Social security, education, retirement and growth*

Transcription

1 Hacenda P úblca Espa ñola / Revsa de Econom ía P úblca, 198-(3/2011): , Insuo de Esudos Fscales Socal secury, educaon, reremen and growh* CRUZ A. ECHEVARR ÍA AMAIA IZA** Unversdad del Pa ís Vasco Absrac Recbdo: Novembre, 2010 Acepado: Ocubre, 2011 Ths paper analyzes, frsly, he expeced effecs of socal secury reforms ha have been mplemened n Span afer 2004 (and, secondly, he expeced effecs of reducons n he mnmum penson) on reremen decson and human capal accumulaon (and hence on growh and on ncome nequaly). Indvduals n our model economy dffer n her nnae ably and growh s a by-produc of he mos sklled ndvduals producvy. Accordng o our model, ) ncreases n he mnmum and normal reremen ages are expeced o have a srong effec, no only on ndvduals reremen decsons, bu also on her educaon nvesmen; ) augmened ncenves o lae reremen are no expeced o have any effec; ) reducons n he mnmum penson are no expeced o have a sgnfcan effec unless s compleely elmnaed. Keywords: Socal Secury; Pay-as-you-go; Volunary Reremen; Human Capal; Mnmum Penson. JEL classfcaon: O4, H3. 1. Inroducon A grea deal of leraure has analyzed he effec of pay-as-you-go socal secury on workers volunary reremen age. The avalable emprcal evdence suggess ha, a leas for he US economy, socal secury s relevan for reremen age ssues, despe he lack of We would lke o acknowledge useful commens a he II REDg Workshop on Dynamc General Equlbrum Macroeconomcs held n 2007 n Sanago de Composela, n parcular commens by Omar Lcandro, and a XXXII Smposo de Análss Económco held n 2007 n Granada, n parcular commens by Vrgna Sánchez. We wsh o hank Mnsry of Scence and Technology, hrough Projec ECO , and Basque Governmen gran IT for her fnancal suppor. ** Amaa Iza. Deparameno de Fundamenos del Análss Económco II. Avda. Lehendakar Agurre 83, Blbao (Span). Tel. (34) Fax. (34) E-mal address: amaa.za@ehu.es

2 10 CRUZ A. ECHEVARR ÍA AND AMAIA IZA oal agreemen on he effec of changes n he payou from he socal secury program. (See, e.g., Damond e al., 1997, Cole e al., 2000, Fabel, 1994, Fenge and Peseau, 2005, Kaleml-Ozcan, 2002.). However, very few papers sudy he effec of a mnmum penson upon workers volunary reremen age. For nsance, accordng o he calculaons n Jménez-Marín and Sánchez-Marín (2007) and Sánchez (2010) on he role of mnmum pensons n posponng early reremen n Span, oal early reremen was almos 50% larger wh mnmum pensons. As hey were consderng an exogenous growh model, hey could no, of course, analyze he effec of mnmum (or maxmum) pensons on growh. Some papers have explored he mpac of a pay-as-you-go socal secury sysem on human capal nvesmen ncenves, and hence on endogenous growh. For nsance, Echevarría and Iza (2006) obaned a dscouragng effec of he sze of socal secury on human capal accumulaon and reremen age. Furhermore, he relaonshp beween he sze of socal secury and he per capa GDP growh rae ha hey found was mosly negave, excep for very low values for he socal secury conrbuon rae. The explanaon les n he dscouragng effec ha socal secury mposes on educaon and, n parcular, reremen age, whch causes a fall n he share of he workng populaon n he economy. However, hey dd no consder he effec of mnmum (or maxmum) pensons on educaon and growh. In hs paper, we focus on he effecs of a mnmum penson paymen n a pay-as-yougo socal secury sysem on human capal (educaon) nvesmen ncenves, and hence, on growh and ncome nequaly. Addonally, reremen age s endogenously deermned, so we also analyze he effecs of penson polces on early reremen ncenves. We buld up a wo-perod, OLG model economy n whch penson benefs are earnngs-relaed and populaed by ex ane heerogeneous ndvduals who dffer n her nnae (learnng) ably. Indvduals n her frs perod of lfe choose her level of educaon. Those born wh hgher ably are expeced o nves more n her educaon. Assumng [whch s he case, among ohers, of he Spansh socal secury sysem] ha penson paymens are earnngs-relaed, he reurn on human capal nvesmen s no consraned o labor ncome whle workng, bu n fac exends o pensons durng reremen 1. Therefore, when ndvduals choose her opmal level of educaon, hey ake no accoun no only he effec on fuure labor earnngs, bu also on fuure penson benefs. Consequenly, socal secury nroduces an ncenve for hgher nvesmen n human capal 2. Ths ncenve, however, mgh break down because he penson scheme ncludes a mnmum penson paymen 3. For nsance, as he mnmum penson ncreases, so does he hreshold for he nnae ably for whch ndvduals end up recevng he mnmum penson. Mnmum pensons, herefore, have a dscouragng effec on educaon nvesmen for hose ndvduals wh suffcenly low nnae ably. In her second perod, ndvduals supply labor elascally (.e. opmally choose her reremen age). Therefore, volunary reremen

3 Socal secury, educaon, reremen and growh 11 age also depends on he ncenves embedded n he publc penson sysem: no only mnmum pensons, bu also penales for early reremen whch ake he form of reducons n he ne penson paymen f reremen occurs before normal reremen age, and he ncenves for rerng afer normal reremen age. Mnmum pensons work n he oppose drecon o penales and ncenves as hey promoe early reremen. In shor, socal secury n hs economy nfluences boh he sze of he workng populaon n he economy and s producvy. We calbrae he model and consruc a benchmark case whch farly reproduces some sylzed facs of he Spansh economy n Sarng from hs baselne case, frsly, we solve for a new balanced growh pah economy akng no accoun socal secury polces already n place or projeced o be mplemened n he near fuure (hgher mnmum and normal reremen ages and hgher ncenves o lae reremen). We analyze her effecs on reremen, educaon nvesmen (and hence, on nernal rae of reurn of he penson sysem, ncome nequaly and growh) under wo scenaros: ) assumng he same educaonal dsrbuon of workers and dependency rao as observed n 2004, and ) consderng a new saonary educaonal dsrbuon of workers and dependency rao o whch he Spansh economy s expeced o converge by Thrdly, we analyze he effecs of reducons n he mnmum reremen penson benefs under he above menoned wo scenaros. Our man resuls follow: I. Our model predcs ha more sklled ndvduals enjoy a hgher reurn on her nvesmen n educaon and, consequenly, spend more on educaon. II. The exsence of a mnmum penson, however, may reduce low skll ndvduals ncenves o nves n human capal. III. Increases n he mnmum and normal reremen ages are expeced o have a srong effec, no only on ndvduals reremen decsons, bu also on her educaon nvesmen. IV. Augmened ncenves o lae reremen are no expeced o have any effec. V. Reducons n he mnmum penson are no expeced o have a sgnfcan effec un- less s compleely elmnaed. VI. Polces enhancng human capal nvesmen for he cleveres workers ncrease growh as a by-produc of hese workers producvy. The paper s organzed as follows: Secon 2 descrbes he economy. The calbraon and he correspondng numercal exercse are carred ou n Secon 3. Secon 4 presens he conclusons. A mahemacal Appendx s ncluded a he end. 2 The economy Ths economy s characerzed by he behavor of households and frms whch ac n perfecly compeve markes for one unque (aggregae) commody and wo producon facors (physcal capal and human capal) n he presence of socal secury. Tme s dscree.

