Aggregate Instability under Labor Income Taxation and Balanced-Budget Rules: Preferences Matter

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Aggregate Instability under Labor Income Taxation and

Balanced-Budget Rules: Preferences Matter

Nicolas Abad, Thomas Seegmuller, Alain Venditti

To cite this version:

Nicolas Abad, Thomas Seegmuller, Alain Venditti. Aggregate Instability under Labor Income Taxation

and Balanced-Budget Rules: Preferences Matter. 2013. �halshs-00793213v2�

Working Papers / Documents de travail

Aggregate Instability under Labor Income Taxation

and Balanced-Budget Rules: Preferences Matter

Nicolas Abad

Thomas Seegmuller

Alain Venditti

Taxation and Balan ed-Budget Rules: Preferen es Matter

∗

Ni olas ABAD

Aix-MarseilleUniversity(Aix-MarseilleS hoolofE onomi s),CNRS-GREQAMand EHESS

ThomasSEEGMULLER

Aix-MarseilleUniversity(Aix-MarseilleS hoolofE onomi s),CNRS-GREQAMand EHESS

AlainVENDITTI

Aix-MarseilleUniversity(Aix-MarseilleS hoolof E onomi s),CNRS-GREQAM,EHESS andEDHEC

November 2013

Abstra t: Weinvestigatetheroleofpreferen esintheexisten eofexpe tation-driven instabilityunderabalan edbudgetrulewhere governmentspendingsarenan edbya taxonlaborin ome. Consideringaone-se tor neo lassi algrowthmodelwith alarge lassofpreferen es, wend thatexpe tation-drivenu tuations aremorelikelywhen onsumptionandlaborare Edgeworth substitutes. Under thisproperty, an intermedi-aterange oftaxrates anda su ientlylowelasti ity ofintertemporal substitutionin onsumption leadtoinstability. Numeri alsimulationsof themodelsupportthe on- lusionthat labor in ometaxation isa plausible sour e ofinstability in most OECD ountries.

Keywords: Indetermina y, expe tation-driven business y les, labor in ome taxes, balan ed-budgetrule,innite-horizon model.

JournalofE onomi LiteratureClassi ationNumbers:C62,E32,E62.

∗

WethankStefanoBosi,RaoufBou ekkine,HippolyteD'Albis,Jean-Mi helGrandmont, CuongLeVan,CarineNourry,XavierRauri h,ChrissyGiannitsarouandRoger Farmerfor useful ommentsandsuggestions. Thispaperalsobenetedfrompresentationsatthe Con-feren einhonorofCuongLeVan, MSE-PSE-UniversityofParis1,Paris,De ember2011; OLGDays2012,Marseille;PET2012,TaipeiandLAGV2012,Marseille.

During the Summer 2011, the European sto k market downswing leads poli y-makerstoimplementrulesthataimtobalan egovernmentbudget. This revival of balan ed-budget rules an be understood asamean to redu e risks ofanexplodingpubli debtbutalsoasasignalsenttotheratingagen iesthat publi nan esare kept sane. Indeed, in ountries fa inga toolarge debt or un ontrolled publi a ount, long-term growth may be rowded out and bor-rowingon nan ial market may be hardersin e risinginterestrates in reases theburdenofpubli debt. Thisleadsthereforetounsustainablesituationasin Gree e, Spain orPortugal in the re ent years. In this ontext, the European Unionenfor estheEuropeanFis alCompa t.

Althoughanextensiveliteratureaddressesthequestionofs alrulesthrough itspro y li alee t,akeyargumenthasbeenstressedintheseminal ontribu-tionofS hmitt-GrohéandUribe[21℄. Theyshowthatinastandardone-se tor Ramseymodelwitha onstantstreamof wastefulgovernmentexpenditures -nan edbyadistortionarytaxonlaborin ome,abalan ed-budgetrulemaybe asour eofaggregateinstability. Indeed,taxrateslargerthanthe apitalshare ofin omeandlowerthanthetaxrateasso iatedtothepeakoftheLaer urve involveexpe tation-driven u tuations. The me hanismbehindinstability re-liesonthevolatilityofagents'expe tationsandgoesasfollows. Anin reasein theexpe ted tax rateimplies aredu tionin future employment andtherefore of apitalreturns. Consequently,investmentde reases andhouseholdsneedto workless. Thetaxratebeingde reasingin hoursworked,thegovernementhas toin reasethetax ratetomaintainthebudget balan edandexpe tationsare thereforeself-fullling.

Several ontributionsextendthisframework,butprovide ontradi tory on- lusions. GhilardiandRossi[5℄generalizethete hnologywithaCESprodu tion fun tionassumingthesamepreferen esasS hmitt-GrohéandUribe[21℄. They showthat instabilityis morelikelywhen apital and laboraresubstitutes. In onstrast, Linnemann [14℄ keeps a Cobb-Douglas te hnology, but onsiders a parti ular lass of non-separable preferen es and showsthat instabilityis un-likely. In this paper,wepropose to reexamine thedestabilizingee t of labor in ometaxationunderamoregeneralapproa handemphasizeparti ularlythe roleof preferen es. Ourinvestigation ismotivated bytwopoints. Onthe one hand,theroleofpreferen esintheo uren eofindetermina yisa ornerstone forseveral ontributions. Forinstan e,Nishimuraetal. [17℄ onsiderthe o ur-ren eofindetermina yinamodelwithaggregateexternalities. Theyshowthat the onditionsfortheemergen eofinstabilitystronglydependontheproperties

indetermina y. Ontheotherhand,thereisagrowinginterestintheimpa tof s alpoli iesin presen eofspe i preferen es, espe iallywhennon-separable utilityfun tionsareassumed. Theempiri alinvestigationofTrabandtand Uh-lig[22℄ examinestheshapesoftheLaer urvewithlinearlyhomogeneousand King-Plosser-Rebelo [12℄ (KPR) preferen es. Bilbiie [2℄ investigates how non-separabilityofpreferen esexplainstheobservedin reaseinprivate onsumption inresponseto s alsho ks.

Weexamine theinterplaybetweenpreferen es and te hnology onthe exis-ten e of tax ratesthat generate indetermina y. This is explored in a neo la-ssi alinnite-horizon growth modelembedding mostpopularpreferen esused byma roe onomists. We onsiderthree lassesofutilityfun tion: i)additively separablepreferen es withnon-unitary elasti ity ofintertemporal substitution in onsumptionandelasti laborsupply,ii)alinearlyhomogeneousutility fun -tion, iii) aJaimovi h-Rebelo [9℄ (JR) formulationwhere thedegree of in ome ee t an be ontrolled and admits two polar ases. In absen e of in ome ef-fe t,aGreenwood-Her owitz-Human[7℄(GHH)utilityfun tionis onsidered. Onthe ontrary,withamaximizeddegreeofin omeee t,thepreferen esare hara terizedbyaKing-Plosser-Rebelo[12℄(KPR) formulation. Ageneralized produ tionfun tiondes ribesthete hnologyofthermssoastoen ompassthe resultsof GhilardiandRossi[5℄oninputssubstitution. Governementse toris nally hara terizedbythesamebalan ed-budgetrule onsideredby S hmitt-GrohéandUribe[21℄ forwhi h thetaxrateis ounter- y li alwithrespe tto thetax base.

Our generalinvestigation identiesa robustpropertyto obtain indetermi-na y. Indeed, wend that indetermina y is morelikely when thepreferen es exhibitEdgeworthsubstitutabilitybetween onsumptionandlabor,su hthata marginalin reasein labor de reasesthemarginalutilityof onsumption. This propertyis alwayssatisediftheutilityfun tion is linearlyhomogeneous,but requiresalargeenoughdegreeofin omeee tforJRpreferen es. Furthermore, withthethreespe i ationswe onsider,alowenoughelasti ityof intertempo-ralsubstitutionin onsumptionisne essarytogetindetermina y. Underthese properties, an intermediate range of tax rates is destabilizing. The intuition behind our results omes from the fa t that intertemporal and intratemporal ee ts need to be in a ordan e. Coming ba k to the intuition of S hmitt-GrohéandUribedes ribedabove,whenhouseholdsde reasetheirlaborsupply in urrentperiod,atthesametime,theymustde reasetheir onsumptionsin e theyhavelessin ome. However,this isonly ompatible with amarginal

util-between onsumptionandlabor.

