Is So cial Security Really Bad For Growth?
Giorgio Bellettini
Univ ersit y of Pennsylv ania
and
Univ ersita' di Bologna
Carlotta Berti Ceroni
Colum bia Univ ersit y
and
Univ ersita' di Bologna
May 22, 1995
Abstract
This pap er develops a model of endogenous gro wth with o verlapping
generations toinvestigate thejointdetermination ofsocialsecurity,public
investment and gro wth in a small op en economy. We argue that a pure
pay-as-you-gosystempro videsthetaxpa yerswiththeincentivestosupp ort
gro wth-oriented p olicies, which increase the future pro ductivity of lab or.
We nd thatoutcomes characterizedbyp ositivelevels ofintergenerational
redistribution, publicinvestmentand long rungro wthcanb e sustainedas
subgame-p erfect Nashequilibria ofan in nitelyrep eated intergenerational
game, if and only if the marginal pro ductivity of public capital is large
enough. Furthermore, w e show that transfers either como ve with public
investment and gro wth or display a non-monotonic relation, where they
initially increase along with public investment and gro wth and then
de-crease. ( JEL, E62, H55)
3
Weare greatly indebted to Roberto Perottiand Jose-Victor Rios-Rull for their help and
encouragement. Weha vealsob ene tedfromcommentsandsuggestionsbyAlessandraCasella,
VincenzoDenicolo',Gio vanniForni,LorenzoGarbo,RogerLaguno ,AndreaMoro,andStephen
Most of the recen t economic growth models predict that purely redistributiv e
p olicies, such as so cial securit y programs, should depress growth through
nan-cial crowding out and the adverse incen tive e ects associated with distortionary
taxation. On the other hand, accum ulation ma y b e p ositiv ely associated with
gov ernment expenditure on education and with public in vestment in
infrastruc-ture or other pro ductiv e activities. A t times where industrialized countries are
confronted with the issue of establishing and main taining a sound scal p olicy
whilefosteringeconomic growth,thetrade-o b et weenpro ductiv eand
redistribu-tiv e gov ernment expenditures b ecomes esp ecially harsh and p olicy prescriptions
calling for a rede nition of the natureand scop e of the Welfare State gain
advo-cates.
However,thesepredictionsare not en tirelysupp orted bydata. Some evidence
can in fact b e providedregarding the existence of apositive association b et ween
redistributiv e expenditures as a p ercen tage of GDP and the long run growth
rate. The follo wingtable rep orts the correlation co ecientsb et weenthe average
gov ernmentredistributiv eexpenditureasap ercen tageofGDP
TR
Y
,theaverage
gross public domestic in vestmentas a p ercen tageof GDP
I g
Y
and the average
annual growth rate of real GDP (x ), in a sample of 72 countries for whic h the
relevantobservations are available,in the p eriod1970-85.
Table 1- CorrelationMatrix
x TR=Y I g =Y x 1 TR=Y 0.16 1 I g =Y 0.17 0.10 1 mean 0.01 0.05 0.03 max 0.08 0.20 0.08 min -0.04 0.00 0.00
Theredistributiv evariableTR iscalculated asthe di erenceb et weengov
ts alone. Both redistributiv e expenditure and public in vestment app ear to b e
p ositiv elycorrelated withgrowth. Also,redistributiv eexpenditure and public
in-v estmentarep ositiv elycorrelated. When other relevante ects are controlledfor,
including measures of the initiallevelofdev elopment,domestic in vestment
(pub-licand private),scholarization,gov ernmentconsumptionand p oliticalinstability,
evidence of a p ositiv e relation b et ween redistributiv e expenditure and growth is
main tained. 1
? ? ?
We b eliev e that the p ositiv e e ect of redistributiv e p olicies on accumulation
suggested by data can b e explained incorp orating p olitics in a standard growth
model. The bulkofthe recen tp oliticaleconom ymodelsof growtharguethat the
composition of so cial con icts ma y lead to extensive redistribution whic h has a
depressiv e e ect on growth. 2
Though this in tuition is app ealing, we think that
it needs quali cations. As long as redistribution plays a role in buying social
consensus for growth-oriented activities, it may well foster rather than depress
growth. 3
Redistribution ev ens up the costs and b ene ts of growth across so cial
classes. Ifredistribution mak esgrowthso ciallypalatable,theabsenceofadequate
redistributiv eprograms ma yfueladeep so cial con ictresulting inp o oreconomic
and growth p erformance. The presen t pap er explores this in tuition and puts
forward amotivationfor whyredistributiv eand growth-orientedp olicies, though
competing for scarce tax rev enues, migh t go hand in hand and bring ab out fast
economic growth.
The situation we hav e in mind is one where sustained growth is generated
1
Regressions displa ying the a verage gro wth rate as the dep endent variable and including
various components of go vernmentexp enditure among regressors ha veb een prop osed, among
others,byBarroandSala-y-Martin[ 5],Easterly[ 15], Perotti[ 22 ]andSala-y-Martin[ 29].
Sala-Y-Martin [ 29 ]explicitelymen tionsthesurprisinglyp ositiv eco ecientoftransfers.
2
Seeforinstance AlesinaandRodrick[ 1 ], Bertola[ 7], Krusell,Quadrini andRios-Rull[ 20 ],
Perotti[ 23],PerssonandTab ellini[ 24].
3
Sala-y-Martin[ 29] providesadi erentexplanation,where p ensionsma yincreasegro wthif
by public in vestment in capital go o ds. Public in vestment needs to b e nanced
through tax rev enues. The feasible level of scal pressure and the allo cation of
the gov ernementbudget todi eren texpenditure componentstends to re ectthe
in terestsof a majority of the p opulation and is agreedup on collectiv ely.
Hetero-geneousagentsb ear di eren tcostsandenjoydi eren tgainsinthegrowthpro cess.
Redistribution ma y help at ev eningout such di erences, create p olitical supp ort
for taxation and b o ost the tax rev enues availablefor gov ernmentexpenditure.
Inparticular,considerasmallop eneconomyandassumethatwhilenoimp
ed-iments exist to the mobilit y of private nancial wealth across national b orders,
other factors of pro duction, such as lab or and public capital, are non-tradable.
Heterogeneit y is in tro duced byassuming that the econom y isp opulated byov
er-lapping generations of nitely-liv ed and non-altruisticagents living for twop
eri-o ds. Except forage, agentsare iden tical. Agen ts work, save and paytaxeswhen
y oungandreceiv earetiremen tp ensionb ene tontopofprivatesavingswhenold.
Taxationfallsonlab orincomealone. Sincelab or supplyisinelastic,this amoun ts
tolump-sum taxation. However,taxrev enuescollection iscostly: this isthe only
source of distortion in our setup.
In this context, we argue that the en titlement to indexed-to-wages p ension
paymentsatretiremen tin apurepay-as-you-gosystem, nancedout of
contribu-tory taxation, mayprovide the taxpay erswith the incen tivesto supp ort
growth-orien ted p olicies, since it mak es them able to reap some of the b ene ts deriving
from increased taxation that would otherwise b e inaccessible to them. In the
absence ofin tergenerationalredistribution, sustained growth wouldb ep olitically
impracticable, thoughtechnically and economically feasible. Indeed, the b ene ts
deriving from (public in vestment driv en) growth are appropriable by taxpay ers
through twoc hannels: the increased marginal pro ductivityof private capitaland
the growth ofunit wages. Thelatteris lost atretiremen t. Theformer isnot ev en
atwork,inasmallop eneconom ywithcompletecapitalmobilit y,sincethein terest
rate is xed at the world-wide lev el. 5
Therefore the agreemen t to allo cate some
4
Public in vestmentin capital goo ds includes in vestmentin infrastructure, health facilities
and humancapital.
5
Inthecontextofano verlappinggenerationmodelwherepartiallyaltruisticagentsvoteon
render otherwise impraticable p ositiv e levels of taxation and public in vestment
p olitically viable and allowthe economyto take-o .
Noticethat afullyfundedsystem wouldnot achievethesame result,sincethe
rate of return fromthat system is exogenousand equal to the worldwidein terest
rate.
? ? ?
