Working Paper Social security systems, human capital, and growth in a small open economy

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1 econsor Der Open-Access-Publikaionsserver der ZBW Leibniz-Informaionszenrum Wirschaf The Open Access Publicaion Server of he ZBW Leibniz Informaion Cenre for Economics Kaganovich, Michael; Meier, Volker Working Paper Social securiy sysems, human capial, and growh in a small open economy CESifo working paper, No Provided in Cooperaion wih: Ifo Insiue Leibniz Insiue for Economic Research a he Universiy of Munich Suggesed Ciaion: Kaganovich, Michael; Meier, Volker (2008) : Social securiy sysems, human capial, and growh in a small open economy, CESifo working paper, No This Version is available a: hp://hdl.handle.ne/10419/26533 Nuzungsbedingungen: Die ZBW räum Ihnen als Nuzerin/Nuzer das unengelliche, räumlich unbeschränke und zeilich auf die Dauer des Schuzrechs beschränke einfache Rech ein, das ausgewähle Werk im Rahmen der uner hp:// nachzulesenden vollsändigen Nuzungsbedingungen zu vervielfäligen, mi denen die Nuzerin/der Nuzer sich durch die erse Nuzung einversanden erklär. Terms of use: The ZBW grans you, he user, he non-exclusive righ o use he seleced work free of charge, erriorially unresriced and wihin he ime limi of he erm of he propery righs according o he erms specified a hp:// By he firs use of he seleced work he user agrees and declares o comply wih hese erms of use. zbw Leibniz-Informaionszenrum Wirschaf Leibniz Informaion Cenre for Economics

2 Social Securiy Sysems, Human Capial, and Growh in a Small Open Economy MICHAEL KAGANOVICH VOLKER MEIER CESIFO WORKING PAPER NO CATEGORY 1: PUBLIC FINANCE DECEMBER 2008 An elecronic version of he paper may be downloaded from he SSRN websie: from he RePEc websie: from he CESifo websie: Twww.CESifo-group.org/wpT

3 CESifo Working Paper No Social Securiy Sysems, Human Capial, and Growh in a Small Open Economy Absrac We consider a small open economy in which he level of public educaion funding is deermined by popular voe. We show ha growh can be enhanced by he inroducion of pay-as-you-go pensions even if he growh rae of aggregae wages falls shor of he ineres rae. The reason is ha he PAYG sysem allows fuure reirees o parially inernalize posiive exernaliies of public educaion due o he posiive effec of higher fuure labor produciviy on heir pension benefis. The majoriy suppor for educaion funding will be especially srong when he PAYG benefi formula is fla, i.e. progressively redisribuive. This means ha if a fla benefi PAYG pension sysem is in place hen he economy will achieve he highes growh rae relaive o he alernaive pension sysem designs. We argue furhermore ha while such PAYG pension sysem may be opposed by he majoriy of working individuals due o inferior reurns o heir pension conribuions relaive o a funded scheme, i is likely o be poliically susained by he coaliion of reirees and lower income workers. JEL Code: D90, H23, H55, I28. Keywords: pay-as-you-go pensions, social securiy, public educaion, growh, majoriy voing. Michael Kaganovich Deparmen of Economics Indiana Universiy Wylie Hall Bloomingon, IN USA [email protected] Volker Meier Ifo Insiue for Economic Research and CESifo a he Universiy of Munich Poschingersrasse Munich Germany [email protected] November 2008 We graefully acknowledge financial suppor by CAEPR a Indiana Universiy, CESifo, and he Siferverband fuer die deusche Wissenschaf.

4 1. Inroducion Public pay-as-you-go (PAYG) pension schemes in which pensions are financed by conribuions from curren workers have ofen been criicized as derimenal o growh. According o Feldsein (1974) such pension schemes have a negaive effec on capial accumulaion since hey discourage privae saving and, unlike in he case of a funded pension sysem, he paymens ino he PAYG scheme do no conribue o he naional saving. Moreover, he implici rae of reurn on conribuions o a PAYG scheme ypically falls shor of he ineres rae. Therefore according o such analysis, PAYG pension sysems reduce per capia income. This sandard argumen is focused on physical capial accumulaion and fails o ake noice of he effec of PAYG pension sysems have on he accumulaion of human capial, paricularly hrough public educaion. Primary and secondary educaion is now overwhelmingly publicly financed in all OECD counries, and universiies also receive subsanial funding from public sources. Since PAYG pension schemes ypically ie pension benefis of fuure reirees o he produciviy of heir conemporary workers, hey creae an incenive o suppor greaer invesmen in he human capial of hose fuure workers. Therefore if a PAYG pension scheme is inroduced, is fuure beneficiaries may become supporive of higher funding of public educaion via axes since hey will ake ino accoun he posiive impac of aggregae human capial accumulaion on heir pensions. The resuling wage income growh may more han offse losses o economic growh due o he negaive effec on physical capial accumulaion. This seems paricularly likely in a world of inegraed capial markes, where inernaional capial flows subsanially reduce he impac of naional saving on facor prices and he supply of capial. In fac, boh he size of public PAYG pension schemes and he share of expendiure on public educaion in GDP have grown subsanially over he firs decades afer he Second World War in almos all OECD counries. This paper provides a possible explanaion for his co-movemen and offers a model represening an inergeneraional compac where a PAYG sysem suppored by reirees compels he working generaion, i.e. he fuure reirees, o suppor greaer funding of public educaion, consisen wih he analysis by Becker and Murphy (1988). To illusrae he poenially beneficial impac of a PAYG scheme in is simples form, we analyze a small open economy in which savings do no affec he ineres rae or he supply of capial. We argue ha wage income growh in such an economy will generally be sronger under PAYG pension schemes, in comparison o he fully funded pension 1

