Ve ical ex e nali ies wi h lump-sum axes: how much
di¤e ence does unemploymen make?
Diego Ma inez
Depa men o Economics, Uni e si y Pablo de Ola ide
Tomas Sjög en
Depa men o Economics, Umeå Uni e si y
This e sion: Augus 7, 2012
Abs ac
This pape analyses how he exis ence o unemploymen a¤ec s he con en ional ap-
p oach o e ical ex e nali ies. We discuss he op imali y ule o he p o ision o public
inpu s bo h in an uni a y and a ede al coun y. Ou …ndings show ha decen alizing
he spending esponsabili y on public inpu s can b ing i s op imali y ule close o he
p oduc ion e¢ ciency condi ion. Mo eo e , we desc ibe he inabili y o he ede al go -
e nmen , beha ing as S ackelbe g leade , o eplica e he uni a y ou come, unless o ha e
new policy ins umen s a go e nmen ’s disposal.
Keywo ds: Public inpu s, unemploymen , e ical ex e nali ies.
JEL Classi…ca ion: J2, H4, H7
The au ho s would like o hank he Spanish Minis y o Science (P ojec s ECO2010-15553 and ECO2010-
21706), Jun a de Andalucia (P ojec s SEJ-02479 and SEJ-6882), he Bank o Sweden Te cen ena y Founda ion
(S i elsen Riksbankens Jubileums ond), he Swedish Council o Wo king Li e and Social Resea ch (FAS) and
he Na ional Tax Boa d (Ska e e ke ) o esea ch g an s. The iews exp essed by he au ho s a e no neces-
sa ily hose o he ins i u ions hey a e a¢ lia ed wi h. Co esponding au ho : Diego Ma inez, Depa men o
Economics, Uni e si y Pablo Ola ide, C a. U e a Km. 1, 41013 Se ille. Spain. Email: dma [email protected]
1
1 In oduc ion
The usual app oach o e ical ex e nali ies es ablishes ha sha ing axes be ween di¤e en
le els o go e nmen has an impac on e¢ ciency. F om he semina con ibu ion by Keen
(1998), a numbe a pape s has deal wi h his issue, o¤e ing a ious solu ions o in e nalize
his p oblem as well (see, o ins ance, Boadway and T emblay, 2006). A common issue in
all hese con ibu ions is assuming dis o iona y axa ion. In ac , i is clea ha e ical ax
ex e nali ies only appea as households’decisions a e in‡uenced by dis o ing axes; o he wise,
he ma ginal cos o public unds is no a¤ec ed by lump-sum axes decided by one le el o
go e nmen and, consequen ly, he impac o …scal policies ac oss di¤e en ie s o go e nmen
does no ake place.
Ano he common ea u e in his li e a u e is ha he labo ma ke is compe i i e, wi h he
labo o ce ma ching exac ly he demand o labo . Pape s such as Dahlby and Wilson (2003)
and Ko sogiannis and Ma inez (2008) gi e a cen al ole o he labo supply and demand o
labo in de e mining equilib ia bu always wi h labo ma ke clea ing. In such a wo ld, he e
is no scope o one o he con en ional …scal policies aimed a …gh ing agains unemploymen ,
namely he p o ision o public inpu s. In ac , o he bes o ou knowledge, no pape so a has
deal wi h e ical expendi u e ex e nali ies (caused by he p o ision o p oduc i i y-enhancing
public expendi u es in a ede al con ex ) in he p esence o unemploymen . This has no been
he case when ho izon al ex e nali ies a e in ol ed; Ogawa e al (2006) s udy he implica ions
o labo ma ke impe ec ions on capi al ax compe i ion a he same le el o go e nmen s.
This pape p ecisely combines e ical ex e nali ies and labo ma ke impe ec ions in a
single model. Indeed, we build a heo e ical amewo k in which he ede al go e nmen is in
cha ge o unemploymen bene… s and he s a es p o ide a public inpu wi h posi i e e¤ec s on
demand o labo . Taxes a e assumed o be lump-sum because we a e in e es ing in ocussing
on he e¢ ciency implica ions de i ed om he expendi u e side o go e nmen decisions a he
han on e ical ax ex e nali ies. Anyway, we will show ha igno ing dis o iona y axa ion
as a policy a iable may play a c ucial ole o co ec ing he e ical ex e nali y.
