Neumä ke , Be nha d; Weinel, Je e Leonie
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The implica ions o UBI on he u ili y unc ion and ax
e enue: Fu he calib a ing o basic income e ec s
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FRIBIS
F eibu g Ins i u e o Basic Income S udies
The implica ions o UBI on he U ili y
Func ion and Tax Re enue: Fu he
Calib a ing o Basic Income E ec s
Be nha d Neumä ke
Je e Weinel
DOI:10.6094/FRIBIS/DiscussionPape /12/02-2024
Gö z We ne Chai o Economic Policy & Cons i u ional Theo y [GWP]
Uni e si y o F eibu g
Con ac : [email p o ec ed]
29 h May, 2024
FRIBIS Discussion Pape Se ies
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The Implica ions o UBI on he U ili y Func ion and Tax Re enue:
Fu he Calib a ing o Basic Income E ec s
By Be nha d Neumä ke *,** & Je e Weinel*,**
Abs ac
Economic modeling o Uni e sal Basic Income (UBI) o en ails o conside how indi iduals'
u ili y calcula ions shi wi h uncondi ional ans e s. In his pape we u he de elop he model
o ou p e ious pape - The Implica ions o UBI on U ili y Func ions and Tax Re enue
(Neumä ke , B., Weinel, J., 2022). We con end ha , while adi ional iscal models ely on an
addi i ely sepa able ela ionship be ween consump ion and labo , he u ili y calcula ion o
indi iduals in luenced by UBI is be e ep esen ed by a mul iplica i e ela ionship. This shi
a ises om he ime so e eign y a o ded by UBI, empowe ing indi iduals o become sel -
de e mined, c ea i e, and in insically mo i a ed. We explo e he implica ions o he UBI-
adap ed u ili y unc ion on ax e enue. Speci ically, we analyze he consump ion ax e enue
cu e unde UBI (mul iplica i e p e e ences) e sus a means- es ed wel a e sys em (addi i e
sepa able p e e ences).
P o . D . Be nha d Neumä ke is P o esso o Economics (in pa icula Economic
Policy) and Di ec o o he Gö z We ne Chai o Economic Policy and Cons i u ional
Economic Theo y. Fu he mo e, he is a ounding membe and di ec o o he F eibu g
Ins i u e o Basic Income S udies.
Je e Weinel s udied economics a he Uni e si y o F eibu g and is eam coo dina o o
he FRIBIS Basic Money Team. He esea ch a he Gö z We ne Chai o Economic
Policy and Cons i u ional Economic Theo y ocuses on he implica ions o he Basic
Income on he u ili y unc ion and ax e enue.
*Gö z We ne Chai o Economic Policy and Cons i u ional Theo y
**F eibu g Ins i u e o Basic Income S udies
Keywo ds
Basic Income, La e Cu e, U ili y Func ion, Consump ion Tax, Time so e eign y, In insic Mo i a ion
2
3
1. In oduc ion
The concep o an uncondi ional basic income (UBI) is highly con o e sial in academic ci cles.
While some economis s ehemen ly ad oca e he concep , o he s c i icize he lack o incen i es
and inancing p oblems. This pape add esses he in e sec ion o hese wo c i icisms. I is
a gued ha he lack o incen i es in an uncondi ional subsis ence secu ing ans e sys em leads
indi iduals o d op ou o he labo o ce (e.g. Jaimo ich e al., 2022). This a gumen ollows
he s anda d economic heo y and he p incipal agen heo y. The agen will no wo k i he
p incipal who hi ed he o pe o m a job does no (o canno ) su icien ly moni o he . In he
con ex o UBI, his ansla es in o indi iduals who a e no willing o wo k i hey a e no
exposed o he app op ia e ex insic incen i es o wo k. Hence a ises he a gumen o
unsus ainable unding o he UBI, as an UBI mus be inanced by axes. Howe e , i people
s op wo king o d as ically educe hei wo king hou s, he s a e’s ax e enue is also educed
and, consequen ly, i canno sus ainably inance he ans e sys em. Why do schola s
ne e heless a gue o he in oduc ion o a UBI? P oponen s o he UBI assume ha indi iduals
a e willing o wo k no only because o ex insic incen i es bu also because o in insic
mo i a ion. I is a gued ha in insic mo i a ion is becoming inc easingly impo an in a
changing wo ld o wo k and ha he pe cep ion o wo k is e ol ing (S aubhaa , 2017).