5 sysems. This can be rue even if he reurn on PAYG conribuions is below he ineres rae. The concep ha PAYG schemes provide an incenive for invesing in public educaion has already been noed by Pogue and Sgonz (1977) as well as Becker and Murphy (1988) who also poined ou ha his incenive is sronger for he middle and lower income populaions. This concep has become a common feaure in he analysis of inergeneraional ransfers in various conexs (Konrad, 1995; Kaganovich and Zilcha, 1999; Kemniz, 2000; Boldrin, 2005; Pouvaara, 2006; Soares, 2006). For he environmens in which parens finance he educaion of heir children, including hose where privae and public channels coexis, i has been argued ha posiive growh effecs of inroducing a PAYG scheme can be expeced because i helps relax parens liquidiy consrains: see Kaganovich and Zilcha (1999), Lambrech e al. (2005), Glomm and Kaganovich (2008). 2 The laer paper which focuses on he role of income heerogeneiy underscores he poin ha he effec of relaxing liquidiy consrain on educaing children is paricularly srong in lower income families when PAYG pension benefis are redisribuive. Anoher srand of he lieraure is concerned wih normaive aspecs of linking public educaion and PAYG schemes. One efficiency argumen for providing schooling publicly lies in he impossibiliy o enforce ransfer paymens by children in exchange for financing educaion wihin he family (Sinn, 2004). Bu if alruism oward he young is weak, he level of public educaion may be chosen where he marginal reurn o invesmen in human capial exceeds he ineres rae. In such a siuaion, PAYG pensions will enhance efficiency by fosering human capial accumulaion. Rangel (2003) sudies an inergeneraional conrac wih public educaion and PAYG pensions aimed a resoring efficiency. Boldrin and Mones (2005) argue ha hese wo pars of governmen inervenion can be used o replace he missing marke for financing invesmen in human capial in an efficien fashion. Our paper can be seen as complemening his lieraure by pursuing a posiive approach. I argues ha he possible efficiency gains of increased educaion funding will be a leas parly realized in a poliical economy framework. 2 Pecchenino and Uendorf (1999) reach an opposie conclusion for a model where educaion is exclusively privaely funded: PAYG social securiy will depress invesmen in boh physical and human capial. This is so due o he negaive income effec of social securiy payroll ax, under he assumpion ha he reurns on social securiy conribu i- ons are inferior o reurns on privae savings for all individuals. I is imporan o observe, in paricular, ha when educaion is funded only privaely by parens, hey fail o inernalize he impac of children s educaion on he aggregae produciviy of fuure workers and hrough i on he magniude on he fuure PAYG reiremen benefis. 2

6 The conribuions mos closely relaed o ours are hose by Kaganovich and Zilcha (2008) and Soares (2006). The former analyzes a closed economy seing where, similar o Boldrin (2005), voers ake ino accoun posiive exernaliies of human capial accumulaion for he fuure reurn on savings and pension conribuions. I urns ou ha negaive growh effecs of inroducing a PAYG scheme due o reduced physical capial accumulaion will generally ouweigh, in a fully closed economy, he posiive growh effecs of he enhanced human capial accumulaion. Our presen paper shows ha hese resuls may be urned around when one allows for inernaional mobiliy of capial, such ha he posiive human capial effec dominaes. Indeed, Soares (2006) noes ha educaion ax raes end o be higher in an open economy, for a calibraed model wih a differeniaed age srucure, a mixed educaion sysem wih public and privae schooling, and labor-leisure choice. By considering a simpler framework, we are able o generalize his resuls, o demonsrae he exisence of excepions o his conclusion, o address he effecs of alernaive pension benefi formulas on educaion funding and growh, and o characerize he implicaions for he choice of he pension scheme. In our comparaive analysis of growh implicaions of PAYG and fully funded pension sysems we furher disinguish beween conribuion-relaed and redisribuive pension formulas which affec individuals differenly depending on heir posiion in he disribuion of income. When he level of public educaion funding is poliically deermined and he income of he median voer falls shor of he mean income, hen his children are subsidized hrough public educaion. Under cerain condiions his implies ha lower income voers exhibi sronger suppor for educaion funding. On op of ha, he median voer may benefi hrough a public pension scheme if i is also redisribuive, e.g. fla. We show ha his subsidizaion effec of a fla pension formula can elici majoriy poliical suppor for a relaively higher level of public educaion funding. 3 We noe ha he growh implicaions of conribuion-relaed vs. fla pension benefi formulae depend sarkly on he ax srucure. If educaion is financed by a payroll ax, hen he median voer ends o be a worker earning below average wage. However, if educaion is funded via an income ax or a consumpion ax, hen a par of ax liabiliy falls on he reirees who have lile or no incenive o suppor educaion funding; here- 3 This conrass he conclusions in he lieraure dealing wih growh impacs of pension sysems when educaion is privaely funded, which argues ha conribuion-relaed pension formulas, raher han hose wih fla benefis, paricularly encourage human capial formaion and growh (Kemniz and Wigger, 2000; Docquier and Paddison, 2003). 3

7 fore he median voer in his siuaion is likely o be among he above average wage earners. Consequenly, he median voer would end o be harmed by a redisribuive benefi formula in a pension scheme, and would herefore suppor lower level of public educaion funding han under he corresponding conribuion-relaed pensions. Our view ha he elderly have a paricularly low preference o finance public educaion is in line boh wih he empirical evidence (Poerba, 1997 and 1998) and heoreical sudies (Gradsein and Kaganovich, 2004; Holz-Eakin e al., 2004; Pouvaara, 2006). Thus our paper provides an analysis of iner- and inra-generaional redisribuional effecs of alernaive pension schemes when educaion is publicly funded, and of heir implicaions for poliical suppor for educaion funding and growh. One of our main resuls (Theorem 2), obained under condiions characerisic of mos developed economies, is ha a PAYG pension sysem wih redisribuive benefi formula yields he highes growh rae relaive o he alernaive pension sysem designs. Furhermore, we argue (Theorem 3 and he Corollary) ha while such PAYG pension sysem may be opposed by he majoriy of working individuals due o is inferior reurns on heir pension conribuions relaive o a funded scheme, i is likely o be poliically susained by he coaliion of reirees and lower income workers. Once in place, he PAYG sysem compels a majoriy of working individuals o suppor higher public funding of educaion, and hence enhances economic growh. The remainder of our paper is organized as follows. Afer inroducing he model in Secion 2, we invesigae he impac of differen benefi formulas wihin a given pension scheme on educaion funding and growh in Secion 3. We build on his analysis in Secion 4 o obain our main resuls comparing he implicaion of funded and PAYG schemes for educaion funding and growh. In Secion 5, we consider individual raes of reurn o social securiy conribuions arising in differen pension schemes and heir implicaion for preferences for and poliical susainabiliy of a pension scheme. The final Secion 6 summarizes and discusses our findings. 2. The basic model We consider an overlapping generaions small open economy where perfecly compeiive firms produce a homogeneous good using human and physical capial as inpus in a consan reurns neoclassical echnology. Physical capial is inernaionally mobile and herefore he ineres facor is exogenous from he vanage poin of he small open economy. For simpliciy bu wihou essenial loss of generaliy we posulae he ineres 4

