Population Ageing, Social Security and Fiscal Limits

Transcription

1 Populaion Ageing, Social Securiy and Fiscal Limis Burkhard Heer y Vio Polio z Michael R. Wickens x Augus 26 Absrac We sudy he susainabiliy of social securiy sysems for older people using a life-cycle model where disorionary axaion consrains scal policy acions by seing an upper limi on he real value of ax revenue relaive o he size of he economy. This limi implies an endogenous hreshold dependency raio ha ideni es a criical level in he cross-secion disribuion of he populaion of an economy beyond which ax revenue canno longer susain he planned level of ransfers o reirees. We quanify he hreshold using a compuable life-cycle model calibraed on he Unied Saes and 4 European counries wih dependency raios among he highes around he world.. We are graeful for very helpful commens o John Hudson, Nikos Kokonas and paricipans a he 26 Barcelona GSE Summer Forum, he 26 York Fiscal Policy Symposium, and he 26 Money, Macro and Finance Annual Conference. y Universiy of Augsburg, CESifo; Burkhard.Heer@wiwi.uni-augsburg.de. z Universiy of Bah; v.polio@bah.ac.uk. x Universiy of York, Cardi Universiy, CEPR; mike.wickens@york.ac.uk.

2 Conens Inroducion 3 2 Populaion Ageing 5 2. Demographic Trends Relaed Lieraure The Model 9 3. Demographic Environmen Threshold Dependency Raio An illusraive analyic example Disance-o-Defaul and Defaul Probabiliy Tax Revenue Maximizaion Problem 6 4. Primal Represenaion Soluion Quaniaive Resuls 2 5. Assumpions Calibraion Unied Saes EU4 Counries Conclusions 3 7 References 3 A Fiscal Policy 35 A. Ineremporal Governmen Budge Consrain A.2 Equivalen Tax Revenue Formulaion B Non Equivalence Proposiion 37 C Primal Represenaion 38 C. Implemenabiliy Consrain C.2 Tax Revenue D Analyical model 4 2

3 Inroducion Populaion ageing is a major challenge for he public nances of boh advanced and developing economies. Increasing life expecancy and declining birh raes are causing dependency raios (he number of reirees as a proporion of he labour force) o rise world-wide. This is generaing an increasing burden of axaion on he working populaion. I raises he issue of he long-erm susainabiliy of exising social securiy nes for older people and he welfare cos of policy changes ha may be required o ensure susainabiliy. This paper invesigaes his problem using a modi caion o he heory of opimal axaion in dynamic general equilibrium models wih disorionary axes ha akes ino accoun he possibiliy ha here is an upper limi on he real value of ax revenues. This upper limi may exis because ax revenues may be subjec o La er e ecs on he disorionary axaion of he facors of producion. We refer o his upper limi as he " scal limi", following a erminology due o Davig, Leeper and Walker (2) and Cochrane (2). The exisence of a scal limi implies ha here is a consrain on he maximum value, or hreshold, on he dependency raio. This hreshold ideni es a criical poin in he age disribuion of he populaion beyond which ax revenue can no longer susain he planned level of ransfers o reirees. We refer o his as he hreshold dependency raio. This is deermined by he srucure of he economy, he design of scal policy and evolves over ime due o changes in he age disribuion of he populaion. The economic lieraure on scal limis is ypically based on models wih in niely-lived agens. As we are concerned wih a generaional issue, he susainabiliy of publicly funded suppor for older people, insead we use a life-cycle, muli-period, overlapping generaions model in he radiion of Auerbach, Koliko, and Skinner (983) and Auerbach and Koliko (987). We show ha he hreshold dependency raio is derived from a subse of he compeiive equilibria achievable in a life-cycle economy. This subse includes all compeiive equilibria in which he governmen chooses ax policy o maximize ax revenue. The dependency raio is hen endogenously derived from he soluion of he governmen budge consrain. We formulae he problem of maximizing ax revenue in primal raher han dual form in which he governmen chooses he allocaion of consumpion, labour and capial ha yields he highes ax revenue in presen value erms insead of he ax raes hemselves. This formulaion allows a number of aspecs of he soluion o be illusraed analyically. We derive a saisical measure of he disance beween he acual or projeced dependency raio and he hreshold dependency raio. We refer o his as he disance from he hreshold dependency raio. This is also a measure of he scal space, as i indicaes o wha exen he governmen can exploi is abiliy o raise revenue hrough he ax sysem in order o mainain curren levels of publicly funded suppor for older people. Once he dependency raio reaches he hreshold - he disance is hen zero - he governmen can no longer susain he social securiy ne for older people hrough he ax sysem. I can hen eiher parially defaul on is social securiy commimens, reduce oher ypes 3

4 of public spending, or reform he pension sysem, including making people work longer and/or reire laer. The exisence of his hreshold herefore a ecs boh he bene ciaries of and he conribuors o he social securiy ne. The demographic projecions on which our resuls are based possess a degree of uncerainy. This can be exploied o derive a probabiliy disribuion of he disance from he hreshold. We refer o his as he defaul probabiliy of he social securiy sysem for older people. This provides a measure of he likelihood ha he governmen will defaul on is promised paymens o older people. I can also be viewed as an indicaor of he probabiliy of a governmen exhausing is scal space. Populaion ageing is more likely o a ec advanced economies. Our quaniaive analysis herefore focuses on he Unied Saes and 4 European counries. These are some of he counries in which dependency raios reached he highes values in he world by 25, and are projeced o increase fases over he nex 85 years. Using available populaion projecions, we compue he likelihood ha each of hese counries will reach is hreshold dependency raio a some poin in he fuure and hence how likely i is ha each counry will be able o mainain is curren suppor for older people in he shor, medium and long runs. We also examine how di eren feaures of he economy a ec our resuls. I is assumed ha governmen spending is eiher consan a is curren level or follows a pre-de ned projecion pah. A number of issues concerning paricular feaures of exising social securiy schemes, are beyond he scope of his paper. These include () why we have he social securiy sysems ha we do, (2) wheher here is a socially opimal level of redisribuion from workers o older people, (3) wha scal (and non- scal) policy changes can be implemened o mainain he social securiy ne for older people. Our analysis is limied exclusively o he nancial susainabiliy of a social securiy sysem in he presence of scal limis. The paper is organised as follows. Secion 2 provides a summary of global demographic rends. I highlighs heir hisoric evoluion, projecions and regional di erences. This moivaes our focus on advanced economies. We also relae our work o he exising economic lieraure on compuable life-cycle models, La er curves and scal limis. Secion 3 describes a sylized life-cycle model suiable for he dynamic analysis of scal policy and derives of he hreshold dependency raio. Secion 4 formulaes he revenue maximizaion problem in primal form, derives is soluion and discusses is properies. This secion employs a resriced version of he model of Secion 3 in order o derive a closed-form soluion for he hreshold dependency raio and examine is deerminans. We also explain how we compue he disance from he hreshold and he probabiliy of reaching he hreshold a some poin in he fuure. In Secion 5 we provide some quaniaive resuls based on a calibraion of he model. We also conduc Diamond (24) and Diamond and Orszag (25) discuss various economic argumens underpinning he exisence of social securiy conribuions. Shiller (25) and Beesma e al. (2) discuss advanages and disadvanages of individual savings accouns for social insurance. Volume 9, issue 2, of he Journal of Economic Perspecives has a series of di eren views on social securiy and reforms of social securiy sysems. 4