Finally,westudytheempiri alrobustnessofthemodel. A alibratedversion ofthemodel basedonplausible estimatesofstru turalparametersemphasizes that labor in ome taxes under a balan ed-budget rule are a potential sour e of instability for most OECD ountries. This on ernsparti ularly European ountriessin etheyexperien ethehighesttaxratesandstandwithintherange ofdestabilizingtaxratesformostofthe alibrations onsidered.

Inthenextse tion, wepresentthemodelandderivetheoptimal hoi esof householdsandrms. Se tion3isdevotedtoprovetheexisten eofanormalized steady state. In Se tion 4, we provide the dynami analysis with our main resultsandadis ussionwiththerelatedliterature,whileempiri alillustrations are given in Se tion 5. E onomi interpretations are dis ussed in Se tion 6. Finally,Appendix presentsalltheproofs.

2. The model

Inthisse tion,wedes ribeoure onomywithastandardneo lassi algrowth model. First,wedenethepoli y ruleimplementedbythegovernment. Then, westatehowtheagents hoosetheamountofgood onsumedandhoursworked, andnally,wedes ribethete hnologi alstru ture.

2.1. Government

Following S hmitt-Grohé and Uribe [21℄, we assume that the government hoosesa onstantlevelofpubli spendings

G

,thatneitherae tthepreferen es northe te hnology. Sin e the budget is balan ed,it is equalto the total tax revenue

Ω(t)

generated by atax rate,

τ (t)

, applied on laborin ome,

w(t)l(t)

, with

w(t)

thewagerateand

l(t)

thelaborsupplied:

G = Ω(t) = τ (t)w(t)l(t)

(1) Equivalently,thebalan ed-budgetrule anbewritten as:

τ (t) =

_w(t)l(t)

G

(2) Sin epubli spendingsare onstant,thetaxrateis ounter- y li alwithitstax base,i.e. ade reaseinthelaborin omeendsupinanin reaseinthetaxrate.

1

1

Giannitsarou[6℄ onsidersthesametypeofbalan ed-budgetrulebutfo useson onsump-tiontaxes.

We onsiderane onomypopulatedbyalargenumberofidenti al innitely-livedagents. Weassumewithoutlossofgeneralitythatthetotalpopulationis onstant and normalized to one. Atea h period an agent supplies elasti ally anamountoflabor

l ∈ [0, ¯l]

, with

¯l > 1

his timeendowment. He thenderives utilityfrom onsumption

c

andleisure

L = ¯l− l

a ordingtotheinstantaneous utilityfun tion

U (c, L/B)

,where

B > 0

isas alingparameter,whi hsatises: Assumption1.

U (c, L/B)

is

C

r

over

R

++

×(0, ¯l)

for

r

largeenough,in reasing with respe t to ea h argument and on ave. Moreover,

U

cL

U

c

L

B

−

U

LL

U

L

L

B

6= 1

,

lim

X→0

XU

X

(c, X)/U

c

(c, X) = 0

and

lim

X→+∞

XU

X

(c, X)/U

c

(c, X) = +∞

, or

lim

X→0

XU

X

(c, X)/U

c

(c, X) = +∞

and

lim

X→+∞

XU

X

(c, X)/U

c

(C, X) =

0

.

This assumptionwill ensureexisten eof anormalizedsteadystate. In ad-dition to these general properties, we introdu e the denition of Edgeworth substitutabilitybetween onsumptionandlabor:

Denition1. Ifthemarginalutilityof onsumptionisin reasinginleisuresu h that

U

cL

(c, L/B) > 0

,then onsumptionandlabor areEdgeworth substitutes.

Following Denition 1, Edgeworth omplementarity between onsumption andlaborisobviouslyobtainedwhen

U

cL

(c, L/B) < 0

.

In our investigation, we will onsider three dierent spe i ations of preferen es ommonlyused intheliterature:

i)Anadditivelyseparableutilityfun tion

U (c, (¯l− L)/B) =

c

1−

1

εcc

1 −

_ε

1

_cc

−

((¯l− L)/B)

1+

εll

1

1 +

_ε

1

_ll

(3)

where

ε

cc

is the elasti ity of intertemporal substitution in onsumption and

ε

ll

is the inverse of the wage elasti ity of labor. S hmitt-Grohé and Uribe [21℄ onsider this formulation with

ε

cc

= 1

and

ε

ll

= +∞

. Obviously, these preferen es exhibit neither Edgeworth substitutability nor omplementarity sin e

U

cL

(c, L/B) = 0

.

ii) A linear homogeneous utility fun tion

U (c, L/B)

hara terized by the shareof onsumptionwithintotalutility

α(c, L/B) ∈ (0, 1)

denedby:

α(c, L/B) =

U

c

(c,L/B)c

U(c,L/B)

while the share of leisure is given by

1 − α(c, L/B)

. A parti ular property of these preferen es is that onsumption and labor are always Edgeworth substitutes

U

cL

(c, L/B) > 0

.

iii)AJaimovi h-Rebelo[9℄formulationsu hthat

U (c, L/B) =

(c + (L/B)

1+χ

_c

γ

₎

1−θ

1 − θ

(5)

with

θ, χ > 0

and

γ ∈ [0, 1]

. These preferen es are hara terized by the parameter

γ

that ontrols the degree of in ome ee t and en ompass two standardformulations. Onthe onehand,in absen eof in omeee t (

γ = 0

), the GHH formulation is obtained and yields a labor supply independent of onsumption. Ontheotherhand,whenthein omeee tisthelargest(

γ = 1

), the utility fun tion is a KPR formulation whi h is ompatible with balan ed growth and stationnary hours worked. A ording to Denition 1, Edgeworth substitutabilitybetween onsumptionandlaborrequires

γ > θ

whileEdgeworth omplementarityisobtainedwhen

γ < θ

. Linnemann[14℄ onsidersaparti ular restri tionof this spe i ation assuming a KPR formulation with

θ > 1

that impliesEdgeworth omplementarity.

Finally, all these utility fun tions satisfy normality of onsumption and labor. Inaddition, additivelyseparableand linearhomogeneousspe i ations also satisfy on avity but this is notne essarily the asewith JRpreferen es when

γ 6= 0

(seeSe tion 4.3forfurtherdetails).

Theintertemporalmaximizationprogramoftherepresentativeagentisgiven by:

max

c(t),l(t),K(t)

Z

+∞

t=0

e

−ρt

U c(t), (¯l− l(t))/B

s.t.

c(t) + ˙

K(t) + δK(t) = r(t)K(t) + (1 − τ (t))w(t)l(t)

K(0) > 0

given (6)

where

r(t)

istherentalrateof apital,

ρ > 0

thedis ountrate,

K(t)

the apital sto kand

δ > 0

thedepre iation rateof apital. Moreover,weassumein the following that ea h household onsiders as given the tax rate

τ (t)

on labor in ome.

H = U (c(t), (¯l− l(t))/B) + λ(t)

h

r(t)K(t) + (1 − τ (t))w(t)l(t) − c(t) − δK(t)

i

with

λ(t)

theshadowpri eof apital

K(t)

. Consideringthepri es(11)-(12)and thetax rate

τ (t)

asgiven,wederivethefollowingrstorder onditions:

U

c

(c(t), (¯l− l(t))/B) = λ(t)

(7)

(1/B)U

L

(c(t), (¯l− l(t))/B) = λ(t)(1 − τ (t))w(t)

(8)

˙λ(t) = −λ(t)[r(t) − ρ − δ]

(9)

Anysolutionneedsalsotosatisfythetransversality ondition:

lim

t→+∞

e

−ρt

_λ(t)K(t)

₌

₀

(10) 2.3. Theprodu tion stru ture

Consideringa ompetitivee onomy, a ontinuumof rmsof unit size pro-du es asingle good

Y

using apital

K

and labor

l

. The rms' te hnology is a onstantreturns to s ale produ tion fun tion

Y = AF (K, l)

, with

A > 0

a s aling parameter. We dene the intensive sto k of apital

a = K/l

for any

l > 0

andtheintensiveprodu tionfun tionwrites

Y /l = Af (a)

.

Assumption2.

f (a)

is

C

r

over

R

++

for

r

largeenough,in reasing

(f

′

_{(a) > 0)}

and on ave

(f

′′

_{(a) < 0)}

.

