Taxing versus subsidizing debt under financial frictions

Schabe , And eas
A icle — Published Ve sion
Taxing e sus subsidizing deb unde inancial ic ions
Economic Theo y
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Schabe , And eas (2024) : Taxing e sus subsidizing deb unde inancial
ic ions, Economic Theo y, ISSN 1432-0479, Sp inge , Be lin, Heidelbe g, Vol. 79, Iss. 4, pp.
1383-1420,
h ps://doi.o g/10.1007/s00199-024-01615-3
This Ve sion is a ailable a :
h ps://hdl.handle.ne /10419/323259
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Economic Theo y (2025) 79:1383–1420
h ps://doi.o g/10.1007/s00199-024-01615-3
RESEARCH ARTICLE
Taxing e sus subsidizing deb unde inancial ic ions
And eas Schabe 1
Recei ed: 3 July 2024 / Accep ed: 25 Sep embe 2024 / Published online: 22 Oc obe 2024
© The Au ho (s) 2024
Abs ac
We examine op imal c edi ma ke policies in wo models wi h du ables/capi al as
colla e al. Pecunia y ex e nali ies a ionalize ex-an e deb axes as mac op uden ial
egula ion, achie ing cons ained e iciency. Ex-pos deb subsidies can implemen
i s -bes by s imula ing colla e al demand. Due o he same e ec , deb subsidies
ha a e cons an o e ime can be supe io o deb axes. Sa ing subsidies can u -
he enhance e iciency by add essing dis ibu i e e ec s o pecunia y ex e nali ies
ia in e es a e educ ions. The analysis shows ha deb -inc easing subsidies can
ou pe o m mac op uden ial egula ion, and ha cons ained ine iciency caused by
colla e al ex e nali ies is insu icien o es ablish deb axes as op imal c edi ma ke
policies.
Keywo ds Financial s abili y ·Pecunia y ex e nali ies ·Colla e al cons ain ·
Mac op uden ial egula ion ·Dis ibu i e e ec s
JEL Classi ica ion E44 ·G18 ·H23
1 In oduc ion
Pecunia y ex e nali ies unde colla e al cons ain s can lead o inancial ampli ica ion
and c ises. The mechanism elies on p ice-dependen bo owing limi s o ma gin con-
s ain s ha igh en when asse p ices all. Agen s do no in e nalize he impac o hei
decisions on asse p ices, such ha co ec i e policies can enhance e iciency. Mac o-
p uden ial egula ion, in o m o ex-an e deb axes and capi al con ols, can es o e
"cons ained e iciency" – de ined in he adi ion o S igli z (1982) – by add ess-
ing "o e bo owing", as shown by Jeanne and Ko inek (2010,2019,2020), Bianchi
(2011), Benigno e al. (2016), Ko inek and Sand i (2016), Schmi -G ohe and U ibe
(2017), o Ko inek (2018). These s udies a e based on a speci ic class o models whe e
in e es a es a e exogenously de e mined and agen s ake bo owing limi s as gi en.
Thus, nei he c edi supply no asse s’ colla e alizabili y, which bo h seem o play a
BAnd eas Schabe
[email p o ec ed]
1Cen e o Mac oeconomic Resea ch, Uni e si y o Cologne, Albe us-Magnus-Pla z, 50923
Cologne, Ge many
123
1384 A. Schabe
cen al ole o he build-up o inancial c ises (see Geanakoplos 2010, o Jus iniano
e al. 2019), a e aken in o accoun o he analysis o policies aimed o mi iga e hese
c ises. Is hei neglec i ele an ?
This pape shows ha endogenous in e es a es and colla e al p emia, i.e. he al-
ua ion o asse s o se e as colla e al, a e in ac decisi e o op imal c edi ma ke
policy. We apply wo ini e ho izon models ha exclusi ely con ain con en ional ea-
u es; one model is aken om Da ila and Ko inek (2018).1Lack o commi men
induces bo owing o be limi ed by bo owe s’ holdings o du ables o capi al, se ing
as colla e al. Pecunia y ex e nali ies wi h ega d o he colla e al p ice and o he in e -
es a e gi e ise o "colla e al ex e nali ies" and "dis ibu i e ex e nali ies" (see Da ila
and Ko inek 2018). The o me a e esponsible o he main mechanism in he abo e
ci ed s udies, whe eas he la e a e u ned o he ein. While s a e-con ingen ex-pos
c edi ma ke in e en ions can achie e i s bes , we pa icula ly ocus on policies ha
a e less challenging o be implemen ed han policies ha a e ully s a e-con ingen . As
he main no el con ibu ion, we show ha dis ibu i e e ec s and colla e al p emia a e
esponsible o non-s a e-con ingen c edi ma ke subsides ha s imula e bo owing
o be supe io o a deb - educ ion policy, in pa icula , o mac op uden ial egula ion in
he o m o an ex-an e deb ax.2A la ge, ine iciencies due o ex e nali ies induced by
colla e al cons ain s can be mos e ec i ely add essed by sa ing/deb subsidies ha
educe in e es a es and aise colla e al p ices. This pape speci ically highligh s he
e ec i eness o non-s a e-con ingen policies ha p omo e sa ings and lowe bo ow-
ing cos s. Real-wo ld examples include home loan sa ing con ac s o ax deduc ions
o mo gage in e es paymen s. The analysis implies ha combining hese ypes o
policies wi h ex-an e egula ion ha limi he build-up o deb , which e lec s ac ual
go e nmen p ac ices, can p incipally enhance wel a e e en u he and can eplica e
s a e-con ingen c edi ma ke policies.
In a laissez ai e equilib ium (o bo h models), agen s do no in e nalize ha he
colla e al p ice is oo low and he in e es a e is oo high when bo owing cons ain s
bind in a non-emp y se o s a es. Following Da ila and Ko inek (2018) classi ica ion,
we dis inguish colla e al e ec s, which e e o unin e nalized changes in he p ice o
pledgeable asse s a ec ing he colla e al alue, om dis ibu i e e ec s, which e e
o di e en ial e ec s o unin e nalized in e es a e changes on he e ogenous agen s.
In con as o he abo e ci ed s udies, whe e agen s ake bo owing limi s as gi en, he
p ice o pledgeable asse s is posi i ely a ec ed by a colla e al p emium, i.e. he al-
ua ion o asse s o se e as colla e al.3Co ec i e policies can add ess unin e nalized
changes in he colla e al p ice, le e aging he ac ha he colla e al p ice inc eases
wi h consump ion and wi h agen s’ willingness o bo ow, which aises he colla e al
p emium. Agen s u he do no in e nalize ha equilib ium in e es a es ela e o
hei consump ion/sa ing choices, which exe s a ele an impac on he equilib ium
1We i s de elop and analyze a model wi h du ables as colla e al unde unce ain y. Da ila and Ko inek
(2018) model wi h capi al o ma ion unde ce ain y is subsequen ly analyzed in Sec .5.
2Th oughou he pape , we iden i y mac op uden ial egula ion wi h an ex-an e deb ax, ollowing Bianchi
and Mendoza (2018): " he mac op uden ial deb ax [...] is le ied in good imes when colla e al cons ain s
do no bind a da e bu can bind wi h posi i e p obabili y a +1" (p. 591).
3This asse p ice componen is also known as he "colla e al alue" (see Fos el and Geanakoplos 2008)
o he "colla e alizabili y p emium" (see Ai e al. 2020).
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Taxing e sus subsidizing deb unde inancial ic ions 1385
alloca ion unde di e en ma ginal a es o subs i u ion be ween da es/s a es (MRSs)
o bo owe s and lende s (see also Da ila and Ko inek 2018). Co ec i e policies can
add ess his ex e nali y and educe he in e es a e, which aises bo owe s’ consump-
ion and na ows he dis ance be ween he MRSs. Appa en ly, in e es a es canno be
educed when hey a e assumed o be exogenously de e mined, and colla e al p emia
do no exis when agen s ake bo owing limi s as gi en, like in he abo e ci ed s ud-
ies. In his special case, o e bo owing p e ails and a co ec ion o he colla e al p ice
elies on deb educ ion ia ex-an e deb axes.
To iden i y op imal c edi ma ke policies, we assume ha he policy make ac s
unde ull commi men and we apply he Ramsey app oach o op imal policy, whe e
he policy p oblem depends on he se o a ailable ins umen s.4I s a e-con ingen
c edi ma ke ins umen s a e a ailable, i s bes is implemen able ia a Pigou ian
deb subsidy ha is in oduced ex-pos (i.e., in s a es whe e colla e al cons ain s bind).
The subsidy inc eases incen i es o bo ow and hus he willingness o pay o colla -
e al, measu ed by he colla e al p emium. An ex-pos deb subsidy can he eby aise
he colla e al p ice and he bo owing limi such ha bo owing can in p inciple e en
ge uncons ained, which has also been shown by Ka agi i e al. (2017). Based on hei
quan i a i e analysis, hey conclude ha he op imal deb subsidy is p ac ically in ea-
sible, gi en he size and he equency o equi ed in e en ions. S a e-con ingency
in ac demands policies o be ine- uned in esponse o any change in he s a e o
he economy (see e.g. Bianchi and Mendoza 2018). Gi en ha he equi emen o
accu a ely ack ele an condi ions and o imely adjus policy ools can ha dly be
ul illed in p ac ice (see e.g. Coch ane 2013), ou analysis ocusses on Pigou ian
policies ha a e less complex and easie o implemen han ully s a e-con ingen poli-
cies.5Speci ically, we examine ex-an e policies, which a e imposed be o e bo owing
cons ain s migh become binding, and cons an policies, whe e he ax/subsidy a e
is held cons an ega dless o he s a e o pe iod (see columns o Table 1). These
policies ake he o m o axes/subsidies on bo owing o sa ing (see ows o Table
1), which a e non-equi alen unde po en ially binding bo owing cons ain s. Linea
p e e ences o lende s u he imply ha deb policies do no al e he in e es a e and
can only add ess colla e al e ec s, enabling di ec compa isons wi h ela ed s udies
(see Sec .2). In con as , sa ing policies can endogenize in e es a es and can he eby
add ess dis ibu i e e ec s o pecunia y ex e nali ies (see a ows in Table 1).