5 World Unied Saes Europe EU4 Counries Figure : Evoluion of hisoric and projeced dependency raios ((age 65+)/(age 2-64)) over he period Source: Unied Naions (25). a number of sensiiviy analyses. Secion 6 concludes, summarizing he paper and re ecing on is possible exensions. 2 Populaion Ageing 2. Demographic Trends Figure shows he hisoric and projeced evoluion of he dependency raio around he world, in he Unied Saes (4.4 percen of 25 world populaion) and in Europe (abou percen of 25 world populaion) over he period The gure also shows he speci c rend of 4 European (EU4) counries, covering abou half of he European populaion, ha had, and are projeced o have, some of he highes dependency raios over his period of ime. Table repors he corresponding numerical values in he four regions, and in each of he EU4 counries (hese are ranked according o heir dependency raio in 25). 3 A number of clear rend emerge. Ageing, as re eced in increases in he dependency raios, is a worldwide phenomenon. Ye his is more relevan for advanced han developing economies, as shown by dependency raios for he Unied Saes and Europe being well above he world rend. Dependency raios in hese four regions are generally projeced o double over he period The dependency raio is calculaed as he number of people in he populaion of age 65 and above relaive o he number of people wih age ranging beween 2 and The daa are from he Unied Naions (25). We used he le POP/3-B o exrac he old-age dependency raio for he World, he Unied Saes, Europe and each of he EU4 counries and he le POP/7- o calculae he average old-age dependency raio for he EU4 area. 5

6 World US Europe EU ITA GRE FIN GER POR SWE FRA DEN BEL NET AUS SPA UK IRE Table : Summary of hisoric and projeced dependency raios ((age 65+)/(age 2-64)) over he period Source: Unied Naions (25). The daa on individual EU4 counries also reveal signi can di erences in he levels and projeced raes of change of dependency raios. Ialy, Greece, Finland, Germany, Porugal, Sweden and France have he highes dependency raios in 25. Those of Ialy, Greece, Germany and Porugal are projeced o remain above he EU4 counries average by 2. Ireland has a dependency raio of 22.% in 25, well below he European and EU4 counries averages. However, his is projeced o almos riple by 2, reaching a level above he projeced European average and closer o he EU4 average. Spain, Ausria and Porugal also are projeced o have large increases in heir dependency raio over he 25-2 period. Greece is projeced o have he highes dependency raios by 2. Demographic rends are rouinely updaed as new esimaes of feriliy and moraliy raes become available. A he same ime, projecions need o be inerpreed wih some cauion as hey are driven by he assumpions underlying he expeced evoluion of key deerminans of demographic rends, noably, feriliy raes, moraliy raes, migraion ows and labour force paricipaion. For example, he daa in Figure and Table are based on he Unied Naions medium-feriliy scenario. Figure 2 illusraes he signi cance of he uncerainy surrounding demographic projecions. The projeced dependency raios of Figure are shown (bold lines) ogeher wih projecions based on alernaive low (upper doed line) and high (lower doed line) feriliy scenarios. The uncer- 6

7 W orld Unied Saes Europe EU4 Counries Figure 2: Evoluion of projeced dependency raios ((age 65+)/(age 2-64)) based on medium (solid line), low (upper doed line) and high (lower doed line) feriliy scenarios. Source: Unied Naions (25). ainy abou he demographic projecions appears o increase over ime and o be more signi can in advanced economies. The gap is wider he higher is he dependency raio. 2.2 Relaed Lieraure Our analysis brings ogeher wo disinc srands in he economic lieraure on dynamic scal policy. Firs is he quaniaive lieraure using life-cycle simulaion models, ypi ed by he seminal works of Auerbach, Koliko and Skinner (983) and Auerbach and Koliko (987). These are dynamic general equilibrium models feauring, among oher hings, hree secors (households, rms and he governmen), endogenous labour supply and disorionary axaion. They are characerised by a household secor ha includes a large number of overlapping generaions of individuals. These can be calibraed o replicae he observed demographic srucure of an economy, hereby providing a naural environmen o quanify he macroeconomic and policy implicaions of di eren ypes of demographic changes, including hose brough abou by populaion ageing. Since he early sudies of Auerbach, Koliko and Skinner (983) and Auerbach and Koliko (987), quaniaive life-cycle models have been exensively applied o sudy he welfare e ecs of changes in social securiy sysems, paricularly ha of he Unied Saes. Examples include sudies on he ransiion from unfunded o funded pension sysems by Huang e al. (997), Conesa and Krüger (999) and Fuser e al. (27); and on he welfare e ecs of reforms of pay-as-you-go sysems, by Imrohoro¼glu e al. (995) and De Nardi e 7