The relation b et ween redistribution, so cial cohesion and growth we come to
describehasb een prop osedbyp oliticalscien tistsand economistsindi eren t
con-texts. The dev elopmentof so cial securit y systems in continentalEurop e and the
U.S. b et ween the nineteen th and twentieth cen turies has b een explained as an
atteimpt of the ruling class to bind the workers'to the State, in resp onse to the
so cial unrestassociated tofasteconomicgrowthandthe spreadingofthe So cialist
mo vement (Rimlinger [ 25]). This idea also p ermeates the Solidaristic approach
to dev elopment of the small North-European countries, where the expansion of
the Welfare State has b een an important p olitical concomitan t of liberal trade
and growth-orientedp olicies (Katzenstein [ 16], [ 17]). Besides, the p o or economic
p erformance of some Latin Americancountries is sometimes explained as a
con-sequence ofthe deepso cial con ictthat, inthe virtual absenceofaWelfareState
ev ening out the e ects of economic growth, leads to frequen t upsurges of so cial
discontemptthat put growth and mark et-orientedp olicies at stake (Sachs[ 26]).
Our in terpretation of the so cial security system as an institution capable of
enlarging the (p olitical) supp ort forgrowth-orientedp olicies, isalso v eryclose in
spirit to the notion of Welfare State delineated by Gary Bec ker in his Treatise
on the family . As Bec ker puts it (pp. 370, [ 6]): "...expenditures on the elderly
are part of a so cial compact b et ween generations. Taxes on adults help nance
ecien t in vestments in c hildren. In return, adults receiv e public p ensions and
wheregro wthisledbyhumancapitalaccum ulation,Boldrin[ 10]showsthatno-gro wthequilibria
canarisealsointhecaseofclosedeconomies,atleastattheinitialstagesofdevelopment. This
happ ens b ecause the return from in vestment in public education, from the p oin tof view of
taxpa yers,is lo wwhenthe share of incomedevoted to in vestmentin physicalcapital is small
public in vestment, can ll the v oid left by the breakdown of so cial norms in
modernso cieties..." thatimposedonthe adultmem b ersof thefamilythe burden
of in vesting in the o spring's assets and of supp orting the elderly. Becker do es
not model formally howsuch in tergenerationalagreemen t should emerge and b e
enforced, nor do es heexplain indetail its c haracteristicsor the consequences for
growthof its in tro duction,whic h isindeed our scop e.
? ? ?
Wethinkof scalp oliciesasendogenousvariableswhic haredeterminedthrough
theaggregationofindividualpreferencesbysomemec hanismsuchasmajorityv
ot-ing. In the context of dynamic models, a major conceptual problem arises. The
dicult y has to do with the relation b et ween curren t and future p olicy c hoices,
the in teractionb et weenp olicyand state variablesand with the wayexpectations
on future p olicies are formed.
On onehand,itis wellknownfromthe publicc hoiceliterature onso cial
secu-rit y, that, if the future levelof p ension b ene ts is b eliev ed to b e independen tof
the curren tone (agentstakefuture p olicy c hoicesas giv en,when deciding onthe
curren tones), ap ositiv elev elof redistribution can nev erarise asa p olitical
equi-librium, unless the old are p olitically predominant. Since there are no incen tives
to payp ensions atanyp oint intime, the onlyrationalexpectation onthe future
level of transfers is zero. In our setup this implies that public in vestmentis also
set equal tozero.
Inordertoobtainarelationb et weentransfers,publicin vestmentandgrowth,
we need to extend the agents' rationalit y to encompass strategic b ehaviour. In
otherwords,weneedtoassumethattheagentsrecognizethatfuturep olicyc hoices
dep end onthecurren tones. However,the in teractionb et weenstatevariablesand
the agents' strategic incentives quickly mak es the analysis v ery complicated in
general,ev eninstandard dynamicmodels. 6
Thisiswhereoursimplesetuphelps.
In fact, most complications can b e avoided in our model due to the small op en
economyassumption and preferences homotheticit y.
6
For a rigorous analysis of dynamicp olitico-economic equilibria, see Krusell, Quadrini and
dev elop edbySjob olm [ 28 ]inaOLGmodelwithoutpro ductionandaccumulation
where b ene ts and contributions are determined through a majority v ote rule.
The most preferred taxrate by the median v oterisshown tob e sustainable as a
subgame-p erfect Nash equilibrium ina rep eated game. 7
Veryrecen tly,BoldrinandRustic hini[ 11]extendedthe analysistoshowthata
pay-as-you-gosystem can b e supp orted asthe subgame-p erfectNashequilibrium
ofanin nitelyrep eatedgameinastandardOLGmodelwithcapitalaccumulation
where the levelof so cial securit y is c hosen through majority v oting. Within this
model, theyin vestigatethe dynamic prop erties of the so cial securit y system and
the impactof c hangesinthe exogenousgrowth rate ofp opulation.
Our contribution uses a similar approach in a OLG model with endogenous
growth tostudythejoint determinationofpro ductiv eexpendituresand transfers,
inordertoshedligh tontherelationshipb et weenthepay-as-you-gosystemandthe
rateofgrowthoftheeconom y. Inparticular,attheb eginningofeac hp eriod(that
is, ineac h stage of the game),the y oung, who act as dictators, c ho osethe shares
of theirlab orincomethattheywanttodev otetop ensionsforthe curren toldand
to in vestmentin public capital. We can showthat, whenev er the p otentialgains
from growth are large enough, an outcome c haracterized by sustained balanced
growthcan b e supp orted as asubgame-p erfect Nashequilibrium ofthe in nitely
rep eatedin tergenerationalgame. Thecrediblethreattob edeniedtheen titlement
to p ensions in old age by the follo wing generation detersthe curren ty oung from
defecting bynot payingp ensions tothe co existing old. 8
This equilibrium canb e in terpretedasthe creation by theso ciet yof an
infor-mal constraint(so cialnorm)suchthatthe y oungtransferresources totheoldand
carryout in vestmentsthatwill b ene tthefuture generations,inexchangeforthe
old's previous in vestmentand expecting the future generationto follo wthe same
7
The rst author toanalyzea median votermodelofso cial securitywas Browning[ 12]. A
surveyofmodelsofvotingforso cialsecuritythatfollo wedandextendedBrowning'scontribution
can b efoundinBoadwa yandWildasin[ 8]. SeealsoTab ellini[ 30].
8
Noticethattheagentsthatarecalledontoactineachstageofthegameareineveryrespect
identical. Therefore,in order tocharacterizetheiroptimalcourseof action,wesimplyneedto
The main resultsof our model can b e summarized asfollo ws:
If the strategies of the players are history-indep enden t, that is, the y oung
do not take in to account the past histories when c ho osing their actions,
the only subgame-p erfect Nash equilibrium is such that there is no public
in vestment,noredistribution and the econom yexperiencesnogrowth.
If and only if the marginal pro ductivity of public capital is large enough,
stationary outcomes c haracterizedbyp ositiv e levelsof redistribution,
pub-lic in vestment and a p ositiv e rate of long-run growth can b e sustained as
subgame-p erfect Nashequilibriaof the in nitely rep eatedin tergenerational
game. Inparticular,westudythose scalp oliciesthatmaximizethewelfare
of the y oung, who b ehav easdictators at eac h stage ofthe game.
In aneigh b orho o d of the equilibrium withp ositiv egrowth: (a) the shareof
publicin vestmentinlab orincomeandtherateofgrowthcomo veinresp onse
to c hanges in the exogenous variables; (b) the share of transfers in lab or
income either como ves with growth or displays a non-monotonic relation
where transfers initially increasealong with growth and then decrease.
The structure ofthe pap er is asfollo ws. Section2sets out the model and the
p olicygameanddiscussesb oththecompetitiveequilibriumandtheequilibriumof
the game. Section3presen tsthecomparativ estaticsresults. Section4concludes.
2. The model
2.1.Theeconomicenvironment
We analyze asmall op en econom ywith two-p eriod liv edov erlapping generations
agents. Population is assumed to b e constant: in eac h p eriod, an equal mass
of y oung and old is aliv e, whic h we b oth normalize to one. Labor is supplied
inelastically bythe y oung.
U t c t t ;c t t +1 =log c t t + log c t t +1 (2.1) where c t s
is the consumption attime s of anagent b orn at time t:
Outputis pro duced according tothe follo wingpro duction function:
Y t =1 L t K 1 0 t g t (2.2)
where L denotes aggregate lab or, K denotes aggregate private capital, g =G=L
is the amoun tof publiccapital p er workerand 1 isthe total factor pro ductivity
whic h is assumed to b e constant. Following Barro [ 3], we assume that public
services are rivaland excludable.