From the protmaximisation ofa rm,weobtain thewage rate

w(t)

and therentalrateof apital

r(t)

as:

r(t)

=

Af

′

_(a(t))

(11)

w(t)

=

A[f (a(t)) − a(t)f

′

_(a(t))]

(12) Wealso omputetheshareof apitalintotalin ome:

s(a) =

af

′

(a)

f (a)

∈ (0, 1)

(13)

andtheelasti ityof apital-laborsubstitution:

σ(a) = −

(1−s(a))f

′

(a)

af

′′

(a)

> 0

(14)

Assumption3. Capital andlaborare su ientlystrongsubstitutes,su hthat

σ(a) > s(a)

.

This last assumption implies that labor in ome is in reasing with the quan-tityof hoursworked. Extendingthe analysisof S hmitt-Grohéand Uribe[21℄ whoassume a Cobb-Douglas produ tion fun tion, Ghilardi and Rossi[5℄ also study the impa t of substitutability between apital and labor on the range ofdestabilizingtax rates. However,we generalizebothprevious ontributions theyassumealogarithmi utilityfor onsumptionandaninnitelyelasti labor withinadditively-separablepreferen es.

Inordertoderivetheintertemporalequilibrium,let

τ ≡ ˜

τ (K, l) =

G

w(K(t)/l(t))l(t)

andsubstitute

τ (K, l)

˜

and thewagerate(12) in therst order onditions(7) and (8). Given

K

and

λ

, the system obtained an be solved to express the onsumptiondemandandlaborsupplyfun tions

c(K(t), λ(t))

and

l(K(t), λ(t))

. Pluggingthelatterintheexpressionofthetax rate,one obtains:

˜

τ (K(t), l(K(t), λ(t))) ≡ τ (K(t), λ(t))

(15) Using(11)-(12), weget theequilibrium valuesfor therental rateof apital

r(t)

andthewagerate

w(t)

with

a(t) = K(t)/l(K(t), λ(t))

:

r(t) = Af

′

_{(a(t)) ≡ r(K(t), λ(t))}

w(t) = A[f (a(t)) − a(t)f

′

_{(a(t))] ≡ w(K(t), λ(t))}

(16)

Substituting theexpressionsobtainedfor pri es,taxrate, onsumption de-mandandlaborsupplyin theequation of apital a umulation(6)and in the Euler equation(9), weobtainthe following systemof dierentialequations in

K

and

λ

:

˙

K(t) = r(K(t), λ(t))K(t) + (1 − τ (K(t), λ(t)))w(K(t), λ(t))l(K(t), λ(t))

− δK(t) − c(K(t), λ(t))

˙λ(t) = −λ(t) [r(K(t), λ(t)) − ρ − δ]

(17)

An intertemporal equilibrium is apath

{K(t), λ(t)}

t≥0

, with

K(0) > 0

, that satisesequations(17)andthetransversality ondition(10).

3. Normalizedsteady state Asteadystateisa4-tuple

(a

∗

_{, l}

∗

_{, c}

∗

_{, τ}

∗

₎

,with

a

∗

_{= K}

∗

_/l

∗

,satisfying:

δ + ρ = Af

′

_(a

∗

₎

(18)

c

∗

_{= l}

∗

_[(Af

′

_(a

∗

_{) − δ)a}

∗

_{+ (1 − τ}

∗

_{)A(f (a}

∗

_{) − a}

∗

_f

′

_(a

∗

_))]

(19)

U

L

(c

∗

, (¯l− l

∗

)/B)

BU

c

(c

∗

, (¯l− l

∗

)/B)

= (1 − τ

∗

_{)A[f (a}

∗

_{) − a}

∗

_f

′

_(a

∗

_)]

(20)

τ

∗

₌

G

A[f (a

∗

_{) − a}

∗

_f

′

_(a

∗

_)]l

∗

(21)

Weusethes aling parameters

A > 0

and

B > 0

toensuretheexisten eof anormalizedsteady state(NSS),

a

∗

_{= 1}

and

l

∗

_{= 1}

, whi h remainsinvariant withrespe ttopreferen esandte hnologi alparameters.

Proposition 1.LetAssumptions 1-2hold. Then there existuniquevalues

A

∗

and

B

∗

su hthatwhen

A = A

∗

and

B = B

∗

,

(a

∗

_{, l}

∗

_{) = (1, 1)}

isaNSS. Proof: SeeAppendix 8.1.

Remark: Usinga ontinuityargument,we on ludefromProposition1that there exists an intertemporalequilibrium forany initial apitalsto k

K(0)

in theneighborhoodof

K

∗

.

Letusintrodu ethefollowingelasti ities:

ε

cc

=

−

_U

U

_cc

c

_(c,L)c

(c,L)

,

ε

lc

=

−

_U

U

L

L

c

(c,L)

(c,L)c

,

ε

cl

=

−

U

U

cL

c

(c,L)

(c,L)l

, ε

ll

=

−

U

L

(c,L)

U

LL

(c,L)l

(22)

Normalityof onsumptionandleisurestatesthat

1

ε

cc

−

1

ε

lc

≥ 0

and

1

ε

ll

−

1

ε

cl

≥ 0

and holds for any preferen es we onsider. Con avity of preferen es implies

1

ε

cc

1

ε

ll

−

1

ε

cl

1

ε

lc

≥ 0

. This property is satisedfor both additive separableand linearhomogeneouspreferen es. However,JR formulationrequires further re-stri tions to satisfy on avity. Sin e we are interested in the lo al dynami s, Lemma 2 in Se tion 4.3 provides additionnal restri tions to ensure on avity intheneighboorhoodofthesteadystate. Moreover,a ordingtoDenition 1, notethat when

1

ε

cl

and

1

ε

lc

are negative(positive), onsumption andlaborare Edgeworthsubstitutes( omplements).

Finally, in the rest of the paper, we evaluate all the shares and elasti i-tiespreviouslydened at theNSS. From(4), (13) and(14), wedenote indeed

α(c

∗

_{, (¯l− 1)/B}

∗

_{) = α}

,

s(1) = s

and

σ(1) = σ

.

4. Instabilitywith balan ed-budgetrules and labor in ometaxes Thisse tioninvestigatesthepropertiesofpreferen esthatenhan ethe like-lihoodof indetermina ywhenageneralte hnologyis onsidered. InAppendix 8.2,welinearizethedynami systemin theneighborhoodoftheNSSand om-pute the tra e and the determinant of the asso iated Ja obian matrix. As thedynami system(17)hasonepredetermined andoneforwardvariable, in-determina y requires a negative tra eand a positive determinant. From the expressionof the tra egiven in Appendix 8.2, we an dire tly derivea ne es-sary onditionforthe existen eofindetermina ywhateverthe spe i ationof preferen es:

mina yof theNSS is

τ > τ

,with:

τ =

s

σ

+ ε

cc

1

ε

cc

1

ε

ll

−

1

ε

cl

1

ε

lc

1 + ε

cc

_ε

1

_cc

_ε

1

_ll

−

_ε

1

_cl

_ε

1

_lc

(23)

Proof: SeeAppendix8.2

Giventhat theterm

1

ε

cc

1

ε

ll

−

1

ε

cl

1

ε

lc

measuresthedegreeof on avityof the utilityfun tion,we on ludethat strongerdegreesof on avityimply that in-determina y requires higher tax rates on labor in ome. Lemma 1 therefore underlinesthe importan e of preferen es for thedestabilizing impa t of labor in ometaxes. Besides,aweakfa torsubstitutabilityin reasesthelowerbound ontaxrate. Thislastpointholdsforanygivenspe i ationofpreferen esand is parti ularly dis ussed in Ghilardi and Rossi[5℄ who onsider the restri ted aseofadditively-separablepreferen eswithlogarithmi utilityfun tionof on-sumptionandinnitelyelasti labor. Theirrelated onditionis

τ >

s

σ

while,in S hmitt-GrohéandUribe's[21℄framework withaCobb-Douglas te hnology,it be omes

τ > s

.

4.1. Additively separablepreferen es

We rst fo us on a generalized version of S hmitt-Grohé and Uribe [21℄ withanadditivelyseparableutilityfun tion. Morepre isely,wedonotrestri t theelasti ityof intertemporal substitution in onsumption ortheelasti ityof apital-labor substitution to be unitary (i.e.