1
Digi aliza ion and globaliza ion a e d i ing he impo ance o educa ion (Vogle -Ludwig &
K iechel, 2013). This, in conjunc ion wi h he high le el o p ospe i y, leads people o choose
hei jobs in line wi h in insic mo i es such as sel - ul illmen (Pende gas , 2008). A he same
ime, his de elopmen equi es a ce ain inancial and ime eedom o ind he igh p o ession
o onesel and o achie e he necessa y le el o educa ion. This is whe e UBI becomes ele an .
Means- es ed social secu i y sys ems a e o en cha ac e ized by a lo o bu eauc acy and a
lawed incen i e s uc u e. The incen i e s uc u e o a means- es ed social secu i y sys em is
lawed i i encou ages people o choose jobs ha do no ma ch hei skills. This can hen lead
o ising misma ch unemploymen in he long un (Sach e s ändigen a Wi scha , 2019). An
UBI, on he o he hand, is non-bu eauc a ic and allows e e y ci izen a highe deg ee o eedom
in choosing a p o ession (e.g., Liebe mann,2012)
Thus, we a gue ha he in e sec ion be ween he wo c i ics desc ibed may be caused by he
lack o alignmen o economic heo y wi h he basic UBI easoning, as i is a leas pa ially a
odds wi h mains eam economics. Mo eo e , i is based on a undamen ally di e en
concep ion o humankind han mos economic models, and i is he e o e inhe en ly p oblema ic
1
Especially, he “Gene a ion Z” seem o be an indica o o hese changes in wo k pe cep ion and labo e hics.
4
o apply hese models o an UBI. Thus, while he e is g owing in e es in he concep among
policy make s and academics due o he changing landscape o labo , we a gue ha his
di e ence has no been aken in o accoun in he economic modeling o he UBI.
The e o e, his pape ocuses on he compa ison o consump ion ax e enue wi h an
uncondi ional ans e o a UBI scheme and a means- es ed social secu i y sys em. In doing so,
i is closely o ien ed on ou p e ious pape (Neumä ke & Weinel, 2022). Howe e , while he
ocus he e was on he no ma i e aspec o he conside a ion, he e i is on he model and i s
u he de elopmen . No e, howe e , ha his is he beginning o a longe esea ch p og am.
The model is he e o e o be unde s ood as a baseline model, as i is a s a ic economy wi h a
ep esen a i e indi idual and a single ax.
The ocus on he consump ion ax is mo i a ed on he one hand by he li ely deba e abou a
consump ion- o VAT- inanced UBI (e.g. Wakolbinge e al., 2020 o Neumä ke & Pale mo,
2018), and on he o he hand by he ac ha his pape builds on he esul s o Hi aga and
Nu ah a (2016, 2018, 2019, 2021, 2022) as well as T aband and Uhlig (2011, 2013).
Th oughou his pape , we p esen ou main indings. Fi s , a di e en ia ion in he u ili y
calcula ion be ween a UBI scheme and a means- es ed social secu i y sys em is no only
necessa y bu also possible. The u ili y unc ion in economics is based on a ew axioms and is
mean o e lec he basic p e e ence s uc u e. We a gue ha al eady a his unde lying elemen
a dis inc ion can be made be ween he s anda d economic heo y and he UBI d i en economic
logic. We de elop a u ili y unc ion ha , by accoun ing o ime so e eign y ha indi iduals
gain om an UBI and in eg a ing in insic mo i a ion eleased by UBI, is adap ed o he
assump ions o an UBI sys em. Hence, we de elop a u ili y unc ion which is endogenous o
social policy s a egies. Second, we illus a e in a simple s a ic model ha accoun ing o hese
di e ences in unde lying assump ions, by adjus ing he u ili y unc ion, leads o signi ican ly
di e en esul s o consump ion ax e enues. While he UBI u ili y unc ion leads o an
inc easing consump ion ax e enue cu e, he u ili y unc ion we assign o he means- es ed
social secu i y sys em (MT sys em) leads o a classical consump ion ax e enue La e cu e.