The i s op imal non-s a e-con ingen policy is an ex-an e deb ax. I implemen s
a cons ained e icien alloca ion, as de ined in S igli z (1982) o Da ila e al. (2012).
This alloca ion is chosen by a social plane who de e mines bo owing and maximizes
social wel a e subjec o budge and bo owing cons ain s, condi ional on main aining
equilib ium p ice ela ions unde laissez ai e. Gi en ha ex-an e deb axes lea e he
ele an p ice ela ions unde laissez ai e unchanged, he op imal Ramsey policy in
4This p ope y o he Ramsey app oach is also ele an in Benigno e al. (2023), who show ha a se
o ins umen s ha suppo s cons ained e iciency can also implemen an uncons ained alloca ion. In
con as o Bianchi and Mendoza (2018), who ocus on op imal policy unde disc e ion, we abs ac om
ime inconsis ency o policy plans.
5An al e na i e would be o adjus policy ins umen s wi h changes in a iables ha can easily be obse ed,
like gdp o deb (see e.g. Bianchi and Mendoza 2018). In ou main model, he la e a e howe e no co ela ed
o he shock ha cause bo owing cons ain s o bind, namely, an unexpec ed change in income inequali y.
123
1386 A. Schabe
Table 1 Non-s a e-con ingen Pigou ian policies (nume ical examples in bold)
Ex-an e Fixed o e ime
Deb 1. ax/subsidy
→colla e al e ec s
2. ax/subsidy
→colla e al e ec s
Sa ing 3. ax/subsidy
→colla e al & dis ibu i e e ec s
4. ax/subsidy
→colla e al & dis ibu i e e ec s
bo h models implemen s cons ained e iciency (like in Da ila and Ko inek 2018).
Conc e ely, i enhances e iciency by aising he colla e al p ice ia a deb educ ion
ha inc eases he amoun o unds a ailable o consump ion in s a es whe e he
bo owing cons ain binds. In con as o ex-an e deb axes, he o he policies unde
conside a ion al e he p ice ela ions o he in e es a e and he colla e al p ice, which
would no be possible when agen s ake bo owing limi s as gi en and in e es a es
a e exogenous.
The second non-s a e-con ingen policy is a ax/subsidy on deb ha is cons an
o e ime and in luences bo owing ega dless whe he he cons ain is binding o no .
As shown by Bianchi and Mendoza (2018) analysis o op imal deb policy, a policy
make can alle ia e cu en ly binding bo owing cons ain s by an ex-pos subsidy
and u u e bo owing cons ain s by an ex-an e ax. Howe e , a cons an deb subsidy,
which ends o s imula e bo owing, educes esou ces a ailable o consump ion when
he cons ain binds, while i aises agen s’ willingness o pay o colla e al. I hus
combines he in e se e ec s o an ex-an e deb ax on consump ion wi h he e ec s o
an ex-pos deb subsidy on he colla e al p emium. A cons an deb subsidy is supe io
o a cons an deb ax i he e ec on he colla e al p emium domina es. We show ha
his holds uncondi ionally in Da ila and Ko inek (2018) model as well as in he model
wi h du ables i he loan- o- alue a io is su icien ly la ge (including alues ypically
used in quan i a i e s udies). Acco dingly, a cons an deb ax can be supe io o a
subsidy in he la e model unde smalle loan- o- alue a ios, which educe he e ec
ia he colla e al p emium. This ela es o Bianchi and Mendoza (2018) inding o
ela i ely small wel a e gains o cons an deb axes in a model whe e he loan- o-
alue a io alls in a c isis, and o Bianchi (2011), who epo s sizable wel a e e ec s
o cons an deb axes o a model wi hou a colla e al p emium.
The emaining wo non-s a e-con ingen policies impose axes/subsidies on lende s.
The hi d policy is an ex-an e sa ing ax/subsidy, which di ec ly al e s he p ice ela ion
o he equilib ium in e es a e. Agen s do no in e nalize he e ec s o hei con-
sump ion plans on he in e es a e, which canno be add essed by axing/subsidizing
bo owe s, gi en ha he in e es a e equals he in e se o he lende s’ discoun ac o .
By subsidizing sa ings, lende s demand a lowe in e es a e, which aises bo owe s’
cu en ela i e o u u e consump ion. The educ ion in bo owing cos s he e o e na -
ows he dis ance be ween he MRS o lende s and bo owe s; he la e engaging in
p ecau iona y sa ing unde po en ially binding bo owing cons ain s. Inc eased ou -
s anding deb , howe e , ends o educe consump ion when he colla e al cons ain
binds and hus o lowe he colla e al p ice. Hence, he e is a ade-o be ween he
e ec s on he p ices o deb and o colla e al. A policy make decides o subsidize
123

Taxing e sus subsidizing deb unde inancial ic ions 1387
sa ing ex-an e and o educe he cos s o bo owing o po en ially cons ained agen s
when dis ibu i e e ec s domina e colla e al e ec s. This is pa icula ly he case in
bo h models when he weal h dis ibu ion is su icien ly unequal. The ou h policy
is a sa ing ax/subsidy ha is cons an o e ime. In con as o he ex-an e sa ing
subsidy, i ends o s imula e bo owing as well as consump ion be o e and while he
colla e al cons ain is binding by educing he in e es a e and by aising he colla e al
p emium. I can he eby simul aneously add ess dis ibu i e and colla e al e ec s.
To un eil he ole o he colla e al p emium and o econs uc indings o he s ud-
ies on mac op uden ial egula ion ci ed abo e, we e e o an al e na i e speci ica ion
whe e he bo owing limi is assumed o depend on he agg ega e s ock o pledgeable
asse s. Fo his speci ica ion o he bo owing cons ain , which is no consis en wi h
he unde lying impe ec ion (i.e. limi ed commi men ), he p ice ha al e s he bo -
owing limi is no a ec ed by he colla e al p emium, such ha deb /sa ing subsidies
can nei he implemen i s bes no add ess ad e se colla e al e ec s. I dis ibu i e
e ec s a e u he dis ega ded, op imal ex-an e and cons an policies a e deb axes,
implying ha agen s o e bo ow.
While he analy ical esul s e eal he main p inciples, we u he p o ide nume ical
esul s o he less s ylized model wi h du ables o illus a i e pu poses. The ou
op imal policies a e (1) an ex-an e deb ax, (2) a cons an deb subsidy, (3) an ex-an e
sa ing subsidy, and (4) a cons an sa ing subsidy (see Table 1). Excep o he ex-an e
deb ax, all policies end o aise deb be o e he bo owing cons ain binds, and
he cons an policies induce he la ges inc eases in he colla e al p ice, e ealing he
ele ance o colla e al p emia. The ex-an e deb ax has he leas impac on bo owe s’
consump ion and leads o he smalles wel a e gains ela i e o laissez ai e, which a e
i ually negligible based on he dis ance o i s bes . Sa ing policies exe ela i ely
la ge edis ibu i e and social wel a e e ec s ia in e es a e educ ions.6An op imal
cons an sa ing subsidy, which leads o he la ges wel a e gains ela i e o laissez
ai e, can he eby educe wel a e losses by abou a hal compa ed o i s bes .
By con ining ou analysis o models employing a linea u ili y unc ion o lende s,
a decision ha acili a es he eplica ion o es ablished esul s on mac op uden ial
egula ion, we abs ac om dis ibu i e e ec s unde deb policies. I lende s’ u il-
i y we e ins ead a non-linea unc ion o consump ion like bo owe s’ u ili y, in e es
a es would depend on agen s’ endogenous MRSs. An ex-an e deb ax would lowe
bo owe s’ cu en ela i e o u u e consump ion, such ha lende s’ cu en consump-
ion would inc ease ela i e o u u e consump ion and he pe iod-1 in e es a e would
unambiguously all. The educ ion in he in e es a e would mi iga e (bu no in e )
he deb ax e ec on bo owe s’ cu en ela i e o u u e consump ion. The dis ibu-
i e e ec s would howe e demand an inc ease o bo owe s’ cu en consump ion,
which is dep essed in a laissez ai e equilib ium due o p ecau iona y sa ing. The
ecommenda ion ega ding an ex-an e deb policy is he e o e less clea -cu when
conside ing he ele ance o dis ibu i e e ec s unde non-linea lende s’ u ili y. In
con as , sa ing subsidies can add ess dis ibu i e e ec s unde non-linea lende s’
u ili y, since hey would educe lende s’ cu en consump ion as well as in e es a es.
6Op imal sa ing policies lead o a edis ibu ion o unds om lende s o bo owe s, despi e he absence
o a edis ibu i e mo i e o he social planne , owing o quasi-linea p e e ences.
123
1388 A. Schabe
Bo h e ec s would induce bo owe s o aise consump ion ela i e o lende s, na -
owing he dis ance be ween hei MRSs. This mechanism is in p inciple ele an
whene e bo owing cons ain s bind wi h a non-ze o p obabili y in a he e ogeneous
agen economy.