4 12 CRUZ A. ECHEVARR ÍA AND AMAIA IZA 2.1. Households A any me hs economy s populaed by wo overlappng generaons of ndvduals, young and old. Indvduals consume boh when young and old, and supply nelascally one un of labor when young. In her second perod, however, ndvduals choose her opmal lesure consumpon: hgher lesure consumpon s nerpreed as workers choosng o rere earler 4. Second perod lesure s modeled as a connuous varable choce, bounded boh above and below, so ha a whole range of nermedae choces s possble. A smlar seup s used n Garrga and Manresa (1999). Addonally, ndvduals n her frs perod mus choose her opmal level of educaon: hs choce wll affec no only her labor ncome, bu also her reremen penson benefs. Ths s so because we assume ha ) socal secury s non-funded, and ) pensons are earnngs-relaed (and, herefore, defned-benef ype). We assume one unque source of heerogeney among ndvduals. Thus, we assume ha here are four ypes of ndvduals ( = 1, 2, 3, 4) who dffer by her nnae ably, θ (where θ1 < θ2 < θ3 < θ4): hgher nnae ably means hgher learnng ably and hgher reurn on nvesmen n human capal and, herefore, hgher educaon expendure n prncple (whch, n urn, mples hgher economc growh). Types 1, 2, 3 and 4 represen ndvduals wh no prmary sudes, ndvduals aanng prmary, secondary (hgh school) and college educaon, respecvely. Ths heerogeney drves he ncome nequaly n hs economy, parally mgaed by he socal secury sysem 5. As wll be seen, he exsence of mnmum penson benefs, along wh he earnngs-relaed naure of penson benefs, mgh pose an ncenve problem: low-skll ndvduals mgh fnd opmal o reduce her nvesmen n educaon for a suffcenly hgh mnmum reremen penson. The preferences of an -h ype ndvdual born a me are represened by he uly funcon uc ( y,, c o, +1, +1 ) = ln c + β ln c +ξ ln, [1] y, ( o, ) where β > 0 sands for he dscoun facor, cy, and co,+1 denoe frs perod and second perod consumpon, ξ > 0 represens he second perod relave preference of lesure upon consumpon, and o,+1 [ L, U ] denoes second perod lesure. We assume ha second perod lesure s bounded below ( L > 0),.e. workers are legally forced o rere a some me before a maxmum age; and, also, bounded above ( U < 1),.e. a mnmum reremen age exss 6. Whenever an ndvdual choce varable s affeced by wo subscrps, he frs one denoes he ndvdual s age (y for young and o for old, respecvely), and he second one denoes calendar me. We assume he producvy level, h, hs ndvdual aans s a funcon of hs/her nnae ably, θ, and hs/her expendure on educaon, e, once normalzed by he oal facor producvy a me, A. More precsely, we assume ha

5 Socal secury, educaon, reremen and growh h =θ 1+ ( ) γ e / A, γ ( 0, 1). 13 [2] As n Bouzahzah, De la Crox and Docquer (2002) [BDD hereafer], he engne of growh of hs economy wll be gven by he aggregae sae of knowledge n he economy. Even hough our model s very close o he one n BDD, he way n whch we separae he ndvdual human capal level from he sae of knowledge s n fac closer o Romer (1990), as we dsngush beween he prvae knowledge aaned by an ndvdual who lves a fne lfe, h, and he non-rval knowledge of he economy whch can be accumulaed ndefnely, A. Educaon expendure s normalzed by he sae of knowledge n Eq. (2) n order o oban a balanced growh pah of he economy along whch he wage (per un of labor) grows a he same rae as he oal facor producvy, A. Therefore, he expendure on educaon wll ncrease a he same rae as A, and all ndvduals of ype wll spend a consan share of her ncome on educaon, so ha boh e / A and h reman consan oo; and he oal producvy wll be h A, growng a he same rae as A. A major dfference beween our model and he one n BDD s ha nvesmen n educaon comes from ncome raher han me. A second dfference beween he wo models s ha we rea reremen age as endogenous: we beleve ha a horough undersandng of all he ncenves embedded n socal secury sysems enals consderng he reremen decson as a choce varable. The frs perod budge consran s gven by c y, + s y, + e = wn, h A, [3] where s ss y, denoes savngs, w n, = (1 τ )w denoes he ne of socal secury conrbuon ss wage rae per effcen un, τ denoes he socal secury conrbuon rae (.e. he pay-roll ax rae), and w denoes he wage rae per effcen un 7. The second perod budge consran s gven by c, + = +r 1, + b ) + s, o 1 (1 + ) s y ( w ha 1 n, n s + 1 [4] ss where w n,+1 = (1 τ +1)w +1, r +1 denoes he neres rae beween perods and +1, b +1 sands for he socal secury reremen penson benef (per un of me), and ss +1 denoes he lump-sum ransfer ha old ndvduals receve as a resul of sharng he dfference beween socal secury conrbuons mnus reremen penson paymens 8. Gven he redsrbuve role played by socal secury, where ncome s manly ransferred from he young (workers) o he old, we assume ha he dfference beween conrbuons and pensons n our model s ransferred o ndvduals n her second perod. Noe ha boh he penson benef and he labor ncome ha he ndvdual s pad n hs/her second perod are convenenly weghed by lesure me and labor me, +1 and 1 +1, respecvely 9.