Thi d, he di e ence in ax e enue le els is ema kable. In ou model, he UBI u ili y unc ion
leads o signi ican ly highe ax e enues han he MT u ili y unc ion o each ax a e. Fou h,
and mos signi ican in he con ex o his pape , is di e en ia ion o he e ec s o ime
so e eign y and in insic mo i a ion on ax e enues. I is ound ha he ime so e eign y e ec
has a s onge impac on consump ion ax e enues han he in insic ac o .
5
The s uc u e o he pape is as ollows. In chap e wo we e iew he ela ed li e a u e. The
pa adigm shi in ime so e eign y will be explained in chap e h ee. Chap e ou is de o ed
o he model. Chap e s i e and six p esen and discuss he esul s. The pape closes wi h he
conclusion in chap e six.
2. Rela ed Li e a u e
This pape con ibu es o h ee bodies o li e a u e. Fi s and o emos , we con ibu e o he
exis ing li e a u e on he modelling o a UBI. The li e a u e on UBI models is con inuously
g owing. Wakolbinge e al (2020), o example, analyze a consump ion ax inanced UBI o
Ge many in di e en scena ios. Neumä ke and Pale mo (2018), on he o he hand, model he
ime alloca ion e ec s o he UBI. We con ibu e o his line o li e a u e by p o iding a model
ha accoun s o he e ec s o he UBI a he u ili y calculus le el on he one hand and examines
he e ec s on he ax e enue o he consump ion ax on he o he hand. Thus, we can in eg a e
he adi ional UBI a gumen s om he u ili y side a he han he p oduc ion side.
Second, we con ibu e o he li e a u e on he implica ions o he u ili y unc ion by King and
Rebelo (1999) on ax e enue. He e we a e closely ela ed o he body o esea ch by Hi aga
and Nu aha a (2016, 2018, 2019, 2021, 2022) which consis s o wo main esea ch oci. The
i s esea ch pa h s udies he sensi i i y o he ax e enue cu e wi h espec o he u ili y
unc ion (ibid. 2016, 2019, 2021). The au ho s compa e he e ec s o addi i e sepa able and
mul iplica i e u ili y unc ions on he consump ion ax a e in a neoclassical se ing. The eby,
i is shown ha he consump ion ax e enue cu e canno be hump-shaped o a mul iplica i e
u ili y unc ion, while i can be hump-shaped o an addi i e sepa able u ili y unc ion. The
second esea ch pa h, on he o he hand, ocuses on he di e ence in shape o he ax e enue
cu e o he consump ion- and he labo income ax (ibid. 2018, 2022). Whe eas one can
obse e he classical La e cu e (humped-shaped ax e enue cu e) o he labo income ax
independen o he u ili y unc ion, one canno obse e he same o he consump ion ax
e enue cu e. By manipula ing he mul iplica i e u ili y unc ion o King and Rebelo (1999)
o an UBI and hen applying i o he consump ion ax, we hus con ibu e o he esea ch on
he ela ionship be ween he consump ion ax e enue cu e and he u ili y unc ion.
And hi d, we a e con ibu ing o he body o esea ch on in insic mo i a ion, especially in he
waged labo con ex . Along wi h he ans o ma ion o he wo k en i onmen , a esea ch ocus
has also mo ed in o he di ec ion o in insic mo i a ion in he labo ma ke con ex . Economis s
6
such as Mu dock (2002) o P ende gas (2008) inc easingly emphasize he impo ance o
in insic mo i a ion in highly specialized jobs. Ou con ibu ion o his line o li e a u e ela es
o he inco po a ion o in insic mo i a ion in o he u ili y unc ion. In he con ex o his pape ,
we e e o in insic mo i a ion whene e an indi idual inds pleasu e in a ask o ac i i y o
easons ha lie in he ac i i y i sel and no in i s consequences, hus ollowing F ey's (1994)
de ini ion. O , pu i in ano he way, in a labo ma ke and wo k sys em in which only
compensa ion o he disu ili y o wo k e o is conside ed o be ele an in insic mo i a ion
is “c owded ou ” because “i does no pay”. Wi h UBI, we a gue, his logic will be unde mined.