The emainde is s uc u ed as ollows. Sec ion2discusses he ela ed li e a u e.
Sec ion3de elops he model wi h du ables as colla e al unde unce ain y. Sec-
ion4examines op imal policies. Sec ion4.5 p esen s nume ical illus a ions. Sec ion 5
p esen s analy ical esul s o Da ila and Ko inek (2018) model wi h endogenous capi-
al o ma ion, whe e he bo owing cons ain binds wi h ce ain y. Sec ion 6concludes.
2 Rela ed li e a u e
This pape is ela ed o se e al s udies on co ec i e policies unde colla e al ex e nal-
i ies, like Jeanne and Ko inek (2010,2019), Bianchi (2011), Benigno e al. (2016),
Ko inek and Sand i (2016), Schmi -G ohe and U ibe (2017), Bianchi and Mendoza
(2018), o Ko inek (2018). They ocus on cons ained e icien alloca ions, as de ined
in S igli z (1982), and mac op uden ial policies, like deb axes o capi al con ols,
ha a e imposed when bo owing cons ain s a e no binding. In con as o ou anal-
ysis, hese s udies apply models whe e in e es a es a e exogenously de e mined and
whe e – excep o Bianchi and Mendoza (2018) – agen s ake bo owing limi s as gi en,
implying ha he e a e nei he dis ibu i e e ec s no colla e al p emia on pledgeable
asse s. Bianchi and Mendoza (2018) ocus on ime-consis en policies unde disc e-
ion, such ha commi men o ex-pos policies is no possible and i s bes canno be
implemen ed. They discuss how he colla e al p emium p incipally a ec s he p ice o
colla e al and op imal deb policy. In hei quan i a i e analysis, hey epo esul s o
mac op uden ial deb axes ha a e imposed when he colla e al cons ain does no
bind and he colla e al p emium equals ze o. They u he apply cons an deb axes
and ind ha hey lead o ela i ely small wel a e gains o wel a e losses, consis en
wi h ou esul s on cons an deb policies. Bianchi (2011) inds ha a cons an deb
ax can achie e sizable wel a e gains in a model whe e a colla e al p emium is non-
exis en . In addi ion o deb axes, Benigno e al. (2016) analyze policies in oduced in
o he ma ke s, and show ha an ex-pos ax on non- adables can aise he colla e al
p ice, such ha he bo owing cons ain does no bind. Bianchi (2016) and Jeanne
and Ko inek (2020) ind wel a e gains om ex-pos policies in o m o deb elie s o
liquidi y p o isions, which do no implemen i s bes . In addi ion o hese analyses,
we examine ime- and s a e-in a ian deb /sa ing subsidies, and show ha hey can be
supe io o deb axes.
Ou inding ha he s imula ion o bo owing can enhance social wel a e ela es o
he ollowing s udies: Benigno e al. (2013) examine he cons ained e icien alloca ion
o an economy whe e agen s ake in o accoun ha labo supply al e s he bo owing
limi , which compa es o ou analysis whe e agen s in e nalize ha bo owing limi s
depend on hei holdings o eligible asse s. They show ha one should a he ealloca e
esou ces be ween ( adable and non- adable goods) sec o s o aise bo owing limi s
han subsidize bo owing. In a ela ed model, A ce e al. (2023) show ha an ex-an e
deb ax is desi able e en when ex-pos labo ma ke policies a e applied and bo owing
123
Taxing e sus subsidizing deb unde inancial ic ions 1389
is enhanced in a cons ained e icien alloca ion. Ka agi i e al. (2017) apply a a ian
o Jeanne and Ko inek (2010) model whe e agen s in e nalize colla e al se ices o
eligible asse s. Like in ou models, an ex-pos deb subsidy can implemen i s bes
by aising he colla e al p ice ia he colla e al p emium such ha he bo owing limi
is no binding. Thei analysis nei he examines in e es a e e ec s no non-s a e-
con ingen policies, on which ou analysis ocusses. Schmi -G ohe and U ibe (2021)
es ablish he exis ence o mul iplici y in he model examined by Bianchi (2011), gi ing
ise o equilib ia wi h unde bo owing due o excessi e p ecau iona y sa ings. Fo a
model wi h bank in e media ion, Chi e al. (2022) show ha agen s bo ow less unde
laissez ai e compa ed o equilib ia wi h ex-pos expansions o bank ese es. O onello
e al. (2022) shows ha cons ained ine iciency depends on whe he bo owing limi s
depend on cu en o u u e colla e al p ices, and ha deb subsidies can be op imal in
he la e case.
In con as o ou analysis, none o he abo e ci ed s udies conside s dis ibu i e
e ec s. In a seminal pape , Lo enzoni (2008) shows ha dis ibu i e ex e nali ies
unde inancial ic ions cause agen s o o e in es and o o e bo ow in an un egula ed
economy. Da ila e al. (2012) show in a model wi h an endogenous weal h dis ibu ion
ha dis ibu i e e ec s can ei he lead o o e - o unde accumula ion o capi al. Lan e i
and Rampini (2023) de elop a model o endogenous o ma ion and ealloca ion o
capi al. They show ha dis ibu i e e ec s o pecunia y ex e nali ies wi h ega d o
he capi al p ice a e la ge han colla e al ex e nali ies, such ha a subsidy on new
in es men enhances e iciency. Bo h s udies do no analyze c edi ma ke policies.
Da ila and Ko inek (2018) apply a gene al amewo k wi h capi al o ma ion, o
which hey es ablish colla e al and dis ibu i e e ec s. They show ha pecunia y
ex e nali ies can ei he cause o e - o unde in es men , while hey emphasize ha
"colla e al ex e nali ies gene ally en ail o e bo owing" (p. 354). We show o hei
model ha his conclusion holds only i he analysis is es ic ed o ex-an e deb
policies.
The abo e ci ed s udies ocus on he analysis o cons ained e icien alloca ions,
which can ei he be de i ed om a p oblem o choosing ini ial alloca ions o om a
Ramsey p oblem when equilib ium p ice ela ions a e una ec ed by policy ins umen s
(see also Da ila and Ko inek 2018). In con as , he solu ions o ou policy p oblems
di e om his ype o cons ained e icien alloca ion when he p ice ela ions o
he colla e al p ice o o he in e es a e a e a ec ed by policy. Rela edly, Benigno
e al. (2023) e-examine policy ins umen s used in Benigno e al. (2016), applying
he Ramsey app oach. Complemen a y o ou analysis o di e en policy ins umen s,
hey show ha a se o ins umen s ha can implemen a cons ained e icien alloca ion
can also be used o implemen a supe io alloca ion whe e bo owing cons ain s ne e
bind. This possibili y elies on he use o axes/subsidies ou side he c edi ma ke ,
while we show ha i s bes is implemen able wi h ex-pos c edi ma ke policies.
3 A model wi h incomple e ma ke s and limi ed commi men
In his sec ion, we de elop a ini e ho izon model wi h du ables in ixed supply. Sec-
ion5p esen s Da ila and Ko inek (2018) model wi h capi al o ma ion, which is
123
1390 A. Schabe
sligh ly mo e s ylized (wi hou unce ain y and wi hou discoun ing).7The e exis
wo impe ec ions in bo h models: Only non s a e-con ingen deb is a ailable and
agen s a e no able o commi o deb epaymen . The la e leads o he key inancial
ic ion, i.e. a bo owing cons ain wi h he bo owe ’s asse se ing as colla e al.
3.1 De ails
The e a e wo mass-one g oups {b,l}wi h in ini ely many agen s, who li e o h ee
pe iods =1,2,3. In each pe iod , a household i∈{b,l}de i es u ili y om
consump ion o a non-du able good, ci, , and a du able good (o housing), di, ,as
gi en by he unc ion ui, =u(ci, ,di, ). Agen s maximize hei expec ed li e ime
u ili y, E3
=1β −1u(ci, ,di, ), whe e uis s ic ly inc easing and conca e, Edeno es
an expec a ions ope a o condi ional on in o ma ion in pe iod 1, and β∈(0,1)is a
discoun ac o . In each pe iod, agen s ecei e a po en ially andom endowmen yi, o
non-du able goods and hey exhibi an ini ial endowmen o du ables di,0. Agen s can
bo ow and lend only in e ms o non s a e-con ingen one-pe iod bonds bi, , which
a e issued a he p ice 1/ . The budge cons ain o an agen i o pe iod is gi en
by
ci, +q (di, −di, −1)+(1−τi, )bi, / =bi, −1+yi, +Ti, ,(1)
whe e τi, deno es dis o iona y axes/subsidies on deb /sa ing. Speci ically, we con-
side Pigou ian- ype iscal in e en ions, whe e budge a y e ec s o axes/subsidies
a e (ex-pos ) neu alized in a non-dis o iona y way:
Ti, =−τi, bi, / ,(2)
which is no in e nalized by agen s. The e is no unce ain y in he pe iods 1 and 3, whe e
o al endowmen wi h non-du ables is equally dis ibu ed: yb,1=yl,1=y/2. Agen s b
(l) s a wi h nega i e (posi i e) ini ial ne inancial weal h bb,0<0(bl,0>0) and will
be called bo owe s (lende s). In pe iod 2, endowmen s a e andomly de e mined and
can ei he ake he same alues as in pe iod 1 (s a e L) o can be unequally dis ibu ed
(s a e H). Speci ically, bo h s a es a e equally likely and endowmen o bo owe s in
s a e H(wi h Highe inequali y) is yb,2=y/(1+δH), whe e δH>1.