The ollowing con ibu ions can be summa ized om ou esul s. Fi s ly, we p o e ha ,
in spi e o using exclusi ely lump-sum axes o …nance go e nmen s (and hus no space o
ax ex e nali ies), a e ical expendi u e ex e nali y a ises when unemploymen exis s. This
con… ms a p e ious esul ound in he li e a u e (Dahlby and Wilson, 2003; Ma inez, 2008),
namely, ha bo h e ical ( ax and expendi u e) ex e nali ies a e independen o each o he .
The p o ision o public inpu s c ea es a posi i e e ical impac on ede al e enues as long
2
as his ype o public spending inc eases he demand o labo and, he e o e, i educes he
esou ces needed a ede al le el o paying unemploymen bene… s. And his occu s wi hou
he co-occupancy o elas ic ax bases.
Mo eo e , we also see how he op imali y ule o he p o ision o public inpu s a s a e
le el is close o he p oduc ion e¢ ciency condi ion han he op imal condi ion in a uni a y
coun y wi h a non-clea ing labo ma ke . In a sense, one could say ha mo e ede alism does
no necessa ily leads o mo e ine¢ ciency. Pa icula ly, in he p esence o a dis o ion (in he
labo ma ke , esul ing in unemploymen ), i could be posi i e o e¢ ciency o b ing in a new
dis o ion ( ha coming om he e ical expendi u e ex e nali y).
Secondly, we ha e s udied whe he he ede al go e nmen is able o eplica e he equilib ium
o an uni a y coun y. As usual, we ha e assumed ha he uppe le el o go e nmen knows he
s a es’ eac ion unc ions and, beha ing as S ackelbe g leade , ies o achie e he cen alized
ou come. Ou esul de ia es om p e ious pape s as long as we conclude ha he policy
a iables a ailable o he ede al go e nmen a e no e¤ec i e ins umen s o ge he uni a y
equilib ium. We guess he e ha he ac o using exclusi ely lump-sum axes p e en s om
a¤ec ing decisions aken by go e nmen s and households, in an a emp o in e nalize he e¤ec s
om s a es’policy.
In a sense, his esul can be placed on he discussion ini ia ed by Sa o (2000) abou he
capabili y o ede al go e nmen o eplica e second-bes esul s depending on he ede al in-
s umen s a ailable. P ecisely, as esul o aking in o conside a ion a new policy ins umen ,
i. e., a public inpu p o ided by he ede al go e nmen ha is complemen o ha o¤e ed by
he s a es, he uppe le el o go e nmen is able o eplica e he second-bes ou come o an
uni a y coun y.
The s uc u e o he pape is as ollows. Sec ion 2 desc ibes he main ea u es o he
model and he di¤e en e sions o he op imali y ule o he p o ision o public inpu s, aking
accoun whe he he coun y is ede al o no . Sec ions 3 and 4 e alua e he abili y o he
ede al go e nmen o eplica e he uni a y ou come wi h he policy ins umen s a ailable and
wi h a complemen a y public inpu , espec i ely. Finally, sec ion 5 concludes.
2 The Basic Model
This sec ion aims o show wo poin s. Fi s , o cha ac e ize he equilib ium in a cen alized
coun y wi h unemploymen ; his will allow us no only o see how he op imal ule o he
3
p o ision o public inpu s mus be modi…ed wi h espec o a si ua ion wi h ull employmen ,
bu also ha ing a benchma k scena io o compa e wi h ede al equilib ia. Second, o highligh
ha he …scal decisions aken by one le el o go e nmen (pa icula ly ha wi h spending
esponsabili ies on public inpu s) will a¤ec o he le els o go e nmen ; consequen ly, e ical
expendi u e ex e nali ies will a ise despi e o using exclusi ely lump-sum axes.