8 facor R o be saionary. The consan reurns echnology hen implies ha he wage rae w per efficiency uni of labor (human capial) is also exogenous and saionary. Each individual in his OLG populaion lives for hree periods: childhood, working age, and reiremen. An individual is indexed by his family name ω and his generaion =0,1,2, which is idenified wih he ime period of his working age. A he beginning of his working age each individual gives birh o an exogenously se number n of children, who will bear his family name. By Ω we denoe he se encompassing he populaion of generaion individuals and we le µ ( ω ) be he Lebesgue measure on Ω which we assume is well defined. Human capial endowmens h ( ) 0 ω of he members of he iniial generaion 0 are presumed given. As a child, i.e. during he ime period -1, each member of generaion =1,2, receives public educaion in a uniform amoun e 1 which is funded by a proporional educaion ax on conemporary wage income whose rae is deermined by he parens generaion hrough a poliical process. The human capial is aained by he sar of one s working age according o he educaion producion funcion h ( ω) = Bh [ ( ω)] e, (1) σ 1 σ 1 1 wih B > 0 and 0 < σ < 1, where h ( ) 1 ω is he paren s human capial, hus boh school and family inpus conribue o a child s human capial accumulaion. Noe ha all individuals in generaion who share a family name ω are idenical in heir human capial aainmen. Wherever possible wihou loss of clariy we ll be omiing family idenificaion ω for breviy. During his working age, an individual inelasically supplies his human capial and receives wage income I ( ω) = wh ( ω) (2) He is required o conribue a fixed fracion τ of his wage income o he public pension scheme, and also pays a payroll ax a a rae? o finance public public educaion sys- y em. He allocaes his disposable income o working age consumpion c ( ω ) and savings s ( ω ) such ha 5

9 ( ) y c ( ω) = 1 τ wh ( ω) s ( ω) (3) In his he old age, he generaion individual consumes his enire asses equal o he sum of his public pension π ( ) + 1 ω and he gross reurn on savings. Thus his old-age consumpion is given by c ( ω) = π ( ω) + Rs ( ω) (4) o + 1 y o Preferences are defined by a uiliy funcion Uc (, c, h + 1) ha is sricly increasing and sricly concave in all argumens, where he hird argumen is he level of human capial aained by he children of he individual, which reflecs parenal alruism. For analyical racabiliy we will consider he case of addiively-separable logarihmic uiliy: Uc (, c, h ) = α lnc + β lnc + γ ln h (5) y o y o wih α, β, γ > 0. According o he firs-order condiion of he individual opimizaion problem (2)-(4), he opimal levels of saving, young-age and old-age consumpion are given by, respecively: β απ+ 1( ω) s ( ω) = ( 1 τ ) I ( ω) α + β ( α + β) R y α π+ 1( ω) c ( ω) = ( 1 τ ) I ( ω) + α+ β R o Rβ π+ 1( ω) c ( ω) = ( 1 τ ) I ( ω) + α+ β R (6) Since we assume ha expendiures on educaion are financed by a proporional payroll ax, he school budge equaion is given by ne = I = wh (7) 6

10 where I and h are, respecively, he average individual wage income and he average individual human capial level in he generaion. According o he noaion inroduced earlier h = Ω h ( ω ) dµ ( ω) µ ( Ω ) (8) whereby we noe ha according o he exogenous populaion dynamics defined above µ ( Ω ) = nµ ( Ω ) Combining he expressions (7) and (1) yields he expression for he dynamics of individual human capial: ( w ) 1 ( ) [ ( )] σ σ 1 σ 1 σ 1 ω ω h + = Bh h (9) n Then using (8) we obain he equaion for he dynamics of he average human capial 1 σ w = + 1 h hb M n = µ ( Ω ) dµ ( ω) h (10) Ω where [ ] 1 h ( ) ω M σ Noe ha he expression M in equaion (10) characerizes he dispersion of human capial (equivalenly, of wage income) in he populaion. Is highes value is 1, which obains iff human capial aainmen is compleely homogeneous; M declines wih increasing dispersion of incomes. Thus equaion (10) implies ha given he same educaion funding levels he average human capial will grow a he higher rae across generaions he less dispersed are he wage incomes. We assume ha period educaion ax rae is deermined by majoriy voing. In our model he curren old do no bear he cos nor receive he benefi of educaion spending in period. Therefore hey are assumed o be indifferen o he voe s oucome and o absain in he voe. We will discuss he implicaions of he model s modificaions where he elderly paricipae in he voe, a he end of Secion 3. We will presen a comparaive analysis of pay-as-you-go (PG) and fully funded (FF) public pension schemes, where each can use eiher a conribuion-relaed or a fla pen- 7

11 sion formula. We will use he self explanaory superscrips PGc, PGf, FFc, and FFf in he noaion associaed wih he respecive pens ion sysems. The pension benefis provided under he respecive schemes are given by FFc π ( ) ( ) ( ) + 1 ω = RτI ω = Rτwh ω (11a) FFf π + 1 = RτI = Rτwh (11b) ( ) ( ) π ( ω) = nτi I ( ω)/ I = nτwh h ( ω)/ h (11c) PGc π = nτi = nτwh (11d) PGf Thus under he fully funded schemes pension benefis are financed by he reurns on he invesmen of he beneficiaries conribuions. While he fla version of he scheme is inra-generaionally redisribuive, he conribuion relaed scheme is equivalen o he privae saving, albei a a prescribed rae. The pay-as-you-go schemes, however, consiue an iner-generaional ransfer where he benefis o he reirees are funded by axes on he wage income of he conemporary workers. Thus he effecive reurns o one s conribuions ino he sysem are direcly ied o wage growh, i.e. o he growh of labor produciviy in he nex generaion of workers. Since he facor prices are exogenous and saionary, physical capial is perfecly mobile and he neoclassical echnology exhibis consan reurns o scale, he supply of human capial and he dynamic compeiive equilibrium are uniquely deermined, for a given sequence of educaion ax raes and a specified pension scheme, by he allocaion decisions of individual agens as expressed by he relaionships in (6). As saed above, in each period educaion ax rae is deermined by majoriy vo - ing of he relevan consiuens, i.e. he members of generaion. In he nex secion, we will derive he mos preferred levels of he ax for individual members of he generaion based on heir preferences and resource allocaions in he dynamic compeiive equilibrium. We will also show ha his allows one o uniquely and recursively define he ax rae chosen by he majoriy in each period. We define he recursive dynamic poliical equilibrium (DPE) as he resuling sequence of educaion ax raes { } = combined wih he corresponding dynamic compeiive equilibrium. 0,1,... 8