3. Pa adigm Shi in Time So e eign y
Th oughou his pape , we a gue ha a social policy shi om a means- es ed social secu i y
sys em o an uncondi ional basic income implies a pa adigm shi in ime so e eign y. This
pa adigm shi is due o he ac ha indi iduals in an UBI sys em a e no longe o ced o wo k
and hus become uly ime so e eign. Tha ime so e eign y, in u n, hen allows indi iduals
o ake in insic mo i a ion in o accoun in hei u ili y calculus. This is mos easily illus a ed
by he p incipal agen heo y, al hough i can also be ound in many o he economic models,
such as he Shapi o-S igli z e iciency wages model (1984). In he classic unde s anding o he
employe -employee ela ionship, he employee is only mo i a ed o wo k by ex insic
incen i es and cons an ly ies o asse his own in e es s, which a e inconsis en wi h hose o
he employe . This shi is no jus heo e ical; i impac s how we design social policies and
labo ma ke s. P e ious esea ch, such as s udies on he Nega i e Income Tax, has o en missed
his nuance by ocusing solely on labo - ime educ ion unde he old addi i e amewo k
(Wide quis , 2019, 304 .). No only wo king condi ions bu also he design o he wel a e s a e
is based on hese assump ions. Thus, many social secu i y sys ems consis o nume ous con ol
mechanisms o jobseeke s, which a e in ended o ensu e ha hey ge back in o employmen
as quickly as possible. Howe e , UBI ad oca es assume ha indi iduals aim o be a p oduc i e
pa o socie y on hei own ini ia i e (e.g., S anding 2017, Van Pa ijs & Vande bo gh 2017,
To y, 2019). Thus, he absence o hese con ol and incen i e mechanisms is a key componen
o UBI, as i gi es each pe son absolu e eedom o e hei own ime. We hus a gue ha such
a d as ic shi in social policy a ec s he u ili y calculus o indi iduals and endogenizes he
u ili y unc ion wi h espec o i . The e o e, we a gue ha ou app oach is a igo ous
implemen a ion o UBI logic.
13
Figu e 2: Consump ion Tax Re enue Cu es (Di e en U ili y Func ions)
Figu e 3 illus a es he e ec o he in insic ac o and he ime so e eign y on consump ion
ax e enue. The in insic ac o , 𝜌, in he u ili y unc ion is de ined as he in insic mo i a ion
o wo k. The in insic mo i a ion e ec on he ax e enue, howe e , is he di e ence in
consump ion ax e enue ha a ises h ough he conside a ion o he in insic ac o . The e o e,
i is no he di e ence ha a ises be ween he MT- and he UBI u ili y unc ion bu he di e ence
be ween he mul iplica i e u ili y unc ion wi h 𝜌 and wi hou 𝜌. The ime so e eign y e ec ,
on he o he hand, is he di e ence in ax e enue ha a ises be ween he mul iplica i e u ili y
unc ion (wi hou 𝜌) and he addi i e sepa able u ili y unc ion. In o de o isola e he e ec
om he in insic mo i a ion e ec 𝜌 is no conside ed. Time so e eign y is de ined as a ue
eedom o ime, i.e., no obliga ion o wo k. I is indica ed by he mul iplica i e linkage be ween
consump ion and wo k in he u ili y unc ion. One can see ha , independen o he ax a e, he
ime so e eign y e ec is bigge in magni ude on he ax e enue han he in insic mo i a ion
e ec . This is in line wi h many UBI p oponen s who a gue agains he s o y o he “lazy
indi idual”. In his s a ic model economy wi h a ep esen a i e indi idual, one ax
(consump ion ax), and one uncondi ional ans e , he impac o he modeling o he u ili y
unc ion on he ax e enue is c ucial. Mo eo e , hese esul s indica e ha while in insic
mo i a ion is an impo an d i e in he u ili y unc ion i is no he only one no he mos
impo an one.