8 al. (999). More recenly, Heer and Irmen (24) invesigae he inerplay beween demographic changes, pension sysems and innovaion using a life-cycle model wih endogenous growh. The second srand of he lieraure is concerned wih he implicaions for public nances of disorionary axaion due o La er e ecs. Ireland (994), Bruce and Turnovsky (999) and Novales and Ruiz (22) focus on self- nancing ax cus due o dynamic La er curve e ecs using an endogenous growh model; Schmi-Grohe and Uribe (997) sudy he implicaions of La er curves e ecs when he governmen follows a balanced-budge rule in a neoclassical growh model. More recenly, Traband and Uligh (2) measure La er curves in he Unied Saes and Europe using di eren varians of a neoclassical growh model. Daniel and Gao (25) sudy he e ec on he La er curves of producive governmen spending, whereas D Erasmo, Mendoza and Zhang (26) consider how La er curves are furher a eced by inernaional capial ows. Davig, Leeper and Walker (2) and Cochrane (2) argue ha he presence of La er e ecs implicily ideni es a scal limi, i.e. a poin beyond which he governmen is unable o increase ax revenue o pay for public spending. Bi (22) builds a real business cycle model where he risk premium on sovereign deb is endogenously deermined by he disance of he economy from he scal limi. Polio and Wickens (24, 25) use his approach o derive a deb limi ha can be used o deermine a measure of he probabiliy of sovereign defaul which hey hen conver ino a credi raing. The common denominaor among hese sudies is ha hey are based on in niely-lived agen models. We have been able o idenify only wo sudies on La er curves in overlapping generaions models. Uhlig and Yanagawa (996) employ a wo-period overlapping generaions model wih endogenous growh and balanced-budge rules o sudy he opimal axaion of income from capial. More recenly, Holer, Krueger, Sepanchuk (24) compue La er curves for he Unied Saes using a compuable muli-period overlapping generaions model wih a very deailed demographic srucure. None of hese papers explois he La er curve o sudy he susainabiliy of social securiy sysems or derive a hreshold dependency raio. We make a furher conribuion o he lieraure by applying he model o each of he EU4 counries in Table, and also o he Unied Saes. The Naional Research Council (22) provides a comprehensive and up-odae repor on he macroeconomic implicaions of populaion ageing. Lee (24) gives an e ecive summary. The repor focuses on six key areas of ineres: healh and disabiliy, labour force paricipaion, produciviy and innovaion, naional saving, rae of reurns and scal oulooks. The repor highlighs how he heoreical lieraure is generally no conclusive on he implicaions of populaion ageing for he rs ve areas. In addiion, he empirical evidence ofen suggess ha some of hese are eiher negligible or subjec o signi can regional di erences. The assessmen of he repor is more conclusive wih regard o he scal oulook. As a large share of governmen spending is devoed o social securiy and healhcare for older people, i is clear ha populaion ageing will increase pressures on public nances, hough he pace of hese pressures may vary across counries and over ime. The repor s policy recommendaions are 8

9 ha economies should consume less and/or work more by (i) inducing workers o increase privae saving o prepare for beer reiremen, (ii) increasing he axes paid by workers o nance bene s for older people, (iii) reducing bene s for older people o bring hem in line wih curren ax and saving raes and (iv) making people work longer and reire laer, hus raising heir earnings and oal oupu. The rs hree of hese are aimed a reducing consumpion (he rs wo for workers, he hird for reirees). 3 The Model The economy is described by a sylized life-cycle model comprising a large number of overlapping generaions of households wih a nie life, a represenaive rm and he governmen. Each household includes one individual who makes consumpion, saving and labour supply choices o maximize lifeime uiliy. The reiremen age is exogenous. The rm uses aggregae capial and labour o maximize pro s, while operaing a neoclassical producion echnology. Consumpion and he paymens received by individuals for he supply of capial and labour are subjec o age-dependen linear axes. The governmen uses ax revenue and issues deb o nance he provision of public consumpion goods and he social securiy sysem, which includes ransfers o all individuals and pension paymens Demographic In each period a new cohor of individuals is born and denoed by is dae of birh. Individuals in each cohor live for J + periods, wih J. In =, J cohors of individuals are already alive, each indexed by heir dae of birh ( ; 2; :::; J). We denoe by j he age of an individual in =, so ha for any cohor born in J, j = max f ; g. The probabiliy of an age-j individual born in period surviving unil j + is ;j. 5 Wihou loss of generaliy, we assume ;j = for j 2 j ; J and ;J =. The populaion grows a he rae n >. The share of individuals of age j in he populaion, j, is consan and given by j = = ( + n) j for j 2 (; J), wih P J j= j =. Individuals work in he rs j R periods of heir life and reire from age j R onwards, wih j R 2 (2; J). The dependency raio d - he number of reirees in he economy as a proporion of workers - is given by: d = d j J j= ; n; j R = R W ; () where R = P J j=j R j and W = P j R j= j denoe he relaive shares in he 4 The noaion is adaped from Erosa and Gervais (22). 5 Throughou he paper, unless oherwise indicaed, he rs subscrip denoes he dae in which an individual is born, whereas he second denoes he age of he individual. Thus he sum of he wo subscrips is he curren period. Variables wih only one ime subscrip are no age dependen and he subscrip denoes he period in which are observed. 9

10 populaion of reirees and workers respecively. The dependency raio is deermined by hree facors: he disribuion of age-j individuals in he populaion, he growh rae of he populaion and he reiremen age. The rs wo are a eced by populaion ageing, hrough reducions in birh raes and increases in life expecancy. Given life expecancy, a decline in he birh rae resuls in a reducion of n ha leads o an increase in he number of reirees relaive o workers in he populaion. Given he birh rae, an increase in life expecancy, for example hrough a reducion in he moraliy rae, leads o increase in he dependency raio, as would a change in j for any given n. Wihou loss of generaliy, we absrac from exogenous changes in he cross-secion disribuion of he populaion (due, for example, o migraion). We rea j and n as exogenous alhough, in pracice, hey could be relaed o he economic environmen and policy, and hence be endogenous. The reiremen age j R also a ecs he dependency raio. This is ypically a policy parameer under mandaory pension sysems. Making hese hree variables endogenous would have no qualiaive e ec on our resuls on he susainabiliy of he social securiy nes for he elderly. We appraise he e ec of variaion of he parameers deermining d in he quaniaive analysis. 3.2 Environmen Households. Individuals wihin each cohor are all equal. They are endowed wih an iniial allocaion of asses in he rs period of heir life, a ;, and do no leave bequess. They are also endowed wih one uni of ime a each age of heir life. This is shared beween labour and leisure during working age. No labour is supplied during reiremen. Each uni of ime devoed o labour provides z j unis of produciviy. Individual preferences depend on consumpion and leisure. For any J, hese are ordered by he uiliy funcion: U = JX j j u (c;j ; l ;j ) ; (2) j=j where = ( + ) is he common discoun facor, wih he discoun rae; c ;j and l ;j are he consumpion and he labour supply of an individual of age j born in period, respecively. The uiliy funcion u is sricly increasing in consumpion and leisure, wice coninuously di ereniable, sricly concave and sais es he Inada condiions. Individuals have perfec foresigh. The budge consrains faced by individuals for j 2 j ; J are: q ;j c ;j + a ;j+ = x ;j + r ;j + ( + r ;j ) a ;j, (3)