6 14 CRUZ A. ECHEVARR ÍA AND AMAIA IZA As for he reremen penson, wo cases mus be consdered n urn: ) rerees whose rep penson benef s he resul of applyng a replacemen rae τ +1 o pas earnngs and eher a before-normal-age-reremen penaly, q, or an afer-normal-age reremen ncenve, Φ ; and ) rerees who, oherwse, would be recevng a oo low penson under he prevous scheme, and who are pad a mnmum penson, b mn +1. Thus, hs second ype of rerees penson benef becomes earnngs-unrelaed 10. Formally, reremen penson for an -h ype ndvdual a me +1 s gven by 11 b, for Φ q τ w h A < b b +1 = rep Φ q τ w h A, oherwse. mn rep mn [5] Concernng he frs case, we assume ha he replacemen rae apples o he average labor ncome obaned durng he frs acve perods. If he economy grew a a non-zero per capa rae, for a balanced growh pah o exs, penson benefs should grow a he same rae a whch per capa varables grow. Concernng Eq. (5) wo remarks are n order. Frsly, we assume ha he before-normal-age reremen penaly and he afer-normal-age reremen ncenve only apply o ndvduals whose reremen pensons are earnngs-relaed. Secondly, we assume ha he (absence of) penaly, q, s a lnear funcon decreasng n +1, 1, f +1 q = N N 1 α 1 ( +1 ), f < +1 L N U [6] where α1 = (1 α0) / ( U N ), α0 (0,1), N ( L, U ), denong he lesure correspondng o he normal-reremen-age. Thus, a worker rerng a he mnmum reremen age (so ha = U ) would be pad a fracon q = α0 of he penson ha he/she would oban, oherwse, f he/she rered a or afer normal reremen age (.e. f N ). Addonally, for an early reremen penaly rae per year, π, one has ha α0 s gven by N mn α 0 1 π R R ), [7] = ( where R N and R mn denoe normal reremen age and mnmum reremen age, respecvely 12. Thrdly, n order o evaluae he socal secury reforms mplemened afer 2004, we wll consder he lae-reremen-age ncenves. In parcular, we assume ha he ncenve Φ s a lnear funcon decreasng n +1, 1, f < + 1 Φ = N L 1+α 2 ( +1 ), f +1 N U N [8] where α2 denoes he exra penson paymen for remanng n he labor force afer reachng he normal-reremen-age.

7 Socal secury, educaon, reremen and growh 15 Thus, assumng away borrowng consrans, he problem ha an -h ype ndvdual faces can formally be expressed as he maxmzaon of Eq. (1) wh respec o c y,, co,+1, sy,, +1 and e, subjec o Eqs. (2), (3), (4), (5), (6) and (8). Addonally, mus be he case ha L +1 U. For he sake of clary, he se of frs order necessary condons for hs problem can be presened n wo pars: frsly, we show he condon ha deermnes he opmal e (and b+1), whch s equvalen o ha one ha maxmzes he dfference beween he sum of he dscouned value of frs and second perod earnngs (penson benefs ncluded), mnus he educaon expendure 13. And, secondly, we show he condons for opmal, c y,, c o,+1, s y,, and +1. The opmal values for all choce varables are obaned, of course, by solvng all he condons smulaneously 14. Opmal educaon and reremen penson. The opmal soluon for educaon expendure depends on wheher he reremen penson ha he reree ges pad s earnngs-relaed or no. Thus, can be shown ha f he penson benef does depend on he labor ncome ha he ndvdual obaned when he/she was a worker, he opmal educaon and penson paymen are gven by rep 1 γ ss Φ q τ w + n, +1 (1+ λ +1 1 )(1 +1 ) e = e θ A γθ w 1 τ + + [9] 1, ( ) R +1 R +1 rep and b +1 = Φ q τ +1w h A, respecvely, where λ = (A +1 A ) / A,.e. he growh rae of he oal facor producvy, and R +1 = 1 + r As expeced, and along balanced growh pahs, e 1, ( θ ) grows over me wh A and depends posvely on he ne wage raes per effcency un and he penson replacemen rae. A hgher dscoun facor reduces he dscouned value of reremen pensons and second perod labor ncome, so ha reduces he ncenve o nves n educaon or human capal. Las bu no leas, e 1, ( θ ) depends posvely on θ: more sklled ndvduals enjoy a hgher reurn on her nvesmen n educaon and, consequenly, are expeced o spend more on educaon. Ths s a well known resul n human capal leraure. (See Le Garrec, 2005 and references here n.) Laer we characerze he range of values of θ for whch opmal educaon s gven by (9). The exsence of a mnmum reremen penson, however, mgh break he lnk beween educaon expendure and reremen penson benefs. Indvduals wh a skll level below some lower bound θ mgh fnd opmal o ge pad jus he mnmum penson and nves n educaon accordngly (.e. less). In hs case, can be shown ha he opmal educaon and penson paymen are gven by e = e 2 θ A γ w, w n 1 1 γ ), +1 (1+ λ )(1 +1 [10] R +1, ( ) θ n +, = b mn θ θ Φ q rep and b +1 +1, respecvely, for all <. Of course, as long as τ > 0, mus be he case ha e 1, ( θ ) > e 2, ( θ ) > 0. Thus, our model predcs ha he exsence 1

8 16 CRUZ A. ECHEVARR ÍA AND AMAIA IZA of a mnmum penson may reduce he ncenves of low skll ndvduals o nves n human capal acquson. Noe also ha e 2, ( θ ) s ncreasng n θ. Therefore, a lower bound for he skll parameer mus exs. Once θ θ, he reremen penson becomes earnngs-relaed. Takng no accoun ha [gven he uly funcon n Eq. (1)] oprep mal θ s mplcly gven by Φ +1 s srcly posve, q τ +1w θ = b mn h ( )A +1. For he sake of compleeness, Fgure 1 shows how opmal educaon and penson paymen are relaed o θ and where a dsconnuy of e a θ = θ shows up: sarng a a low θ, when he learnng ably parameer equals θ, educaon expendure jumps upwards 16. e, b b b mn e = e 1 ( θ ) e = e 2 ( θ ) θ θ Fgure 1 The oher opmal decsons: consumpon, savngs and lesure. The res of frs order necessary condons for an neror soluon are gven by 1 βr = +1 [11] c c y, o, +1 where ξc o, +1 b +1 L U n, = w h A b, for < <, [12] L N mn 0, f < +1 < or b +1 = b+1 b +1 α 1 b = +1, f N < < U and b mn < b +1 q α 2 b +1 L N mn, f < < and b < b Φ [13]