12 In our comparaive dynamics experimen we compare he economic oucomes in respecive dynamic poliical equilibria under alernaive social securiy schemes, characerized by benefi formulae (11a) (11d), inroduced in period =0. In cases of a PG scheme his means ha he generaion of reirees presen a =0 receives he benefis as an unanicipaed gif which herefore has no effec on heir young-age poliical and saving decisions. 3. Inra-generaional redisribuion in pension benefi formulas and growh In his Secion, we will compare he effec ha he choice of fla vs. conribuion relaed benefi formula has on growh under eiher pay-as-you-go or fully funded sysems. When he pension benefi formula is fla i is progressively redisribuive wihin each generaion of reirees, as opposed o he conribuion relaed pension benefi formula. We will herefore show how hese pension formulas differ in heir impac on incenives of differen income groups o suppor funding of public educaion, via he choice of he educaion ax rae, and hereby on growh. We will firs show ha individual preferences are single-peaked wih respec o he educaion ax rae. Subsiuing he consumpion funcions given in (6) as well as he expressions (1) and (7) for he human capial producion ino he uiliy funcion (5) we obain he expression for he working age individual s value funcion * π 1 (, ) ( ) ln ( 1 ) ( ) ( ) + ω U ω = α + β τ I ω + + γ ( 1 σ) ln + D R (12) where D summarizes he erms ha do no depend on. Lemma 1: The value funcion is sricly concave in under eiher of he pension formulae (11a)-(11d). Proof. See he Appendix. According o Lemma 1 each individual has a unique mos preferred educaion ax. Using he firs order condiion of maximum of he funcion U * ( ) defined in (12) as well 9

13 as he relaionship (7), we obain he expression for he ax rae preferred by generaion individual in family ω : π 1( ω) + e π+ 1( ω) γ ( 1 σ) 1 τ + ( α β) * RI( ω) + + RI( ω) e ( ω) = α + β + γ σ ( 1 ) (13) π 1 where he erm ( ) + ω e e in general depends on he value of * ( ω ). Under he conribuion-relaed fully funded pension scheme, according o (13) and he benefi formula (11a), he choice of he DPE educaion ax rae is unanimous and consan over ime: FFc FFc = γ ( 1 σ) ( 1 ) α + β + γ σ (14a) Noe ha since his paricular pension scheme is equivalen o a self-financed reiremen, he preferred educaion ax rae given by (14a) is moivaed purely by each individual s alruisic concern for he educaional aainmen of his children in he public school sysem. The unanimiy of he preferred ax rae is a consequence of he logarihmic specificaion of he uiliy funcion. Replacing i by a more general CES form would make he preferred ax raes decrease or increase in individual s income, depending on he elasiciy of subsiuion beween own consumpion and children s educaion. Empirical esimaes of income and subsiuion elasiciies sugges ha he logarihmic (equivalenly, Cobb-Douglas) specificaion of he uiliy funcion is a reasonable approximaion (Gradsein e al., 2005, ch. 4), implying ha he preferred educaion ax should indeed vary lile across incomes under he conribuion relaed fully funded pension sysem in he framework a hand. If a funded scheme uses he fla benefi formula (11b), hen he logarihmic uiliy implies ha he preferred ax rae falls wih increasing wage income. Indeed, according o (11b) pension benefis do no depend on fuure workers human capial, i.e. π ( ) + 1 ω = 0 and herefore according o (13) e 10

14 FFf h γ ( 1 σ) 1 τ + τ h ( ω) ( ω) = α+ β + γ σ ( 1 ) (14b) Thus under his fla pension benefi scheme, similarly o he conribuion relaed case described by (14a), all individuals prefer o devoe he same, alruisically moivaed fracion of lifeime income o public educaion. However, given he redisribuive naure of he fla scheme his implies ha he preferred rae of educaion ax decreases in individuals wage income. FFf Le ( I ) sand for he ax rae preferred by generaion individuals whose wage FFf FFc equals o he economy s average. Then according o (14b) and (14a) ( I ) =. Furhermore, assuming ha he median voer earns less han he average wage, a majoriy of workers under he fla benefi fully funded pension scheme will prefer a higher educaion ax rae han he rae FFc chosen under he conribuion-relaed fully funded pension sysem. Likewise, if he median voer earns an above average wage, hen a majoriy of workers under he FFf sysem will prefer a lower educaion ax rae han FFc. The following Proposiion summarizes he above argumen and shows furhermore ha a similar comparaive resul holds for he conribuion relaed vs. fla PAYG pension schemes. Proposiion 1: (i) Assume ha median voer s wage income is below he average. Then for a given pension conribuion rae and he pension scheme ype (funded or PAYG) he DPE educaion ax rae in each period will be higher under a fla pension benefi formula han under he corresponding conribuion-relaed one. In oher words, he following inequaliies are rue: FFf FFc > and PGf > for all =0,1, PGc (ii) Assume ha median voer s wage income is above he average. Then for a given pension conribuion rae he respecive comparisons yield he resuls opposie o hose in par (i), i.e. he following inequaliies are rue: FFf FFc < and PGf < for all =0,1, PGc Proof. As argued above, he proof for he case of fully funded pensions is esablished by comparing he expressions (14a) and (14b). We will now urn o he case of pay-as- 11