ε

cc

,

σ 6= 1

). Furthermore, labor supplyis assumed to be elasti su h that

1

ε

ll

∈ (0, +∞)

. Note that this lass ofpreferen esis hara terizedby

1

ε

cl

=

1

ε

lc

= 0

meaningthat onsumptionand laborare neitherEdgeworthsubstitutesnor omplements. FromLemma1,we derive

τ =

s/σ+

1

εll

1+

1

εll

andarstresultfollows:

Proposition 2. Under Assumptions 1-3, let

U (c, L/B)

be given by (3) and

τ =

s/σ+

1

εll

1+

1

εll

. There exist

ρ ∈ (0, +∞]

¯

,

ε

¯

cc

> 0

and

τ ∈ (τ , 1)

¯

su hthatthe NSS islo allyindeterminate ifandonly if

ρ ∈ (0, ¯

ρ)

,

ε

cc

< ¯

ε

cc

and

τ ∈ (τ , ¯

τ )

.

Proof: SeeAppendix8.3.

Thispropositionhighlightstheexisten eofanintervaloftaxratesthatleads toindetermina y. Ontheonehand,thelowerboundofthisintervaldependson the apitalshareofin ome

s

, the apital-laborelasti ityofsubstitution

σ

and theinverseofthewageelasti ityoflabor

1

ε

ll

τ

isin reasing in

1

ε

ll

. Asa onsequen e, innitelyelasti labor(i.e.:

1

ε

ll

= 0

), as onsidered in S hmitt-Grohé and Uribe [21℄ and Ghilardi and Rossi [5℄, is the less restri tive ase sin e

τ =

s

σ

. To ensure that

τ < ¯

τ

, the elasti ity of intertemporalsubstitution in onsumption hasto be lowenough. Namely,

ε

cc

hasto belowerthan

ε

¯

cc

with:

¯

ε

cc

=



1−s/σ

1+

1

εll



[(ρ + δ) (1 − s) + sρ]

s

σ



1−s/σ

1+

1

εll



(ρ + δ) (1 − s) + sρ



Itisworthpointingoutthatthisupperboundisde reasingwith

1

ε

ll

. More pre isely,when

1

ε

ll

= 0

,

ε

¯

cc

isthelargest. Thisargumentreinfor esthe on lu-sionofS hmitt-GrohéandUribe[21℄thatwithinadditively-separable preferen- es,instabilityismorelikelywhenthelaborsupplyisinnitely-elasti . 4.2. Linearhomogeneouspreferen es

Alinearhomogeneousspe i ationis hara terizedby

1

ε

lc

,

1

ε

cl

< 0

su hthat onsumption and labor are always Edgeworth substitutes. Moreover, noti e that

1

ε

cc

1

ε

ll

−

1

ε

cl

1

ε

lc

= 0

and weobtainthereforefrom (23)

τ =

s

σ

. Lookingfor onditionsensuring theexisten e of a ontinuum of equilibrium paths around thesteadystate,weobtainthenextproposition:

Proposition 3. Under Assumptions1-3, let

U (c, L/B)

be linearhomogeneous and

τ =

s

σ

. There exist

ρ ∈ (0, +∞]

¯

,

τ ∈ (τ , 1)

¯

and

ε

¯

cc

> 0

su hthat the NSS islo allyindeterminate ifandonly if

ρ ∈ (0, ¯

ρ)

,

ε

cc

< ¯

ε

cc

and

τ ∈ (τ , ¯

τ )

.

Proof: SeeAppendix 8.4.

As in Proposition 2,weshow that theequilibrium is lo ally indeterminate fortaxrateswithinaboundedinterval. Thelowerboundontaxratesin reases with the apital share of in ome and de reases with the elasti ity of apital-laborsubstitution. Notethat in orderto have

τ < ¯

τ

, weimpose alowenough elasti ity of intertemporal substitution in onsumption su h that

ε

cc

< ¯

ε

cc

. This ondition is equivalent to the one obtained in the additively separable ase. Nevertheless, it is not restri tive be ause

ε

¯

cc

tends to +

∞

when the shareof onsumption in total utility

α

tendsto unity. Finally, the restri tion on the elasti ity of intertemporal substitution in onsumption has important impli ationsonthewageelasti ityof labor. Indeed, ombiningequations (35) and (37)in Appendix 8.4, one shows that a su ientlylow

ε

cc

implies alow enoughwageelasti ityoflabor

ε

ll

su hthat

ε

ll

< ¯

ε

ll

with:

¯

ε

ll

=

(1−α)[(ρ+δ)(1−s)+sρ]+α(1−

s

σ

)(ρ+δ)(1−s)

α

2 s

σ

(ρ+δ)(1−s)

(24) This on lusion ontraststhereforewithS hmitt-GrohéandUribe[21℄and Ghilardi and Rossi [5℄ that assume additively-separable preferen es but also with Linnemann [14℄, that onsiders a parti ular KPR utility fun tion. All these ontributions require a large enough wage elasti ity of labor to obtain indetermina y.

4.3. Jaimovi h-Rebelo preferen es

WithJRpreferen es,we an ontrolthedegreeofthein omeee tthrough the parameter

γ ∈ [0, 1]

. It is worthstressing that these preferen es exhibits Edgeworthsubstitutabilitybetween onsumptionandlaborif

γ > θ

and Edge-worth omplementaritywhen

γ < θ

.

In order to ensure on avity in the neighborhood of the NSS, we add the followingrestri tion:

Lemma2. Let

U (c, L/B)

begivenby

(5)

andAssumption1holds. Ane essary and su ient ondition to obtain on avity in the neighborhood of the NSS is

θ ≥ θ(τ, γ, χ)

with:

θ(τ, γ, χ) =

γC(τ )(γ + χ)(1 + χ − (1 − γ)C(τ ))

(1 + χ)

2

_{[χ + γC(τ ) 2 −}

(1−γ)C(τ )

1+χ

]

Proof: SeeAppendix8.5.

Contrary to the previous ases, it is not possible to derive from (23) an expli itexpressionof

τ

sin eitisimpli itlygivenby

τ = h(τ )

with:

h(τ ) =

s

σ

+

θ(1+χ)

2



_{χ+γC(τ ) 2−}

(1−γ)C(τ )

1+χ



−γC(τ )(γ+χ)[1+χ−(1−γ)C(τ )]

θ(1+χ)

2

_{−γ(1−γ)C(τ )[1+χ−(1−γ)C(τ )]}

1 +

θ(1+χ)

2



χ+γC(τ ) 2−

(1−γ)C(τ )

1+χ



−γC(τ )(γ+χ)[1+χ−(1−γ)C(τ )]

θ(1+χ)

2

_{−γ(1−γ)C(τ )[1+χ−(1−γ)C(τ )]}

Asshownin Appendix8.6, thereexists aunique

τ ∈ (0, 1)

su h thatLemma 1 holds. Wegetthenthenextproposition:

Proposition 4. Under Assumptions1-3, let

U (c, L/B)

be given by

(5)

. There is a riti al value

γ ∈ (0, 1)

for whi h for any given

γ ∈ (γ, 1]

, there exist

¯

ρ ∈ (0, +∞]

,

θ ∈ (θ, +∞]

¯

,

σ ∈ (s, +∞)

,

τ ∈ (0, 1)

and

τ ∈ (τ , 1)

¯

su h that the NSS is lo ally indeterminate if and only if

ρ ∈ (0, ¯

ρ)

,

θ ∈ (θ, ¯

θ)

,

σ > σ

and

Thisproposition,jointlywithPropositions2and3,highlightstherobustness oftheexisten e ofanintermediate rangeof destabilizingtaxrates. Moreover, a dire t out ome of Proposition 4 is that indetermina y is more likely when onsumption and labor are Edgeworthsubstitutes. Indeed, we anshow (see Appendix 8.5) that

γ

is always larger than

θ

. As a result, Proposition 4 implies Edgeworth substitutability orweak omplementarity. More pre isely, lo alindetermina yisruled outwithaGHHspe i ation hara terizedbythe absen e of in ome ee t (

γ = 0

) and a strong Edgeworth omplementarity. On the ontrary, with KPR preferen es (

γ = 1

), onsumption and labor are obviously Edgeworth substitutes if

θ < 1

. In this ase, the existen e of a rangeof destabilizingtaxratesis ensured. Otherwise, when onsumptionand labor be ome weak Edgeworth omplements, indetermina y may still hold but requires higher tax rates. This on lusion explainstherefore the resultof Linnemann[14℄aboutthela kofplausibilityofindetermina ysin eheassumes

θ > 1

, i.e. a strong Edgeworth omplementarity between onsumption and labor.