The heo e ical amewo k consis s o … ms, households and wo di¤e en ie s o go e n-
men : he ede al le el and ksubna ional s a es. Fi ms a e iden ical ac oss he coun y and,
o he sake o simplici y, we assume ha hei numbe is no malized o one in each s a e. All
o hem p oduce a single good on he basis o he ollowing p oduc ion unc ion:
F(N; K; G) = NK1G;(1)
whe e Nis labo , Ka …xed ac o and Ga public inpu . Such a p oduc ion echnology allows
us o quali y he public inpu as ac o -augmen ing1. In his con ex , he public spending will
inc ease he e u n o he …xed p oduc ion ac o K, which we no malized o one, in which case
he p o… can be exp essed as:2
=F(N; G)wN; (2)
whe e wis he wage a e. P o… maximiza ion implies o de…ne he … s -o de condi ion w=
FN(N; G), ha implic ly de…nes he ollowing unc ion o labo demand:
N(w; G) = 1
1G
1w1
1(3)
Combining equa ions (2) and (3), he p o… unc ion can be ob ained:
(w; G)(4)
We conside ha all households ha e he same p e e ences o consump ion cac oss he
ede a ion and desc ibed by a u ili y unc ion u(c), which is inc easing in c. Each s a e is
popula ed by h ee ypes o consume s: a … m-owne , employed and unemployed wo ke s,
1An al e na i e app oach would imply a p oduc ion unc ion wi h cons an e u ns o scale in all he inpu s
(p i a e and public). This would be he case o … m-augmen ing public inpu . I would c ea e economic en s
ha , in e ms o he model we de elop he e, would no exhibi subs an ial di¤e ences wi h espec o wha we
ob ain below.
2The e u n o labo is no a¤ec ed by he public inpu , al hough his would be he no mal si ua ion wi h
ac o -augmen ing public inpu s. This is no he case he e because we a e in e es ed in conside ing he impac
o he public inpu on employmen , and he demand o labo we ob ain below implies ha he wage a e is
independen o G. In a model wi h ull-employmen , howe e , we should se up w(M; g).
4
which a e deno ed by supe indices " ", "e" and "u", espec i ely. The … m-owne , endowed
wi h he p oduc ion ac o K, is who ecei es he p o… in e u n o hi ing he …xed ac o
o he … m. His budge -cons ain is de…ned by c = , whe e  is a lump-sum ax.
Rega ding he o he wo ypes o consume s, we inse he e a dis inc ion be ween he o al
labo o ce a ailable o wo king Mand he numbe o households ha e¤ec i ely a e employed
N. Obs iously, ull employmen is cha ac e ized by M=N. The budge cons ain o an
employed wo ke is ce=we, whe e eis a lump-sum ax, while wo ke s wi hou jobs aces
cu=b, whe e bdeno es a ne o ax unemploymen bene… .
In a cen alized coun y, o he policy a iables  ; e; b; G, he go e nmen maximizes
a u ili a ian wel a e unc ion
W=kNue+k[MN]uu+ku (5)
subjec o he ollowing budge cons ain :
kNe+k kG k[MN]b= 0 (6)
In a si ua ion whe e he e is no unemploymen , he … s -o de condi ions a e as ollows:
FOC  :=u 0
(7)
FOC (e) : = (ue)0(8)
FOC (G) : FG= 1 (9)
FOC () : Ne+ G= 0;(10)
whe e is he Lag ange mul iplie . The wo … s equa ions show he usual esul om op imiza-
ion wi h lump-sum axes and ans e s: p i a e ma ginal u ili y (o each ype o consume )
mus be equal o social wel a e cos o axa ion, which is ep esen ed he e by he Lag ange
mul iplie o go e nmen budge cons ain . The equa ion (9) is he s anda d p oduc ion e -
…ciency condi ion in he p o ision o public inpu s. Finally, (10) is he budge cons ain o
cen al go e nmen , whe e he las e m o LHS in (6) has been d opped as M=N.
Le us u n o he equilib ium wi h unemploymen . Fo ins i u ional easons (i. e., he
exis ence o a minimum wage), he a e wage is assumed o exceed he ma ke -clea ing wage
and, consequen ly, M > N. Things d ama ically change o he op imal p o ision o Gwhen
unemploymen appea s; addi ionally, he … s -o de condi ion o he unemploymen bene… b
also mus be aken in o conside a ion:
FOC (b) : = (uu)0(11)
5

FOC (G) : NG(ueuu)
+NGe+NGb+FG= 1:(12)
Le us conside now he case o di¤e en ie s o go e nmen s. We assume ha he ede al
le el is in cha ge o p o iding he unemploymen bene… while he s a es p o ide he public
inpu s3. Bo h le els o go e nmen sha e he ax on employed wo ke s (wi h he ax a es
Teand echosen by he ede al and s a es go e nmen s, espec i ely; e=Te+ e). The
e enues collec ed om he ax on p o… s a e assigned in a p opo ion (which is exogenously
de e mined) o he s a es (0 1), while he ax a e  is exclusi ely decided by he ede al
go e nmen .