0
0.1
0.2
0.3
0.4
0.5
0.6
Consump ion Tax Re enue
Tax Re enue (UBI)
Tax Re enue (MT)
Di e ence in Tax
Re enue (MT & UBI)
No e: MT s ands o means- es ed social secu i y sys em and UBI s ands o
uncondi ional basic income. The y-lab shows he ax e enue. The x-lab shows he ax
a e. Pa ame e alues:
𝜂=0.25, 𝜆=0.25, 𝜅=2.5, 𝜌=1.
14
Figu e 3: In insic and Time So e eign y E ec
6. Discussion
The ollowing discussion is conce ned wi h he implica ions and he plausibili y o he
pa ame e alues o he nume ical example as well as con ex ualizing o he esul s.
The pa ame e alues o he nume ical example a e no gi en by speci ic alues om he
li e a u e. The e a e h ee easons o ha . Fi s , he li e a u e lacks hese es ima es o an UBI
sys em. E en hough he li e a u e on UBI is g owing signi ican ly, he e has ne e been a ull
scale UBI sys em in place which makes i e y di icul o p edic how he labo supply would
eac . Second, while we would ha e been able o use es ima es o he isk a e sion, labo supply
elas ici y and disu ili y o wo k om one s udy (e.g., T aband & Uhlig, 2011), we s ill would
ha e been obliga ed o use an es ima ion o he in insic alue om a di e en s udy. Gi en ha
he li e a u e lacks es ima ions o in insic mo i a ion o he labo o ce, we would be o ced
o use alues ha a e no in line wi h he o he pa ame e alues. The easons o his a e
di e ences in he unde lying da a se s, economic-, and econome ic -assump ions.
Ne e heless, in ou p e ious pape (Neumä ke & Weinel, 2022) one can ind a nume ical
example wi h pa ame e es ima es om he li e a u e.
No e: MT s ands o means- es ed social secu i y sys em and UBI s ands o uncondi ional basic
income. The y-lab shows he ax e enue. The x-lab shows he ax a e. Pa ame e alues: 𝜂=
0.25, 𝜆=0.25, 𝜅=2.5, 𝜌=1.
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
In insic and Time So e eign y E ec
In insic Mo i a ion
E ec on he Tax
Re enue
Time So e eign y E ec
on he Tax Re enue
15
The hi d eason is he es ima ion o in insic mo i a ion o wo k i sel . While he e is a la ge
body o li e a u e on in insic mo i a ion, i ocuses mainly on speci ic ca ego ies o wo k. Ye
e en i we ind an es ima e o in insic mo i a ion o he labo o ce, i is likely o be inadequa e
o ou pu poses because i was ob ained in a speci ic social secu i y sys em. In his con ex ,
he ela ionship be ween ex insic and in insic mo i a ion is pa icula ly p oblema ic. F ey
(1997) o example a gues ha ex insic incen i es can lead o a c owding ou o in insic
mo i a ion. I is he e o e expec ed ha he alue o in insic mo i a ion in a UBI sys em di e s
om ha in a means- es ed sys em.
These conce ns led us o choose hypo he ical pa ame e alues ha a e, howe e , no
un easonable. We se bo h, he ela i e isk a e sion, and he in e se o he labo supply
elas ici y, ela i ely small. This is because hey mus mee he assump ion 𝜂+𝜆<1 o he
means- es ed u ili y unc ion o be humped shaped. As he in e se o he labo supply elas ici y
is se a 0.25, he ac ual labo supply elas ici y is 4. The calib a ion o he labo supply elas ici y
o F isch elas ici y is, gene ally, ela i ely ola ile (Pe e man, 2016). While i is usually highe
in mac oeconomics han in mic oeconomics, 4 is a qui e high alue e en o mac oeconomics.