9 Socal secury, educaon, reremen and growh 17 where h s gven by Eq. (2) and, of course, he opmal e wll n general depend on +1, plus he frs and second perod budge consrans n Eqs. (3) and (4) respecvely. The nerpreaons of Eqs. (11) and (12) are he usual ones. Eq. (13) represens how he choce of he reremen age affecs he penson paymen Aggregae labor force We assume an exogenous, consan populaon growh rae n 0, so ha he proporons of young and old ndvduals are gven by µ y = (1+n)/(2+n), and µ o = 1/(1+n), respecvely. Addonally, we assume ha he exogenous dsrbuon of sklls among he populaon s such ha he proporon of ndvduals of ype s gven by{ψ } 4 =1 0, where 4 =1ψ = 1. Denong by P he oal populaon a me, aggregae labor force supply s gven by 4 4 y =1 =1 L = µ P Ψ h + µ P Ψ h (1. o 1 ) [14] The frs erm on he rgh-hand-sde of Eq. (14) represens he labor force of young ndvduals, and he second erm sands for he labor force of old ndvduals. In hs laer case he reremen decson s crucal Socal secury The socal secury budge equaon a any me s gven by 4 ss τ wla = P µ o Ψ b + s s, [15] =1 where he lef-hand-sde represens oal revenue, and he rgh-hand-sde denoes oal expendure on reremen pensons plus lump-sum ransfers. Boh socal secury revenues and paymens on reremen pensons depend on ) he age srucure of he populaon, ) he dsrbuon of skll levels, and ) (as n he case of he aggregae labor force) he dsrbuon of reremen ages across old ndvduals Frms Regardng he producon secor, we assume he exsence of a represenave, compeve frm whch produces one unque oupu, Y, ou of physcal capal, K, and human capal n effcency uns, A L, and whch maxmzes curren profs. Formally, assumng Cobb- Douglas producon echnology, faces he followng problem α 1 ( A L ) α max ZK w A L r +δ K, {K, L } ( ) [16]

10 18 CRUZ A. ECHEVARR ÍA AND AMAIA IZA where Z > 0 s a scalng facor of he echnology level, α (0,1) denoes he elascy of oupu wh respec o physcal capal, and δ (0,1) sands for he physcal capal deprecaon rae. The producvy of he labor force here depends on wo ndependen facors: ) he sae of knowledge of he economy, A, and ) he ndvduals human capal level 17, h, whch n urn depends on.1) her nnae ably, θ, and.2) her nvesmen n educaon, e, whch allows ndvduals o ncrease her human capal level above her nnae ably. The frs order necessary (and suffcen) condons for he problem n Eq. (16) gve us he facor prce equaons α α 1 (1 ) Z and r δ α k w = α k + = Z, [17] where k K / (A L ),.e. he sock of physcal capal per effcen un of labor Goods marke equlbrum As n Damond (1965), he condon for equlbrum n he goods marke saes ha he aggregae savngs of he young generaon a any me mus equal he sock of physcal capal nsalled n he economy a me +1. Formally, and denong ype- young ndvdual s savngs by s y,, we have ha 4 µ P Ψ s = K. [18] y y, +1 = Growh We assume ha he growh rae a me of he oal facor producvy, A, s gven by λ = ρ h 4 > 0, [19] for some ρ > 0. Thus, we are assumng ha growh s a by-produc of he young cleveres or ype-4 agens ndvdual human capal, h 4. Noe ha Eq. (19) mples ha hs growh model s no of vnage ype. I s an analogous specfcaon o he one n BDD wh wo dfferences: ) we assume heerogeney of nnae ables, and ) we allow for dfferen echnologes for ndvdual and socal human capal accumulaon. A smlar specfcaon o ha n Eq. (19) has been used by Caucu e al. (2003) where growh s a by-produc (an exernaly) of hrng sklled workers. Once he model s se up, we solve he equlbrum for hs economy. In order o do so, some (quany) varables mus be frs redefned relave o effcency uns so ha all hese redefned varables reman consan on a balanced growh pah. We have normalzed he ndvdual varables by he oal facor producvy, A, [whch

11 Socal secury, educaon, reremen and growh 19 on a balanced growh pah grows a a consan rae equal o λ] and he aggregae varables by he aggregae labor force n effcency uns, A L, [whch also grows a a consan rae (1 + λ)(1 + n) 1 on a balanced growh pah]. The saonary seady sae equlbrum can be solved as a sysem of smulaneous nonlnear equaons wh he help of some numercal echnques 18. (See Appendx A.) As a by-produc, our model allows us o sudy he redsrbuonal role played by he socal secury and s evenual conflc wh ndvdual ncenves o labor supply, reremen and economc growh. We focus on one parcular measure of (n)equaly such as Gn s ndex relave o he sum of dscouned lfe-me ne ncome, bˆ (1+ λ) wh n (1+ λ)( 1 ) wh+ n +, R R whch we denoe by I G. 3. A numercal example 3.1. Calbraon The non-lneary of he model and he number of equaons nvolved (n spe of s smple dynamc srucure) preven us from obanng analycal resuls for he soluon o he ndvdual problem, le alone for he general equlbrum problem, so ha unqueness mus be assumed. Therefore, we have o rely on numercal analyss for whch we need some basc values for preferences, echnology, demographcs and socal secury polcy. Our am when choosng values s smply o llusrae qualavely he workng and he man feaures of our model, bu approachng o some exen ceran observed fgures of he Spansh economy n a base year, 2004, akng no accoun he Spansh socal secury polcy feaures n ha year. Demographcs. We assume ha each of he wo perods n he model represens abou 32.5 years. Accordng o he INE 19, and followng Díaz-Gménez and Díaz- Saavedra (2009a,b), we choose he value for he populaon growh rae o mmc he observed dependency rao n 2004, whch gves a value for n equal o Preferences. As for preferences, he subjecve dscoun facor s se a β = 2.0. Ths means ha he yearly preference dscoun facor equals β 1/32.5 = , slghly hgher han ohers found n he leraure 21. For nsance, Conesa and Garrga (1999) se a 0.985, and Garrga and Manresa (1999) a The lesure-relaed parameer n he uly funcon ξ s se equal o Ths value has been chosen so ha ype-1 ndvduals choose o rere a he mnmum reremen age, (.e. a R mn = 60), and ype-2, ype-3 and ype-4 ndvduals choose o rere around normal reremen age (.e. a R N = 65), (64.7, 64.8 and 65.0, respecvely) 22. Jménez-Marín and Sánchez-Marín (2007) and Anón e al. (2007) show ha reremen hazard rae clearly exhbs wo peaks: a 60 and a 65.