Wehavehighlightedtheroleof preferen esontheemergen eof indetermi-na yinaRamseymodelwithabalan ed-budgetrulenan edbyalaborin ome tax. Wendthat Edgeworth substitutability between onsumption and labor andalowelasti ityofintertemporalsubstitutionin onsumptionare ru ialfor theexisten e of arange of destabilizingtax rates. Next se tion isdevoted to ompareour on lusionsto therelatedliterature.

4.4. Comparison with therelatedliterature

S hmitt-Grohé[20℄ andS hmitt-GrohéandUribe[21℄ laim thatthere isa lose orresponden ebetweenindetermina yin modelswith produ tive exter-nalitiesandmodelswithbalan ed-budget. Wearguethatthisequivalen eisnot ageneralpropertywhenoneassumesnonadditivelyseparablepreferen es. The ontributionofNishimuraetal. [17℄ onsiderstheo urren eofindetermina y inamodelwithsmallaggregateexternalities. Theyalsondthatindetermina y requires anin reasing marginalutilityof onsumption with respe tto leisure, i.e.

U

cL

(c, L) > 0

. A ording to Denition 1, this implies that onsumption and labor are Edgeworth substitutes. Nevertheless, additively separable and linearhomogeneous preferen es display indetermina yprovided that the elas-ti ityofintertemporalsubstitutionin onsumptionissu ientlylarge. Sin ein

substitutionin onsumptiontoobtainindetermina y,the lose orresponden e dis ussedbyS hmitt-Grohé[21℄ doesnothold.

Theliteratureonbalan ed-budgetruleshasalsofo usedonthedestabilizing roleof onsumptiontaxes. Assuminganadditiveseparableutilityfun tion, Gi-annitsarou[6℄showthat onsumptiontaxhasstabilizingee tsin esaddle-path stabilityisalwaysensured, ontrarilytoS hmitt-GrohéandUribe[21℄witha la-borin ometax. However,Nourryetal. [18℄dis ussthis on lusion onsidering non-separablepreferen es with varying in ome ee t. Theynd that in pres-en eofanintermediateoralowdegreeofin omeee t, onsumptiontaxeslead toinstability. Inotherwords,in ontrasttolaborin ometax,thekeyelement foradestabilizing onsumptiontaxisEdgeworth omplementaritybetween on-sumptionandlabor. Itfollowsthateventhough onsumptionandlaborin ome taxes introdu e similar distortions in the onsumption-leisure trade-o, they requireoppositepropertiesofpreferen esinordertobedestabilizing.

Inthenextse tion,weinvestigatethenumeri alpropertiesofourresultsin ordertodis usstheirempiri alplausibility.

5. Empiri alillustration

Togivebetter insights of our results, we pro eed to a numeri al exer ise. We rst divide ountries between four groups a ordingto therange of their tax rates on laborin ome. The lassi ationis based on the ontribution of TrabandtandUhlig [22℄that omputestheee tivetaxratesupto2008using themethodologyofMendozaetal. [15℄. Thefourgroupsof ountriesaregiven inTable1.

τ ∈ (0.25, 0.30)

Japan(0.27),U.S.(0.28),U.K.(0.28), Ireland(0.27)

τ ∈ (0.30, 0.40)

Portugal(0.31),Spain(0.36)

τ ∈ (0.40, 0.50)

Belgium(0.49), Denmark(0.47),EU-14(0.41), Fran e(0.46), Germany(0.41),Italy(0.47),Netherland(0.44),

τ ∈ (0.50, 0.60)

Austria(0.50), Sweden(0.56)

Table1: Estimatedlaborin ometaxrates

Weneednowtoxthevaluesofthestru turalparameters. Onthebasisof quarterlydata, we onsider the ben hmark values

(ρ, δ, s) = (0.01, 0.025, 0.3)

. A ordingtotheempiri alliterature,thereisno learagreementonthesizeof theelasti ityof apital-laborsubstitution. Nevertheless,thehigherestimatesof

giou[4℄andKaragiannisetal. [10℄. Thereisalsono onsensusontheelasti ity of intertemporal substitution in onsumption. Several ontributions provides the range (0.2,0.8) (see Campbell [3℄ and Ko herlakota [13℄), while Mulligan [16℄andmorere entlyVissing-JorgensenandAttanasio[23℄showeviden esfor higher estimates with an interval (2,3). Finally, many ontributions onsider that labor is innitely elasti . However,Rogerson and Wallenius [19℄ investi-gateaggregateparti ipationin thelabormarketatthema rolevelandndan intervalofthewageelasti ityof laborthatstandsbetween2.25and 3.

From the dis ussion above, we fo us on the following intervals for

ε

cc

∈

[0.66, 2]

,

ε

ll

∈ [2.5, +∞]

. Wealsoassume

σ ∈ [0.8, 1.4]

. This intervalallowsto onsidertheestimates givenbytheempiri al literaturebutalso extendto the aseof omplementaritybetweeninputs sin ethe on lusions ofthe literature are still un ertain.

2

We rst alibratean additively-separableutility fun tion hara terizedby

ε

cc

= 0.67

and

ε

ll

∈ (2.85, +∞)

.

σ =

0.8

σ =

1.4

ε

ll

= +∞

(0.38,0.87) (0.21,0.87)

ε

ll

=2.86 (0.54,0.74) (0.42,0.74)

Table2:Rangeofdestabilizingtaxrates(

τ

,

τ

¯

)withadditively-separablepreferen es Table 2 reports the intervals of destabilizing tax rates in the additively-separable ase. We observe that the lowerbound

τ

is between 0.21 and 0.54 whiletheupperbound

¯

τ

ishigherthan0.74. A ordingtoTable1,thisstresses the plausibility of our results. Indeed, ex ept when

σ = 0.8

and

ε

ll

= 2.86

, most ountries in Europeare destabilized. This on ernsparti ularlySweden, Austria,Belgium, Denmark, ItalyandFran e. Moreover,thewhole sampleof ountries stands inside the rangeof destabilizingtax rates when laborsupply isinnitelyelasti andthesubstitutabilitybetween apitalandlaborisstrong enough.

Whentheutilityfun tionishomogeneouslinear,we alibratetheparameters

ε

cc

= 0.66

,

α = 0.65

and

ε

cc

= 2

,

α = 0.51

in order to mat h an intervalof

ε

ll

∈ (2.5, 2.9)

giventheadmissiblevaluesof

τ

inTable1.

As shownin Table3,thehomogeneouslinearformulationdisplaysaneven betteroutline. Theintervalof

τ

be omesinthis ase[0.21,0.38℄and

τ

¯

isinside

2

Note that we assume a high enough elasti ity of apital-labor substitution su h that

σ

=0.8

σ

=1.4

ε

cc

=0.66,

α

=0.65 (0.38,0.97) (0.21,0.97)

ε

cc

=2,

α

=0.51 (0.38,0.81) (0.21,0.81)

Table3:Rangeofdestabilizingtaxrates(

τ

,

τ

¯

)withhomogeneouslinearpreferen es

theinterval(0.81,0.97). Consequently, all ountries with tax ratesabove 0.40 inTable1arenowintherangeof thedestabilizingtaxrates.

Finally,with theJRformulation, onsidering that thetax ratesstands be-tween(0.27,0.57)inTable1,

θ = 0.5

and

χ = 0

mat h

ε

ll

∈ (2.3, 2.5)

. Figure1 showsthelowerandtheupperboundon

τ

asafun tionofthedegreeofin ome ee t

γ

.

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

γ

τ

σ = 0.8, θ =0.5, χ =0

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

γ

τ

σ = 1.4, θ =0.5, χ =0

Indeterminacy

Indeterminacy

Total

instability

Determinacy

Determinacy

Determinacy

Determinacy

Total

instability

Figure1: Destabilizingtaxratesinthe asewithJaimovi h-Rebelopreferen es. Dashlines:

τ

,solidlines:

τ

¯

Underour alibration,Figure1showsthattheminimumlevelof

γ

isin the interval [0.51,0.63℄. Moreover, Table 4 reports the interval of

γ

for ountries onsideredinTable1takingtheirtaxratesasgiven.