15 you-go pensions. Under he conribuion-relaed PG pension scheme, he chosen educaion ax rae is independen of income. Indeed, according o relaionships (11c) and (12) i saisfies PGc n h n h γ ( 1 σ) 1 τ + τ ( α β) τ(1 σ) R h + + R h = α + β + γ σ ( 1 ) (14c) Likewise, according o (11d), under he fla benefi PG pension scheme he individual preferred educaion ax raes saisfy he equaion PGf n h + 1 n h+ 1 γ ( 1 σ) 1 τ + τ ( α β) τ(1 σ) Rh( ω) + + Rh( ω) ( ω) = α + β + γ σ ( 1 ) (14d) We noe ha h + 1 on he righ-hand sides of (14c) and (14d) depends on he educaion ax raes. Lemma 1 implies ha (14c) defines a unique unanimously mos preferred educaion ax rae under conribuion-relaed pensions, and ha (14d) has a unique individ- PGf ual-specific soluion ( ω ). Comparing (14c) o (14d) immediaely shows ha PGf PGc PGf ( I ) = where ( I ) is he educaion ax rae preferred, under he corresponding pension scheme, by generaion individuals whose wage equals o he economy s average. Moreover, i is obvious from (14d) ha he preferred rae of educaion PGf ax ( ω ) decreases in individuals wage income, which implies he Proposiion s resuls for he cases of pay-as-you-go pension sysems. I is clear ha he choice of a higher educaion ax rae will resul, ceeris paribus, in speedier accumulaion of human capial. In he case of a small open economy a hand, moreover, i also implies a faser economic growh. Indeed, due o perfec capial mobiliy, neoclassical consan reurns echnology and he assumed saionariy of he ineres rae, physical capial will be supplied in consan proporion o he human capial, and hereby he aggregae oupu will grow a he rae of human capial accumulaion. Le g denoe he growh facor of per capia oupu, hen as explained above i is equal o he growh facor of he average human capial. Therefore according o (10) 12

16 1 σ h+ 1 1 σ w = = h n g B M (15) Le FFc g, FFf g, PGc g, PGf g denoe he respecive per capia growh facors which obain in he dynamic poliical equilibria under he pension schemes defined by he benefi formulae (9a)-(9d). The relaionship (13) along wih Proposiion 1 leads o he following resul showing ha in a small open economy a social securiy scheme which generaes greaer suppor for public educaion funding will also yield relaively speedier economic growh a all imes =0,1,. Theorem 1: (i) Assume ha median voer s wage income is below he average. Then under he provisions of Proposiion 1 he economy s growh rae will be higher in each period under a fla, i.e. redisribuive among reirees, pension benefi formula (wheher PAYG or FF) han under he corresponding conribuion-relaed one. In oher words, he following inequaliies are rue: g FFf > g and FFc g PGf > g for all =0,1, PGc (ii) Assume ha median voer s wage income is above he average. Then under he provisions of Proposiion 1 he respecive comparisons yield he resuls opposie o hose in par (i), i.e. conribuion relaed pension benefi formulas are more conducive o economic growh. In oher words, he following inequaliies are rue: g FFf < g and FFc g PGf < g for all =0,1, PGc Proof. Observe ha according o he relaionships (9) and (10) he expression equaion (15) is invarian of he pension scheme and of he educaion ax rae M in. Therefore he choice of a pension scheme affecs growh facor g only direcly hrough he erm 1 σ in equaion (15), which is deermined by he DPE value of he educaion ax rae corresponding o each of pension sysems. Thus i remains o refer o Proposiion 1. The inuiion for he above resuls is given by he fac ha when he pension conribuion rae is fixed and he pension benefi formula is fla, i progressively redisribues reiremen income wihin each generaion, as opposed o he case of conribuion relaed pension benefis. If he pension scheme is fully funded, his means ha poorer agens will be willing o devoe relaively higher fracion of heir income o educaion funding 13

17 based on he uniform degree of alruisic moive. Under a PAYG pension scheme, he individual rae of reurn i offers on invesmen in public educaion of he nex generaion s workers is likewise higher for he agens wih below average income who will herefore suppor relaively higher levels of educaion funding. Thus, for a given pension scheme (PAYG or funded) and a given wage income disribuion in which mean income exceeds median income (so below average wage earners are in he majoriy) he wage growh will be faser under he fla benefi pension formula, as saed in Proposiion 1(i) and Theorem 1(i). These resuls are reversed when he median voer earns more han he average wage. In such a siuaion, moving from a conribuion-relaed pension scheme o a fla benefi scheme of he same ype a a given conribuion rae reduces public invesmen in educaion and growh. Such a siuaion may easily occur alhough empirical wage disribuions always exhibi median below he mean. When educaion is funded, in addiion o axes on wages also hrough axes on non-wage income and/or consumpion, he elderly are no longer indifferen wih respec o he choice of he ax level. Since hey will no benefi from workers produciviy in he nex period, and since hey (in our model) do no have an operaive alruisic moive, unlike he young parens, heir preferred ax rae is zero. Ordering he individuals by he level of preferred ax raes will hen ypically show ha a median voer will be a working age individual wih above average wage income, even if he wage disribuion is skewed righ. The curren rends in life expecancy and feriliy in OECD counries indeed poin in he direcion of a scenario wih he median voer earning more han he average wage. Since he ax burden of he old is generally no negligible, one may expec ha growh effecs of a given pension scheme will be less favorable under Beveridgean-syle fla benefis. 4. Funded vs. PAYG pensions While he previous Secion focused on he growh effec of inra-generaional redisribuion in a pension benefi formula, we will now urn o he impac of inergeneraional ransfers. Namely, we will compare he growh oucomes under he PAYG sysem, which ransfers pension benefis from he young o he old, o he growh oucomes under he fully funded (FF) sysem which is financed by he reirees hemselves. We will firs compare he dynamic poliical equilibrium educaion ax raes obained under he PAYG and FF. Using he expression (12) for he value funcion of an individual as well as he pension benefi formulae (11c) and (11d), we obain he value funcion 14

18 expressions for he cases of conribuion relaed and fla benefi PAYG sysems, respecively: ( ) ( ) ( ) U ( ) = α + β ln 1 τ + R nh τ / h + γ 1 σ ln + D * PGc (16) ( ) ( ) ( ) U ( ω, ) = α + β ln 1 τ h ( ω) + R nτh + γ 1 σ ln + D * PGf where D 1 and D 2 summarize he erms in he respecive expressions ha do no depend on. (17) Since, according o Lemma 1, boh value funcions are sricly concave he firs order condiions of heir maxima uniquely deermine he respecive individually preferred levels of he educaion ax rae. In paricular, noe ha under he PAYG sysem wih conribuion relaed benefis preferences are unanimous since he value funcion (16) is invarian across conemporary individuals. We differeniae (16) using he expression (10) and afer sraighforward algebraic ransformaion we obain * PGc U ( ) d1 {( 1 σ)( α β γ) BR w σ n σ = + + τ σ M + ( 1 )( 1 ) ( 1 ) } + γ σ τ α + β + γ σ (18) where d 1 is an expression which is posiive for any value of he ax rae. Therefore he DPE ax rae PGc is defined by he firs order condiion 1 1 σ σ 1 σ ( σ)( α β γ) τ γ ( σ)( τ ) α β γ ( σ) BR w n M = 0 Furhermore, hanks o sric concaviy of he value funcion (16), based on he sign of he derivaive DPE ax rae * PGc U ( ) FFc = one can infer he relaionship beween PGc and he FFc given by formula (14a), which corresponds o he fully funded conribuion relaed pension benefi scheme. Thus we subsiue FFc = in he expression (18) and verify is sign. This along wih formula (15) yields he following resul: 15