11 where x ;j = w;j z j l ;j + r W ;j for j 2 j ; j R p ;j + r R ;j for j 2 (j R ; J) ; (4) l ;j = for j 2 (j R ; J) ; (5) a ;J+ =. (6) Furher, q ;j = + c ;j, w ;j = ;j l bw+j and r ;j = ;j k br+j are he afer-ax prices of consumpion, income from labour and income from capial, respecively; c ;j, l ;j and k ;j are he corresponding age-dependen ax raes; bw +j and br +j denoe he pre-ax prices of labour and capial; r;j W and rr ;j are ransfers received by he individual during working and reiremen ages, respecively; p ;j is he pension conribuion received by reired individuals. Wihou loss of generaliy, we do no include explicily a payroll ax. For an individual born in of age j, he soluion o he lifeime maximizaion problem is he sequence of individual allocaions (c ;j ; l ;j ; a ;j+ ) J j=j ha for any J sais es he necessary and su cien condiions: u c;j = q ;j ;j, for j 2 j ; J ; (7) u l;j = ;j w ;j, for j 2 j ; j R ; (8) ;j = ;j+ ( + r ;j+ ), for j 2 j ; J, (9) and he consrains in (3) - (6), where ;j is he Lagrange muliplier associaed wih an individual s budge consrain. Producion. In each period here is a single produced good ha can be used as privae consumpion, public consumpion or capial. Goods are produced by a neoclassical producion funcion wih consan reurns o scale, y = f (k ; l ) k, where y, k and l denoe per-capia ne oupu, capial and e ecive labour, respecively, is he rae of physical depreciaion and f is monoonically increasing, sricly concave and sais es he Inada condiions. Facors of producion are paid heir marginal producs. The before-ax prices of capial and labour are: br = f k () bw = f l ; () respecively. Governmen. The governmen nances an exogenous sequence of public consumpion goods, ransfers and pension paymens, (g ; r ; p ) =, hrough revenue from axaion, (ax ) =, and by issuing public deb, (b ) = (all variables are in per-capia erms). The sequence of governmen budge consrains for is given by: g + r + p + ( + br ) b = ax + ( + n) b + ; (2) where ax revenue in any is given by ax = P J j= (q j;j ) j c j;j + P j R j= ( bw w j;j ) j z j l j;j (3) + P J j= (br r j;j ) j a j;j :

12 Wihou loss of generaliy, we absrac from a separae social securiy budge a his sage. The dependency raio is implicily accouned for in he consrains faced by scal policy hrough equaions (2) and (3). These depend on he J same, n and j j j= R ha deermine d in equaion (). This moivaes he derivaion of he hreshold dependency raio in he nex secion. The ineremporal soluion of he governmen budge consrain can be derived as follows. Denoe he populaion-growh-adjused primary surplus, rae of reurn and discoun facor for as ps = (ax g r p ) = ( + n), Q + br n = ( + br ) = ( + n) and bp = ( + br s ) n, respecively. The forward s= b T =, is soluion of (2), which is subjec o he solvency condiion lim bp T T! b = P = [bp (ps )]. The presen value of ax receips for he governmen is: P V ax = + n 2 X 6 bp 4 = P J j= (q j;j ) j c j;j + P jr j= ( bw w j;j ) j z j l j;j + P J j= (br r j;j ) j a j;j : (4) Appendix A. describes he derivaion of he ineremporal governmen budge consrain. Marke-clearing condiions and Feasibiliy. The equilibrium condiions for per-capia labour, asse holdings and consumpion are: l = P j R j= jz j l j;j ; (5) a = P J j= ja j;j = k + b ; (6) c = P J j= jc j;j ; (7) respecively. The per-capia resource consrain requires y + ( ) k = c + g + ( + n) k + : (8) Transfers and pension paymens per-capia are r = P J j= jr j;j and p = P J j=j R j p j;j, respecively. 3.3 Threshold Dependency Raio Firs we de ne he se of compeiive equilibra. We hen highligh ha he dependency raio can be derived endogenously as he unique number supporing a speci c compeiive equilibrium. The hreshold dependency raio is hen simply a special case. De niion (Compeiive Equilibrium) Given an iniial aggregae endowmen of asses a = k + b, a compeiive equilibrium is a dependency raio d = d(( j ) J j= ; n; j R), a sequence of governmen spending, (g ; r ; p ) =, ax, ((q ;j,w ;j,r ;j ) J ) j=j = J, and borrowing, (b +J+) = J, policies, a sequence of prices (br ; bw ) = and a sequence of individual allocaions ((c ;j ; l ;j ; a ;j+ ) J ) j=j = J such ha: 2