12 20 CRUZ A. ECHEVARR ÍA AND AMAIA IZA As for L and U, assumng ha ndvduals sar solvng her maxmzaon problem a he age of 15 and ha her deermnsc lfe expecancy s 80 = , he upper bound U equals 0.615, whch corresponds wh a mnmum reremen age of 60 years [.e. U = (80 60)/32.5]. The lower bound L s se a 0.308, hus represenng a compulsory reremen age of R max = 70 years [.e. L = (80 70)/32.5]. Heerogeney of ndvduals nnae ably. Concernng he values of nnae ables, we normalze θ1 = 1, and we pck up he values for θ2, θ3 and θ4 akng no accoun ha he hgher he nnae ably, he hgher he educaonal aanmen. In shor, we make a one-o-one correspondence beween ndvduals nnae ables and her educaonal aanmens. We se θ 2 so ha he rao of ype-2 workers hourly wage rae o ha of ype-1 workers farly replcaes he observed rao of he annual earnngs of workers wh prmary educaon o ha of workers earnng he mnmum legal wage (annual earnngs) (.e., ha we are assumng ha annual hours are he same for all workers). In parcular, n Span n , hs rao equals 2.25, and θ 2 s se equal o Concernng he value for θ3 we choose s value such ha he rao of hourly wage rae of ype-3 workers o ha of hourly wage rae of ype-2 workers s equal o 1.38, and θ3 s se equal o Analogously, o se θ4 we consder ha he rao of he hourly wage rae of workers wh college educaon o ha of workers wh hgh school educaon s he same as he observed rao. Ths value was equal o 1.52 n Span n 2002, and θ4 s se equal o Furhermore, hs way we are able o oban n our benchmark case ) he wo ypes of penson benefs: mnmum, for ype- 1 ndvduals, and earnngs-relaed, for ype-2, ype-3 and ype-4 ndvduals; and ) he wo ypes of educaon expendure: e 2, for ype-1 ndvduals, and e 1, for ype-2, ype-3 and ype-4 ndvduals. As for he dsrbuon of he skll parameer, we assume a consan nra-generaonal dsrbuon ha mmcs he observed dsrbuon of he workers regardng her reremen age and penson benefs. We use he projecons of he educaonal dsrbuon provded by Díaz-Gménez and Díaz-Saavedra (2009a,b), whch follow Messeguer (2001) s projecons, o oban he educaonal dsrbuon of workng-age populaon n Span n In parcular, we choose he value for ψ1 so ha he proporon of rerees recevng he mnmum penson s close o he observed value 24. Thus, we assume ha ψ1 = The value for ψ2 = s chosen as he dfference beween he proporons of workers wh prmary educaon and hose pensoners recevng he mnmum penson. The proporon of workng-age populaon wh hgh school, whch s equal o he proporon of ype-3 ndvduals, s ψ3 = Trvally, he value for ψ4 s equal o 1.0 ψ1 ψ2 ψ3 ha mmcs he proporon of workng-age ndvduals wh college n Span n Fnally, as regards human capal producon, we assume ha γ = Socal secury sysem. We assume ha bˆ mn = The mnmum penson s chosen o capure ha he mnmum penson receved by ype-1 ndvduals s approxmaely equal o her hourly wage rae. In Span, he mnmum penson s around 100% he mnmum wage. In our model, ype-1 ndvduals receve he mnmum

13 Socal secury, educaon, reremen and growh 21 wage 26. Jménez-Marín and Sánchez-Marín (2007) menons ha 70% of workers rerng a 60 are low-ncome workers who receve he mnmum penson. As all ype- 1 ndvduals are homogeneous n our model, we oban ha all ype-1 ndvduals receve he same (he mnmum) penson. We se τ ss = hus equang he observed value 27. We do no consder sources of revenues oher han conrbuons (such as ransfers and subsdes, fnancal asse ncome or sale of real esae and fnancal asses). We assume ha for hose workers whose penson benefs are earnngs-relaed and who rere afer he normal reremen age, R N, (.e. a he age nerval [65, 70]), her replacemen rae s equal o one. However, here are ndvduals whose replacemen rae s below one. In parcular, hose who rere before he normal reremen age and are penalzed accordngly. Replacemen rae for ndvdual s obaned as b ˆ +1(1 + λ)/(w h ), so ha he average λ ψ ˆ replacemen rae along balanced growh pahs s gven by ARR = [(1 + )/w] 4 (b / h ). 28 =1 Observed replacemen raes vary dependng on he lfe experence of workers. For an average worker, penson represens 81.2% of average earnngs 29. We oban a value equal o For he sake of comparson, Conesa and Garrga (1999) oban As for nequaly, he Gn ndex equals Defnng he (balanced growh rae) nernal rae of reurn for ndvdual, IRR, as ha rae of reurn for whch he sum of dscouned values of hs/her conrbuons equals he sum of dscouned values of hs/her penson paymens, IRR s gven by ss ˆ 053 wh 1 λ ( 1 ) b ss. τ ( + ) (1+ λ) 053τ. wh + =. 1+ IRR 1+ IRR [20] Followng Jmeno and Lcandro (1999), we adjus he conrbuon rae τ ss by he coeffcen 0.53, because he expendure on reremen pensons approxmaely represens a 53% share of oal conrbuve pensons. Rememberng 1 perod n our model represens 32.5 years, an approxmae measure of he annualzed socal secury nernal rae of reurn for ndvdual can be gven by rr = (1+IRR ) 1/ We oban ha rr 1 = 4.81%, rr 2 = 2.243%, rr 3 = 2.241% and rr 4 = 2.237%; hs yelds a weghed average of 2.956%. Jmeno and Lcandro (1999) clam ha he observed values range beween 3.7% and 5.03% [dependng on he number of acve (conrbued) years, reremen age and lfe expecancy]. As expeced, he nernal rae of reurn s hgher for ype-1 ndvduals (.e. hose recevng he mnmum penson) and lower for ype-4 ndvduals. Be aware ha n hs economy, even hough he conrbuon and replacemen raes are consan, he exsence of he mnmum pensons means ha he Socal Secury sysem s progressve. The penaly and ncenve parameers. As for α0 and α1 n he penaly funcon, Eqs. (6)-(7), seng normal reremen age, R N, equal o 65, and mnmum reremen age, R mn, equal o 60, and an 8% penaly per year of advanced reremen, π, makes α0 = 0.6. On he oher hand, rememberng, once more, our perod convenon and ha ndvduals are assumed o become opmzng agens a 15, N = 1 (R N 47.5)/32.5