Unde such a amewo k, le us assume ha he s a es beha e as Nash playe s, ha is, each
subna ional go e nmen igno es he impac o i s …scal decisions on ede al e enues. The e o e,
he op imiza ion p oblem o be sol ed by he s a es is:
Max W =Nue(we)+(MN)ub(b) + u  (13)
s: : N e+ G+S= 0
N=N(w; G)
wo> we;
whe e Sis a e ical lump-sum om he ede al go e nmen o s a es. Las inequali y e e s o
he dis o sion exis ing in he labo ma ke , which is he eason o unemploymen . Fi s -o de
condi ions o e,Gand gi e:
FOC ( e) : = (ue)0(14)
FOC (G) : NG(ueuu)
+NG e+FG1 = 0 (15)
FOC () : N e+ G+S= 0 (16)
Exp ession (14) se s up an iden ical ule o chosing he op imal ax a e on employed
wo ke s in a cen alized coun y han in a wo ld wi h wo ie s o go e nmen . This is a di ec
consequence o using lump-sum axes. E en in he p esence o ax sha ing be ween di¤e en
le els o go e nmen , i he households’beha io is no a¤ec ed by axes, he e is no scope o
e ical ax ex e nali ies.
By con as , and lea ing aside he discussion on he op imal le els o G(see Ma inez and
Sjong en (2009) o a u he analysis), exp ession (15) shows he main di¤e ence by compa ing
3This dis ibu ion o spending esponsabili ies is no c ucial o he esul s, which would be symme ic wi h
an in e se e ical assignmen o public expendi u es. Anyway, he scheme we ollow he e is in line wi h he
mains eam o heo y o …scal ede alism.
6
i o he exp ession (12). The e m NG edi¤e s om i s equi alen in (12), namely, NG(e+b).
As long as he ede al go e nmen se s up a non-nega i e ax a e Teon employed wo ke s,
he ac o ha ing s a es deciding on Gleads o educe he o e p o ision bias ha he p esence
o unemploymen c ea es in he p o ision o public inpu s. In o he wo ds, exp ession (15) is
close o (9) han equa ion (12).4
In his ega d, and con a y o he con en ional iew in p e ious li e a u e on e ical
ex e nali ies, we guess he e ha mo e ede alism may lead o mo e e¢ ciency in he design o
…scal policies. To see his in an ex eme case, assume ha all en axes acc ue o he s a es
(= 1); he ede al go e nmen needs o be …nanced by a nega i e …scal g an ( om s a es)
and/o by cha ging a posi i e ax a e Teon wo ke s. This la e solu ion in ol es an op imal
ule o he p o ision o public inpu s close o he p oduc ion e¢ ciency condi ion, minimizing
he di¤e en ial e¤ec ha he p esence o unemploymen c ea es in he discussion on op imali y.
Consequen ly, he beha io o ede al go e nmen becomes a c ucial issue o de e mine he
e¤ec o unemploymen on he achi emen o p oduc ion e¢ ciency condi ion in he p o ision
o public inpu s. This is wha we s udy in he nex sec ion.
3 The abili y o ede al go e nmen o eplica e he cen-
alized ou come
A usual way o co ec ing e ical ( ax and expendi u e) ex e nali ies is assuming a ede al
go e nmen beha ing as S ackelbe g leade . In such a con ex , he sequence o he game is as
ollows. Fi s ly, he ede al go e nmen decides on Te, ,Sand, esidually, on b, aking in o
conside a ion he s a es’ eac ion o changes in ede al policy a iables. Secondly, he s a es
choose Gand e, aken as exogenous all he decision a iables o he uppe -le el o go e nmen .
4I is s aigh o wa d o show ha wi h ull employmen no e ical ( ax and expendi u e) ex e nali ies
appea .