19 Proposiion 2: For a given social securiy ax rae and he conribuion relaed reiremen benefi formula, he PAYG pension scheme (11c) will resul in higher DPE educaion ax rae and per capia growh facor han he FF sysem (11a), i.e. he following inequaliies: PGc FFc > and g PGc > g FFc are rue in a period =0,1,, if and only if he following condiion holds: where ( ) ( 1 ) 1 σ σ σ γ σ γ BR w n M > α + β + γ σ α + β + γ M is he expression defined in (10). (19) We now apply a similar analysis o he case of PAYG sysem wih fla benefi formula. Differeniaing he individual value funcion (17) we obain, afer sraighforward algebraic ransformaions and using he relaionship (10): U ( ω, ) h d2 ( 1 )( ) BR w σ n σ = σ α + β + γ τ σ M + h ( ω) * PGf ( 1 )( 1 ) ( 1 ) } + γ σ τ α+ β + γ σ where d 2 is an expression which is posiive for any value of he ax rae. We now subsiue he value (20) FFf = given by formula (14b) in he expression (20) and obain * PGf U ( ) FFf = = 1 σ 1 σ γ 1 1 ( 1 σ ) h σ σ = d3 ( α+ β + γ ) BR w n 1 τ + τ M γ α + β + γ ( 1 σ) h ( ω) where d 3 is an expression posiive for any. (21) While expressions (20) and (21) are individual-specific, noe ha hey are sricly decreasing in individual human capial aainmen h ( ω ). Therefore under majoriy voing he DPE oucomes for generaion will be deermined by he median voer s human capial aainmen, which we denoe by med h. Then similarly o he above analysis we 16

20 med need o examine he sign of expression (21) for h ( ω ) = h in order o deermine he relaionship beween he DPE educaion ax raes FFf and PGf under he respecive fla benefi social securiy sysems. This analysis yields he following resul: Proposiion 3: For a given pension conribuion rae and he fla reiremen benefi formula, he PAYG pension scheme (11d) will resul in higher DPE educaion ax rae and per capia growh facor han he FF sysem (11b), i.e. he following inequaliies: PGf > and g FFf PGf > g FFf are rue in period, if and only if he following condiion holds: ( ) ( 1 ) 1 σ 1 σ h 1 τ τ med h σ σ γ σ γ BR w n + M > α+ β + γ σ α+ β + γ (22) Discussion. According o he equaion (10) and formulas (14a) and (14b) he lef-hand nh+ 1 sides of inequaliies in (19) and (22) are equal o, he real growh facor of he Rh economy s human capial in a DPE under he fully funded pension scheme, wih conribuion relaed benefis in he case of inequaliy (19) and fla benefis in he case of (22). The righ-hand side in boh (19) and (22) represens he degree of each individual s alruisic concern for he human capial aainmen of his children. I is no difficul o see herefore ha under he Cobb-Douglas specificaions of our model, he inequaliies in (19) or (22) compare he alruisic reurn on invesmen in educaion, which is is only moive under an FF pension sysem, o he marginal economic reurn in erms of pension benefis under a PAYG scheme on addiional invesmen in educaion, beyond he FFc ax rae prescribed by he respecive FF sysem ( or FFf respecively). Indeed, a PAYG scheme brings abou an addiional non-alruisic moive o finance public educaion since i increases wage incomes in he nex generaion, of which he reirees will receive a share under PAYG scheme. Specifically, if he inequaliy (19) is saisfied, he reurn o fuure reirees generaed by such addiional educaion funding under he PAYG sysem is high enough; hence he resul of Proposiion 2 ha he PAYG sysem would yield a relaively higher educaion ax rae han he respecive FF sysem, hus PGc FFc >. This resul may be overurned, however, when inequaliy (19) is violaed because he preferred ax based on he alruisic moive declines if he move o a PAYG scheme reduces lifeime income when he rae of reurn on social securiy conribuions falls shor of he ineres rae. 17

21 As poined ou in Secion 2 following he equaion (10), he value of expression M is a mos 1, which obains only if human capial levels are idenical across conemporary individuals, and furhermore M declines wih increasing dispersion of incomes/human capial levels. Thus condiions in (19) and (22) impose a bound on he degree of income inequaliy. Proposiions 2 and 3 show in paricular, according o he condiions (19) and (22), ha he scenario where DPE educaion ax rae and he growh rae of per capia GDP will be higher under a PAYG scheme han under he respecive funded sysem is likely in an environmen wih some of he following feaures: moderae degree of inequaliy, srong public educaion sysem ( B is relaively large), he wage income of a median voer is below he economy s average and a culure where parens give relaively low prioriy o he human capial aainmen of heir children (? relaively small). We noe ha he condiions lised above are characerisic of developed economies. We also remark ha when he median voer s wage is below he economy s average, he inequaliy (19) auomaically implies (22). Combining he resuls of Proposiions 2 and 3 wih hose of Proposiion 1(i) and Theorem 1(i) we obain he main resul of his secion: Theorem 2: Consider he DPE oucomes in oherwise idenical four small open economies, where condiion (19) is saisfied and median voer s wage is below he economy s average, bu which use differen alernaive pension schemes defined by he formulae (11a)-(11d) wih he same pension conribuion rae a =0,1,. The economy using he PAYG social securiy sysem wih fla benefi pension formula (PGf) delivers he highes DPE educaion ax raes and per capia growh raes. We can hus conclude ha under he condiions characerisic of (small open) developed economies he PAYG fla benefi social securiy sysem, i.e. he one mos redisribuive boh beween and wihin generaions is mos conducive, relaive o he alernaives sysems given he same conribuion rae, of public invesmen in human capial and hence economic growh. As we explained earlier, he role of inergeneraionally redisribuive feaure of he PAYG sysem is ha i provides an economic incenive o fund he nex generaion s educaion, hrough he link beween he reiree pension benefis and heir 18

