eal capi al e u ns a e high, and as he p e e ence o he ep esen a i e consume
is much highe o p i a e consump ion han o public consump ion. On he con a y,
e en i consump ion dec eases pe head, public in es men expendi u e appea s o bene i
economic ac i i y: i can inc ease income pe head, p o ided eal e u ns on capi al a e no
oo small. In es men public spending is hen all he mo e bene icial as he capi al sha e
in he p oduc ion unc ion, he capi al dep ecia ion a e, and he popula ion g ow h a e
a e high. Mo eo e , we can heo e ically unde line he exis ence o an in e ed U-shaped
ela ion be ween he a ia ion o public expendi u e and economic g ow h. An inc ease in
public spending would be bene icial o economic g ow h only up o a maximal a ia ion,
which posi i ely depends on eal capi al e u ns. Inc easing public spending could also
mainly be bene icial in coun ies whe e he capi aliza ion (sha e o capi al in he p oduc ion
unc ion) is weak.
As a p ac ical implica ion o he pape o policymake s, ob iously, economic policies
should a o he inc ease in public in es men a he han public consump ion expendi u e.
An excessi e inc ease in public spending could be de imen al beyond a gi en le el,
especially o coun ies whe e he capi al sha e in he p oduc ion unc ion is high, o
whe e he p e e ence o public se ices is weak. Rega ding iscal esou ces, we show ha
policymake s should a o an inc ease in he sha e o iscal esou ces collec ed h ough
consump ion axes. Indeed, such an inc ease could be bene icial o global economic
g ow h, e en i i is de imen al o p i a e consump ion. Then, in pa allel o he inc ease
in consump ion axes, i could be possible o educe capi al axa ion a es, and mainly o
s ongly educe labo axa ion a es.
Ou pape was only in e es ed in he e ec o public expendi u e on economic g ow h.
Howe e , he e a e many o he implica ions o public spending, beyond he scope o he
cu en pape : social equi y and edis ibu ion, quali y o li e and well-being including
en i onmen al conside a ions and secu i y a he na ional and in e na ional le el, e c.
Public spending is no he bes way o inc ease economic g ow h; o en, i is no app op ia e
in e ms o e iciency. Howe e , public spending is usually jus i ied o co ec ma ke
ailu es, ine iciency in he p i a e p oduc ion o some public se ices, o o jus ice o
equi y conside a ions.
The ex ension o he cu en pape would be o con inue o s udy he consequences
o he size o he public sec o on economic g ow h. Theo e ically, my u u e esea ch
di ec ion is o y o e alua e he op imal size o public spending in a ious coun ies,
acco ding o he cha ac e is ics and he mac oeconomic pa ame e s o hese coun ies.
Many econome ical s udies ha e sugges ed such op imal sizes. Howe e , a na u al u u e
p olonga ion o he cu en heo e ical s udy would be o link his op imal size o he
mac oeconomic pa ame e s o he coun ies.
Funding: This esea ch ecei ed no ex e nal unding.
In o med Consen S a emen : No applicable.
Da a A ailabili y S a emen : AMECO da abase.
Con lic s o In e es : The au ho s decla e no con lic s o in e es .
Economies 2025,13, 74 19 o 22
Appendix A
The go e nmen maximizes he u ili y pe head o he ep esen a i e consume in
Equa ion (1) unde h ee cons ain s: he global equilib ium in (8) and (16), he go e nmen
budge cons ain in (14), and he consume budge cons ain in (6). So, wi h a iables pe
popula ion uni , he Hamil onian can be w i en as ollows:
H=
j
∑
m=0
βmθ
(θ−1)αc(c +m)(θ−1)
θ+αg(gc
+m)(θ−1)
θ−αl
1
(1+φ)L(φ+1
θ)
+m(A1)
−λ1










j
∑
m=0
(1+n)m
(1−δ)mA +m(k +m)αgI
+mγ(L +m)γ−
j
∑
m=0
(1+n)m
(1−δ)mgc
+m+gI
+m+(1−δ)k
−(1+n)j+1k +j+1
(1−δ)j−
j
∑
m=0
(1+n)m
(1−δ)mc +m