The laws ofmotion of private and public capitalare giv enby:
K t +1 =(10 K )K t +I p t (2.3) G t +1 =(10 G )G t +I g t (2.4)
Private capital can mo ve b et ween the foreign and the domestic pro duction
sector at no cost. The worldwide in terestrate on nancial assets is denoted by
r 0:
2.2.Thegame
In our model, the y oung consume, sav e and ma y use part of their lab or income
to nance in vestmentin public capital and transfers to the old. We will analyze
the c hoice of these two scal p olicies in the context of an in nitely rep eated
in tergenerationalgame. Let usb egin by describingthe "constituent" stagegame
0. A ttimet ,they oung,whorepresen tthemajorityofthep opulation,decidewhat
fraction of their lab or income to dev olve to public in vestment and to transfers.
These fractions (tax rates)are denoted by
i;t and
tr ;t :
Noticethat,ineac hstageofthegame,theoldplaynoroleinthedetermination
oftheequilibriump olicies. Thisisinlinewiththemainpurp oseofthepap er,that
istoin vestigatethe jointdeterminationof transfersandpro ductiv eexpenditures,
in relation with the rate of growth, in order to nd an explanation for why ev en
redistribution wouldnot add m uchinsight. 9 LetA t =( i;t ; tr ;t
)b etheactionspacewith
i;t ; tr ;t 0and t i;t + tr ;t 1:
We will consider a rep eated game with p erfect information where players can
observe all previous actions. Thus, let h
t = ( i0 ; tr0 ; i1 ; tr1 ;:::; i;t 01 ; tr ;t 01 ) b e
the history of the game atthe end of stage t01:Furthermore,we leth
0 =;:
Inthis setting,astrategy forthe y oung isa contingentplan of howtoplayin
eac h stage t for p ossible history h
t
: If we denote with H
t
the set of all p ossible
histories h
t
, a strategy
t
is a map from the set H
t
to the action space A
t ; that is, t (h t )2A t forall h t .
Forev ery history (
i0 ; tr0 ;:::; i;t ; tr ;t
;:::) of actions, the pay o toeac h player
t is giv en by her lifetime utility, evaluated at the competitive equilibrium with
( i;t ; tr ;t ) and ( i;t +1 ; tr ;t +1 ):
2.3.Theeconomicequilibrium
We will now c haracterize the competitive equilibrium of our econom y, given a
sequence of historiesfh t g 1 t =0 :
An agentb orn at time t solvesthe follo wingmaximization problem:
V t =max c t t ;c t t+1 n log c t t + log c t t +1 o (2.5)
subjectto:
c t t =w t (10 t )0s t c t t +1 =s t (1+r)+TR t +1 (2.6) where TR t +1 = tr ;t +1 (10 i;t +1 0 tr ;t +1 )w t +1 :
Here,weassumethatthereareconv excostsincollecting taxes;ifthey oungat
timetputaside
t
oftheirlab orincomeforpublicexpenditures,only 0 t ( t 0 2 t )
is available to nance public in vestment and/or lump-sum transfers to the old.
9
Ifweassumedap ositiv erateofgro wthofp opulation,ateachp oin tintimetheyoungwould
b ethemajorityofthep opulation. Wecouldthende netheequilibriump oliciesastheoutcome
and transfersare thusgiv enby: I g t = t 0 t w t TR t = (10 t ) 0 t w t (2.7) where t i;t = t and (10 t ) tr ;t = t
. Notice that b oth the level of public
in vestment and the lev el of transfers can b e written in terms of
i;t and tr ;t : In particular, we hav e: I g t =[ i;t (10 i;t 0 tr ;t )]w t (2.8) TR t =[ tr ;t (10 i;t 0 tr ;t )]w t (2.9)
The solutionto problem (2.5) yieldsthe follo wingsavingfunction:
s(r;w t ; t ;TR t +1 )= 1+ w t (10 t )0 TR t +1 (1+ )(1+r) =s t (2.10)
Private nancial wealth at the b eginning of p eriod t+1; A
t +1
, is thus giv en by
the savings of the y oung at time t, namely A
t +1 =s(r;w t ; t ;TR t +1 ). From rst
order conditions for pro t maximization weget:
w t = Y t q t =(10 ) Y t K t (2.11) where q t
is the ren tal rate of capital. Equilibrium conditions on the go o ds and
assets mark ets imply:
1 F t +1 =F t +1 0F t =S t 0I t (2.12) q t = K + r (2.13) whereS t andF t
resp ectiv elydenoteaggregatesavingsandthe sto c kofnet foreign
assets held bythe privatesector attime t and whereF
t =A t 0K t . It can easily
b e v eri ed that the competitiveequilibrium implies 10
:
10
Fromno won,wewill set
G =0 :
K t G t == 1 10 r+ K (2.14) Y t G t = = 10 r+ K 10 1 1 (2.15) Let x Z t
denote the rate of growth of Z. By observation of (2.14) and (2.15) it
should b e clear that, in equilibrium, x K t = x Y t = x G t = x t 8t . The economy's
dynamics inequilibrium isc haracterizedby:
1 G t +1 = i;t (10 i;t 0 tr ;t ) 1 0 G t 1 F t +1 = & t 0 & t 01 1+x t 0x t G t (2.16) where: & t = s(r;w t ; t ;TR t +1 ) G t
De ning the Balanced Growth Path (BGP) as the lo cus where all variables
grow at a constant (p ossibly common) rate and
t
= 8t; it can b e v eri ed
that constancy of p olicy variables over time, i.e.
i;t = i and tr ;t = tr 8t , is
b oth necessary and sucient for the economy to move along a stationary path
with sustained growth. It is also immediate to recognize that, along the BGP,
x F =x K =x Y =x G
=x , where the equation for the rate of growth is giv enby:
x= i (10 i 0 tr ) (2.17)
In our model, growth is driv en by the accum ulation of capital in the public
sector, whic h is nanced out of tax rev enues on lab or income. This implies that,
giv en the share of GDP to b e dev oted to public in vestment at eac h p oint in
time, the equilibrium rate of growth is increasing with the average pro duct of
public capital in pro duction, , whic h is constant over time and is exogenously
determined by the worldwide in terest rate r and the tec hnological parameters,
K
;1and : Notice that, along stationary paths, the ratio of public in vestment
to GDP, i (10 i 0 tr
),is constant,so that the technology for theaccumulation
withtheequilibriummarginalpro ductivityofpubliccapital . Giv en ,c hanges
in the exogenousvariablesthat trigger anincreasein the averagepro ductivityof
public capital, , stim ulategrowth. Instead, the overalle ect of anincrease in
ongrowthisapriori indeterminate. Infact,although increasesin raise taxable
incomeand fostergrowth,thee ect ontheaveragepro duct ma yturnout tob e
negativ e.
Giv en the marginal pro ductivity of public capital, growth is increasing with
the share of public in vestment in lab or income,
i (10 i 0 tr ); whic h in turn
dep ends on p olicyvariables. Anincrease in
i
has a twofolde ect ongrowth: on
one hand, it increases the share of tax rev enues allo cated to public in vestment
and stim ulatesgrowth;ontheotherhand, itdepressesgrowththrough the higher
ineciency of taxation. An increase in redistribution, as implied by an increase
in
tr
, unambiguously depresses growth, since it b oth reduces the share of
rev-en ues allo cated to public in vestment and increases the collection costs of taxes.
Nevertheless,as we will extensiv ely
show,there cannotexist anequilibrium with
p ositiv egrowth and no redistribution.
Letusnowc haracterizetheeconomicequilibriumthatariseswhenallhistories
of previous actions are giv en by the null v ector, that is, h
t
= (0;0;:::::0) for all
t . From (2.17), it is immediate to v erify that, in this case, there is no economic
growthand the equilibrium allo cations in ev eryp eriod are giv enby:
c y = 1 1+ Y 0 c o = 1+ Y 0 ! (1+r) (2.18) where Y 0
is the giv eninitial lev el of output, c y
and c o
denote consumptionwhen
y oung and old, resp ectively.