Ne e heless, King and Rebelo (1999) also calib a ed i as such in hei RBC model. Since he
mul iplica i e u ili y unc ion o igina es om his wo k, i seems jus i ied o choose he alue
acco dingly o he pu pose o he nume ical example. The cons an ela i e isk a e sion is se
o be 0.25. Since his is a s a ic model wi hou ic ions, he isk beha io is i ele an . Howe e ,
𝜂 can be unde s ood as an index o he cu a u e o he u ili y unc ion (Hi aga & Nu aha a,
2019, Meye & Meye , 2005). The disu ili y o labo is se o be 2.5. Compa ed o T aband and
Uhlig (2011), who ake a disu ili y o labo o 3.46 o he benchma k o a balanced g ow h o
labo o 25 pe cen age o he o al ime, and Hi aga and Nu aha a (2022), who se he disu ili y
o labo equal o one in hei nume ical example, in ou example he chosen alue is in a window
be ween hese wo. Ne e heless, please no e ha we s ill accoun he in insic alue and hus
ob ain a lowe o e all alue o (𝜅−𝜌) o he UBI u ili y unc ion compa ed o he wo o he
pape s. The in insic alue is assumed o be one. As we ha e al eady indica ed be o e, he
measu emen o in insic mo i a ion is somewha di icul . Howe e , he e a e wo easons why
we ha e chosen o use his alue in he con ex o his pape . Fo one, o a oid ha he disu ili y
o labo and he in insic alue cancel each o he ou in he calcula ion. Secondly, he e m (κ-
ρ) is o be kep posi i e. Consequen ly, he in insic alue mus be smalle han he disu ili y o
labo . Wi hin hese wo limi s, howe e , i could ha e aken any alue.
16
The esul s o his s udy, and in pa icula hose o he nume ical example, do no ha e any
di ec poli ical implica ions. Howe e , he e a e economic- heo e ical implica ions.
Con e sely, he p oposi ion ega ding he u ili y unc ion implies ha any calcula ion
conce ning he UBI is based on alse assump ions, i i does no ake beha io al changes in o
accoun . Thus, i is pa icula ly in e es ing ha , in he nume ical example, we obse e
signi ican di e ences in ax e enues be ween he schemes ha ake beha io al adjus men
in o accoun and he ones ha do no . Mo eo e , we see ha ime so e eign y is mo e impo an
in he u ili y calculus han he in insic ac o . This could be in e p e ed as a clea a gumen in
a o o an UBI, since i is possible o p omo e a highe in eg a ion o in insic mo i a ion in
he exis ing sys em h ough a ious measu es, bu he same canno be said abou ime
so e eign y.
7. Conclusion
This pape in es iga es he implica ions o he UBI o he u ili y unc ion and consump ion ax
e enues. We assume an endogenous u ili y unc ion wi h espec o social policy. We a gue
ha he addi i e u ili y unc ion exp esses he u ili y calculus o an indi idual in a means- es ed
social secu i y sys em, while he mul iplica i e u ili y unc ion ep esen s he u ili y calculus o
an indi idual in an UBI sys em. In a simple s a ic model wi h a ep esen a i e indi idual, he
consump ion ax e enue cu e o he wo sys ems di e s. While he consump ion ax e enue
cu e ises o he UBI sys em, i shows he classic La e -hill o he means- es ed social
secu i y sys em. Mo eo e , a clea di e ence in ax e enue can be obse ed. In he con ex o
his pape , howe e , he ocus is on he e ec o ime so e eign y and in insic mo i a ion, bo h
aken in o accoun in he UBI u ili y unc ion, on he consump ion ax e enue cu e. Time
so e eign y is accoun ed o by he mul iplica i e link be ween wo k and consump ion in he
u ili y unc ion. In insic mo i a ion o wo k, on he o he hand, is ep esen ed by 𝜌 in he u ili y
unc ion. Bo h aspec s a e shown o ha e a signi ican impac on ax e enues. Howe e , he
ime so e eign y ha indi iduals gain h ough an UBI exe s a g ea e in luence on ax e enue.