We assume ha agen s canno commi o epay deb and ha deb can be enego ia ed
a e issuance in he same pe iod. Bo owe s can make a ake-i -o -lea e-i o e o
educe he alue o deb . I a lende ejec s he o e , she/he can seize a ac ion γo
he bo owe ’s du able goods, which she/he can sell a he ma ke p ice q .O e sa e
he e o e accep ed when he epaymen alue o deb a leas equals he cu en alue
o seizable asse s. Wi hou loss o gene ali y, we assume ha de aul and enego ia ion
ne e happen in equilib ium. When deb is issued, he amoun o deb −bi, is he e o e
7While he analysis o hei model includes an addi ional (capi al in es men ) ax/subsidy, he esul s on
bo owing and sa ing axes/subsidies co espond o he esul o he model wi h du ables.
123
Taxing e sus subsidizing deb unde inancial ic ions 1397
4.1 An ex-an e Pigou ian ax on deb
We i s conside he case, whe e a ax/subsidy on deb migh be in oduced in pe iod
1, whe eas no policy ins umen is applied in pe iod 2. Ex-an e deb axes, which
co espond o capi al con ols in open economies, ha e al eady been examined in
se e al ela ed s udies (see Da ila and Ko inek 2018, o E en e al. 2021, o an
o e iew), es ablishing ha hey can implemen a cons ained e icien alloca ion as
de ined by S igli z (1982). Following Bianchi and Mendoza (2018), we will e e o
an ex-an e deb ax as mac op uden ial egula ion. Unde such a policy, bo owe s’
op imali y condi ion (5) changes o
1−τb,1c−1
b,1/ 1=βEc−1
b,2.(17)
In equilib ium, condi ion (17) and he op imal lending choice 1/ 1=βimply
1−τb,1=cb,1Ec−1
b,2. By axing deb in pe iod 1, τb,1>0, agen s can be induced
o bo ow less, which ends o aise cb,2 ela i e o cb,1and he du ables p ice q2
ia (12).12 Gi en ha bo owe s do no in e nalize he ad e se e ec o pe iod-1-
bo owing on he du ables/colla e al p ice and hus he bo owing limi in pe iod 2,
a policy make can enhance e iciency by add essing colla e al e ec s o pecunia y
ex e nali ies wi h an ex-an e deb ax. This mechanism is well-es ablished in he li e -
a u e on mac op uden ial egula ion and capi al con ols, and has led o he no ion o
“o e bo owing”.
P oposi ion 2 Suppose ha he policy make can apply a Pigou ian ax/subsidy on
deb be o e he bo owing cons ain migh be binding. Then, he op imal alloca ion
is cons ained e icien and associa ed wi h a ax on deb , sa is ying
τb,1=cb,1γd(1−βγ)Eμ b1
2·χc≥0,(18)
whe e μ b1
2≥0deno es he mul iplie on he bo owing cons ain o he policy
p oblem.
P oo See Appendix. 
The op imal ex-an e deb ax desc ibed in P oposi ion 2implemen s he “cons ained
e icien alloca ion”, which is chosen by a social plane espec ing budge and bo -
owing cons ain s and allowing ma ke s o du ables and non-du ables o clea in a
compe i i e way (see S igli z 1982, o Da ila e al. 2012). Conc e ely, a cons ained
e icien alloca ion is chosen by a social plane who de e mines bo owing and maxi-
mizes social wel a e Wsubjec o budge and bo owing cons ain s, while aking he
compe i i e equilib ium ela ions o in e es a es (10) and he du ables p ice (12)
unde laissez ai e in o accoun . The Ramsey op imal ex-an e deb ax leads o he
same ou come, since i lea es he p icing Eqs. (10) and (12) una ec ed.13 In con as ,
12 This posi i e e ec o highe ne wo h o bo owe s on he du ables/colla e al p ice co esponds o he
e ec in Da ila and Ko inek (2018) imposed by hei condi ion 1.
13 Benigno e al. (2016) and Da ila and Ko inek (2018) also show ha his app oach can be equi alen o
a Ramsey op imal policy whe e he policy make chooses axes ex-an e o on i s pe iod alloca ions.
123

1398 A. Schabe
we will examine polices in he subsequen sec ions ha al e (10) and (12), such ha
he p ices q2,1/ 1, and 1/ 2can be al e ed by policy in mo e di ec ways. Unde
al e na i e c edi ma ke polices, compe i i e equilib ium alloca ions can he eby be
implemen ed ha a e supe io o he cons ained e icien alloca ion unde he ex-an e
deb ax.
4.2 A cons an Pigou ian ax/subsidy on deb
In his model, s a e con ingency canno simply be induced by cyclicali y o policy
ins umen s. We he e o e conside ha he deb ax/subsidy τbcan nei he be made
con ingen on speci ic pe iods no on he s a e o he economy, i.e. on he dis ibu ion
o agen s’ endowmen , such ha he deb ax/subsidy is cons an and equally imposed
in he pe iods =1 and =2. In his case, he ax/subsidy has ex-an e and ex-pos
e ec s ela i e o he s a e o he economy whe e he bo owing cons ain migh be
binding. The bo owe s’ op imali y condi ions (5) and (7) hen change o
(1−τb)c−1
b,1/ 1=βEc−1
b,2,(19)
(1−τb)c−1
b,2/ 2=β+μb,2,(20)
whe e 1/ 1=1/ 2=β. Condi ion (19) and (20) imply ha he mul iplie on he
colla e al cons ain sa is ies μb,2=βc−1
b,2cb,1Ec−1
b,2−1, which di e s om
laissez ai e μb,2=(c−1
b,2−1)β. The colla e al p emium ξ2=γμ
b,2and condi ion
(9) hen lead o he ollowing p ice ela ion:
q2= d(d)(1+β)
c−1
b,21−βγcb,1Ec−1
b,2+βγ
,(21)
which simpli ies in s a e L o q2=cb,2 d(d)(1+β). The du ables p ice q2 ends o be
highe unde a la ge colla e al p emium ξ2(see 9), while a cons an deb ax τb>0
ends o educe he mul iplie μb,2and hus ξ2(see 20). Due o his e ec on q2and
he nega i e e ec o he deb ax on non-du ables consump ion cb,1 ela i e o cb,2
(see 19), he p ice q2is he e cha ac e ized by a posi i e ela ion o cb,1in equilib ium
(see 21).
A cons an deb ax ends o induce agen s o bo ow and o consume less in pe iod 1
ela i e o pe iod 2 (as in he case o he ex-an e ax), bu also ends o educe bo owing
and consump ion in pe iod 2 when he bo owing cons ain migh be binding (see 20).
Due o a lowe colla e al p emium, a deb ax can induce a educ ion in he du ables
p ice and in he bo owing limi in pe iod 2. I migh he e o e be p e e able o apply a
subsidy a he han a ax on deb . These wo e ec s o deb policies due o bo owing
cons ain s ha bind in he cu en pe iod and in he subsequen pe iod co espond o
hose discussed in Bianchi and Mendoza (2018) o an op imal s a e-con ingen policy
unde disc e ion. In con as o a policy make unde disc e ion, who can in luence
expec a ions abou u u e policy make s’ choices only ia endogenous s a e a iables,
123
Taxing e sus subsidizing deb unde inancial ic ions 1399
a policy make unde commi men ully accoun s o agen s condi ioning hei expec-
a ions on he policy choices. Fo he policy p oblem unde commi men , he ollowing
p oposi ion e eals when a deb subsidy (o ax) is p e e able14:
P oposi ion 3 Suppose ha he policy make can apply a cons an Pigou ian
ax/subsidy on deb in he pe iods 1 and 2. Then, he op imal alloca ion is associ-
a ed wi h a ax/subsidy a e on deb sa is ying
τb=cb,1γdEμ b
2χc−χξ·2cb,1
cb,2
+βγβ
c2
b,2,(22)
whe e μ b
2≥0deno es he mul iplie on he bo owing cons ain o he policy p ob-
lem, and he a e τbis nega i e i cb,1
cb,2(H)>1−β2γ
2βγ .
P oo See Appendix. 
As e ealed by P oposi ion 3, he colla e al e ec s gi en on he RHS o (22)imply
ha an op imal non-con ingen deb policy can ei he be a ax (τb>0) o a subsidy
(τb<0). The eason is ha a cons an deb subsidy ends o aise q2 ia he colla e al
p emium on du ables (see χξ), simila o an ex-pos deb subsidy (see Sec .3.2.2). A
he same ime, a cons an deb subsidy ends o educe cb,2and hus q2 ia inc eased
deb (see χc), which a e he in e se e ec s o an ex-an e deb ax. I he impac on
he colla e al p emium summa ized by he posi i e e m in he cu ly b acke s in (22)
domina es he la e e ec , a deb subsidy is op imal, τb≤0. The inequali y a he
end o he p oposi ion, e eals ha his holds i he a io o pe iod-1-consump ion
o pe iod-2-consump ion in s a e H,cb,1/cb,2(H), is su icien ly la ge. Recall ha
his a io is equal o one unde i s bes (see 13), while i exceeds one unde laissez
ai e when he colla e al cons ain binds (see 11). The inequali y is he e o e sa is ied
when he alloca ion unde he cons an ax/subsidy is (s ill) cha ac e ized by a binding
colla e al cons ain , inducing he a io cb,1/cb,2(H) o exceed one, and he h eshold
1−β2γ
2βγ is smalle o equal o one. The la e is in ac he case when he liquida ion alue
o colla e al γis su icien ly la ge, i.e. i γ≥1/β2+2β, which can in p inciple be
sa is ied by empi ically plausible loan- o- alue- a ios (e.g. γ=0.8 ). Fo smalle loan-
o- alue- a ios, a cons an deb ax can be supe io o a deb subsidy. Rela edly, Bianchi
and Mendoza (2018) ind ela i ely small wel a e gains o an op imized cons an deb
ax in an in ini e-ho izon model whe e he loan- o- alue a io alls in a c isis, which
ends o educe he bene icial e ec s o a deb subsidy ia he colla e al p emium.