2.4.Equilibria of thegame
Going backtothe game0thatwedescribedab ov e,assumenowthatev eryy oung
generation adopts ahistory-indep enden t strategy. In other words,in eac hp eriod
equilibrium of the in nitely rep eated game 0(1 ) is to set
i;t =
tr ;t
= 0, for all
t . In fact, if future actions will b e independen t of the curren t ones, there is no
incen tive whatsoev er for the y oung to carry the cost of paying transfers to the
old. Consequently, the y oung hav enoin terestto in vestin publiccapital as well,
since they anticipatethat future generationswill b ehav ein the samewaysothat
they will not receiv e p ensions when old. Thus,we can summarize this discussion
as follo ws:
Proposition 1. If ev ery generation adopts a history-indep enden t strategy, the
only subgame-p erfect Nash equilibrium of the in nitely rep eated game 0(1 ) is
3
=(0;0):
In conclusion, if there isno linkb et weenpast and curren tp olicies, the y oung
will neitherin vestinpublic capitalnorpaytransfers tothe old,the econom ywill
experiencenogrowth and the consumptionallo cations will b e equal toc y
and c o
:
Clearly, this equilibrium is not v ery satisfactory. More generally, actions are
functionof thehistoryofthe game. Thus,our nextstep willb etoallowagentsto
adopt history-dep enden tstrategies and toc haracterize the equilibrium outcomes
in this case; in particular, we will show that there are cases where it is p ossible
to construct equilibrium strategies such that the outcome of the game will yield
strictly positive valuesfor b oth
i;t and
tr ;t
,and allgenerations will b eb etter o
than in the equilibrium that we havejustdescribed.
Consider rst the follo wing equilibrium candidate of our game:
( 3 i ; 3 tr )=argmax i ; tr flog [ W( i ; tr ;r;X)w t ]+Cg s:t: W( i ; tr ;r;X)= h (10 i 0 tr )(1+r+ tr )+ i 1 tr (10 i 0 tr ) 2 X i i 0 tr 0 i + tr 1 (2.19)
where X represen ts the marginal pro ductivity of public capital and C is a
function of parameters. Inwords, 3 i and 3 tr
maximizethe indirectutilit yfunctionof they oung,when
function (2.5);furthermore, itis easytoshowthat the couple( 3 i ; 3 tr ) thatsolves
(2.19) is the same that maximizesthe functionW(
i ; tr ;r;X) . Noticethat 3 i and 3 tr
can b eseenastheecien tsteadystatetaxrates,inthe
sense that theymaximizethe welfareofeac hgenerationof taxpay ers. Noticealso
that the solutionto the problem (2.19)is time independen t,giv enthe constancy
ofthein terestraterandoftheequilibriummarginalpro ductivityofpubliccapital
X.
Intheconstructionoftheequilibriumstrategy,wewillin tro duceanotherp olicy
outcome, whic h maximizes the indirect utilit y of the y oung when they do not
pay transfer to (i.e. punish) the old, but nev ertheless expect that the follo wing
generation will select the ecien tp olicies 3 i and 3 tr . 11
We denote this outcome
as i ,where: i =argmax i f(10 i )(1+r)+ 3 tr (10 3 i 0 3 tr )[1+ i (10 i )X] g s:t: 0 i 1 (2.20)
Assume now that the y oung at time t adopt the follo wing strategy, that we
will denoteby 3 t (h t ): 1. if h t =;; 3 t =( i ;0) 2. if h t =(1 ; 3 i ; 3 tr ); 3 t =( 3 i ; 3 tr ) 3. if h t = 1 ; 0 i ; 0 tr ;witheither 0 i 6= 3 i or 0 tr 6= 3 tr
orb oth,countthenumb er
of consecutiv e p eriods up to t 0 1 included with t = 0 excluded, where
i 6= 3 i and/or tr 6= 3 tr
: Let this numb erb e denoted by N :
1. if N isev en, 3 t =( 3 i ; 3 tr ) 2. if N iso dd, 3 t =( i ;0):
In words, we constructed a strategy such that, at any p oint in time t , if the
old did not "co op erate" in the previous p eriod (i.e. they did not play 3 i ; 3 tr ), 11
It iseasy toverify,byobservationof2.19and2.20,thatii 3
were punishing generation t02 for deviating. After the punishmen tp eriod, the
"goo d" equilibrium is immediately restored.
If ev ery generation adopts the strategy 3
t (h
t
), the outcome of the in nitely
rep eated game will b e ( 3
i ;
3
tr
) for all t; expect for t = 0; where the outcome is
(
i
;0). Notice that, there is actually an in nite num b er of stationary outcomes
with p ositiv e
i and
tr
whic hcan b e supp orted asa subgame-p erfect Nash
equi-librium, usingthe strategythatwedescribedb efore. Inparticular, wecanshowa
Folk-theorem-likeresultwhereallindividuallyrationaloutcomescanb esustained
as subgame-p erfect Nash equilibria. However, the selected equilibrium outcome
( i ;0; 3 i ; 3 tr ; 3 i ; 3 tr
;:::) is, by construction, the one that maximizes the welfareof
the agentswho are the majorityof the p opulation ineac h stage of the game.
From now on, we will limit the analysis only to the selected equilibrium
( i ;0; 3 i ; 3 tr ; 3 i ; 3 tr
;:::), whic hb ecomes the equilibrium of our model.
In order toprov e that the strategy 3
t (h
t
) isa subgame-p erfect Nash
equilib-rium of the in nitely rep eated game 0( 1 ), we need a preliminary result ab out
the necessary and sucient condition for the existence of an in terior solution to
the maximizationproblem (2.19).
Notice rst that the rst orderconditions of(2.19) are giv enby:
W i =0 (1+r+ 3 tr )+ 3 tr (10 3 i 0 3 tr )X(103 3 i 0 3 tr )=0 (2.21) W tr =0 (1+r+ 3 tr )+( 10 3 i 0 3 tr )+ 3 i (10 3 i 0 3 tr )X(10 3 i 03 3 tr )=0 (2.22)
These conditions can b e explained in tuitively as follo ws. In our model there
existtwoalternativ eformsofin vestment,inprivateorpubliccapital. Thenetrate
of return onthe former, inequilibrium, is exogenouslydetermined by the
world-wide in terest rate r, while the gross rate of return on the latter is endogenous
and corresp onds to the ratio b et ween the p ension receiv ed and the taxes paid
by eac h generation, that is tr (10 i 0 tr )[1+ i (10 i 0 tr )X] i + tr . Each dollar
levied to nancepublic in vestmentorso cial securit ycorresp ondingly reducesthe
totalamoun tofresourcesavailableforin vestmentinprivatecapital. Themarginal
net return from in vestment in public capital, that is the marginal increase in
p ension income min usone. By conditions (2.21)-(2.22), 3
i
and 3
tr
are such that
the marginal cost of increasing eac h of the tax rates is equal to the marginal
b ene t ofincreasing it,or, equivalently, 3
i
and 3
tr
are suchthat the marginalnet
rate of returnfrom in vestmentin publiccapital is equal tothe net rate of return
from in vestmentinprivatecapital and tothe world-widein terestrate r.
Although(2.21)-(2.22)formanon-linearsystemin
i and
tr
,and closedform
solutions for 3 i and 3 tr
cannot b e found, we are nonetheless able to derive some
results whic hallow usto c haracterizethe solution tothe problem of the y oung.
First ofall, note that 3
i
and 3
tr
b ear the relation:
3 i (10 3 i )X = 3 tr (10 3 tr )X01 (2.23)
asitcanb eeasilyv eri edbysubtracting(2.21)from(2.22). Second,wecanshow
that:
Lemma1. If the solution to the problem of the middle-aged (2.19) lies on the
b oundary, then the solutionis ( 3 i ; 3 tr )=(0;0) . Proof. Set tr = 0. Then from (2.21), W i = 0(1+r) < 0, whic h implies 3 i = 0. Now, set i = 0: Again, from (2.22), W tr = 0(r+2 tr ) < 0, whic h implies 3 tr =0:Finally,if i + tr =1; b oth W i and W tr tend to 01 :
Thein tuition forthis resultisprett yobvious. When thereis noin vestmentin
public capital and consequen tlyno growth, asit is implied bysetting
i
=0, the
economy is dynamically ecien t, as long as r 0, and there is no welfare gain
for the y oung from the in tro duction of a so cial securit y system. Besides, in the
absenceof p ensions(
tr
=0), the y oungare nev erwillingto nancegrowth. This
dep ends on the small op en economy and complete capital mobilit y assumptions
whic himply thatpublicin vestmenthas noe ect ontheequilibrium in terestrate.
aged (2.19) is ( 3 i ; 3 tr )= 1 4 ; 1 4 .