γ ∈ (0.6, 0.75]

Austria,Belgium,Denmark,Fran e,Italy,Sweden

γ ∈ (0.8, 0.97]

Germany,EU-14,Netherlands

γ ∈ (0.90, 0.94]

Portugal,Spain

γ ∈ (0.95, 0.96]

Japan,U.S.,U.K.,Ireland

Table4:InstabilitywithJaimovi h-Rebelopreferen es

Thisshowsthatalargerangeofvaluesof

γ

oversindeterminatetaxrates. Furthermore,theintervalsgiveninTable4tstheupperestimatesofKahnand Tsoukalas[11℄. UsingBayesianestimations,theyreportadistributionof

γ

with mean0.81anda10-90per entilesintervalof[0.69,0.95℄. Ournumeri alexer ise illustrates thereforethat mostOECD ountries may experien e instabilityfor

6. E onomi intuition

Tounderstandthee onomi me hanisms,letusassumethatagentsexpe ta largerfuturetaxrate. Following(8)and(11),futurelaborsupplyde reasesand yieldsalowerinterestratethatredu esin omeinfutureperiod. Consequently, investment de reases and sin e they need to work less, households de rease theirlaborsupply in urrent period. The de rease in the tax base for es the governementtoadjustthebudgetbyin reasingtaxratessu hthatvolatilityin agent'sexpe tations areself-fullling. Neverthelessonequestionremains: why indetermina yo urs undersome lassofpreferen eswhileitisruled outwith others ? A ru ial point to understand our results omes from the fa t that the ross-elasti ity

ε

lc

needstobenegativeorweaklypositive,i.e. onsumption andlaborneedto beEdgeworthsubstitutes orweakEdgeworth omplements. In our interpretation, the de rease in interest rate involves

˙λ > 0

sin e

r <

δ + ρ

. Indetermina y is obtained if it is asso iated with a de rease of labor supplyin urrentperiodthat islargerthanthede reaseoflaborsupplyin the nextperiod.

3

Be ause theyhavelessin omein presentperiod, theymustalso de reasetheir urrent onsumption. Self-fulllingexpe tationsimpliestherefore

˙c

c

> 0

and

˙l

l

> 0

. Sin e apitalis predetermined, wehave

˙

w

w

= −

s

σ

˙l

l

. Taking thenthederivativeof equation(8) withrespe t totime, indetermina yo urs ifthefollowingequalityissatised:

(

s

σ

+

1

ε

ll

)

˙l

l

−

1

ε

lc

˙c

c

=

˙λ

λ

(25) Be ause

(

s

σ

+

1

ε

ll

)

ispositive,thisequationissatisedif

1

ε

lc

isnegativeorpositive butsu ientlylowsu hthatthersttermontheleft-handsidedominatesthe se ondone. Itisstraightforwardtoshowitisalwaysthe aseforanadditively separableandalinearhomogeneouspreferen essin e

1

ε

lc

= 0

intheformerand

1

ε

lc

< 0

in thelatter.

Inthe aseofJaimovi h-Rebelopreferen es,thesignof

1

ε

lc

isambiguousand dependsonthesizeof

τ

,

γ

:

1

ε

lc

=

(θ − γ)(1 + χ) + γ(1 − γ)C(τ )

1 + χ − (1 − γ)C(τ )

(26)

3

with

C

′

_{(τ ) < 0}

.

The denominator in equation (26) being positive, the sign of

1

ε

lc

is given bythe numerator. Considerrstthat

γ > θ

. Sin e

C

′

_{(τ ) < 0}

, the numerator of(26)is negativeforalargeenough

τ

. Consequently, JRpreferen esdisplay Edgeworthsubstitutability and equation (25) is always satised. In ontrast, when

γ < θ

,thenumeratorispositiveand onsumptionandlaboraretherefore Edgeworth omplement. Itfollowsthattheintertemporalme hanismsdes ribed inequation(25)islesslikelytobesatisedandthereforerequiresmu h higher taxratestoobtainself-fulllingexpe tations.

7. Con luding omments

Thispaper ontributestothedebatedealingwiththe(de-)stabilizing prop-erties of balan ed-budget rules nan ed by a labor in ome tax. More parti -ularly, weemphasize theme hanismsin preferen es leading to indetermina y. Fo using onthree ommonlyused utility fun tions, weprove that Edgeworth substitutability between onsumption and labor in reases the likelihood of a destabilizinglaborin ometax. Whenthe elasti ity ofintertemporal substitu-tion in onsumption is su iently low, an intermediate range of tax rates is destabilizing. Finally, anumeri al exer isesupports ourndings a ordingto theempiri aleviden eunderlying theplausibilityof balan ed-budgetruleasa sour eof instabilityin mostOECD ountries.

8. Appendix

8.1. Proof ofProposition 1

To establish the existen e of a normalized steady state

(a

∗

_{, l}

∗

_{, c}

∗

_{, τ}

∗

_{) =}

(1, 1, c

∗

_{, τ}

∗

₎

, we have to prove the existen e and uniqueness of solutions

A

∗

and

B

∗

satisfying:

δ + ρ = A

∗

_f

′

₍₁₎

(27)

τ

∗

=

G

A

∗

_{[f (1) − f}

′

_(1)]

(28)

c

∗

_{= (1 − τ}

∗

_)A

∗

_{[f (1) − f}

′

_{(1)] + A}

∗

_f

′

_{(1) − δ}

(29)

U

L

(c, (¯l− 1)/B

∗

)

B

∗

_U

c

(c, (¯l− 1)/B

∗

)

= (1 − τ

∗

_)A

∗

_{[f (1) − f}

′

_(1)]

(30)

Fromequation(27),wederivethat

A

∗

₌

ρ+δ

f

′

(1)

whi hgives,on esubstituted inequations(28)and (29),aunique

τ

∗

and

c

∗

rewrittenas:

τ

∗

₌

s(1)G

(ρ+δ)(1−s(1))

c

∗

₌

s(1)ρ+(1−τ )(ρ+δ)(1−s(1))

s(1)

Considering

A

∗

,

τ

∗

and

c

∗

,wegetthefollowingequationfrom(30):

˜

g(B) ≡

U

L

(c,(¯

l−1)/B)

BU

c

(c,(¯

l−1)/B)

=

(1−τ

∗

)(ρ+δ)(1−s(1))

s(1)

(31)

Existen e of a unique value

B

∗

satisfying equation (31) requires that the marginal rate of substitution

g(B)

˜

does not have a derivative equal to zero and satises appropriate boundaries onditions. Sin e under Assumption 1,

lim

B→0

˜

g(B) = 0

and

lim

B→+∞

˜

g(B) = +∞

, or

lim

B→0

˜

g(B) = +∞

and

lim

B→+∞

g(B) = 0

˜

, the existen e of

B

∗

is guaranteed. Moreover, as

B˜

g

′

_(B)/˜

_{g(B) 6= 0}

,uniquenessof

B

∗

alsofollows. 8.2. Proof ofLemma1

Toprovidean analysisof lo al stability, welinearize (17)aroundtheNSS. Wethenderivethe hara teristi polynomialby onsideringtheelasti ities eval-uated at the NSS. We need rst to derive a relationship between the ross-elasti ities,

ε

cl

and

ε

lc

. Using (22)and the rst order onditions(7) and (8), we get

ε

cl

=

(1−τ )wL

c

ε

lc

. Using the expression of

w

at the NSS given in (16) together with (13)and (18)wend

wL = K(1 − s)(δ + ρ)/s

. Sin e at NSS,

c = l[ρa + (1 − τ )w]

, itfollows:

ε

cl

=

(1−τ )(δ+ρ)(1−s)+sρ

_{(1−τ )(δ+ρ)(1−s)}

ε

lc

(32) Dierentiating

τ (K(t), λ(t))

asgivenby(15),oneobtainstheelasti itiesof thetax ratewithrespe tto

K

and

λ

:

ε

τ k

=

_dK

dτ

K

_τ

= −

(1−τ )s

_σ

_{(1−τ )σ∆ε}

[σ∆ε

cc

_cc

+σ−s]