22 conemporary workers produciviy. The pension formula which is progressively redisribuive among he reirees enhances he majoriy s suppor for higher raes of educaion axes, provided ha he elecorae s income disribuion is righ-skewed. An essenial assumpion in Theorem 2 is ha he median voer is a worker earning a below average wage, which makes Theorem 1(i) applicable. If insead he median voer s wage was above he average, hen Theorem 1(ii) would apply, and combining i wih Proposiions 2 and 3 one would obain, given ha boh (19) and (22) remained saisfied, ha i is he conribuion-relaed PAYG scheme ha delivers he highes educaion ax and he fases growh. Alhough wage disribuions are ypically righ-skewed, noe ha as argued in he Discussion a he end of Secion 3, under a public finance sysem where reirees are also subjec o he educaion ax, hey would be in opposiion o i, so he median voer on he issue would likely be a worker earning more han he average wage. The above resuls on he effecs of pension schemes on educaion funding and growh raise a quesion wheher hey are associaed wih echnological inefficiencies in resource allocaions. In his even, oupu growh is no a is maximum for given levels of aggregae invesmen in human and physical capial. We noe ha an ineremporal allo- h + 1 caion of resources o human capial invesmen is no efficien if w> R holds a e he margin. Indeed, when he marginal produc of public educaion exceeds ha of physical capial, an inergeneraional Pareo improvemen can be achieved by invesing more in public educaion raher han in physical capial, where he governmen deb can be repaid by axing he acive generaion in he nex period. If such a scenario occurs, an obvious quesion is why he governmen does no pursue he policy of greaer funding of educaion anyway. In he real world economies, he governmen insiuions responsible for funding public educaion face hard fiscal consrains, including he caps on borrowing. Besides, in many counries educaion funding is provided a a regional or even local level making i difficul o capure he produciviy gains resuling from jurisdicion specific educaion invesmen. The level of educaion funding is echnologically efficien for a paricular suden if h ( ) + 1 ω w= R is rue. If conracs wih minors could be enforced, parens would simply e ransfer he necessary amoun o finance he efficien level of educaion. The children would repay heir deb wih ineres in he nex period. Alruisic parens could hen 19

23 make some addiional ransfer in he form of physical capial or he final good, eiher iner vivos or as a beques. Therefore, if parens are indifferen beween giving he same amoun of money as educaion funding or in cash, overinvesmen in educaion can occur only if he pension scheme redisribues oward he median voer. Since he alruisic moive may well be oo weak o ensure a sufficienly high level of public educaion, he PAYG scheme can ac as a device o enhance welfare by promoing invesmen in human capial. 5. The choice of a pension scheme The resuls of he previous secion show ha for a wide range of parameers characerisic of developed economies a PAYG pension scheme ends o generae suppor for higher public funding of educaion han a comparable FF sysem would. The above resuls characerize a comparison of oherwise idenical four open economies where he alernaive social securiy sysems (FFc, FFf, PGc, PGf ) are in place and are no hemselves subjec o public choice. In order o approach he laer issue, we need o compare hese alernaive pension sysems in erms of raes of reurn hey generae for heir consiuens. The clear benchmark for his comparison is he FF sysem wih conribuion relaed benefi formula (11a) which provides each reiree he benefi reurn on his conribuions equal o he reurn on privae savings under he ineres facor R. S Le ρ ( ω ) be he rae of reurn a pension sysem S (i.e., one of he four alernaive arrangemens FFc, FFf, PGc, PGf under consideraion) provides for generaion individual in family ω on his conribuions ino he sysem. As menioned above, ρ FFc ( ω) R Therefore, according o (11b), ρ FFf h ( ω) = R h ( ω) and hereby he median voer s reurn o his social securiy conribuions is given by FFf med h ρ ( ω ) = med R (24) h Furher, according o formulae (11c) and (10) he PAYG sysem wih conribuion relaed benefi formula provides each individual in generaion he rae of reurn nh ρ ω ρ = = (25) ( ) 1 σ PGc PGc σ σ PGc ( ) Bw n M h (23) 20

24 on his social securiy conribuions, where as before ax rae chosen by generaion if his pension sysem is in place. PGc sands for he DPE educaion The educaion ax rae ˆ ha induces he rae of reurn on social securiy conribuions under a conribuion-relaed PAYG scheme o become equal o he ineres facor R is 1 1 σ ˆ 1 R = w σ (26) Bn M Similarly, according o (11d) and (10), he rae of such reurn for a generaion individual in family ω under PAYG sysem wih his redisribuive fla benefi formula will be nh h ρ ω = = (27) PGf ( ) 1 σ PGf σ σ ( ) Bw n M h( ω) h( ω) We will firs focus on he comparison of his rae of reurn o he reurn on privae savings, i.e. he ineres facor R. According o (27) for hese facors o be equal he educaion ax rae chosen under he PGf pension sysem would have o saisfy he equaion h h ( ω) i.e. o be equal o 1 σ σ 1 σ Bw n M R 1 h ( ω) R ( ω) = w σ h Bn M = (28) 1 1 σ (29) Since according o (27) he rae of reurn on pension benefis is an increasing funcion PGf of he educaion ax rae, he comparison of ρ ( ω ) o R is equivalen o ha of ( ω ) o ( ω ). The resul of he laer comparison is obained similarly o our PGf proof of Proposiion 3, i.e. by subsiuing = ( ω) ino he formula (20) of he marginal value U ( ω, ) * PGf and verifying is sign. Namely, due o sric concaviy of he * PGf U value funcion he inequaliy PGf (, ω) ( ω) < ( ω) is equivalen o < 0. A sraighforward applicaion of his analysis o he median voer and proceeding analogously for he conribuion-relaed-formulas yields he following resuls. 21