13 . The sequence of individual allocaions sais es (3) - (9), for J, 2. The sequence of prices sais es () and (), for, 3. The dependency raio and he sequence of governmen spending, ax and borrowing policies saisfy (2) and (3), for, 4. Feasibiliy (8) holds, for, 5. All markes clear, i.e. (5) - (7) hold, for. A compeiive equilibrium can be compued in wo sequenial sages. The rs sage consiss in deermining he sequence of individual allocaions and he sequence of prices ha describe he privae secor s opimal choices, given he age of reirmen j R. In he second sage, he dependency raio and governmen policy are deermined subjec o he consrains se by hese privae secor choices and he governmen budge consrain. Crucially, one degree of freedom is missing a his sage, as he dependency raio and he governmen policy have o saisfy he sequence of governmen budge consrains in (2). As a resul, here are many compeiive equilibria, each indexed wih a di eren dependency raio and governmen policy. This mulipliciy implies ha for any given scal policy he dependency raio can be derived as a residual from he soluion of he governmen budge consrain. This however, would no uniquely idenify J d, unless eiher he share of individuals of age j in he populaion,, or j j= he growh rae of he populaion, n, are xed. In boh cases, he soluion is highly nonlinear. The hreshold dependency raio is a special case, being he dependency raio d obained when ax policy is se o maximize ax revenue given governmen spending and borrowing policy. In oher words, i measures he number of reirees per worker ha he governmen could susain hrough ax policy alone. A maximum dependency raio susainable hrough changes in ax policy emerges naurally in a life-cycle model as long as here is an upper bound on ax revenue. This is provided by he La er curve e ec. The upper bound can be exploied in conjuncion wih he governmen budge consrain o give he hreshold dependency raio, d. De niion implies ha here is a compeiive equilibrium where d = d, bu d is no uniquely deermined unless eiher he share of individuals of age j in J he populaion, j, or he growh rae of he populaion, n, are xed, as j= illusraed by equaion (). 3.4 An illusraive analyic example We illusrae analyically he deerminans of he hreshold dependency raio and how his depends on direc and indirec axaion using a life-cycle model wih closed-form soluion ha is resriced in he following ways. 6 The resricions imposed on he general model in Secion 3 are: (i) individuals live for wo 6 Appendix C has all he seps for he derivaion of he resuls in his secion. 3

14 periods (J =, hus j = ; ), work in he rs and reire in he second period (j R = ); (ii) labour produciviy is normalized o one, z = ; (iii) here is no aggregae saving in he economy: a ;j = for j = ; and ; (iv) here is no governmen consumpion g = ; (v) echnology and uiliy are y =!l and U (c ; ; c +; ; l ) = ln c ; + ln ( l ) + ln c +;, respecively, wih! and. As a resul, he dependency raio is d = = = ( + n) ; he governmen budge consrain (in per-capia erms) is p + r = l w l + c ( c ; + c ; ) or, equivalenly, dp + r = l w l + c (c ; + dc ; ); and he individual budge consrains reduce o ( + c ) c ; = l w l + r when j = and + + c c+; = p + when j =. Under hese assumpions, he dependency raio in he economy can be solved from he governmen budge consrain as d = l w l + c c ; r p c : c ; The soluion o he individual and rm problems yield he following opimal condiions for consumpion and labour supply for : c ; = [ l!+ r ]=[( + c ) ( + )], c +; = p=(+ c +) and l = ( + ) r =[( + ) l!], and he equilibrium ax revenue is: ax = l! ( + ) l r ( + ) l + c The La er curve for he labour income ax peaks when l = r ( + c )! l! + r + ( + ) p d ( + c : ) ( + ) 2 : This shows ha he ax rae on income from labour ha maximises ax revenues is negaively relaed o he preference parameer, he level of ransfers and he consumpion ax. I is posiively relaed o he rm s produciviy. The derivaive of ax revenue wih respec o he consumpion ax c = ( + c ) 2 " # l! + r + ( + ) p d > : ( + ) This implies ha, as long as c is nie, he governmen can generae an ever increasing revenue from he axaion of consumpion. Traband and Uligh (2) also nd ha here is no La er e ec on he revenue raised hrough he axaion of consumpion. Consequenly, in our quaniaive analysis we ake he ax rae on consumpion as given, hough we evaluae he impac of changes in his policy insrumen on he hreshold. For any given level of indirec axaion, he hreshold dependency raio susainable by he economy is herefore given by: h r(+ c ) i 2 ( + c )! 2!! r d = : ( + ) p 4

15 The hreshold is enirely dependen on he parameers of he economy and he design of scal policy. In paricular, i unambiguously reduces he higher is he level of governmen expendiure, wheher hrough ransfer or pension paymens. The derivaive of he hreshold wih respec o he consumpion ax rae is ( + ) p (! r! r ( + c ) which is always posiive for any nie value of he consumpion ax rae, as long as c 6= r!. 3.5 Disance-o-Defaul and Defaul Probabiliy The disance beween he projeced and he hreshold dependency raios can be measured given a forecas of he dependency raio. The dependency raio in + can be wrien as d + = E d where E d + is he expeced value of he dependency raio by he end of period + condiional on informaion available in, and + = + is he corresponding innovaion in period +, wih + being an independen and idenically and normally disribued disurbance, + i:i:d: (; ). The h- period ahead dependency raio is herefore d +h = E d +h + +h ; where +h is he h-period ahead innovaion. I follows ha +h i:i:d: ; h 2 where +h = P h j= +j and he h-period ahead condiional variance of he dependency raio is V ar +h = P h j= V ar +j = h 2. The forecas error associaed wih he h-period ahead dependency raio can be wrien as 2 +h = P h j= +j = P h j= +j = u +h : The probabiliy ha he h-period ahead dependency raio exceeds he dependency raio hreshold is p +h = Pr d +h d E d +h = Pr d u +h : If N (; ), hen u +h is also normally disribued, and so he probabiliy of defaul is a cumulaive normal disribuion: E d +h d p +h = N u +h : 5 )