Proof. NotethatW(
i ; tr ;r;X)=8( i ; tr ;r)+X9( i ; tr ),where8( i ; tr ;r)= (1+r+ tr )(10 i 0 tr ) and 9( i ; tr ) = i 1 tr (10 i 0 tr ) 2 . It is easily v
er-i ed that the function 9(
i ;
tr
) has a unique maxim um at 1 4 ; 1 4 , or
equiva-len tly, that there exists " > 0 such that 9 1 4 ; 1 4 0 9( i ; tr ) C " , ( i ; tr ) 2= D 1 4 ; 1 4 ;"
. Nownote thatW 1 4 ; 1 4 ;r;X X 9 1 4 ; 1 4 ,since 8( i ; tr ;r)0. MoreoverW( i ; tr ;r;X) max 8( i ; tr ;r)+X9( i ; tr ) and W 1 4 ; 1 4 ;r;X 0 W( i ; tr ;r;X)X h 9 1 4 ; 1 4 09( i ; tr ) i 0 max8( i ; tr ;r),8( i ; tr )2= D 1 4 ; 1 4 ;" .
Note also that the LHS term in the last inequalit y is strictly p ositiv e for X
large enough. Then, for X large enough, W 1 4 ; 1 4 ;r;X 0W( i ; tr ;r;X) >0, 8 ( i ; tr )2= D 1 4 ; 1 4 ;" , or, equivalently,W( i ; tr
;r;X) has a unique maxim um
at 1 4 ; 1 4
, for X large enough.
By observation of (2.19), it is clear that the y oung can b ene t from output
growthonlythrough the consequen tgrowth inp ension b ene ts. The growthrate
of b ene ts is indeed equal to the output and wages growth, since the ratio of
transfers to wages is constant ov er time. The utility of the y oung is therefore
increasing inthe growth rate of output. Giv enp olicy variables, this isincreasing
withtheequilibriumvalueofthemarginalpro ductivityofpubliccapitalX. Using
this in tuitionit is p ossibleto establishthe follo wingimportantresult:
Proposition 2. Thereexistsavalueofthe marginalpro ductivityofpublic
capi-tal X,suchthat, if and onlyif X >X the maximization problem (2.19)yieldsan
in teriorsolution, ( 3 i ; 3 tr ), with 3 i 2(0; 1 4 ] , 3 tr 2(0; 1 4 ] .
Proof. By Lemma 2 we know that ( 3 i ; 3 tr ) = 1 4 ; 1 4
is the unique absolute
maximum of (2.19), as X ! +1 . Since the rst term in W is decreasing in
i
and
tr
, it turns out that in teriorsolutions are such that 3 i 2(0; 1 4 ] ; 3 tr 2 (0; 1 4 ].
A sucientcondition for W(
i ;
tr
;r;X) to admitin teriorsolutions is:
W 1 4 ; 1 4 ;r;X = 1+ r 2 + 1 8 + 1 64 X >W(0;0;r; ; K )=(1+r)
X
to(2.19)willb eanin teriorone. Notethat,asX !0,theuniquesolutionto(2.19)
is(0;0),since(10 i 0 tr )(1+ r + tr )<(1+r) 8( i ; tr )6=(0;0). NowtakeX
suciently small to obtain( 3
i ;
3
tr
)=(0;0) and considerdecreasing values of X.
Noticethatneither 3
i nor
3
tr
canev ertakevaluesab ov ezeroasXdecreases,since
the rst term in(2.19)is monotonicallydecreasing in
i and
tr
. Therefore,there
exists a threshold X X such that for X X , the solution to (2.19) is (0;0) ,
while forX >X , thesolutionto(2.19)is ( 3 i ; 3 tr )with 3 i 2(0; 1 4 ] ; 3 tr 2(0; 1 4 ].
The ab ov eresults mak eclear that an equilibrium with p ositiv etaxation and
growthma yorma ynot exist,dep ending onwhether the gains fromgrowth,that
the y oungenjoyinpresence of p ensionb ene ts,are sucien tlylargesothat they
prefer the equilibrium with p ositiv egrowth tothe no-growth equilibrium.
Forgiv enp olicyvariables,thesizeofsuchgainsincreaseswiththeequilibrium
value of the marginal pro ductivity of public capital, X. The role of parameters,
(
K
;1 ; ),and the in terestrate, r, indetermining howlarge isX will b ein
vesti-gatedinthefollo wingsection. NoticealsothattheconstancyofX atequilibrium,
as impliedby equation(2.15), ensures that the conditions for the existence of an
equilibrium with sustained p ositiv egrowthare not put atstakebythe accum
ula-tion pro cess.
Moreover, in our set up, p ositiv e growth rates can arise only if some share
of the total tax rev enue is allo cated to so cial securit y. In fact, p ositiv e growth
requiresp ositiv epublicin vestmentateac hp ointintime. Thisinturnrequiresthe
shareof publicin vestmentinaftertaxincometob ep ositiv e. Sincethey oungcan
reap part of the b ene ts derivingfrom in vestmentinpublic capital onlythrough
therevaluationoffuturep ensionspayments,publicin vestmentinaftertaxincome
can only b e p ositiv eif the share of transfers inafter tax income is also p ositiv e,
since this ensures that p ositiv elevelsof p ensions will b e paid in the future.
Weare now ready to provethe main result of this section:
Proposition 3. If and only if X > X ; 3
t (h
t
) is a subgame-p erfect Nash
librium outcome ( i ;0; 3 i ; 3 tr ; 3 i ; 3 tr
;:::)? First of all, note that the sequences of
equilibrium lev els of public in vestmentand so cial securit y expenditure are giv en
by: I g 0 = i (10 i )w 0 TR 0 =0 I g t =i 3 w t fort =1;2;:::1 TR t =tr 3 w t for t =1;2;:::1
Second, notethat, sincet =1, the economymo vesalong astationarypath where
all economic variablesgrowat the common constant rate:
x 3 =i 3 X (2.24) where i 3 = 3 i (10 3 i 0 3 tr ) and tr 3 = 3 tr (10 3 i 0 3 tr
) are the constant
equilib-riumsharesoflab orincomewhic haredev olved,resp ectiv ely,topublicin vestment
and transfers, since t = 1. Note that tr 3
can also b e in terpreted as the
equilib-rium p ension replacemen t rate on earnings. Moreover, consumption allo cations
are equal to:
c 0 0 = w 0 (1+ ) " (10 i )+ tr 3 (1+x) (1+r) # c 0 1 = w 0 1+ [(1+r)(10 i )+tr 3 (1+x)] c t t = w t (1+ ) " (10 3 )+ tr 3 (1+x 3 ) (1+r) # fort =1;2;:::1 c t t +1 = w t 1+ [(1+r)(10 3 )+tr 3 (1+x 3 )] for t =1;2;:::1 (2.25) where 3 = 3 i + 3 tr
is the constant contributory tax rate, or, equivalently, the
constant level of scal pressure, since t =1, and x =
i (10
i
)X is the growth
rate realized b et weent=0and t =1.
Note that our so cial norm is such that expenditure on public in vestment is
initiated (one p eriod)b efore expenditure on so cial securit y,so that the rst
tries, where the birth of so cial securit y systems follo wed the institution of other
programs, suchas publiceducation and public in vestmentininfrastructure.
Itis worthwhiletostress that theadoption of theequilibrium strategy 3
(h
t )
yieldsaneciencygain,b ecausethecorresp ondingcompetitiveequilibriumyields
alev elofutilityforall generationshigherthanintheequilibriumthatweanalyzed
in the previous subsection, where
i =
tr
= 0 and there is no growth. Indeed,
whenX >X;they oungwouldratherputasideafraction 3
oftheirlab orincome
and receiv e a fraction 3
tr
of the future income, than not pay any taxes and not
receiv e any transfer. Furthermore, the old generations clearly b ene t from the
in tro duction of the so cial securit ysystem.
Going back to the motivation of this work, we would lik e to in terpret our
equilibrium as the emergence of a so cial norm, whereb y p ensions are paid in
ev ery p eriod to the old generations in exchange for their previous in vestment
in activities whic h b ene ted the curren t y oung. This so cial norm arises as the
equilibrium outcome of the in nitelyrep eated game,where the crediblethreatof
b eing punished by notreceiving anytransfer,incase the norm isviolated, mak es
all generationsb etter o , byfollo wingit.
3. Comparativ e Statics
In the subgame-p erfect Nash equilibrium that arises if the equilibrium strategy
3
(h
t
) is playedby eac h subsequent generation of taxpay ers, public in vestment,
so cial securit y expenditure and growth are endogenously and sim ultaneously
de-termined variables,whose size dep ends crucially on the equilibrium value of the
marginalpro ductivityofpubliccapitalinpro duction,X,whic hinturnisuniquely
determined, giv enthe in terestrate, r, and the tec hnologyparameters,
K
;1and
.