14 22 CRUZ A. ECHEVARR ÍA AND AMAIA IZA = As a resul of he values assgned o α 0, U and N, one obans ha α 1 = 2.6. As for α 2, he exra penson paymen for remanng n he labor force afer reachng he normal-reremen-age, hs s se o 0.0. Producon echnology parameers. Concernng producon echnology, he parcpaon of capal ncome n oal ncome s se equal o α = 0.375, as n Conesa and Garrga (1999). The deprecaon rae of physcal capal s se a δ = 1 ( ) 32.5 = 0.778, as n Conesa and Kehoe (2004). The scalng facor Z s se a Growh. Fnally, wh respec o he growh parameer, we assume ρ = We are herefore able o replcae he observed yearly per capa growh rae of 0.02 [wha mples ha λ = (1+0.02) = 0.903]. Table 1 summarzes he parameers calbraed for he benchmark case, and compares smulaed and observed values for he magnudes calbraed. Table 2 summarzes hose parameers no calbraed. Table 1 BENCHMARK CASE: PARAMETERS CALIBRATED 31,32 Parameer Value Targe Varable Targe Value Model Value Sample sze n (re/ac) a Preferences ξ [R 1,R 2,R 3,R 4 ] b [60,65,65,65] [60,64.7,64.8,65] Ably Socal Secury θ2 θ3 θ4 ψ1 ψ2 ψ3 ψ4 bˆ mn α0 α1 α 2 L (wh 2 / wh 1 ) c (wh 3 / wh 2 ) d (wh 4 / wh 3 ) d share 1 e share 2 f share g 3 share h 4 bˆ mn / (wh 1 ) α0 π α 2 R max [0.85,1.14] U R mn N R N Growh ρ λ annual The calbraed srucural parameers n col. 2 are hose nroduced n he ex. Col. 3: values assgned o such parameers. Col. 4: argeed varable. Col. 5: argeed value n he benchmark economy (.e. year 2004) and, fnally, Col. 6: smulaed value n he model economy. a : rao of populaon older han 65 o workng age populaon. b : ype- ndvdual s opmal reremen age. c : rao of he gross hourly wage rae of ype-2 ndvduals o ha of ype-1 ndvduals, he laer assumed o be pad he mnmum wage; d : rao of he gross hourly wage rae of ype- j ndvduals o ha of ype- ndvduals, for j = 3, 4 and = 2, 3; e : proporon of rerees who are pad he mnmum penson; f : proporon of rerees wh prmary school educaonal aanmen and penson above he mnmum; g : proporon of rerees wh secondary school educaonal aanmen, and h : proporon of rerees wh college educaonal aanmen. : λ annual: annualzed per capa GDP growh rae.

15 Socal secury, educaon, reremen and growh 23 Tabla 2 BENCHMARK CASE: PARAMETERS NON CALIBRATED Parameer Values Preferences Ably Learnng Technology β θ1 γ α Z Socal Secury δ ss τ rep τ Fndngs We perform some numercal exercses o see wha our heorecal model predcs abou he response of he economy (n erms of ncenves of human capal nvesmen and early reremen) upon reducons n he mnmum penson. Gven he socal secury budge balance consran n Eq. (15), whenever he mnmum penson s reduced, he dfference beween socal secury conrbuons and reremen pensons, ss, s convenenly adjused whle he socal secury conrbuon rae s kep consan a he benchmark case value 33. Resuls are summarzed n able 3. Four caveas concernng he way n whch we presen our resuls are n order. Frsly, we have evaluaed he effecs of any decrease n he mnmum penson assumng ha 1) he new mnmum reremen age s 63 years old, 2) he new normal reremen age s 67 years old, and ha 3) he premum for each exra year workng afer he normal reremen age s 3%, as hese are new socal secury polces eher already mplemened afer he benchmark year, or projeced o be mplemened n he near fuure 34. Secondly, we evaluae he ndvdual and aggregae effecs of decreasng he mnmum penson under wo scenaros: assumng ha he educaonal dsrbuon of he workng-age populaon and he dependency rao reman a her nal value n he benchmark balanced growh pah [scenaro 1]; and, alernavely, ha hese are close o hose expeced values n a hypohecal new balanced growh pah [scenaro 2]. The fgures for he laer scenaro are aken from Díaz-Gménez and Díaz-Saavedra (2009a, 2009b) who use he projecons n Messeguer (2001). In parcular, hey consder ha he educaonal dsrbuon n Span wll converge o a new seady sae by, approxmaely, Ths new educaonal dsrbuon wll be gven by: ψ1 = , ψ2 = NBGP NBGP NBGP NBGP NBGP NBGP 0.20, ψ3 = 0.38 and ψ4 = In order o oban ψ1 and ψ2, we have assumed ha he rao ψ1 / ( ψ1 + ψ2 ) remans consan across boh balanced growh pahs, as he projecons n Messeguer provde fgures for he sum of ψ1 + ψ2. The new value se for n s equal o zero, whch gves a dependency rao hgher han n 2004, bu lower han he one expeced for 2050, as he expeced populaon growh rae for ha year s negave (whch s no compable wh he exsence of a balanced growh pah).