Appa en ly, he e m in he cu ly b acke s in (22) is equal o ze o i he e we e
no colla e al p emium, like unde a bo owing cons ain ha does no depend on he
indi idual s ock o du ables (see 16). In his case, he RHS o (22) would be s ic ly
posi i e, such ha he op imal cons an policy imposed on bo owe s would be a deb
ax. This co esponds o he wel a e-enhancing cons an deb ax in Bianchi (2011),
whe e a colla e al p emium is non-exis en .
14 Bianchi and Mendoza (2018) do no analy ically o quan i a i ely iden i y he condi ions unde which
a deb subsidy is op imal.
123
1400 A. Schabe
4.3 An ex-an e Pigou ian ax/subsidy on sa ing
We now conside a ax/subsidy on sa ing as a closely ela ed policy ins umen , which
is howe e imposed on lende s. Gi en ha bo owe s and lende s s uc u ally di e
wi h ega d o p e e ences and cons ain s, he impac o a ax/subsidy on sa ing
will in gene al no be equi alen o he impac o a ax/subsidy on deb . Speci ically,
he analysis will e eal ha dis ibu i e e ec s o pecunia y ex e nali ies play an
impo an ole o he policy make ’s choice unde a sa ing policy, which di ec ly
al e s he in e es a e. No ably, he absence o a edis ibu i e mo i e o he social
planne unde quasi-linea p e e ences (see Sec .3.2) implies ha edis ibu i e e ec s
o in e es a e changes exclusi ely s em om he mi iga ion o pecunia y ex e nali ies.
In con as , he in e es a e was exogenous unde he linea u ili y unc ion o lende s
(see Sec s.4.1 and 4.2) as long as only bo owe s we e axed.
Unde an ex-an e ax/subsidy on sa ing, he in e es a e in pe iod 1 can di ec ly be
al e ed by policy, as shown by he lende s’ op imal sa ing decision
(1−τl,1)/ 1=β. (23)
Combining (23) wi h he bo owe s’ op imali y condi ion (5), gi es 1/(1−τl,1)=
cb,1Ec−1
b,2, implying ha bo owe s’s pe iod-1 non-du ables consump ion cb,1 ends o
dec ease ela i e o cb,2wi h a sa ing ax, τl,1<0. Gi en ha he in e es a e now
becomes endogenous ia he policy make ’s op imal choice, he ele an p ice ela ion
is gi en by
1=c−1
b,1/βEc−1
b,2,(24)
while he colla e al p ice sa is ies he laissez ai e p ice ela ion (12), like unde he
ex-an e deb ax. An ex-an e ax/subsidy on sa ing can indi ec ly al e he bo owing
limi ia he e ec o cb,2on he colla e al p ice simila o he ex-an e deb ax, while
i can addi ionally a ec he in e es a e in a di ec way ia (23). The social plane
can u ilize he la e e ec and lowe he in e es a e o add ess dis ibu i e e ec s
o pecunia y ex e nali ies. In ac , he dis ibu i e e ec s call o a subsidy on sa ing
and he colla e al e ec s o a ax on sa ing. The sign o he op imal ax/subsidy a e
he e o e depends on he ela i e magni udes o bo h e ec s.
P oposi ion 4 Suppose ha he policy make can apply a Pigou ian ax/subsidy on
sa ing be o e he bo owing cons ain migh be binding. Then, he op imal alloca ion
is associa ed wi h a ax/subsidy a e on sa ing sa is ying
τl,1=−bb,1 1E(μ l1
2)E∂φb
1
∂cb,1− 1E∂φb
1
∂cb,2
 
≥0
− 1βγd
(1−βγ)−1E(μ l1
2χc)
 
≥0
,(25)
123
Taxing e sus subsidizing deb unde inancial ic ions 1401
whe e ∂φb
1/∂cb,1>0,∂φb
1/∂cb,2<0, and μ l1
2≥0deno es he mul iplie on he
bo owing cons ain o he policy p oblem and φb
1=β(cb,1/cb,2) he s ochas ic
discoun ac o .
P oo See Appendix. 
The condi ion o he op imal ex-an e ax/subsidy a e (25) in P oposi ion 4 e eals
ha he sign o he ax/subsidy a e depends on wo opposing e ec s: The i s e m
(in cu ly b acke s) on he RHS is s ic ly posi i e and summa izes he dis ibu i e
e ec s induced by he bo owing cons ain ha is binding wi h a posi i e p obabili y
(see Assump ion 2).15 These e ec s, which would be inexis en wi hou pecunia y
ex e nali ies (see e ms in he squa e b acke s), call o a sa ing subsidy, τl,1>0,
inducing a lowe in e es a e. Due o he highe deb p ice 1/ 1, bo owe s can inc ease
hei consump ion o non-du ables in pe iod-1 ela i e o pe iod 2 compa ed o laissez
ai e (see 11). The second e m (in cu ly b acke s) on he RHS is also s ic ly posi i e
and summa izes he colla e al e ec s, which can be add essed by educing bo owing
ia a sa ing ax, τl,1<0, ha ends o educe he supply o deb (like an ex-an e
deb ax ends o educe he demand o deb , see P oposi ion 2). E iden ly, he policy
make applies a sa ing subsidy, τl,1>0, when colla e al e ec s a e domina ed by
dis ibu i e e ec s, which is mo e likely o highe le els o deb −bb,1(see 25); he
la e p ima ily depending on he exogenously gi en ini ial deb le el −bb,0.
4.4 A cons an Pigou ian ax/subsidy on sa ing
Now suppose ha he ax/subsidy on sa ing can nei he be made con ingen on pa ic-
ula pe iods no on he s a e o he economy, such ha he ax/subsidy a e is equally
imposed in he pe iods =1 and =2. This policy egime would e en be non-
equi alen o a cons an ax/subsidy on deb i all agen s we e ex-an e iden ical, because
o he asymme y o agen s’ p oblems in pe iod 2 induced by he bo owing cons ain .
The lende s’ op imali y condi ions a e hen gi en by
(1−τl)/ =β, whe e 1= 2= ,(26)
ins ead o (10), implying ha he cons an sa ing ax/subsidy al e s he in e es a e
in bo h pe iods, 1 and 2. These in e es a e e ec s o he cons an sa ing ax/subsidy
u he a ec he bo owing decisions in pe iod 1 and 2
c−1
b,1/(1−τl)=E[c−1
b,2],(27)
c−1
b,2β/(1−τl)=β+μb,2.(28)
The condi ions (27) and (28) indica e ha a cons an sa ing subsidy τl>0 ends o
aise bo owe s’ non-du able consump ion in pe iod 1 and 2. Simul aneously, i al e s
he alua ion o he bo owing cons ain , measu ed by he mul iplie on he bo owing
15 No ably, he mul iplie μ l1
2now depends on he di e ence be ween he ma ginal u ili ies o bo owe s
and lende s (see p oo o P oposi ion 4).
123
1402 A. Schabe
cons ain μb,2and he eby he colla e al p emium ξ2. Combining (27) and (28), gi es
μb,2=c−1
b,2cb,1βE[c−1
b,2]−β, which can be used o subs i u e ou he mul iplie μb,2
in (6). Then, he du ables p ice ela ion di e s om he laissez ai e e sion (12) and
sa is ies (21), like unde he cons an deb ax/subsidy. Wi h hese changes in he p ice
ela ions o du ables and deb , he policy make can use a cons an ax/subsidy on
sa ing o simul aneously add ess colla e al e ec s ia he du ables p ice q2as well as
dis ibu i e e ec s ia he in e es a e .
P oposi ion 5 Suppose ha he policy make can apply a cons an Pigou ian
ax/subsidy on sa ing in he pe iods 1 and 2. Then, he op imal alloca ion is asso-
cia ed wi h a ax/subsidy a e on sa ing sa is ying
τl=+, wi h (29)
=β−bb,1 βEμ l
2
φbE∂φb
∂cb,1− E∂φb
∂cb,2
−Ebb,2
μ l
2
φb∂φb
∂cb,1
− ∂φb
∂cb,2≥0
=βγdEμ l
2χξ·1+2 cb,1
cb,2γβc−2
b,2− χc,
and ∂φb/∂cb,1>0,∂φb/∂cb,2<0, and μ l
2≥0deno es he mul iplie on he
bo owing cons ain o he policy p oblem and φb=β(cb,1/cb,2) he s ochas ic
discoun ac o . The e m is posi i e i cb,1
cb,2(H)>1−βγ −1
2βγ .
P oo See Appendix. 
Acco ding o P oposi ion 5, dis ibu i e e ec s, which a e summa ized by he e m 
in (29), can be add essed by a cons an sa ing subsidy, which ela es o he indings
in P oposi ion 4. Compa ed o an ex-an e sa ing subsidy, a cons an sa ing subsidy
addi ionally educes he in e es a e in pe iod 2, whe e he bo owing cons ain migh
be binding. This addi ional e ec is cap u ed by he las e m in he squa e b acke s
o . In con as o he e ms e e ing o he dis ibu i e e ec , he sign o he e m
, which summa izes he colla e al e ec s, is ambiguous and depends on he e ec s
on he colla e al p emium summa ized in he cu ly b acke s in . Gi en ha a highe
willingness o bo ow inc eases he alua ion o colla e al, he colla e al e ec s also
call o a sa ing subsidy i he impac on he colla e al p emium is su icien ly la ge.