I seems ema kable ha he igo ous inco po a ion o UBI assump ions in o he u ili y unc ion
no only wo ks, bu also has a posi i e e ec on consump ion ax e enues ha exceeds hose
o means- es ed social secu i y sys ems.
Based on ou esul s, we can al eady de i e some impo an implica ions. Fi s , i should be
no ed ha he assump ions o UBI a e no consis en wi h he classical assump ions o
economics. Consequen ly, he s ic applica ion o he classical addi i e u ili y unc ion in his
17
con ex will always lead o ( ollowing his a gumen a ion) w ong esul s. Accoun ing o hese
assump ions in he u ili y unc ion, on he o he hand, leads o undamen ally di e en esul s
wi h espec o consump ion ax e enues. The inancing o an UBI is epea edly poin ed ou as
a coun e a gumen . Ou model shows ha sus ainable inancing could be possible, o a leas ,
easie . Ne e heless, i should be no ed ha his is he beginning o he esea ch wo k. We will
expand he model u he in he nex s eps by in oducing di e en ax egimes, he e ogeneous
agen s and adding quasi-dynamic cons ain s o he “new o dolibe alism”, namely,
enego ia ion-p oo ness o a cons i u ional UBI con ac .
3
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20
Appendix
A.
To de i e he consump ion ax e enue in he model, we apply he condi ions by Hi aga and
Nu aha a (2016, 2018, 2019 a & b, 2021).
The household’s consump ion-labo choice is gi en by
−𝑈𝑛
𝑈𝑐=𝑅𝑃𝐿.
(16)
The ela i e p ice o leisu e, RPL, is
𝑅𝑃𝐿≡ 1
(1+𝜏𝑐)×𝑤.
(17)
The equilib ium elas ici y o consump ion wi h espec o he consump ion ax (𝜏𝑐) is
|𝑑𝑐/𝑐
𝑑𝜏𝑐/𝜏𝑐|=|𝑑𝑅𝑃𝐿/𝑅𝑃𝐿
𝑑𝜏𝑐/𝜏𝑐|×(−𝑐𝑈𝑐𝑐
𝑈𝑐+𝑛𝑈𝑛𝑛
𝑈𝑛+𝑐𝑈𝑐𝑛
𝑈𝑛−𝑛𝑈𝑐𝑛
𝑈𝑐)−1,
(18)
whe e he elas ici y o he ela i e p ice o leisu e, RPL, wi h espec o 𝜏𝑐 is gi en by
|𝑑𝑅𝑃𝐿/𝑅𝑃𝐿
𝑑𝜏𝑐/𝜏𝑐|= 𝜏𝑐
1+𝜏𝑐.
(19)
Hi aga and Nu aha a (2018) ha e also shown ha a necessa y condi ion o he consump ion
La e cu e o be humped shaped is ha ,
𝑑𝑐/𝑐
𝑑𝜏𝑐/𝜏𝑐<−1
(20)
holds.
B.
Mul iplica i e U ili y Func ion
In he case o he UBI u ili y unc ion, (16) leads o he ollowing,
21
1
(1+τc)×w=η(λ+1)(c(𝜅−𝜌)nλ
1−(𝜅−𝜌)(1−η)nλ+1).
(21)
Gi en ha c is equal o n in his model, we ge he ollowing esul when ea anging he
equa ion wi h espec o c,
c=[(𝜅−𝜌)(η((τc+1)λ+τc)+1)]−1/(1+λ).
(22)
The o al ax e enue is gi en by,
T=τc×[(𝜅−𝜌)(𝜂((τc+1)𝜆+τc)+1)]−1/(1+λ).
(23)
The de i a i e o he ax e enue is
∂T
∂τc=ηλ(τc+1)+1
(η((λ+1)τc+λ)+1)((𝜅−𝜌)(η((λ+1)τc+λ)+1))1/(λ+1).
(24)
To de i e he elas ici y o consump ion wi h espec o he consump ion ax, we ob ain he
ollowing in e media e esul .