Inthissection,westudythee ectsonpublicin vestment,so cialsecurit y
expen-diture andgrowthof ceterisparibus variations inr;
K
;1and ,by in vestigating
the comparativ e statics prop erties of this equilibrium, for t 1. The main
K
with 1;(2) theequilibrium shareof publicin vestmentinlab or incomei 3
and the
equilibrium income growth rate x 3
resp ond p ositiv elyto c hangesin
K
;1 and
that trigger increases in X and are decreasing with r; (3) the share of transfers
in lab or income tr 3
resp onds p ositiv ely to c hanges in
K
;1 and that trigger
increases inX and isdecreasing with r, if r0: 5 and/or X isb elo w athreshold
value f X. When r < 0: 5 and X > f X, tr 3
ma y resp ond negativ ely to c hanges in
K
;1and that trigger increasesinX and itmayb e increasing with r.
In ordertodeterminehowi 3
,tr 3
andx 3
resp ondtoc hangesinr;
K
;1and ,
we use the follo wingpro cedure. First,we study the function X(r;
K
;1 ; ) and,
in particular, we determine the sign of the partial derivatives of X with resp ect
to r;
K
;1 and . Second, we derive a sucien t condition for 3 i and 3 tr to b e
di eren tiable in r and X. Whenev er this condition is satis ed, the sign of the
partial derivatives of 3 i and 3 tr
with resp ect to
r
and X can b e determined by
applying the Implicit FunctionTheorem (IFT). Third,we com binethese results
toestablishthe direction ofadjustmen tof 3 i and 3 tr tovariations in r; K ;1and
. Finally,wepin downthe e ects of c hangesinr;
K ;1 and oni 3 , tr 3 and x 3 .
Our rst result, whic hwe state without pro of, provides some information on
thelimitingb ehaviorofthefunctionX(r;
K ;1 ; ). Inparticular,itisimmediate to showthat: Lemma3. lim K!0 X =X 00 0 >0 lim 1!0 X = 0 limX !0 =+1 lim K!1 X =X 00 1 <0 lim 1!1 X =+1 lim !1 X = 0
Consider nowthe subset of v ectors of exogenous variablessuch that the
nec-essary and sucient condition for the existence of in teriorsolutionsto the
maxi-mizationproblem (2.19)issatis ed,thatisrestrictattentiontoc hoicesofr;
K ;1
and suchthat X > X.
Thenwe can provethe follo wing:
Lemma4.IfX>X,theequilibriumvalueofthemarginalpro ductivityofpublic
@r <0 @ K <0 @1 >0
Proof. From (2.21) it can b e v eri ed that in terior solutions to (2.19) can b e
obtained only if X > 1. It is immediate to c heck that X > 1 implies @X @r < 0, @X @ K <0and @X @1 >0.
The in tuition for the ab ov eresult is prett yobvious. Recall that the marginal
pro ductivityofpubliccapitalisthepro ductoftwofactors: theelasticityofpublic
capital in pro duction, , and the average pro ductivity of public capital, . For
giv en , the latter isan increasing function of the private to public capitalsto c k
ratio, , whic hin turn is decreasing with
r and
K
and isincreasing with 1.
Thesign ofthe partialderivativeofX with resp ectto ,isam biguousapriori
and dep ends onthe exogenousvariables. Asucien tcondition to obtaina
nega-tiv eov eralle ect of onX isgiv enby1(10 )>r+
K
,whic h wehenceforth
assume to b esatis ed.
Before applying the IFT to determine the sign of the partial derivatives of
3 i and 3 tr
with resp ect to r and X, we need to mak e sure that 3 i and 3 tr are
di eren tiable inr and X. The follo wingresult guarantees that this isindeed the
case, atleast for X large enough.
Proposition 4.Thereexists avalueof themarginal pro ductivityof public
cap-ital, c
X >X, such that, if X > c
X, the maximization problem (2.19) yields an
in terior,unique and di eren tiable solution, ( 3 i (r;X); 3 tr (r;X)), with 3 i 2(0; 1 4 ] and 3 tr 2(0; 1 4 ] .
Proof. See App endixB.
Prop osition 4 implies that all the di eren tiable solutions to (2.19) are also
in teriorsolutions. For c
X >X >X,theremayexistin teriorandnon-di erentiable
solutions to(2.19).
Henceforth,werestrictourattentiontoc hoicesoftheexogenousvariablessuch
that X> c
and of the equilibrium so cial securit y tax with resp ect to the in terest rate is
strictly negativ e,whilethepartialderivativeofthe equilibriumpublic in vestment
tax and of the so cial securit y tax with resp ect to the marginal pro ductivity of
public capital isstrictly p ositiv e,for X <1 . Namely,for X <1 , we have:
@ 3 i (r;X) @ r <0 @ 3 i (r;X) @X >0 @ 3 tr (r;X) @r <0 @ 3 tr (r;X) @X >0 Moreover: lim X!1 @ 3 i (r;X) @ r =0 lim X!1 @ 3 i (r;X) @X =0 lim X!1 @ 3 tr (r;X) @ r =0 lim X!1 @ 3 tr (r;X) @X =0
Proof. See App endixB.
To grasp the in tuition b ehind these results, some observations are in order.
First, recallthat,inequilibrium, themarginal b ene tof increasinganyofthe tax
rates, whic h is the marginal net return from public in vestment,or, equivalently,
the marginalincreaseinp ensionincomemin usone,m ustb eequaltothemarginal
opp ortunity cost of increasing any of the tax rates, that is the net rate of return
from other forms of in vestment, r. Second, note that, in equilibrium, the total
tax rev enue is increasing with b oth tax rates, while the marginal tax rev enue
is decreasing with them. In fact, the marginal increase in rev enue follo wing an
increase in anyof the tax rate is equal to [102( 3
i +
3
tr
)], whic h is p ositiv efor
( 3 i ; 3 tr ) 1 4 ; 1 4
and is decreasing with 3
i
and 3
tr
, due to the conv excollection
cost of taxes. Third,note that, giv en 3
tr
, the marginal b ene t of increasing 3
i is
diminishing, since tr 3
b ecomes progressively smaller and the total and marginal
collection costs of taxes b ecome progressively larger as 3 i increases. Giv en 3 i ,
the marginal b ene t of increasing 3
tr
, is also diminishing, since i 3
b ecomes
pro-gressiv ely smaller and the total and marginal collection costs of taxes b ecome
progressivelylarger as 3
tr
increases. Fourth,note thatan increasein 3 tr ( 3 i )also
has a cross e ect onthe marginal b ene t of increasing
i (
tr
). The sign of such
e ect isam biguous: anincreasein 3
tr
implieshighertotal andmarginalcollection
costs, on one hand, and a larger transfer share, on the other hand, while an
in-crease in 3
i
implies higher total and marginal collection costs, on one hand, and
a larger public in vestmentshare, on the other hand. However,our results imply
that thesee ects, if negativ e,are not v ery large inabsolutevalue.
Starting from an equilibrium situation, an increase in X, giv en r, increases
the rateofgrowth oflab orincomeand thereturnonin vestmentinpubliccapital,
for giv en tax rates. As a consequence, the marginal b ene t of increasing any of
the taxrates b ecomes larger than the marginal opp ortunity cost of increasing it.
In order to restore the equilibrium, b oth tax rates must b e increased. In fact,
this impliesa reductionof themarginal b ene t of increasinganyofthe tax rates.
On the contrary, an increase in r, giv en X, pushes the opp ortunity cost of eac h
additional dollar paid out in taxes up. In order to restore the equilibrium, b oth
tax rates m ust then b e reduced.
Putting together Lemma 4 and Prop ositions 5, it is immediate to conclude
that:
Corollary 1. The equilibrium value of b oth tax rates is strictly increasing with
the total factor pro ductivityandstrictly decreasingwith the depreciation rateof
privatecapital,theelasticit yofpubliccapitalinpro ductionand thein terestrate,
for X <1 . Namely,for X <1 ,we have:
d 3 i dr <0 d 3 i d K <0 d 3 i d1 >0 d 3 i d <0 d 3 tr dr <0 d 3 tr d K <0 d 3 tr d1 >0 d 3 tr d <0 Moreover: lim X!1 d 3 i dz =0 z=r;1 ; K ; lim X!1 d 3 tr dz =0 z =r;1 ; K ;
higher equilibrium values of b oth tax rates. Instead, ceteris paribus increases in
the in terest rate, r, decrease the equilibrium value of b oth tax rates, since b oth
the direct e ect, through the increased opp ortunity cost, and the indirecte ect,
through the reducedvalue of X, push in this direction.