7
Consequen ly, he op imiza ion p oblem o he ede al go e nmen is:
Max W =kNue(we) + k(MN)ub(b) + ku  (17)
s: : kNTe+k(1 ) k(MN)bkS = 0
G=G(Te; ;  ; S; M; N)(18)
e= e(Te; ;  ; S; M; N)(19)
N=N(w; G)
wo> we:
Exp essions (18) and (19) a e he s a es’ eac ion unc ions. The e o e, o sol ing he ede al
p oblem, some in o ma ion on compa a i e s a ics o hese eac ion unc ions is equi ed. To
do ha , we s a om he … s -o de condi ions o s a es (15) and (16)5. Di¤e en ia ing o ally
and (and igno ing supe index "e" o sake o simplici y in he no a ion), we ha e:
GdG +  d + TdT + SdS +  d = 0
GdG +  d + TdT + SdS +  d = 0
This wo-equa ion sys em can be exp essed using a ma icial o m as ollows (and a e sol ing
o dG and d ):
dG
d =0
@G
G
1
A
10
@TS
TS
1
A0
B
B
@
dT
dS
d
1
C
C
A(20)
Ma icial manipula ion on (20) shows ha :
dG
dT =GT=A( T T)(21)
dG
dS =GS=A( S S)(22)
dG
d =G =A(    )(23)
d
dT = T=A(gTgT)(24)
d
dS = S=A(gSgS)(25)
d
d =  =A(g g )(26)
5The … s -o de condi ion (14) can be igno ed in his analysis. In a sense, his exp ession does no admi
any in‡uence om ede al a iables and, consequen ly, i does no ma e a his poin . Anyway, exp ession
(14) can be easily inse ed in (15) wi hou modi ying subs an ially he analysis below.
8
whe e Ais 1
G G .
Tu ning back o he ede al p oblem, i is clea ha i s budge cons ain can be w i en
as b=NT e+(1) S
(MN). Plugging his in o he objec i e unc ion (17), we ob ain he … s -o de
condi ions o he policy a iables o he ede al go e nmen :
FOC(Te) : N(ue)0(1 + T)+(MN)(ub)0(bT+bT T+bGGT)(27a)
+(u )0(FNNGGTwNGGT)ubNGGT= 0
FOC( ) : N(ue)0(  )ubNGG + (MN)(ub)0(b +b  +bGG )(27b)
+(u )0(FNNGG wNGG )(u )0= 0
FOC(S) : N(ue)0( S)+(MN)(ub)0(bS+b S+bGGS)ubNGGS(27c)
+(u )0(FNNGGSwNGGS) = 0
FOC() : kNTe+k(1 ) k(MN)bkS = 0 (27d)
Taking in o accoun ha bcan be esidually ob ained om he abo e ou equa ion-sys em,
we sympli y (27a)-(27d) and he ollowing esul is achie ed:
T= 0 (28)
 =
N(29)
S=1
N;(30)
whe e w=FN(N; G), (21)-(23) and he co esponding pa ial de i a i es o and (acco ding
o (15) and (16)) ha e been used. Wha is implici ly es ablished in (28)-(30) is he inabili y o
ede al go e nmen o a¤ec s a es’beha io . In ac , no only he ede al ax a e on employed
wo ke s Tehas no e¤ec on he equi alen s a e ax a e e(equa ion (28)), bu also none o he
policy a iables o uppe le el o go e nmen has any impac on he s a e p o ision o public
inpu s. Indeed, om exp essions (21)-(23), i is clea ha GT=G =GS= 0, ha is, he e is
no way h ough which he ede al go e nmen can modi y he p o ision o public inpu s. The
unique impac o he ede al policy a iables ( and Son e) is i ial: an inc ease (dec ease)
in some o hem educes ( ises) he s a e ax a e in a magni ude gi en by he numbe o
employed wo ke s N. The e o e, he highes le el o go e nmen is no able o eplica e no
only he … s -bes ou come o (9) bu also he op imali y ule o he p o ision o public inpu s
in an uni a y coun y wi h unemploymen 6.
6Anyway, we mus be awa e ha he … s -bes alues o Teand ea e gua an eed in each scena io as long
as hey a e lump-sum axes.
9