O he wise, he e m is nega i e and calls o a sa ing ax, which ends o inc ease
consump ion by educing deb (see χc). Fo >0, he inequali y cb,1
cb,2(H)>1−βγ −1
2βγ
has o hold, which di e s om he co esponding condi ion o he cons an deb
subsidy by −1 eplacing β(see P oposi ion 3). As in Sec . 4.2, his inequali y is
likely o be sa is ied when bo owing emains cons ained e en unde he op imal
policy (such ha cb,1exceeds cb,2(H)) and o su icien ly high loan- o- alue a ios
γ, which s eng hen he e ec ia he colla e al p emium.
I he bo owing cons ain we e howe e independen o indi idual holdings o
du ables (see 16), such ha he e would be no colla e al p emium e ec on q2, he
123

Taxing e sus subsidizing deb unde inancial ic ions 1403
Fig. 1 Ins umen s and p ices (benchma k alues γ=0.8andδH=1.1)
e m would be s ic ly nega i e. Ye , e en in his case he policy make would apply
a sa ing subsidy i he dis ibu i e e ec s domina e; he la e being mo e likely
unde highe ini ial deb le els −bb,0.
4.5 P ices, alloca ion and wel a e
We now aim a illus a ing he impac o co ec i e policies on p ices, he alloca ion and
social wel a e, and he possibili y o imp o e on he (cons ained e icien ) alloca ion
implemen ed by an ex-an e deb ax ia al e na i e non s a e-con ingen policies. We
in oduce a unc ional o m o (db, ): (db, )=κlog db, . We u he assign
alues o model pa ame e s ha induce he – admi edly s ylized – model o gene a e
meaning ul alues o a ge ed a iables. Speci ically, we no malize yand se i equal
o 2, and u he se d/y=1.5, bb,0/y=−0.25, κ=0.1, and β=0.9, leading
o a housing- o-GDP a io, deb - o-income a ios, in e es a es, and ax/subsidy a es
wi hin easonable anges. The benchma k alues o he inequali y measu e δHand
o he sha e o seizable colla e al γa e 1.1 and 0.8, espec i ely; he la e ela ing o
commonly applied loan- o- alue a ios. We hen examine he sensi i i y o he e ec s
by al e ing he igh ness o he bo owing cons ain γand income inequali y δH.
The solu ions o he equilib ium objec s unde he ou non-s a e-con ingen policies
summa ized in Table 1and unde laissez ai e a e p esen ed in he Figs.1,2,3.The
i s ow in all Figu es e e s o a a ia ion in γ, whe e an inc ease in γ educes he
igh ness o he bo owing cons ain and he eby he s eng h o he inancial ic ion,
which de-emphasizes he colla e al e ec . The second ow in all Figu es e e s o
a a ia ion in δH, whe e an inc ease in δHinc eases he inequali y o agen s’ non-
du ables endowmen in s a e Hin pe iod 2 and he eby he ele ance o he inancial
ic ion as well as o dis ibu i e e ec s.
123
1404 A. Schabe
Fig. 2 Deb and consump ion (benchma k alues γ=0.8andδH=1.1)
Fig. 3 Social wel a e (benchma k alues γ=0.8andδH=1.1)
The i s column o Fig.1shows he ax and subsidy a es unde all i e egimes. The
laissez ai e case (black do ed lines) exhibi s ze o ax/subsidy a es. The i s policy
egime (solid black lines wi h c osses) is he op imal ex-an e ax on deb τb,1>0
(see P oposi ion 2), which dec eases wi h γand inc eases wi h δH. The second policy
egime ( ed dashed lines wi h c osses) is he op imal cons an subsidy on deb τb<0
(see P oposi ion 3). The hi d (blue solid lines wi h ci cles) and he ou h egime (g een
dashed lines wi h ci cles) a e he op imal ex-an e and he op imal cons an sa ing
subsidy, τl,1>0 and τl>0, as cha ac e ized in P oposi ions 4and 5. The second
123
Taxing e sus subsidizing deb unde inancial ic ions 1405
column shows he du ables (colla e al) p ice in pe iod 2, which is sligh ly inc eased
compa ed o laissez ai e unde he ex-an e deb ax. The cons an deb subsidy, which
ends o aise bo owing in bo h pe iods 1 and 2 (see Fig.2), also leads o highe
du ables p ices due o i s impac on he colla e al alue. In con as , he ex-an e sa ing
subsidy, which aises deb in pe iod 1 and educes non-du ables consump ion cb,2
in pe iod 2 (see Fig.2), leads o lowe du ables p ices q2. Simul aneously, i educes
he in e es a e in pe iod 1 below i s laissez ai e alue (see hi d column), such ha
bo owing unds equi es issuance o less deb bb,1. The cons an sa ing subsidy leads
o he mos p onounced inc ease in he du ables p ice q2. I u he leads o a educ ion
in he in e es a e 1in pe iod 1 ha is la ge han unde he ex-an e sa ing subsidy and
i equally educes he in e es a e 2in pe iod 2. Figu e 2 u he shows ha all h ee
subsidies aise deb −bb,1and lead o highe le els o non-du ables consump ion in
pe iod 1 compa ed o laissez ai e, which in con as dec eases unde he ex-an e deb
ax. The cons an deb subsidy and he ex-an e sa ing subsidy educe consump ion
cb,2due o a highe deb bu den in pe iod 2. The opposi e esul o cb,2is induced
by he ex-an e deb ax and by he cons an sa ing subsidy, which lowe s bo owing
cos s in bo h pe iods 1 and 2.
Figu e3p esen s he wel a e e ec s o he policy egimes. The wel a e measu e is
based on W(see 4) and exp essed in e ms o equi alen s o bo owe s’ non-du ables
consump ion in pe iod 1. The i s column shows wel a e e ec s o he ou policy
egimes ela i e o he laissez ai e case. E iden ly, he deb policies (ex-an e deb ax
and cons an deb subsidy) lead o much smalle wel a e gains han he sa ing subsi-
dies. This esul is simply due o he ac ha he o me policies can – by cons uc ion
– no add ess dis ibu i e e ec s by changes in he in e es a e.16 In con as , sa ing
policies induce subs an ial in e es a e educ ions compa ed o laissez ai e, which
leads o a edis ibu ion o esou ces in a o o bo owe s (see also Fig.4). The second
column o Fig.3zooms in in o he wel a e e ec s o he deb policies, e ealing ha
he ex-an e deb ax leads o he smalles wel a e gains unde he benchma k pa ame-
e alues. I u he shows ha he ex-an e deb ax can p incipally be supe io o he
cons an deb subsidy o igh e bo owing cons ain s, i.e. o lowe loan- o- alue
a ios γ(see also P oposi ion 3), which educe he posi i e impac o he cons an deb
subsidy on he colla e al p ice ia he colla e al p emium, ξ2=γμ
b,2.
A la ge income inequali y δH educes bo owe s’ pe iod-2 consump ion cb,2unde
laissez ai e (see Fig.2), lowe ing he colla e al p ice acco ding o (12). The wel a e
gains o ex-an e deb axes, which aise cb,2 ia a deb educ ion, he e o e inc ease
mono onically wi h δH(see Fig.3). Co espondingly, he wel a e loss o an ex-an e
deb subsidy, which lowe s cb,2, would inc ease wi h δH. The o al wel a e e ec o
a cons an deb subsidy does howe e u he depend on he e ec on he colla e al
p emium, which inc eases wi h he mul iplie on he colla e al cons ain , gi en by
μb,2=c−1
b,2cb,1βEc−1
b,2−β. When income ge s mo e unequal, he ad e se e ec
o he deb subsidy on cb,2(see Fig.2) domina es he posi i e e ec on he colla e al
p emium, such ha he o al wel a e gain alls (see Fig.3). The cons an deb subsidy
can he e o e be ou pe o med by he ex-an e deb ax o high δH alues. The las
16 This would in p inciple be possible unde al e na i e speci ica ions o lende s’ u ili y, o example,
loga i hmic u ili y, which is neglec ed he e o keep he exposi ion anspa en using pola cases.
123
1406 A. Schabe
Fig. 4 Wel a e o bo owe s and lende s (benchma k alues γ=0.8andδH=1.1)
column o Fig.3p esen s wel a e losses compa ed o i s bes . The alues o laissez
a e and he ex-an e deb ax a e i ually iden ical, indica ing ha he o al wel a e
gains o an ex-an e deb ax a e negligible ela i e o i s bes . In con as , he cons an
sa ing subsidy can subs an ially educe he wel a e loss in a compe i i e equilib ium
compa ed o i s bes . Fo he benchma k alues, i educes he wel a e loss by abou
a hal . Finally, Fig. 4con i ms he exis ence o edis ibu i e e ec s unde sa ing
policies. Bo owe s gain and lende s lose compa ed o laissez ai e due o lowe in e es
a es. The i s and he second column o Fig.4show ha hese edis ibu i e e ec s
mono onically inc ease wi h he igh ness o he colla e al cons ain (lowe γ) and he
inequali y o income (highe δH). Likewise, hese wel a e e ec s inc ease wi h ini ial
deb −bb,0(no shown). In con as , deb policies solely a ec bo owe s’ wel a e ia
in e empo al subs i u ion, which is e ealed in he las column o Fig.4, showing he
same e ec s as o agg ega e wel a e (see second column o Fig.3).