Having c haracterizedthe e ect of c hangesin the exogenous variables on the
equilibrium tax rates,we nowturn our attentiononthe e ect of suchc hangeson
the equilibrium shares of lab or income whic h are dev oted to public in vestment
and so cial securit yand onthe equilibrium growth rate of income.
As for the equilibrium public in vestment share and the equilibrium income
growthrate, wecan provethe follo wing:
Proposition 6.The equilibrium share of public in vestment in lab or income is
strictly increasing with the total factor pro ductivity and strictly decreasing with
the depreciationrate of private capital,the elasticityof public capitalin pro
duc-tion and the in terestrate, for X <1 . Namely,forX <1 , we have:
di 3 d1 >0 di 3 d K <0 di 3 d <0 di 3 dr <0 Moreover: lim X!1 di 3 dz =0 z =r;1 ; K ;
Proof. See App endixB.
By putting together Prop osition 6 and Lemma 4, it is then immediate to
conclude that:
Corollary 2. Theequilibriumrateofgrowthofincomeisstrictly increasingwith
the total factor pro ductivityandstrictly decreasingwith the depreciation rateof
privatecapital,theelasticit yofpubliccapitalinpro ductionand thein terestrate.
That is: dx 3 d1 >0 dx 3 d K <0 dx 3 d <0 dx 3 dr <0
order.
First,byProp osition 5,weknow thatc hangesinthe exogenousvariablesthat
mak e in vestment in public capital more rem unerative, relativ e to in vestment in
privatecapital, imply astronger scal pressureatequilibrium, since b oth 3 i and 3 tr
increase asa consequence of such c hanges.
Second, note that, giv en the level of scal pressure, such c hanges also imply
that alargershare oftaxrev enueisallo catedtopublicin vestmentinequilibrium.
To v erify this statemen t, let 3 = 3 i + 3 tr
denote the equilibrium level of scal
pressure, and 3
denote the equilibrium share of tax rev enueallo cated to public
in vestment. Pensionincome, can then b e rewritten as:
3 (10 3 )(10 3 )[ 1+ 3 3 (10 3 )X] (3.1) whic h, giv en 3 , is maximized for 3 = 1 2 0 1 2 3 (10 3 )X . As X increases,
the marginal b ene t of increasing 3
b ecomes larger. In fact, the partial
deriva-tiv eof the av eragep ension b ene t with resp ect to 3 ,[ 3 (10 3 )] 2 (102 3 )X0 3 (10 3 ), isincreasingwith X,if 3 1 2
,asitisindeedthecase, since 3 i 3 tr , in equilibrium.
Third,note thatthe marginalb ene t ofincreasing 3
isdiminishing,asitcan
b ev eri edbyobservingthat thesecondpartialderivativeofp ensionincome with
resp ectto 3
isnegativ e. Infact,as 3
increases,themarginalb ene tofincreasing
3
,interms of increasedpublicin vestment,falls,since the transfershare inlab or
incomeb ecomesprogressivelysmaller,whilethe marginal costofincreasing 3
,in
terms of reduced transfers, increases, since the public in vestment share in lab or
income b ecomesprogressivelylarger. The last twoobservationsimply that, as X
increases, 3
m ustincrease inorder to restorethe equilibrium.
Starting from an equilibrium situation, increases in1, or decreases in
K , ,
or r, increase X. This implies that b oth 3
and 3
increase. This explains whyi 3
and x 3
b oth increase, follo wing suchc hanges.
The fact that i 3
increases, follo wing increases in 1, or decreases in
K , ,
or r, do es not necessarily imply that tr decreases, as a consequence of such
c hanges. Our in tuition go es as follo ws. We know that, for small X and 3
, the
total and marginal collection costs of taxes are also small. Then, the increase
in tax rev enue associated to increases in 1, or decreases in
K
, , or r, will b e
large enoughtocompensateforthe reductioninthe equilibriumshareof rev enues
allo catedtotransfersandtr 3
willb eincreasingwith1anddecreasingwith
K , ,
and r. AsX and 3
b ecomeprogressivelylarger,thetotalandmarginalcollection
costs of taxes increase. For some critical value of X, they ma y b ecome so high
that the increase in tax rev enueno longer compensates for the reduction in the
equilibrium share ofrev enuesallo catedtotransfers andtr 3
ma ystart todecrease
with 1 and to increase with
K
, , and r. This argumen t suggests that tr 3
should increase ordecrease, follo wingceteris paribus variations inthe exogenous
variables, dep ending on the equilibrium value of the marginal pro ductivity of
public capital.
Although the in tuition is prett y clear, a general result regarding the sign of
the variation of tr 3
in resp onse to c hangesin the exogenous
variables is hard to
establish. Still,we can provethe follo wing:
Proposition 7. If
r
0: 5, the equilibrium share of transfers in lab or income
is strictly increasing with the total factor pro ductivity and strictly decreasing
with the depreciation rate of private capital, the elasticit y of public capital in
pro duction and the in terestrate. That is:
dtr 3 d1 >0 dtr 3 d K <0 dtr 3 d <0 dtr 3 dr <0 if r 0: 5
If r < 0: 5, there exists a v ector of values of the marginal pro ductivity of public
capital, h f X z i , z=r; K ;1 ; , c X f X z <1 , suchthat: dtr 3 dz 0 c X < f X z z =r; K ; dtr 3 d1 0 c X < f X 1
lim X! e X 0 1 dtr 3 d1 >0 lim X! e X 0 z dtr 3 dz >0 z =r; K ; lim X!1 dtr 3 d1 <0 lim X!1 dtr 3 dz >0 z =r; K ;
Proof. See App endixB.
Summing up, our model implies that x 3
,i 3
and tr 3
will como vein resp onse
toceterisparibus c hangesinthe exogenousvariablez,if r0: 5 orif r<0: 5and
c X <X < f X z . When r<0: 5and X f X z , tr 3
will moveinthe opp ositedirection
with resp ecttox 3
andi 3
,inresp onsetoceterisparibus variationsofz,ifX tends
to f
X
z
from ab ov eor if X tends toin nit y.
Evenifthe resp onseoftr 3
toceterisparibus variationsof zcan notb ec
harac-terized analitically for f
X
z
<X <1and
r
<0: 5, our in tuitionis that tr 3
should
mo vein the opp osite direction withresp ect to x 3
and i 3
inthis range. This in
tu-ition issupp orted bynumerical sim ulationsof the model, whic hindicatethat the
derivativesof the transfer share function with resp ect to the exogenous variables
c hangesign atmost once asX varieswithin the range ofvalues such that (2.19)
admits in teriorand di eren tiable solutions.
Before concluding this section, we want to stress once again the in tuition
b ehind the resultswe obtained.
Theemergenceofacredibleso cialnorm(WelfareState),whereb yp ensionsand
publicin vestmentare nancedinev eryp eriodbylab orincometaxation,allowsall
generationstoenjoythe b ene tsassociated to(public-investmentdriv en)growth.
Theso cialnormre ectsthein terestsofthecurren ttaxpay ersandwillb emodi ed
iftheen vironmentc hangesduetoexogenoussho c ks. Inparticular,ifthemarginal
pro ductivityofpubliccapitalinpro ductionincreases,itseemsreasonablethatthe
so cial normshouldb emodi edinordertoallowthetaxpay erstoreap the b ene t
associated to the larger growth p otential. Namely, it seems reasonable that the
so cial norm should b e modi ed by increasing the lev el of scal pressure so as to
scal pressure and the total and marginal collection costs of taxes are not v ery
large, the so cial normshould also b e modi edbyincreasing the replacemen trate
on earnings, that is the share of transfers in lab or income, since this allows for
further increases in future p ensions and provides further b ene ts for the curren t
taxpayers. On the other hand, if the marginal pro ductivity of public capital is
largeandsoarethetotalandmarginalcollectioncostsoftaxes,increasesinpublic
in vestment and future p ensions can only b e obtained by reshuing the curren t
public expenditure from so cial securit y to public in vestment, that is byreducing
the replacemen trate on earnings. Note that curren t p ensions will increase along
with public in vestmentinthe former case, while theywill decrease inthe latter.
Inotherwords,ourmodelprovidesanin tuitiveargumen tforwhyanexpanding
piecanmak ethedistributioncon ictb et weenthey oungandtheoldlessstringent.