_{+τ (s−σ)}

ε

τ λ

=

_dλ

dτ

λ

_τ

= −

_{(1−τ )σ∆ε}

(1−τ )(σ−s)ε

_cc

_{+τ (s−σ)}

cc



1

ε

cc

−

1

ε

lc



Using (22), the Impli it Fun tion Theorem gives thepartial derivativesof thefun tions

c(K(t), λ(t))

and

l(K(t), λ(t))

evaluatedat theNSS:

dc

dK

=

c

K∆ε

cl



s

σ

−

τ ε

τ k

1−τ



,

_dλ

dc

= −

_λ∆

c

h

_ε

1

_ll

− (1 −

τ ε

τ λ

1−τ

)

1

ε

cl

+

s

σ

i

dl

dK

=

l

K∆ε

cc



s

σ

−

τ ε

τ k

1−τ



,

_dλ

dl

=

_λ∆

l

h

(1 −

τ ε

τ λ

1−τ

)

1

ε

cc

−

1

ε

lc

i

with

∆ =

1

ε

cc



1

ε

ll

+

s

σ



−

1

ε

cl

ε

lc

. Fromthese resultsand(16)wealso deriveat theNSS:

dr

dK

= −

r(1−s)

Kσ

h

1 −

1

∆ε

cc

s

σ

−

τ ε

τ k

1−τ

i

,

dr

dλ

=

r(1−s)

λ∆σ

h

(1 −

τ ε

τ λ

1−τ

)

1

ε

cc

−

1

ε

lc

i

dw

dK

=

ws

Kσ

h

1 −

_∆ε

1

_cc

_σ

s

−

τ ε

τ k

1−τ

i

,

dw

_dλ

= −

_λ∆σ

ws

h

(1 −

τ ε

τ λ

1−τ

)

1

ε

cc

−

1

ε

lc

i

Linearizingthesystem(17)aroundtheNSS,using(32)andtheaboveresults, gives:

d ˙

K

dK

= ρ −

(δ+ρ)(1−s)

s

n

τ

h

ε

τ k

+

_σ

s

1 −

_∆ε

1

_cc

(

_σ

s

−

τ ε

_1−τ

τ k

)



i

−

_∆ε

1−τ

_cc

(

_σ

s

−

τ ε

τ

k

1−τ

)

o

−

(1−τ )(1−s)(δ+ρ)

_s∆ε

_cl

(

_σ

s

−

τ ε

τ k

1−τ

)

d ˙

K

dλ

=

(1−τ )(1−s)(δ+ρ)K

s∆λ

h

1

ε

ll

+

s

σ

− 1 −

τ ε

τ λ

1−τ

1

ε

cl

i

+ (1 − τ )

h

1 −

τ ε

τ λ

1−τ

1

ε

cc

−

1

ε

lc

i

+

(δ+ρ)(1−s)K

_sλ



τ

h

∆ε

τ λ

−

_σ

s

(1 −

τ ε

_1−τ

τ λ

)

_ε

1

_cc

−

_ε

1

_lc



i



d ˙λ

dK

= −

λ(δ+ρ)(1−s)

Kσ

h

∆ +

1

∆ε

cc

s

σ

−

τ ε

τ k

1−τ

i

d ˙λ

dλ

= −

(δ+ρ)(1−s)

∆σ

h

(1 −

τ ε

τ λ

1−τ

)

1

ε

cc

−

1

ε

lc

i

Aftertedious omputationsandstraightforwardsimpli ations, using(32),the expressionsof

ε

τ k

,

ε

τ λ

asgivenabove,wegetthefollowing hara teristi poly-nomial:

P(λ) = λ

2

− T λ + D = 0

(33) with

T = ρ −

_{στ −s−(1−τ )σε}

(ρ+δ)(1−s)τ

cc

[

_εcc

1

_εll

1

−

_εcl

1

_εlc

1

]

and

D =

(ρ+δ)(1−s)ε

cc

h

[(1−τ )(ρ+δ)(1−s)+sρ]

h

(1−τ )

1

εcc

−

1

εlc

+

1

εll

−

1

εcl

−τ

i

+τ (1−τ )(ρ+δ)(1−s)

1

εcc

−

1

εlc

i

sσ



στ −s−(1−τ )σε

cc

[

_εcc

1

_εll

1

−

_εcl

1

_εlc

1

]



where

T

and

D

arerespe tivelythetra eandthedeterminantoftheasso iated Ja obianmatrix. Lo alindetermina yrequires

T < 0

and

D > 0

. Ane essary onditiontohaveanegativetra eis

τ > τ

with:

τ =

s

σ

+ ε

cc

1

ε

cc

1

ε

ll

−

1

ε

cl

1

ε

lc

1 + ε

cc

_ε

1

_cc

_ε

1

_ll

−

_ε

1

_cl

_ε

1

_lc

8.3. Proof ofProposition 2

In the ase of additivelyseparable preferen es, the expression of the tra e is:

T = ρ −

τ (ρ + δ)(1 − s)

στ − s −

(1−τ )σχ

_ε

_ll

FollowingLemma1,wederivedire tlythelowerbound on

τ

:

τ =

s

σ

+

1

ε

ll

1 +

_ε

1

_ll

(34)

pressiontobesatised:

ρ(στ − s − (

1 − τ )σ

ε

ll

) − (ρ + δ)(1 − s)τ > 0

Thisleadstoanupperboundon

ρ

su hthat: i)

ρ < ¯

ρ =

δ(1−s)τ

στ −s−

(1−τ )σ

εll

−s−(ρ+δ)(1−s)τ )

if

στ − s −

(1−τ )σ

ε

ll

− (ρ + δ)(1 − s)τ ) > 0

ii)

ρ ∈ (0, +∞)

,otherwise

Consideringthedeterminant,wegetthefollowingexpression:

D =

(δ+ρ)(1−s)

σs στ −s−

(1−τ )σ

εll

P (τ )

where

P (τ ) =

h

sρ + (δ + ρ)(1 − s)

i

1

ε

cc

+

1

ε

ll

−τ

h

sρ + (δ + ρ)(1 − s)

1

ε

cc

+

1

ε

ll

+

(ρ + δ)(1 − s)

ε

ll

+sρ + (δ + ρ)(1 − s)

i

+ τ

2

(1 − s)(ρ + δ)(1 +

1

ε

ll

)

Thedenominator of

D

is positive when

τ > τ

. Consequently, the sign of the determinantis givenbythe signof

P (τ )

. Thelatterfun tion ispositivewhen

τ = 0

whilenegativewhen

τ = 1

. Sin e

P (τ )

isstri tlyde reasingin

τ ∈ (0, 1)

, thereexists thereforeaunique

τ ∈ (0, 1)

¯

su h that

P (τ ) > 0

when

τ < ¯

τ

. The ondition

τ ∈ (τ , ¯

τ )

impliesthereforeapositivedeterminant.

Finally, weneedto ensurethat

τ < ¯

τ

. Thisis the aseifand onlyif

P (τ )

evaluatedat

τ

ispositive. Aftersomesimpli ations,wederive:

P (τ ) =



1−s/σ

1+

1

εll



[(ρ + δ) (1 − s) + sρ]

1

ε

cc

−

s

σ



1−s/σ

1+

1

εll



(ρ + δ) (1 − s) + sρ



> 0

whi hissatisedifandonlyif

ε

cc

< ¯

ε

cc

with:

¯

ε

cc

=



1−s/σ

1+

1

εll



[(ρ + δ) (1 − s) + sρ]

s

σ



1−s/σ

1+

1

εll



(ρ + δ) (1 − s) + sρ



Notethatwithlinearhomogeneity,allthepreferen eselasti itiesarewritten asfun tion of

ε

cc

su h that

ε

lc

= −ε

cc

(1−α)

_α

,

ε

cl

= −ε

cc

(1−α)

_α

(1−τ )(δ+ρ)(1−s)+sρ

_{(1−τ )(1−s)(ρ+δ)}

,

ε

ll

= ε

cc

(1−α)

2

_{[(1−τ )(ρ+δ)(1−s)+sρ)}

α

2

_{(1−τ )(ρ+δ)(1−s)}

(35)

Sin e linearhomogeneity yields

1

ε

cc

ε

ll

−

1

ε

lc

ε

cl

= 0

, thelowerbound ontax rateisgivenfrom (23)by:

τ =

s

σ

(36)

Underthis ondition,we on ludethat

T < 0

when: i)