16 24 CRUZ A. ECHEVARR ÍA AND AMAIA IZA Thrdly, as we have poned ou above, some reforms have already been mplemened snce 2004 because hey are par of he projeced reform for 2011 (he delay n he normal reremen age, he ncrease n he mnmum reremen age [fully effecve n 2027] and he 3% bonus). Consequenly, before sarng wh he analyss of he effec of reducng he mnmum penson, we analyze wha our model would predc abou he new balanced growh pah f no more reforms ook place. And aferwards, we look a wha our model would predc f, addonally, reducons n he mnmum penson were also se n place. Fourhly, we dsngush beween paral and general equlbrum effecs. The general equlbrum effecs wll come from wo sources. On he one hand (even n he absence of any change n he mnmum penson), he changes n he mnmum legal and normal reremen ages and n he lae-reremen ncenve wll make penson expendures decrease and conrbuons ncrease. Ths wll be he case snce workers wll end up rerng laer. Consequenly, ss (whch we wll refer o as he socal secury surplus n he sequel) wll rse, hereby makng ndvduals second perod ncome hgher wha, n urn, decreases savngs. On he oher hand, penson paymens for ype-2, ype-3 and ype-4 workers reman earnngs-relaed. As we wll see, penales for early reremen wll be hgher for hose ndvduals (despe posponng reremen!). Therefore, hese ndvduals wll have ncenves o ncrease her savngs. We wll see ha ne effec on young ndvduals savngs (and k and facor prces), wll crcally depend on he educaonal dsrbuon. Thus, for he n and ψ s n he nal balanced growh pah, urns ou ha k s predced o fall; whle for hose n he expeced new balanced growh pah, k s predced o rse, as he proporons of ype-3 and ype-4 ndvduals go hgher. As already menoned above, we frs analyze he ndvdual and aggregae effecs of he new mnmum legal and normal reremen ages and he new premum for lae reremen whou any change n he mnmum penson whasoever, because hese reforms have already been mplemened afer And, nex, we analyze he addonal effecs of changes n he mnmum penson. In boh cases, we evaluae such effecs under he wo above menoned scenaros. Resuls are shown n able 3. New balanced growh pah wh reforms already mplemened afer Scenaro 1: Inal educaonal dsrbuon and dependency rao. (See able 3, column 4) Type-1 ndvduals respond by rerng a he new mnmum legal reremen age, so ha her ncenve o educaon ncreases. Type 2 o 4 ndvduals end up rerng laer han n he benchmark economy, bu que before he new normal reremen age. Even hough ype-1 ndvduals educaon nvesmen rses, her nernal rae or reurn falls as so does her reremen lengh. Concernng ype 2 o 4 ndvduals, he response of her educaon nvesmen and nernal rae of reurn are (manly) explaned by general equlbrum effecs. As menoned above, he resulng hgher socal secury surplus (lump-sum dsrbued o old age ndvduals) causes ndvduals frs perod savngs decrease. Bu, a he same me, ype 2 o 4 ndvduals ncrease her frs perod savngs as penson paymens are earnngs-relaed and her penales for early reremen rse. As urns ou, he ne effec on k s negave, so ha boh

17 Socal secury, educaon, reremen and growh 25 he ne wage rae and he dscoun facor ge lower, hereby decreasng he ncenves o educaon nvesmen. (See Eqs. (9) and (10).) Ths mples a negave effec on he ê s, for all ype of workers excep for ype-1 ndvduals, for whch he menoned above paral effec domnaes. The nernal raes or reurn for ype 2 o 4 ndvduals decrease more han for ype-1 ndvduals. Ths s so because penson paymens for ype 2 o 4 are lower (reremen pensons are earnngs-relaed, labor ncome falls, penales for early reremen are hgher and, fnally, penson paymens are more heavly dscouned). As for ype-1 ndvduals, however, hey receve he same (mnmum) penson, bu for a shorer me span. Income nequaly decreases relave o he benchmark case. Labor ncome rses for ype-1 ndvduals, snce her educaon nvesmen rses, bu her penson benefs reman consan. However, for he res of ndvduals, educaon nvesmen becomes lower and so do her labor ncomes and penson paymens. Therefore, he dsperson of labor earnngs (pensons nerpreed as deferred labor earnngs) s necessarly reduced. Fnally, he fall n ype-4 ndvduals educaon fully explans he fall n he growh rae per capa oupu. Scenaro 2: New educaonal dsrbuon and dependency rao. (See able 3, column 8) Type-1 ndvduals rere a he new mnmum legal reremen age, of course. Concernng he res of ndvduals, hey rere laer han n he benchmark economy, as expeced, and slghly laer han n scenaro 1, bu, once agan, before he new normal reremen age. Comparng resuls wh hose n scenaro 1, ype-1 ndvduals ncrease her educaonal nvesmen n a much hgher proporon. Ths s so because he general equlbrum effec now renforces he posve paral effec. The fnal explanaon comes from he fac ha wh a hgher proporon of qualfed workers n scenaro 2, he ne effec on k s posve now, hereby ncreasng he educaon nvesmen ncenves for all ypes of workers. Concernng he nernal raes of reurn, he falls for ype 2 o 4 ndvduals are smlar o hose n he scenaro 1. The decrease n he nernal rae for ype-1 ndvduals now s hgher: even hough he same penson paymen s less dscouned, socal secury conrbuons are hgher as wages are also hgher due o he posve effec on k. As for he ncome nequaly, hs falls more han n he frs scenaro, because he proporons of lower ncome ndvduals ( θ 1 and θ2) fall and, of course, he proporons of hgher ncome ndvduals ( θ 3 and θ 4) rse. Lasly, he ncremen n ype-4 ndvduals educaon explans he predced ncremen n per capa growh. Reducons n mnmum penson. Scenaro 1: Inal educaonal dsrbuon and dependency rao. Consder, frsly, reducons n he mnmum penson so ha ype- 1 workers sll fnd opmal o rere a he (new) mnmum legal reremen age, and (of course) be pad he mnmum penson. In parcular, when mnmum penson drops 10.0% or 15.0%. (See able 3, columns 5 and 6). Ths polcy reform affecs drecly ype-1 ndvduals. The (que smlar) predced ncremen n hese ndvduals educaonal nvesmen s explaned he same way as above. Concernng general equlbrum effecs, here s an addonal source semmng from he reducons n b ˆ mn. On he one hand, socal secury penson expenses are reduced and, consequenly, he socal secury surplus mus ncrease more (See how res s hgher for hgher b ˆ mn drops along las row n able 3), hereby decreasng savngs. Bu, on he oher