(A2)
−λ2(j
∑
m=0gC
+m+gI
+m−
j
∑
m=0
τc
+mc +m−
j
∑
m=0
τl
+m
W +m
P +m
−
j
∑
m=0
τk
+m
R +m
P +m
k +m)(A3)
−λ3{c
+
j
∑
m=1
(1+n)m(1+τc
+m)
(1+τc
)[(1−τk
+1)R +1
P +1+(1−δ)]...[(1−τk
+m)R +m
P +m+(1−δ)] c +m
−1
(1+τc
)[(1−τk
)R
P + (1−δ)]k −(1−τl
)
(1+τc
)
W
P
−
j
∑
m=1
(1+n)m(1−τl
+m)
(1+τc
)[(1−τk
+1)R +1
P +1+(1−δ)]...[(1−τk
+m)R +m
P +m+(1−δ)]
W +m
P +m
+(1+n)j+1k +j+1
(1+τc
)[(1−τk
+1)R +1
P +1+(1−δ)]...[(1−τk
+j)R +j
P +j+(1−δ)] }
(A4)
We suppose ha nominal wages and in e es a es a e gi en. By de i ing he Hamil o-
nian in (A1), we can ob ain he ollowing i s -o de condi ions:
∂H
∂gC
+j
=αgβjgC
+j−1
θ+λ1(1+n)j
(1−δ)j−λ2=0(∀j≥0)(A5)
∂H
∂gI
+j
=−λ1"γ(1+n)j
(1−δ)jA +jk +jαgI
+jγ−1L +jγ−(1+n)j
(1−δ)j#−λ2=0(∀j≥0)(A6)
∂H
∂τc
=∂H
∂τl
=∂H
∂τk
=0 all imply λ2=λ3
1
(1+τc
)(A7)
∂H
∂τc
+j
=∂H
∂τl
+j
=0. (A8)
Using alos Equa ion (A4), hese equa ions imply
1−τk
+1R +1
P +1+(1−δ). . . "1−τk
+j R +j
P +j!+(1−δ)#=(1+n)j(∀j≥1)(A9)
•∂H
∂τk
+j
=
0. Wi h Equa ions (A4) and (A5), his equa ion is always e i ied acco ding o
he beha io o he ep esen a i e consume in Equa ions (3) and (4).
Economies 2025,13, 74 20 o 22
•∂H
∂c +j=0. Conside ing Equa ions (A2), (A4) and (A5), his implies
gC
+j=αg
αcθ
c +j(∀j≥0)(A10)
∂H
∂L +j=−αlβj(1+φθ)
(1+φ)θL((φθ+1−θ)
θ)
+j
−λ1γ(1+n)j
(1−δ)jA +jk +jαgI
+jγL +jγ−1=0
(A11)
The e o e, using Equa ion (A2) and (A3) o de ine (λ1), we ob ain
gI
+j=αl
αg(1+φθ)
(1+φ)θL
(1+φθ)
θ
+jgC
+j
1
θ=αl
αc(1+]φθ)
(1+φ)θL
(1+φθ)
θ
+jc +j
1
θ(∀j≥0)(A12)
αcθ
(θ−1)c +j(θ−1)
θ=αl
1
(1+φ)L +j(1+φθ)
θ(A13)
acco ding o he alue o he u ili y unc ion. So, we ha e
L +j(1+φθ)
θ=αc
αlθ(1+φ)
(θ−1)c +j(θ−1)
θ(A14)
By combining Equa ion (A7) and (A8), we ob ain
gI
+j=(1+φθ)
(θ−1)c +j(∀j≥0)(A15)
Equa ions (3)–(5) o he budge cons ain o he ep esen a i e consume , (12) o he
eal wage, and (14) o he public deb imply
c +j+gC
+j+gI
+j=1
α R +j
P +j!k +j−(1+n)1+.
k +j+1k +j+(1−δ)k +j(A16)
The e o e, using Equa ions (A6), (A9) and (A10), we ob ain
c +j=h1
αR +j
P +j−(1+n)1+.
k +j+1+(1−δ)i
αg
αcθ+θ(1+φ)
(θ−1)k +j(∀j≥0)(A17)
gC
+j=h1
αR +j
P +j−(1+n)1+.
k +j+1+(1−δ)i
1+αc
αgθθ(1+φ)
(θ−1)k +j(∀j≥0)(A18)
gI
+j=h1
αR +j
P +j−(1+n)1+.
k +j+1+(1−δ)i
αg
αcθ(θ−1)
(1+φθ)+θ(1+φ)
(1+φθ)k +j(∀j≥0)(A19)
y +j=1
α R +j
P +j!k +j(∀j≥0)(A20)

Economies 2025,13, 74 21 o 22
F om he poin o iew o he ep esen a i e consume (i
λ1=λ2=
0 and
wi hou conside ing he global equilib ium o he budge cons ain ), using Equa ion
(A5), ∂H
∂c and ∂H
∂c +jimply
c +j=βθj(1+τc
)θ
1+τc
+jθc (∀j≥0)(A21)
The e o e, using Equa ion (24), we ob ain
.
c =.
k ="βθ1+τc
−1θ
(1+τc
)θ−1#(A22)
Equa ion (A14) implies
.
y =.
k +1+.
k 
.
R
P (A23)
Mo eo e , Equa ion (A11) implies
1+.
c =hR
P −α(1+n)1+.
k +1+α(1−δ)i1+.
k 
hR −1
P −1−α(1+n)1+.
k +α(1−δ)i(A24)
The e o e, using Equa ion (24) [ .
c =.
k ], we ob ain
.
R
P
R −1
P −1
=α(1+n).
k +1−.
k (A25)
Then, by combining Equa ions (A17)–(A19), we ob ain
.
y =βθ1+τc
−1θ
(1+τc
)θ+α(1+n)β2θ
R −1
P −1"1+τc
−1θ
1+τc
+1θ−1+τc
−12θ
(1+τc
)2θ#−1 (A26)
As he p ima y de ici should be null acco ding o he op imiza ion cons ain o he
ep esen a i e consume , Equa ions (3)–(5), and (12) imply
c =1
(1+τc
)1
α−τk
−1
α−1τl
R
P
−(δ+n)−(1+n).
k +1k (A27)
The e o e, Equa ions (A19) and (26) and [ .
c =.
k ] imply
α
.
τk
τk
−1+(1−α)τl
−1
.
τl

=τc
−1
.
τc
(1+τc
−1)


α(1+n)βθ1+τc
−11+.
τc
θ
1+τc
−11+.
τc
1+.
τc
+1θ−α(1+n)βθ(1+τc
−1)θ
1+τc
−11+.
τc
θ+R −1
P −1


{α(1+n)βθ"(1+τc
−1)
1+τc
−11+.
τc
#θ
+1+τc
−11+.
τc

τc
−1
.
τc
hατk
−1+(1−α)τl
−1iR −1
P −1
−α(1−δ)−R −1
P −1} − hατk
−1+(1−α)τl
−1i
(A28)
Economies 2025,13, 74 22 o 22
No e
1
These a e: Educa ion, Heal h, O he wages and in e media e consump ion (i.e.,: de ense, jus ice), Old-age and su i o pensions,
Sickness and disabili y, Unemploymen bene i s, Family and Child en, Subsidies, In es men , O he p ima y expendi u e.
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