5 A model wi h capi al o ma ion
To assess he obus ness o ou indings and o acili a e compa isons, we u he
apply a model wi h endogenous capi al o ma ion, like Bianchi and Mendoza (2018)
o Da ila and Ko inek (2018). Conc e ely, we use Da ila and Ko inek (2018) model
applied o colla e al ex e nali ies and eplica e hei esul s on an ex-an e deb policy
ha implemen s he cons ained e icien alloca ion. In addi ion, we examine he o he
h ee policy egimes gi en in Table 1, like in he analyses o he p e ious model (see
Sec .4).
The e is no unce ain y and he e a e no du able consump ion goods in his economy.
Agen s’ li e ime u ili y sa is ies ul=cl,1+cl,2+cl,3and ub=log cb,1+log cb,2+cb,3,
which acco ds o Assump ion 1wi hou du ables ( =0) and implies no discoun ing
123
Taxing e sus subsidizing deb unde inancial ic ions 1413
λ l1
b,1= 1βEλ l1
b,2,(49)
βλ l1
b,2=βc−1
b,2−1−λ l1
b,1bb,1
∂φb
1
∂cb,2
+βμ l1
2γ∂q2
∂cb,2
d,(50)
μ l1
2=βλ l1
b,2≥0.(51)
Applying expec a ions and subs i u ing ou he mul iplie s λ l1
b,1and λ l1
b,2in (48)-(50),
gi es
1
1+bb,1E∂φb
1/∂cb,1
1+ 1bb,1E∂φb
1/∂cb,2=c−1
b,1−1
β(Ec−1
b,2−1)+βγdEμ l1
2∂q2/∂cb,2.
Combining he la e wi h (23) and (24) and using ∂q2(cb,2)
∂cb,2=(1−βγ)χc(see 12),
leads o he ollowing condi ion o he ex-an e ax/subsidy a e on sa ing
τl,1=−bb,1 1Eμ l1
2E∂φb
1/∂cb,1− 1E∂φb
1/∂cb,2
− 1βγdEμ l1
2(1−βγ)χc,
whe e ∂φb
1/∂cb,1=β/cb,2>0 and ∂φb
1/∂cb,2=−β(cb,1c−2
b,2)<0. Combin-
ing (48), (50), and (51), shows ha μ l1
2sa is ies Eμ l1
2= −1
1(c−1
b,1−1)/(1+
bb,1E[∂φb
1/∂cb,1])≥0. 
P oo o P oposi ion 5Fo he policy make ’s p imal p oblem unde commi men in
Lag angian o m, we de ine φb
1(cb,1,cb,2)=βcb,1/cb,2and φd
2(cb,1,cb,2)=
d(d)(1+β)
c−1
b,2−(cb,1c−2
b,2−1)βγ , and use he goods ma ke clea ing condi ions o ew i e he wel a e
unc ion, o con enience:
L=E{log cb,1+ (d)+(y−cb,1)+βlog cb,2+ (d)+y−cb,2
+β2[y+ (d)]+λ l
b,1bb,0+yb,1−cb,1−bb,1φb
1(cb,1,cb,2)
+βλ l
b,2bb,1+yb,2−cb,2−bb,2φb
1(cb,1,cb,2)
+βμ l
2[γφd
2(cb,1,cb,2)d+bb,2]},
leading o he i s o de condi ions
λ l
b,11+bb,1E∂φb
1/∂cb,1 (52)
=c−1
b,1−1−βEλ l
b,2bb,2∂φb
1/∂cb,1+βEμ l
2γd∂φd
2/∂cb,1,
βλ l
b,21+bb,2∂φb
1/∂cb,2 (53)
=βc−1
b,2−1−λ l
b,1bb,1∂φb
1/∂cb,2+βμ l
2γd∂φd
2/∂cb,2,
123

1414 A. Schabe
λ l
b,1= βEλ l
b,2,(54)
μ l
2=φbλ l
b,2≥0,(55)
whe e we used Eφb
1(cb,1,cb,2)=1/ . Taking expec a ions and subs i u ing ou he
mul iplie s λ l
b,1and λ l
b,2in (52)–(54), leads o
Eμ l
2/φb
11+bb,1E∂φb
1/∂cb,1−Eμ l
2/φb
11+ bb,1E∂φb
1/∂cb,2
=1
βc−1
b,1−1−Ec−1
b,2−1−1
Eμ l
2/φb
1bb,2∂φb
1/∂cb,1
+1
γdEμ l
2∂φd
2/∂cb,1+Eμ l
2/φb
1bb,2∂φb
1/∂cb,2
−γdEμ l
2∂φd
2/∂cb,2,
and by applying (26), o he ollowing condi ion o he cons an ax/subsidy a e
τl=−bb,1 βEμ l
2/φb
1E∂φb
1/∂cb,1− E∂φb
1/∂cb,2
+βE−bb,2μ l
2/φb
1(∂φb/∂cb,1)− (∂φb
1/∂cb,2)+, (56)
whe e ∂φb
1/∂cb,1>0, ∂φb
1/∂cb,2<0, and =βγdE[μ l
2∂φd
2/∂cb,1− ∂φd
2/
∂cb,2].The e mon he RHS o (56) can by using ∂φd
2
∂cb,1=χξγβ
c2
b,2
and ∂φd
2
∂cb,2=
χc−χξ2γβcb,1
c3
b,2
be ew i en as
=βγdEμ l
2χξ
γβ
c2
b,21+2 cb,1
cb,2− χc,(57)
Fu he applying χξ=c2
b,2χc(see 9) o ew i e (57)as=βγd E[μ l
2(χc{γβ( −1+
2cb,1c−1
b,2)−1})], shows ha ≥0i cb,1
cb,2(H)>1−βγ −1
2βγ .
P oo o P oposi ion 6Conside he economy wi h capi al o ma ion. Unde a Pigou-
ian deb subsidy in pe iod 2, he capi al p ice sa is ies qk=A3[c−1
b,2(1−φ)+
φ+τb,2·φc−1
b,2]−1. Thus, he colla e al cons ain is slack unde he i s bes alloca-
ion, −b b
b,2≤φqkk b, i he subsidy a e sa is ies τb,2≤A3kb,2[−b b
b,2]−1−φ−1,
whe e we used c b
b,1=c b
b,2=1 and k b =(A2+A3)/α and b b
b,2is gi en by
b b
b,2=yb,2+yb,1−2−(A2+A3)2/(α2)+A2(A2+A3)/α.
P oo o P oposi ion 7Conside he economy wi h capi al o ma ion. In equilib ium,
whe e capi al is en i ely held by bo owe s, he budge cons ain s can be w i en
as cb,1+αk2/2+bb,1/ 1=yb,1,cb,2+bb,2/ 2=yb,2+bb,1+A2k,cb,3=
yb,3+bb,2+A3k,cl,1+bl,1/ 1=yl,1,cl,2+bl,2/ 2=yl,2+bl,1, and cl,3=yl,3+bl,2.
123
Taxing e sus subsidizing deb unde inancial ic ions 1415
The social wel a e unc ion (4) can hus o β=1/ 1=1/ 2=1 be ew i en as
W=log cb,1+yl,1+log cb,2+yl,2+yb,3+bb,2+A3k+yl,3.
To es ablish he claims made in he i s pa o he p oposi ion, conside ha he
policy make in oduces an in es men ax/subsidy τk,1and an ex-an e deb ax/subsidy
τb,1, which a e ully compensa ed (ex-pos ) by ype-speci ic lump-sum ans e s (like
2). The bo owe s’ op imali y condi ions hen sa is y
(1−τb,1)=cb,1/cb,2,(58)
(1−τk,1)αk1/cb,1=1/cb,2A2+qk.(59)
The p imal policy p oblem o he policy make is iden ical o he p oblem o a
social plane who de e mines pe iod-1-bo owing as well as he capi al in es men
decision and maximizes social wel a e Wsubjec o budge and bo owing con-
s ain s aking he equilib ium p ice ela ion (31) unde laissez ai e in o accoun ,
leading o a cons ained e icien alloca ion. The p oblem can be summa ized as
max Ww. . . cb,1,cb,2,bb,1,bb,2,and ksubjec o cb,1+αk2/2+bb,1=yb,1,
cb,2+bb,2=yb,2+bb,1+A2k, and bb,2+φqk(cb,2)k≥0, whe e qk(cb,2)sa -
is ies (31) and hus ∂qk/∂cb,2>0. The Lag angian can be w i en as
L=log cb,1+yl,1+log cb,2+yl,2+yb,3+bb,2+A3k+yl,3
+λ 1
1yb,1−cb,1−αk2/2−bb,1+λ 1
2yb,2+bb,1+A2k−cb,2−bb,2
+μ b1
2bb,2+φqk(cb,2)k,
leading o he i s o de condi ions o cb,1,cb,2,bb,1,bb,2,and k
λ 1
1=1/cb,1,λ 1
2=(1/cb,2)+μ b1
2φk∂qk/∂cb,2,λ
1
1=λ 1
2,(60)
μ b1
2=λ 1
2−1≥0,(61)
λ 1
1αk=A3+λ 1
2A2+μ b1
2φqk(cb,2). (62)
Subs i u ing ou he mul iplie s λ 1
1and λ 1
2using he h ee condi ions in (60), gi es
1/cb,1=1/cb,2+μ b1
2φk∂qk/∂cb,2.Using(58) o subs i u e ou 1/cb,2in he
la e , leads o he ollowing condi ion o he ex-an e deb ax/subsidy a e τb,1:
τb,1=μ b1
2cb,1φk∂qk/∂cb,2≥0,
whe e μ b1
2=(c−1
b,1−1)≥0. Fu he subs i u ing ou he mul iplie s wi h λ 1
1=
λ 1
2=1/cb,1and μ b1
2=1/cb,1−1in(62), gi es αk1/cb,1=A3+1/cb,1A2+
1/cb,1−1φqk. Rew i ing i wi h he capi al ading decision qk(1/cb,2)=A3+
κb,2φqkas (1+τk,1)αk1/cb,1=1/cb,2A2+A3+1/cb,2−1φqkand
combining wi h (59), implies ha he in es men ax/subsidy a e sa is ies
τk,1=−
1/cb,1−1/cb,2A2+φqk
A3+1/cb,1A2+μ b1
2φqk.