16 In order o beer capure he probabiliy of defaul, i may desirable o replace he Normal disribuion wih a disribuion ha has much wider ails (eiher an empirical or a -disribuion). In corporae nance he disance-o-defaul is de ned as he number of sandard deviaions ha he rm is away from defaul. By analogy, we can de ne he disance from he hreshold, DT, as he number of sandard deviaions ha he economy is away from he dependency raio hreshold. This is given by: DT (; + h) = E d +h The probabiliy of exceeding he dependency hreshold raio increases as he gap beween he expeced and he hreshold dependency raio widens and he uncerainy surrounding he dependency raio projecion increases. This probabiliy changes over ime as changes in he base year and in informaion aler he projecion of he dependency raio, is uncerainy and he hreshold. 4 Tax Revenue Maximizaion Problem We describe he problem of he governmen choosing a ax policy ha maximizes he presen value of ax revenue subjec o he privae secor consrains. The problem is formulaed in primal form, in which he governmen chooses allocaions of consumpion, labour supply and capial direcly raher han ax raes or afer-ax prices. The primal formulaion allows an explici analyical characerizaion of he ype of scal policies he governmen implemens in deriving he hreshold dependency raio. For any scal policy, he privae secor s opimaliy condiions can be used o generae a compeiive equilibrium allocaion. The se of allocaions from which he governmen can choose - he se of implemenable allocaions - is resriced o saisfy he opimaliy condiions of he privae secor ( and 2 in De niion ). Hence, he se of implemenable allocaions includes all possible allocaions ha can be implemened as a compeiive equilibrium hrough scal policy. Based on his de niion, a well-known resul in he Ramsey analysis of scal policy is ha wo scal policies can implemen he same allocaion as long as hey do no aler he real wage, he relaive price of curren versus fuure consumpion and he value of asse holdings all expressed in erms of he aferax price of consumpion, see Escolano (99) and Erosa and Gervais (22). 7 This resul is ofen advocaed in he analysis of Ramsey scal policy in order o highligh he redundancy of eiher consumpion or labour income axes. In paricular, a scal policy wihou consumpion axes can implemen he same allocaion as a scal policy wihou labour axes by a suiable rede niion of he oher policy insrumens. As a resul, wihou loss of generaliy, many 7 This holds when he se of policy insrumens includes axes on consumpion, income from labour and deb. d : 6

17 analyses of Ramsey scal policy eliminae eiher he consumpion ax or he labour income ax. This is no however wihou of loss of generaliy if he objecive of he governmen is o evaluae he hreshold dependency raio. While wo di eren scal policies and sequences of asse holdings can implemen he same allocaion, hey yield di eren ax revenues. The inuiion is sraighforward, as ax revenue is derived from he implemenaion of wo di eren policies on he same allocaion of consumpion, labour supply and capial. More formally, appendix B shows ha he ax revenue obained from wo differen scal policies, ((q ;j,w ;j,r ;j ) J ; b j=j +J+ ) = J and ((eq ;j, ew ;j,er ;j ) J ; e b j=j +J+ ) = J, and wo di eren sequences of asse holdings, (a ;j ) J j=j +) = J and (ea ;j) J j=j +) = J, ha implemen he same allocaion is he same only if ( + br ;j )(a ;j ea ;j ) = a ;j+ ea ;j+. This condiion is no me unless he wo sequences ((a ;j ) J j=j +) = J and (ea ;j ) J j=j +) = J are equal, which conradics he earlier assumpion ha hey are di eren. 4. Primal Represenaion We now formulae he governmen problem in a primal form. Appendix B. shows how he budge consrain of he household in equaion (3) is solved forward and he condiions in equaions (4) - (9) are replaced o derive he implemenabiliy consrain of he generaion born in period J as: JP j=j j u c;j c ;j = j RP j=j j j R P u l;j l ;j + JP j q ;j u c ;j j=j R j=j j q ;j u c ;j r W ;j + (9) p ;j + r;j R + q u ;j c;j a ;j : Equaion (9) capures all he consrains imposed by he privae secor opimal choices on he governmen decision. The governmen s objecive of maximizing he presen discoun value of ax revenues can be formulaed in equivalen erms as maximizing he presen discouned value of ax receips from each generaion. Appendix A.2 shows ha he presen value of governmen ax revenue in equaion (4) can be formulaed as 2 3 P V ax = X 6 + n 4 = J Q s= j 7 ( + br s n 5 ; ) wih ( + br s n ) = for s 2 ( J; :::; ), where is he value of lifeime axes paid by any generaion born in J and discouned a he dae of birh. Appendix B.2 shows ha for J: P = J c ;j c ;j + k ;j br +ja ;j Q j + j RP l ;j bw +jz j l ;j Q j j=j s=j ( + br +s ) j=j s=j ( + br +s ) : (2) 7

18 Using he rs-order condiions from he soluion o he individual opimizaion problem, he presen value of ax paymens from each generaion born in J can be formulaed as h i P = J (q ;j ) c ;j + + br +j uc ;j q;j u l;j u c;j q ;j a ;j + f k+s P f l + q ;j u c;j z j l ;j +jr + f k+s : j=j Q j s=j j=j Q j s=j (2) Sricly speaking, boh he implemenabiliy consrain (9) and he presen value (2) are in a semi-primal form since hey sill include he afer-ax price of consumpion q ;j. These can be combined o obain for any J: 8 >< e = + >: j R P j=j q ;j u c;j a ;j ; JP j=j j j R P u c;j c ;j j q ;j u c ;j r;j W j=j j u JP j q ;j u c ;j j=j R l;j l ;j p ;j + r;j R where is he Lagrange muliplier aached o he implemenabiliy consrain. 4.2 Soluion Using he resuls in he previous secion, he governmen s problem may be formulaed as X max (c ;j;l ;j) J j=j ;k +J+ = J = J subjec o he feasibiliy consrain in equaion (8), where = bp e +j, wih Q bp = for 2 ( J; ) and bp = ( + br s ) n for. s= This problem has a srucure similar o ha of he Ramsey problem in primal form when he economy includes overlapping generaions of households (Chari and Kehoe (999), Erosa and Gervais (22)). In he Ramsey problem he governmen wans o nance a given level of spending by choosing an allocaion ha maximizes is objecive funcion subjec o he resource consrain of he economy. The Ramsey planner s objecive is formulaed in erms of a pseudo-uiliy funcion obained from he combinaion of lifeime uiliy and each generaion s implemenabiliy consrain. The Ramsey planner discouns he pseudo-uiliy of each generaion according o a subjecive discoun facor ha re ecs he weigh aached by he planner o curren and fuure generaions. In he ax revenue maximizaion problem he governmen wans o nance a given level of spending having available he same se of choice variables, he allocaion (c ;j ; l ;j ) J j=j ; k +J+, and being subjec o he same feasibiliy consrain as he Ramsey planner. The objecive funcion also = J includes 9 >= >; 8