−𝑐𝑈𝑐𝑐
𝑈𝑐=𝜂,
𝑛𝑈𝑛𝑛
𝑈𝑛=𝑛𝜆+1(𝜂−1)(𝜂(𝜆+1)−1)(𝜅−𝜌)+𝜆
𝑛𝜆+1(𝜂−1)(𝜅−𝜌)+1 ,
𝑐𝑈𝑐𝑛
𝑈𝑛=1−𝜂,
−𝑛𝑈𝑐𝑛
𝑈𝑐=𝑛𝜆+1(𝜅−𝜌)(1−𝑛𝜆+1(𝜅−𝜌)(𝜂−1))𝜂−1(1−𝜂)𝜂(𝜆+1)
(1−𝑛𝜆+1(𝜅−𝜌)(1−𝜂))𝜂.
Due o condi ion (18) his yields he ollowing esul s,
𝑑𝑐/𝑐
𝑑𝜏𝑐/𝜏𝑐=𝜏𝑐
(1+𝜏𝑐)×(𝑛𝜆+1(𝜂−1)(𝜂(𝜆+1)−1)(𝜅−𝜌)+𝜆
𝑛𝜆+1(𝜂−1)(𝜅−𝜌)+1
+𝑛𝜆+1(𝜅−𝜌)(1−𝑛𝜆+1(𝜅−𝜌)(𝜂−1))𝜂−1(1−𝜂)𝜂(𝜆+1)
(1−𝑛𝜆+1(𝜅−𝜌)(1−𝜂))𝜂
+1)−1.
(25)
Hence, he elas ici y o consump ion o he consump ion ax a e is gi en by
22
|𝑑𝑐/𝑐
𝑑𝜏𝑐/𝜏𝑐|= 𝜏𝑐
(1+𝜏𝑐)×(𝑛𝜆+1(𝜂−1)(𝜂(𝜆+1)−1)(𝜅−𝜌)+𝜆
𝑛𝜆+1(𝜂−1)(𝜅−𝜌)+1
+𝑛𝜆+1(𝜅−𝜌)(1−𝑛𝜆+1(𝜅−𝜌)(𝜂−1))𝜂−1(1−𝜂)𝜂(𝜆+1)
(1−𝑛𝜆+1(𝜅−𝜌)(1−𝜂))𝜂
+1)−1
(26)
o 𝜆>0 and 𝜏𝑐>0.
I 𝜏𝑐=0, hen |𝑑𝑐/𝑐
𝑑𝜏𝑐/𝜏𝑐|=0. I 𝜏𝑐 is inc easing, hen |𝑑𝑐/𝑐
𝑑𝜏𝑐/𝜏𝑐| inc eases.
C.
Addi i e sepa able U ili y unc ion
In he case o he addi i e sepa able u ili y unc ion (i.e., means- es ed social secu i y sys em)
we obse e he ollowing esul s.
The consump ion labo supply condi ion is gi en by
1
(1+𝜏𝑐)×𝑤=𝜅𝑛𝜆𝑐𝜂.
(27)
Sol ing his condi ion o c yields o
𝑐=[𝜅(1+𝜏𝑐)]−1/(𝜂+𝜆).
(28)
The o al ax e enue is
𝑇=𝜏𝑐[𝜅(1+𝜏𝑐)]−1/(𝜂+𝜆)
(29)
and he pa ial de i a i e is gi en by,
𝜕𝑇
𝜕𝜏𝑐=(𝜆+𝜂−1)𝜏𝑐+𝜆+𝜂
(𝜆+𝜂)(𝜏𝑐+1)(𝜅(𝜏𝑐+1))1(𝜆+𝜂)
⁄.
(30)
Fo elas ici y o consump ion o he consump ion ax a e we de i e he ollowing in e media e
esul ,
−𝑐𝑈𝑐𝑐
𝑈𝑐=𝜂, 𝑛𝑈𝑛𝑛
𝑈𝑛=𝜆, 𝑐𝑈𝑐𝑛
𝑈𝑛=0, −𝑛𝑈𝑐𝑛
𝑈𝑐=0.