123
1416 A. Schabe
Since 1/cb,1=1/cb,2+μ b1
2φk∂qk/∂cb,2implies 1/cb,1≥1/cb,2, he policy
make subsidizes capi al τk,1≤0i A2+φqk≥0. This es ablishes he claims made
in he i s pa o he p oposi ion.
Fo he second pa o he p oposi ion, we conside a cons an deb ax/subsidy
and an in es men ax/subsidy, which a e ully compensa ed (ex-pos ) by lump-sum
ans e s. Agen s’ bo owing and in es men decisions hen sa is y
(1−τb)/cb,1=1/cb,2,(63)
(1−τb)/cb,2=1+κb,2,(64)
and (59). Subs i u ing ou κb,2in he capi al ading condi ion qk(1/cb,2)=A3+
κb,2φqkwi h (64) and hen he ax/subsidy a e τbwi h (63), gi es he p ice ela ion
qk=A3cb,2
1−cb,1/cb,2φ+cb,2φ, (65)
implying ha qk ela es o cb,1and cb,2by ∂qk/∂cb,1=φA3c2
1φcb,1−cb,2−φc2
1−2
>0 and ∂qk/∂cb,2=1−2φcb,1/cb,2∂qk/∂cb,1. The Lag angian o he policy
make ’s p oblem can be w i en as
L=log cb,1+yl,1+log cb,2+yl,2+yb,3+bb,2+A3k+yl,3
+λ b
1yb,1−cb,1−αk2/2−bb,1
+λ b
2yb,2+bb,1+A2k−cb,2−bb,2+μ b
2[bb,2+φqk(cb,1,cb,2)k],
whe e qk(cb,1,cb,2)sa is ies (65). The i s o de condi ions o cb,1,cb,2,bb,1,bb,2,
and ka e
λ b
1=1/cb,1+μ b
2φk∂qk/∂cb,1,λ
b
2=1/cb,2+μ b
2φk∂qk/∂cb,2,λ
b
1=λ b
2,
(66)
μ b
2=λ b
2−1≥0,(67)
λ b
1αk=A3+λ b
2A2+μ b
2φqk(cb,1,cb,2). (68)
Subs i u ing ou he mul iplie s λ b
1and λ b
2using he i s h ee condi ions in (66),
1/cb,1+μ b
2φk∂qk/∂cb,1=1/cb,2+μ b
2φk∂qk/∂cb,2, and subs i u ing ou
1/cb,2wi h (63), gi es he ollowing condi ion o he deb ax/subsidy a e τb:
τb=μ b
2cb,1φk[(∂qk/∂cb,2)−(∂qk/∂cb,1)]. Using ha he capi al p ice qksa is ies
∂qk/∂cb,2=1−2φcb,1/cb,2∂qk/∂cb,1and ∂qk/∂cb,1>0 (see 65), he la e can
be ew i en as
τb=−μ b
2cb,1φk2φcb,1/cb,2∂qk/∂cb,1≤0.
Fo he hi d pa o he p oposi ion, we conside an ex-an e ax/subsidy on sa ing
and an in es men ax/subsidy, which a e ully compensa ed (ex-pos ) by lump-sum
123
Taxing e sus subsidizing deb unde inancial ic ions 1417
ans e s (see 2). Agen s’ sa ing and in es men decisions hen sa is y
(1−τl,1)/ 1=1,(69)
and (59). Gi en ha (69) endogenizes he in e es a e o he policy make , 1/ 1=
cb,1/cb,2is a ele an es ic ion o he policy p oblem. Using he esou ce cons ain s
o subs i u e ou cl, , he Lag angian o he policy make ’s p oblem can be w i en as
L=log cb,1+(y1−cb,1)+log cb,2+y2−cb,2+[y3+A3k]
+λ l1
1yb,1−cb,1−αk2/2−bb,1(1/ 1)
+λ l1
2yb,2+bb,1+A2k−cb,2−bb,2+μ l1
2bb,2+φqk(cb,2)k,
whe eweusedy =yb, +yl, , and qkand 1/ 1sa is y (31) and 1/ 1=cb,1/cb,2,
espec i ely. The i s o de condi ions o cb,1,cb,2,bb,1,bb,2,and ka e gi en by
λ l1
11+bb,1∂(1/ 1)/∂cb,1=1/cb,1−1, (70)
λ l1
1bb,1∂(1/ 1)/∂cb,2+λ l1
2=(1/cb,2)−1+μ l1
2φk∂qk/∂cb,2,(71)
λ l1
1(1/ 1)=λ l1
2≥0, (72)
μ l1
2=λ l1
2,(73)
and (62). Subs i u ing ou he mul iplie s λ l1
1and λ l1
2in (70)–(72), leads o
1
(1+bb,1∂(1/ 1)/∂cb,1)
1+ 1bb,1∂(1/ 1)/∂cb,2=1/cb,1−1
(1/cb,2)−1+μ l1
2φk∂qk/∂cb,2
.
Fu he using (69)aswellas1/ 1=cb,1/cb,2and ea anging e ms, gi es
τl,1=−bb,1 1μ l1
2(∂ (1/ 1)/∂cb,1)− 1∂(1/ 1)/∂cb,2
− 1μ l1
2φk∂qk/∂cb,2,(74)
whe e ∂(1/ 1)/∂cb,1>0 and ∂(1/ 1)/∂cb,2<0.
Fo he ou h pa o he p oposi ion, we conside a cons an ax/subsidy on sa ing
and an in es men ax/subsidy, which a e ully compensa ed (ex-pos ) by lump-sum
ans e s (see 2). Agen s’ sa ing and in es men decisions hen sa is y
(1−τl)/ =1,(75)
whe e = 1= 2, and (59). Subs i u ing ou he in e es a es in agen s’ bo owing
decisions wi h (75), c−1
b,1/(1−τl)=1/cb,2and c−1
b,2/(1−τl)=1+κb,2, and combining
he la e o cb,1/cb,2=cb,21+κb,2, implies ha qksa is ies he p ice ela ion (65).
P oceeding as abo e, he Lag angian o he policy make ’s p oblem can be w i en as
123
1418 A. Schabe
L=log cb,1+(y1−cb,1)+log cb,2+y2−cb,2+[y3+A3k]
+λ l
1yb,1−cb,1−αk2/2−bb,1(1/ )
+λ l
2yb,2+bb,1+A2k−cb,2−bb,2(1/ )+μ l
2bb,2+φqkcb,1,cb,2k,
whe e qkand 1/ sa is y (65) and 1/ =cb,1/cb,2, espec i ely, leading o he ol-
lowing i s o de condi ions o cb,1,cb,2,bb,1,bb,2,and k
λ l
1(1+bb,1∂(1/ )/∂cb,1)+λ l
2bb,2∂(1/ )/∂cb,1
=1/cb,1−1+μ l
2φk∂qk/∂cb,1, (76)
λ l
1bb,1∂(1/ 1)/∂cb,2+λ l
21+bb,2∂(1/ )/∂cb,2
=(1/cb,2)−1+μ l
2φk∂qk/∂cb,2,(77)
λ l
1(1/ )=λ l
2, (78)
μ l
2=λ l
2(1/ )≥0,(79)
and (68). Subs i u ing ou λ l
1and λ l
2in (76)-(78) and aking di e ences, leads o
μ l
2bb,1∂(1/ )/∂cb,1+μ l
2bb,2∂(1/ )/∂cb,1
− μ l
2bb,1∂(1/ )/∂cb,2+ μ l
2bb,2∂(1/ )/∂cb,2
= −11/cb,1−1−(1/cb,2)−1+ −1μ l
2φk∂qk/∂cb,1
−μ l
2φk∂qk/∂cb,2,(80)
whe e ∂(1/ )/∂cb,1>0 and ∂(1/ )/∂cb,2<0. Combining (80) wi h c−1
b,1/ =
1/cb,2and (1−τl)= , leads o he ollowing condi ion o he ax/subsidy a e
τl= μ l
2bb,1+μ l
2bb,2∂(1/ )/∂cb,2−(μ l
2bb,1+1
μ l
2bb,2)∂(1/ )/∂cb,1
+1
μ l
2φk∂qk/∂cb,1[2φ+1− ]/ ,(81)
whe eweused∂qk/∂cb,2=1−2φcb,1/cb,2∂qk/∂cb,1 o de i e he las e m in
(81). 
Acknowledgemen s The au ho is g a e ul o Felix Bie b aue , Emanuel Hansen, and Joos Roe ge o
help ul commen s and sugges ions, as well as o An on Ko inek o insigh ul commen s on se e al de ails
o he analysis.
Funding Open Access unding enabled and o ganized by P ojek DEAL. Financial suppo was ecei ed
om he Deu sche Fo schungsgemeinscha (DFG, Ge man Resea ch Founda ion) unde Ge many’s Excel-
lence S a egy – EXC 2126/1 – 390838866.
123

Taxing e sus subsidizing deb unde inancial ic ions 1419
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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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