19 he implemenabiliy consrain of each generaion. There are, however, wo di erences from he Ramsey problem. Firs, he presen value of ax revenues received from each generaion replaces he lifeime uiliy of hose generaions. Second, in he ax revenue maximizaion problem he governmen discouns using he pre-ax rae of reurn on capial raher han he subjecive discoun rae of he Ramsey planner. A furher observaion concerns he appropriae comparison of he wo problems soluion. This requires he level of spending faced by he governmen in he wo problems being he same, eiher in aggregae or per-capial erms. For any given sequence of (aggregae or per-capia) spending, he wo soluions can always be compared if deb is adjused o saisfy he governmen budge consrain given he di eren sequences of ax raes implied by each of hem. The compuaion of he hreshold dependency raio, however, requires ha he governmen chooses ax raes o maximize ax revenue and he addiional ax revenue o be used for nancing a greaer number of reirees, a he curren pension rae. Consequenly, a correc comparison of he wo soluions can only be done if he governmen nancing consrain is in erms of spending per capia. The analysis of his secion implicily assumes his form of comparison beween he wo soluions. 8 If he Lagrange muliplier associaed wih he resource consrain in period + j is denoed as " +j, he rs-order necessary condiions for are: c ;j " +j ;j = l ;j + " +j ;j z j f l;j = " ( + n) + " + + f k+ + k+ = : Eliminaing " and ;j gives he dynamic inra-emporal and iner-emporal necessary equilibrium condiions: l ;j c ;j = z j f l;j, for j 2 j ; j R (22) c ;j = + f k+ + k+, for j 2 j ; J ; (23) c ;j+ In saionary equilibrium hese reduce o: c j l j c j = z j f lj, for j 2 (j; j R ) ; (24) c j+ = ( + f k ) + k, for j 2 (j; J ) : (25) 8 Alernaively, he governmen could choose spending and nancing in boh problems, wih he addiional consrains on ax revenue implied by he hreshold compuaion. 9

20 Afer di ereniaing he objecive funcion wih respec o consumpion and labour we obain c j = W cj + u cj cj l j = W lj + u lj lj ; where: W cj = u cj + + H c j W lj = u lj + + H l j cj = c j u cj u cj lj = l j u lj u lj H c j = u c jc j + u ljc j l j u cj H l j = u c jl j c j + u ljl j l j u lj : The erms W cj and W lj denoe he derivaive of he Ramsey planner objecive wih respec o consumpion and labour, respecively, wih Hj c and Hl j indicaing heir general equilibrium elasiciies, as in Akeson, Chari and Kehoe (999). The wo erms cj and lj measure he proporional di erences of he marginal e ecs on ax revenues from changes in consumpion and labour relaive o heir e ecs on household marginal uiliies. As in he lieraure on opimal axaion, i is of ineres o derive under wha condiions he allocaion chosen by he governmen resuls in zero axaion on eiher consumpion, labour or capial. We herefore compare hese necessary condiions wih hose from he individual maximizaion problem obained from (7) - (9) in a dynamic equilibrium, u l;j = w ;j l ;j = z j f l;j, u c;j q ;j + c for j 2 j ; j R ; ;j u c;j + c ;j = u c;j+ + c + k ;j+ br;j+, for j 2 j ; J, ;j+ and in he seady sae u lj = w j l j = z j f lj, u cj q j + c for j 2 (j; j R j ) ; (26) = + k j+ brj+, for j 2 (j; J ). (27) We focus on he seady sae. Afer combining he inra-emporal necessary condiion from he governmen problem in equaion (24) wih ha of he privae 2

21 secor in (26), for he nonrivial case of nonzero labour supply, we obain l j + + H c j + cj + c = ; j + + H l j + lj which shows ha he allocaion chosen by he governmen o maximize ax revenue gives an inra-emporal wedge di eren from ha of he Ramsey planner unless cj = lj =, i.e. cj = u cj and lj = u lj, implying ha he marginal e ecs on ax revenues of changes in consumpion and labour equal hose on uiliy. In he special case when c j =, he soluion for he ax rae on labour is l j = Hl j Hj c + lj cj + + Hj l : + lj The ax rae on labour is herefore zero a each dae if Hj l Hj c = cj lj. The condiion for a seady-sae zero labour ax rae by he Ramsey planner is Hj l = Hc j. Only if c j = lj - i.e. if he proporional di erences in he marginal e ecs of consumpion and labour on ax revenues from hose on marginal uiliy are he same - does a ax-revenue maximizing governmen and a Ramsey planner boh choose a zero labour ax rae. Neiher condiion is likely o hold. A more signi can issue is he axaion of capial. Afer combining he ineremporal necessary condiion from he governmen s problem in equaion (25) wih ha of he privae secor in (27) we obain + c j + + H c j + cj + br + k + c = j+ + + H c j+ + cj+ + k : ;j+ br Consequenly, a each dae he ax rae on capial income is di eren from zero unless axes on consumpion, he general equilibrium elasiciy of consumpion and he di erence in he objecive relaive o he Ramsey are all age independen, ha is c j = c j+, Hc j = Hc j+ and c j = cj+ for all j 2 (; J ), and k =. An imporan special case is when ax raes are age independen and consumpion does no exibi life-cycle behaviour. As cj = c in his case, he previous equaion reduces o = + br + k : + r This implies ha he ax rae on income from capial in seady sae is generally nonzero. 5 Quaniaive Resuls We presen here he resuls from a quaniaive analysis of he hreshold dependency raio for he Unied Saes and each of he EU4 counries described in 2