Since the y oung can b ene t fromhigher growth only by increasing p ensions, the
replacemen trate onearningstr 3
ma ywind upb eing increased, alongwith public
in vestmentandgrowth,sothatb othcurren tandfuturep ensionsincrease, inorder
toexploitanexpandinggrowthp otential. Thisislesslik elytohapp en,the larger
the costs of runningthe Welfare State.
4. Conclusion
Thepresen tpap erputsforwardanexplanationforwhyredistributiv eand
growth-orien tedp olicies,thoughcompetingforscarcetaxrev enues,migh tgohandinhand
and bringab out fasteconomic growth.
Ourmodel analyzesthecase ofasmallop en econom ywheresustainedgrowth
isgenerated bypublic in vestmentincapitalgo o ds nanced through taxrev enues.
In this context, a mechanism of in tergenerational transfers, such a pure
pay-as-y ou-goso cial securit ysystem, thatisone wherep ension b ene tsare fullyindexed
to wages, mayprovide the taxpayerswith the rightincen tivestosupp ort
growth-orien tedp olicies, such asin vestmentin infrastracture orpublic education.
We think it is important to stress that a fully-funded so cial securit y system
to a fully-funded system, whic h is equal to the mark et in terest rate, is xed at
the worldwide lev el and is independen tfrom the accum ulationof domestic
capi-tal. Thus, in this case, agentscannot expect tob ene t from the growth pro cess
through an increase of the future return on their savings. Instead, the accum
u-lation pro cess will certainly a ect the return on their contributions to a
pay-as-y ou-gosystem, throughtheincreasedpro ductivityoflab orand, consequen tly,the
higher lev elof future wages.
Two are the main results of the pap er. First, we showed that, whenev er the
marginal pro ductivity of public capital in the private sector is large enough, an
outcome c haracterized by p ositiv e lev els of redistribution and public in vestment
and by sustained p ositiv e growth can b e supp orted as a subgame-p erfect Nash
equilibrium of anin nitely rep eated in tergenerationalgame, whereat eac hstage
the y oungc ho ose the levelof scalp olicies, taking in toaccountthe past histories
of the game and the consequences of theiractions onthose of future generations.
The credible threat to b e denied the en titlement to p ensions in old age by the
follo wing generationdeters the y oung from defaultingon the so cial norm.
Second, we showed that in a neigh b orho o d of the equilibrium with p ositiv e
growth, giv en a c hange in the exogenous variables: (a) the share of public
in-v estment in lab or income and the rate of growth move in the same direction;
(b) dep ending on whether the marginal pro ductivity of public capital is b elo w
or ab ov e a certain threshold, the share of transfers in lab or income (that is, the
replacemen t rate) and the rate of growth either como ve or move in the opp osite
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5. Appendix A
Proof of Proposition 3
Inthisapp endix,wewanttoshowthat,giv enanyp ossiblehistoryofthegame,
if and only if X >X, no generation has an incen tivetounilaterally deviate from
strategy s 3
t
, that is, this strategy represen ts a subgame-p erfect Nashequilibrium
the co existing old haveset tr ;t 01 = 3 tr and i;t 01 = 3 i , that ish t =(1 ; 3 i ; 3 tr ). If
the y oungplayaccording tostrategy s 3
t
; their lifetime incomewill b e equal to:
W t F = w t (1+r) h (1+ r + 3 tr )( 10 3 i 0 3 tr )+ 3 tr 3 i (10 3 i 0 3 tr ) 2 X i (5.1)
On the other hand, expecting all other generations to adopt the strategy s 3
t ,
the b est p ossible deviation for the y oung is to set b oth public in vestment and
transfers equal tozero. In this case, their lifetime incomeis giv ensimply by:
W t B =w t (5.2)
whic h isalways smallerthan W t
F
if the necessary and sucient condition for the
existence of an in teriorequilibrium holds, i.e. X >X.
Let us now consider the case where h
t =(1 ; 0 i ; 0 tr
) and N is ev en. Clearly,if
the y oung follo ws the equilibrium strategy, their utilit y is equal to W t
F
, whereas
if theydeviate, theywillb e punished bythefollo winggenerationand the highest
levelof utilit ythat they can obtainis W t
B .
Finally,consider the case where h
t =(1 ; 0 i ; 0 tr
) and N is o dd. Here, we m ust
showthatthey ounghav ealwaysanincen tivetopunishtheco existingold. Indeed,
if the y oungadopt the strategy s 3
t
theirutilitywill b e equal to:
W t P = w t (1+r) [(10 i )(1+r)+ 3 tr (10 3 i 0 3 tr )[ 1+ i (10 i )X] ] (5.3) By construction of i
, this is the maxim um level of utility that the y oung can
reac h when the future p olicies are ( 3 i ; 3 tr ).
Thelast argumen tcan b e also usedtoshowthat the rst generationdo es not
haveanyincen tivetodeviate fromthe strategy 3
t
Proof of Proposition 4
We decided to omit this pro of from the text, since it is not relevant for the
follo wing discussion. This pro of isavailablefrom the authors up on request.
Proof of Proposition 5
Denote the matrix of second partial derivatives of W(
i ; tr ;r;X), calculated at ( 3 i (r;X); 3 tr (r;X) ), by 2 22
. The genericelemen tof
222 is then: W ab = @ 2 @ a @ b W( 3 i (r;X); 3 tr (r;X);r;X)
where a=(i;tr)and b =(i;tr).
For X > c
X, the solutions to the maximization problem (2.19) are in terior
and di eren tiable, by Prop osition 4. Then, we can apply the IFT to derive the
expressions for the partial derivatives of 3
i
and 3
tr
with resp ect to
r
and X. In
particular, these are giv enby:
@ 3 i @ r = W trtr 0W itr j j @ 3 tr @ r = W ii 0W itr j j @ 3 i @X = 0 W iX W trtr +W trX W itr j j @ 3 tr @X = 0 W trX W ii +W iX W itr j j (6.1) where: W ii =0 2 3 tr X(203 3 i 02 3 tr ) W trtr =0 2[ 1+ 3 i X(202 3 i 03 3 tr )] W itr =W tri =[ (10 3 i 0 3 tr )(103 3 i 03 3 tr )+2 3 i 3 tr ]X01 W iX =(10 3 i 0 3 tr ) 3 tr (103 3 i 0 3 tr ) W trX =(10 3 i 0 3 tr ) 3 i (10 3 i 03 3 i ) j j=det()=W ii W trtr 0(W itr ) 2
and allfunctions and derivativesare evaluatedat(
i
(r;X);
tr
(r;X) ).
By Prop osition 4, we know that W
ii
< 0, W
trtr
< 0, j j > 0. Moreover,
Lemma 2 implies lim
X!1 ( 3 i (r;X); 3 tr (r;X))= 1 4 ; 1 4
. Some algebraalso p ermits
to v erifythat (W itr 0W trtr ) and (W itr 0W ii
) are strictly p ositiv e at all in terior
solutions.
It is then immediate to v erify that @ 3 i @r <0, @ 3 tr @r
< 0, for X < 1 and that
lim X!1 @ 3 i @X =0, lim X!1 @ 3 tr @X
=0. Byusing numerical calculations,it isalso p ossible
to c heckthat lim
X!1 @ 3 i @r =0 and lim X!1 @ 3 tr @r =0.
Nowuse (2.23) and (6.1) toobtain:
@ 3 i @X = 3 i (10 3 i 03 3 tr )(W itr 0W trtr )0 W trtr X 1(10 3 i 0 3 tr ) j j @ 3 tr @X = 3 i (10 3 i 03 3 tr )(W itr 0W ii )+ W itr X 1(10 3 i 0 3 tr ) j j
from whic hwe can immediately conclude that @
3
i
@X
>0, for X <1 .
By making use of (2.21) and (2.22), the numerator of @ 3 tr @X can b e rewritten as: 0 3 tr (103 3 i 0 3 tr )+(1+r+ 3 tr ) " (103 3 i 03 3 tr )+ 2 3 i 3 tr 10 3 i 0 3 tr # + +2 3 tr (203 3 i 02 3 tr ) 3 i +r+2 3 tr 10 3 i 0 3 tr !
whic h,bynumericalcalculations,canb ev eri edtob estrictlyp ositiv efor( 3 i ; 3 tr ) < 1 4 ; 1 4
,so that we can conclude that @ 3 tr @X >0,for X <1 . Proof of Proposition 6
Prop osition 4and Corollary1imply that(103 3 i 0 3 tr ) d 3 i dz <0, z =r; K ; and (10 3 i 03 3 tr ) d 3 i d1 >0, if c X <X<1 .