25 PGf med PGf med Proposiion 4: (i) The inequaliy < ( ω ) and equivalenly ρ ( ω ) < R are rue, i.e. he median voer s reurn on pension conribuions under PAYG fla benefi social securiy sysem falls shor of he reurn on privae savings, iff he following condiion holds: ( ) ( 1 ) 1 σ 1 σ h γ σ σ σ α + β + γ BR w n 1 τ τ M 1 med + < h α + β + γ σ γ (30a) PGc (ii) The inequaliy < ˆ PGc and equivalenly ρ < R are rue, i.e. he reurn under PAYG conribuion-relaed social securiy sysems falls shor of he reurn on privae savings, iff he following condiion holds: BR 1 w 1 σ n σ γ α + β ( 1 σ ) + γ ( 1 σ ) 1 σ α + β + γ 1 τ + τ γ 1 σ M < 1 (30b) I can be easily seen ha inequaliies (30a) and (30b) are broadly consisen wih he condiions (19) and (22), i.e. boh ses of condiions are saisfied for a wide subse of he range of parameers. Specifically, (30a) and (30b) impose reasonable upper bounds on feriliy n, he produciviy parameer B of he public educaion sysem, as well as skewness of wage disribuion in he elecorae h h med. Recall ha according o Theorem 2, if condiion (19) is saisfied, hen growh will be he fases under he PGf pension formula. Therefore, noing ha when he disribuion of wages is righ-skewed redisribuive pension formulas are preferred o he conribuion-relaed formulas by a majoriy of voers, we confine our aenion o he choice beween pension schemes wih a redisribuive (fla) benefi formula. We now couple he resul of Proposiion 4 wih ha of Proposiion 1(i) and based on formulas (23)-(25) and (27) obain he following: Theorem 3: Consider a small open economy which saisfies he parameric condiions (19), (22) and (30a). Assume ha a pension scheme is o be chosen in period among he alernaives defined by he formulae (11a)-(11d) wih a given pension conribuion rae, and ha he corresponding DPE educaion ax rae is deermined accordingly (as described by (14a)-(14d)). Then he corresponding raes of reurn o pension conribuions of a median worker in generaion will compare as follows: 22

26 ρ FFf ( ω med ) > ρ FFc R > ρ PGf ( ω med ) > ρ PGc (31) regardless of wha pension sysem, if any, was in place prior o period. Since according o (31) he reurn o social securiy conribuions is higher under he funded scheme, hen oher hings equal including he same level of educaion ax rae, he welfare level of every working age individual will be higher under a fully funded han under he corresponding PAYG scheme. Proposiions 2 and 3 sae, furhermore, ha he majoriy of young agens will choose lower educaion ax raes under fully funded schemes. Thus his revealed preference argumen proves he following: Corollary: Under he provisions of Theorem 3 a majoriy of young voers in each generaion will prefer a fully funded sysem over a PAYG arrangemen. I is obvious, however, ha reirees in each period will be unequivocally in favor of a PAYG arrangemen. 4 Even if here are fewer old people in he economy han hose in he working age, he median voer in he combined elecorae on he issue of choosing a social securiy sysem will be, as also poined ou by Conde-Ruiz and Galasso (2005), a low income young agen, such ha he inequaliy (30a) can be violaed. 5 In his likely scenario he PGf social securiy sysem will be susained despie he wishes of he majoriy of he working populaion. The imporan implicaion of our resuls semming from Theorem 2 is ha his scenario is characerized by consisenly higher raes of economic growh. Therefore, while according o he Corollary he majoriy of young agens always prefer o deviae from he PGf social securiy sysem based on heir lifeime welfare maximizaion, his individually raional incenive ignores he benefi hese agens have received from he pas higher growh raes due in par o he PAYG sysem which was in place in prior periods. I similarly ignores he longer-erm benefis of economic growh for he fuure generaions, if he PAYG sysem is susained. 4 In a more realisic OLG framework wih muliple coexising age cohors older cohors of workers would join he reirees in heir suppor of a PAYG scheme (see e.g. Browning, 1975). 23

27 We can hus conclude: While he inergeneraional ransfer sysem wih PGf pension scheme is susained due o he suppor of all of he old and a minoriy of he young, i causes a majoriy of he young o suppor higher public funding of educaion and hereby ensures he overall highes long-erm growh. 6. Conclusions This paper challenges he common concern ha PAYG schemes harm growh. This is believed o be he case due o he negaive effec of PAYG sysems on physical capial formaion. We absrac away from his concern by considering he case of a small open economy wih open capial markes and focus insead on he effec of pension sysems on human capial accumulaion. We consider an economy where human capial accumulaion is publicly financed, which seems o be a valid descripion of he siuaion in mos OECD counries. We show, using a poliical economy framework, ha he inroducion of PAYG pension schemes can have a posiive effec on economic growh. The reason is ha a majoriy of voers will suppor higher axes or finance public educaion oday when hey can expec o receive higher PAYG pensions omorrow in reurn. In his way, PAYG pension scheme can serve as a device o overcome a endency oward underinvesmen in public educaion. Furhermore, an inra-generaionally redisribuive PAYG scheme (for example he one where benefis are fla raher han conribuion relaed) may lead o an even faser growh. Indeed, invesmen in public educaion will be higher if he median voer benefis from redisribuion in PAYG pension benefis. One should, however, balance his agains a couneracing effec of a labor supply response o he redisribuive policy. An obvious alernaive o our framework would be a specificaion in which public educaion can be supplemened by privae educaion, as a perfec subsiue. In such a framework, he preferred educaion ax of all individuals wih more han average income would be zero when a funded pension scheme is in place. Should he preferred educaion ax of poorer voers rise wih increasing income, an ends-agains-he-middle coaliion may be formed as in Epple and Romano (1996), where he median voer has less han he median wage. Alhough some privae schooling is crowded ou by increas- 5 When voers choose he rae of conribuion o a redisribuive scheme, he winning coaliion ofen consiss of reirees and workers wih medium wages, who prefer higher conribuion raes han rich or poor workers (Casamaa e al., 2000). 24

28 ing public schooling in such a mixed educaion regime, he endency of a posiive impac of exending PAYG schemes on aggregae invesmen in educaion will coninue o hold. The main cavea of our analysis obviously lies in he assumpion ha he choice of a pension sysem has no effec on he level of physical capial in a small open economy. I is imporan o poin ou, however, ha even if each economy under consideraion is small compared o he res of he world, a symmeric choice of pension policies in all counries, which may be favorable o growh from he perspecive of each individual counry, can under cerain condiions have a negaive impac on global physical capial formaion. 25

29 Appendix Proof of Lemma 1. Differeniaing he value funcion wih respec o yields * U = 1 π i + 1 ( α + β ) R ( 1 τ ) w i w i π + R i + 1 γ + ( 1 σ ). (A1) Since ( 1 ) γ σ is decreasing in, i suffices o show ha also falls wih rising. I urns ou ha V( ) = 1 πi R ( 1 τ ) + 1 w i w i π + R i + 1 sgn 2 V( ) 1 π = sgn R i + 1 π + 2 i 1 π R ( ) i + 1 i 1 1 τ w + w. R i 2 (A2) 2 π Thus, i is enough o esablish ha i Under any funded scheme we have 2 + π i 1 π = 0 for any, while under any PAYG scheme i + 1 < 0 holds

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