22 secion 2. 9 Firs, we calibrae he model on daa for he Unied Saes o compue numerically he seady-sae equilibrium, La er curves on he labour and capial income ax and he hreshold dependency raio. We use he Unied Saes daa as a laboraory o evaluae he sensiiviy of his benchmark compuaion o changes in he consumpion ax rae, he reiremen age, pension paymens and he demographic srucure of he populaion. We also provide a projecion of he hreshold dependency raio ha can be compared wih he o cial projecion of he dependency raio. A his sage we also illusrae how measures of he uncerainy surrounding he o cial demographic forecass/projecions can be used o compue he disance of he economy from he hreshold (DT) and he likelihood of reaching he hreshold (LT). We hen proceed by exending he quaniaive analysis of he hreshold dependency raio o he EU4 counries. As noed in secion 3, he dependency raio is no uniquely ideni ed from he soluion o he governmen budge consrain, since his depends on boh he growh rae of he populaion and survival probabiliies. The analysis presened here is based on xing he survival probabiliies and adjusing he growh rae of he populaion. 5. Assumpions Demographics and Timing. A period,, corresponds o ve years. Newborns have a real-life age of 2-24 denoed by j =. All generaions reire a he end of age j R = 8 (corresponding o real-life age 64) and live up o a maximum age of j = J = 4 (real-life age 94). The number of periods during reiremen is equal o J j R = 6. The survival probabiliy is nonzero, oher han in he las period. Households. Households maximize he expeced ineremporal lifeime uiliy U = JX j=j j j jq ;s u(~c ;j ; l ;j ), s= where ;s denoes he condiional probabiliy of surviving up o age + s and he insananeous uiliy is speci ed as a funcion of consumpion and labour as in Traband and Uhlig (2): u(~c ;j ; l ;j ) = ~c ;j h ( )l +=' ;j i, 9 We use sandard numerical mehods in order o compue he seady sae of he quaniaive large-scale OLG model. In essence, we have o solve a high-dimensional non-linear equaions problem in he individual and aggregae equilibrium condiions of he seady sae. The main challenge in his exercise is o come up wih a good iniial value for he Newon- Rhapson algorihm for he individual and aggregae sae variables. Therefore, we sared from a simple wo-period OLG model wih exogenous labor and wo cohors of workers and solved his model. Thereafer, we added one addiional cohor in each sep and used he soluion of he model in he previous sep as an inpu for he iniial value of he nex sep. Nex, we inroduced endogenous labor, pensions, and governmen deb successively. The Gauss compuer programs are available from he auhors upon reques. 22

23 where ~c = c =A is saionary consumpion, wih A denoing he echnology level. In he benchmark case, labour produciviy z j is se o for all j 2 (; J). In our sensiiviy analysis, we also consider he e ecs of age-dependen labour produciviy pro les where daa are available. Equaion (3) includes hree modi caions for he purpose of he numerical analysis. Firs, all axes are ageindependen. Second, he household pays a separae conribuion o he pension sysem a he rae p levied on his wage income; hus afer-ax labour income is ( p w ) bw l, wih l = p + w, for. Third, all non-pension ransfers are also age-independen, hus r W = r R in any period. In equilibrium he household is indi eren beween holding asses in he form of physical capial or governmen deb, since boh yield he same (cerain) aferax reurn. If we only had one household living for wo periods, his would pose no problem because he proporion of asse holdings would be he same a he individual and he aggregae level. Wih many periods, he porfolio allocaion is however indeerminae. Therefore, we assume wihou loss of generaliy ha each household holds he wo asses in he same proporion. This is deermined as he share of capial in oal aggregae asses, e.g. k =(k + b ). Producion. The producion echnology is described by a Cobb-Douglas funcion wih labour-augmening echnological progress, y = k (A l l ). Technology grows over ime a he exogenous rae g A, which is also equal o he balanced-growh rae of he economy. Governmen. The governmen expendiures (consumpion and ransfers) grow a he exogenous balanced-growh rae. The governmen revenue is augmened o include all accidenal bequess from households ha do no survive. There is a separae budge for pensions. This is balanced in every period, so ha aggregae expendiure on pensions is equal o he aggregae revenue raised P hrough he social securiy ax on income from labour: p w l = p = J j=j R j p j;j. Saionary Equilibrium. All individual and aggregae variables are made saionary by expressing hem as a proporion of echnological progress. 5.2 Calibraion Table 2 repors he numerical values used for he calibraion of he seady-sae parameers for he Unied Saes and each of he EU4 counries. The calibraion is based mainly on daa averages for he period used by Traband and Uhlig (2) for he numerical deerminaion of La er curves using in niely-lived-agen models. We choose heir se of numbers for wo reasons. Firs, hey give a useful benchmark for comparing he La er curves derived from our life-cycle model. Second, he quani caion of saionary equilibria is no a eced by he large swings of scal variables during, and in he afermah of, he 28-2 global nancial crisis. We use he age-produciviy pro les esimaed by Hansen (993) for he Unied STaes and Heer and Maussner (29) for Germany. To he bes of our knoledge, here is no daa available on age-produciviy pro les for oher European counries. 23

24 n d g=y r=y b=y p=wl l k c p US AUS BEL DEN FIN FRA GER GRE IRE ITA NET POR SPA SWE UK Noes: Parameer values equal for all counries are: br = 4%, = 2, ' = and g A = 2%. All numbers are in percenage, oher han hose for. The survival probabiliies are averages over Age-produciviy z j in he Unies Saes is aken from Hansen (993) and equal o in all oher counries. Daa source is described in he main ex. Table 2: Parameers calibraion All numerical values for he calibraion of he parameers describing preferences and producion are equal o hose used by Traband and Uhlig (2). Wih regard o he demographic, we calibrae in each counry he seady-sae growh rae of he populaion and he 5-year survival probabiliies for he 5 di eren age groups using he averages from he Unied Naion (25) s daa during he years The seady-sae dependency raio repored in Table 2 is herefore derived as an implied residual. Wih regard o he scal policy variables, all daa for also aken from Traband and Uhlig (2), wih he excepion of governmen ransfers (r=y) and he social securiy ax rae ( p ) ha are endogenously deermined in our model o saisfy he general and social securiy governmen budge consrains, respecively. Pension paymens are compued using daa on gross replacemen raios for pensions from he OECD (25), based on percenage of pre-reiremen income for men. Since Traband and Uhlig (2) employ e ecive ax raes on income from labour ha already include he social securiy conribuion ax rae, we resric our l = w + p o be equal o he same l repored in Traband and Uhlig (2) Unied Saes Fiscal Space. We provide an iniial quani caion of he scal space available o he governmen secor in he Unied Saes. Saring from he benchmark 24