ρ < ¯

ρ =

δ(1−s)τ

στ −s−(1−s)τ

if

στ − s − (1 − s)τ > 0

ii)

ρ ∈ (0, +∞)

otherwise

Considering

D

andusingtheexpressionsin(35),thedeterminantiswritten:

D =

(1−τ )(δ+ρ)(1−s)

_{(1−α)sσ(στ −s)}

P (τ )

with

P (τ )

= [(ρ + δ)(1 − s) + sρ] +

α

1−α

(1 − τ )(ρ + δ)(1 − s)

−

τ [(1−τ )(ρ+δ)(1−s)+sρ](1−α)ε

cc

(1−τ )

Moreover,wederive:

∂P (τ )

∂τ

= −

α

1 − α

(ρ + δ)(1 − s) −

(1 − α)ε

cc

(1 − τ )

2

[(1 − τ )

2

_{(ρ + δ)(1 − s) + sρ] < 0}

Thepolynomial

P (τ )

is positivewhen

τ = 0

and negative when

τ = 1

. Sin e

P (τ )

ismonotoni allyde reasingin

τ

, there existstherefore auniquesolution

¯

τ ∈ (0, 1)

su hthat

P (τ ) > 0

if

τ < ¯

τ

. Sin ethedenominator ispositivewhen

τ > τ

,thedeterminantispositiveifandonlyif

τ ∈ (τ , ¯

τ )

.

Finally, the ondition

τ < ¯

τ

has to be ensured. Substituting

τ =

s

σ

into

P (τ )

, the interval (

τ , ¯

τ

) is non-empty ifand onlyif

P (τ ) > 0

, i.e.

ε

cc

is low enoughsu hthat:

ε

cc

< ¯

ε

cc

=

(1−s/σ)[(ρ+δ)(1−s)+sρ+(1−s/σ)(ρ+δ)(1−s)

α

(1−α)

]

(1−α)

s

σ

[(1−s/σ)(ρ+δ)(1−s)+sρ]

(37)

Inthe aseofJaimovi h-Rebelopreferen es,theelasti itiesin (22)write:

1

ε

cc

= θ

c−γ

(l/B)1+χ

_1+χ

c

γ

c−

(l/B)1+χ

_1+χ

c

γ

− γ(1 − γ)

(l/B)1+χ

1+χ

c

γ

c−γ

(l/B)1+χ

_1+χ

c

γ

,

1

ε

ll

= θ

(l/B)1+χ

1+χ

c

γ

c−

(l/B)1+χ

_1+χ

c

γ

+ χ,

1

ε

cl

=

(l/B)1+χ

1+χ

c

γ

c−γ

(l/B)1+χ

_1+χ

c

γ



θ

c−γ

(l/B)1+χ

1+χ

c

γ

c−

(l/B)1+χ

_1+χ

c

γ

− γ



,

_ε

1

_lc

= θ

c−γ

(l/B)1+χ

1+χ

c

γ

c−

(l/B)1+χ

_1+χ

c

γ

− γ,

(38)

Using these expressions and the relationship between

ε

cl

and

ε

lc

at NSS givenbyequation(32),onederives:

(l/B)

1+χ

1+χ

c

γ−1

1 − γ

(l/B)

_1+χ

1+χ

c

γ−1

=

(1 − τ )(ρ + δ)(1 − s)

(1 − τ )(ρ + δ)(1 − s) + sρ

.

Let

C(τ ) =

(1−τ )(ρ+δ)(1−s)

(1−τ )(ρ+δ)(1−s)+sρ

andsolvethepreviousequationsu hthat:

(l/B)

1+χ

1 + χ

c

γ−1

₌

C(τ )(1 + χ)

1 + χ + γC(τ )

Then,thefollowingexpressionsholds:

c−γ

(l/B)1+χ

_1+χ

c

γ

c−

(l/B)1+χ

_1+χ

c

γ

=

1+χ

1+χ−(1−γ)C(τ )

,

(l/B)1+χ

1+χ

c

γ

c−γ

(l/B)1+χ

_1+χ

c

γ

=

(1+χ)C(τ )

1+χ−(1−γ)C(τ )

Theelasti ities rewritetherefore:

1

ε

cc

= θ

1+χ

1+χ−(1−γ)C(τ )

− γ(1 − γ)

C(τ )

1+χ

,

1

ε

ll

= θ

(1+χ)C(τ )

1+χ−(1−γ)C(τ )

+ χ,

1

ε

lc

= θ

1+χ

1+χ−(1−γ)C(τ )

− γ,

1

ε

cl

=

C(τ )

ε

lc

,

(39)

A ordingtothis,lo al on avityoftheutilityfun tionisensuredwhen

1

ε

cc

≥ 0

and

1

ε

cc

1

ε

ll

−

1

ε

cl

1

ε

lc

≥ 0

. Straighforward omputations show that these two inequalitiesaresatisedifandonlyif:

θ ≥ θ(τ, γ, χ) ≡

γC(τ )(γ + χ)(1 + χ − (1 − γ)C(τ ))

(1 + χ)

2

_{[χ + γC(τ ) 2 −}

(1−γ)C(τ )

1+χ

]

(40)

Finally, note that Edgeworthsubstitutability holds if

γ ≥ θ(τ, γ, χ)

. This inequalityissatisedwhen

γ = 0

and

γ = 1

. It followsthat itwillbesatised forany

γ ∈ (0, 1)

if

1 ≥ ˜

θ(τ, γ, χ)

with :

˜

θ(τ, γ, χ) =

C(τ )(γ + χ)(1 + χ − (1 − γ)C(τ ))

(1 + χ)

2

_{[χ + γC(τ ) 2 −}

(1−γ)C(τ )

1+χ

]

UsingthegeneralexpressionsfortheTra eandtheDeterminant,weobtain withJRpreferen es:

T = ρ −

(ρ+δ)(1−s)τ

στ −s−(1−τ )σε

cc

h

1

εcc

εll

1

−

εcl

1

εlc

1

i

and

D =

(ρ+δ)(1−s)ε

cc



γ(1−τ )[1+χ−(1−γ)C(τ )]

1+χ

[(ρ+δ)(1−s)+sρ]+[(1−τ )(ρ+δ)(1−s)+sρ][γ(1−τ )C(τ )+χ−τ (1+χ)]



sσ

h

στ −s−(1−τ )σε

cc

h

1

εcc

εll

1

−

εcl

1

εlc

1

i

i

with

ε

cc

h

1

ε

cc

1

ε

ll

−

1

ε

cl

1

ε

lc

i

=

θ(1+χ)

2



_{χ+γC(τ ) 2−}

(1−γ)C(τ )

1+χ



−γC(τ )(γ+χ)[1+χ−(1−γ)C(τ )]

θ(1+χ)

2

_{−γ(1−γ)C(τ )[1+χ−(1−γ)C(τ )]}

Considering

γ = 0

(GHH ase),thetra eand thedeterminantaregivenby:

T = ρ −

τ (ρ + δ)(1 − s)

στ − s − (1 − τ )σχ

(41)

D =

(δ + ρ)(1 − s)ε

cc

[(1 − τ )(ρ + δ)(1 − s) + sρ]

σs[στ − s − σ(1 − τ )χ]

χ − τ(1 + χ)

(42)

From (41),ane essary onditiontoobtainanegativetra eis that thetax rateissu ientlylargesu hthat

τ > τ

0

with:

τ

0

=

s

σ

+ χ

(1 + χ)

Inequation(42),the ondition

τ > τ

0

impliesapositivedenominator. The signofthedeterminantisthereforedeterminedbythese ondfa torof(42),i.e.

χ − τ (1 + χ)

. Thisexpressionispositiveifandonlyif

τ < ¯

τ

0

with:

¯

τ

0

=

χ

1 + χ

whi hislowerthan

τ

0

. Sin e

τ

¯

0

_{< τ}

0

,itisnotpossibletoobtainsimultaneously anegativetra eand apositivedeterminant. Indetermina y istherefore ruled out.

Whenwe onsider

γ = 1

(KPR ase),thetra eandthedeterminantbe ome:

T = ρ −

(ρ+δ)(1−s)τ

_{σG(τ )}

and

D =

(ρ+δ)(1−s)ε

cc

sσG(τ )

P (τ )

with

G(τ ) = τ −

_σ

s

− (1 − τ )χ − (1 − τ )C(τ )(2 −

1

_θ

)