18 26 CRUZ A. ECHEVARR ÍA AND AMAIA IZA hand, he mnmum penson receved by ype-1 ndvduals s lower, wha makes hem ncrease her savngs. In sum, ype-1 ndvduals are experencng wo oppose sgn equlbrum effecs: as we move n able 3 from column 4 o 6, he decremen n k s lower 35. Consequenly, he fall n ncenves o educaon nvesmen wll be lower oo, whch mples a lower negave effec on he ê s for all ypes of workers. Therefore, he ncrease n he educaon nvesmen for ype-1 workers s hgher han wh no reducons n he mnmum penson, and he decrease for he res of ndvduals s lower. Regardng he nernal rae or reurn, he one for ype-1 ndvduals drops more as he reducons of he mnmum penson are larger. For ype 2 o 4 ndvduals, as expeced, her reurn raes are hardly affeced by drops n he mnmum penson. The nequaly ndex decreases less as he mnmum penson s reducng. Ths seemngly counernuve resul, however, has a nea explanaon: despe mnmum pensons fall, ype-1 ndvduals ncrease her human capal nvesmen so ha her labor ncomes are hgher. Consder, secondly, reducons n he mnmum penson so ha ype-1 workers do no fnd opmal o rere a he mnmum reremen age any more. Once he mnmum penson s no bndng, he reform would be equvalen o one n whch he mnmum penson s removed. Therefore, we evaluae he effec of elmnang he mnmum penson (See able 3, column 7). Needless o say, ype-1 ndvduals are by far he mos affeced. A sharp ncrease n ê 1 shows up: he mnmum penson dsappears and he penson paymen becomes earnngs-relaed; once hs happens, he reurns o (he ncenves of) educaon nvesmen and he ncenve o pospone reremen rse. For he res of ndvduals, he reremen age decson does no change, and he educaon nvesmen decreases more han n he case of no reducons n he mnmum penson. Ths s so because he socal secury surplus becomes larger, hereby nducng he by now famlar (negave) general equlbrum effecs on savngs, capal per worker and facor prces. Addonally, he ncome nequaly decreases less han n he case ha he reducons n he mnmum penson reman bndng for ype-1 ndvduals, of course: ype-1 ndvduals grealy ncrease no only her labor ncomes, bu also her (now) earnngs-relaed penson benefs; and, moreover, ype 2 o 4 ndvduals producves and labor ncomes decrease n a hgher magnude. Fnally, snce growh s deermned as a by-produc of ype-4 workers producvy, growh decreases more as he mnmum penson dsappears. Scenaro 2: New educaonal dsrbuon and dependency rao. As n he case wh he nal educaonal dsrbuon and dependency rao, we mgh frsly consder reducons n he mnmum penson such ha ype-1 workers would sll rere a he mnmum reremen age and, accordngly, be pad he mnmum penson (See able ˆ 3, columns 9 and 10) and, nex, furher reducons n b mn such ha ype-1 ndvduals would sar rerng afer he mnmum legal reremen age, her pensons becomng earnngs-relaed (See able 3, column 11). However, once we have horoughly analyzed ) he specfc effecs nduced by he change n educaonal dsrbuon and dependency rao, and ) he effecs semmng from he reducon on he mnmum pensons (even s elmnaon) n he benchmark case, he argumens used o explan he resuls would be compleely redundan, so hey wll no be repeaed here.

19 Socal secury, educaon, reremen and growh 27 Table 3 COMPARE STATICS RESULTS Predced NBGP (Secenaro 1) Predced NBGP (Secenaro 2) bˆmn R R R R R R R R ê ê ê ê ê ê ê ê rr rr rr rr rr rr rr rr λannual 0.02 λ annual I G 0.25 I G k k res res x: % change n x, where x sands for, bˆ mn, ê, λ, k, rr, and I G. Columns 4-7 (resp. 8-11) show he resuls under he assumpon ha he dsrbuon of he workng-age populaon by educaon and he proporon of rerees wh respec o he workng-age populaon reman a her benchmark value n 2004 (resp. he expeced for 2050). bˆmn Benchmark 0% 10% 15% 100% 0% 10% 15% 100% 4. Conclusons and fnal remarks Ths paper has analyzed he expeced effecs of changes n he Spansh socal secury sysem ha have been mplemened afer 2004 and, addonally, reducons n he mnmum penson. We have bul a wo-perod, OLG economy populaed by ex-ane heerogeneous ndvduals, who dffer n her nnae ably and decde endogenously her reremen age and her human capal nvesmen, whch, n urn, wll affec her producvy n he labor marke. In hs economy, endogenous growh s a by-produc of mos sklled workers producvy. We ake no accoun some of he specfc feaures of he Spansh socal secury sysem such as ha penson paymens are earnngs-relaed, ha here s a mnmum penson and ha early reremen s penalzed and lae reremen promoed. Gven ha penson paymens are earnngs-relaed, when ndvduals choose her opmal level of educaon, hey ake no accoun no only he effec on fuure labor earnngs, bu also on fuure penson benefs. Consequenly, socal secury nroduces an ncenve for hgher nvesmen n human capal. Ths ncenve, however, parly breaks down due o he mnmum pensons. Indvduals second perod labor supply s elasc. Therefore, he volunary reremen age depends on he ncenves embedded n he publc penson sysem: no only mnmum pensons, bu also penales for early reremen (and ncenves for lae reremen).

20 28 CRUZ A. ECHEVARR ÍA AND AMAIA IZA We have calbraed he model and consruced a benchmark case whch farly reproduces some sylzed facs of he Spansh economy n Sarng from hs baselne case, frsly, we solve for a new balanced growh pah economy akng no accoun socal secury polces whch are currenly n place n he Spansh economy or projeced o be mplemened n he near fuure (hgher mnmum and normal reremen ages). And, secondly, we analyze he effecs of reducons n he mnmum reremen penson benefs. When presenng our resuls, we have dsngushed paral equlbrum effecs (hose exered on a parcular ype of ndvduals n a drec manner) from general equlbrum effecs (hose exered on all ndvduals nduced by changes n he socal secury surplus whch, n urn, nduces changes n aggregae prvae savngs and facor prces). We conclude ha ncreases n he mnmum and normal reremen ages, whch have sared o be mplemened n 2011, are expeced o have a srong effec, no only on ndvduals reremen decsons, bu also on her educaon nvesmen n he resulng new balanced growh pah. However, reducons n he mnmum penson are no expeced o have a sgnfcan effec unless s compleely elmnaed. And, of course, polces enhancng human capal nvesmen for he cleveres workers ncrease growh as a by-produc of hese workers producvy. Fnally, one of he assumpons upon whch we buld our model s ha all ndvduals ener he labor marke a he same age,.e. regardless of her educaonal aanmen. Ths s so because educaon n our model s no made ou of me. In ha case, educaon would also represen an opporuny cos n whch more sklled workers would be wllng o ncur more. In ha seup one would expec ha, as n our economy, more sklled workers makng a hgher educaonal nvesmen would also reac by posponng her reremen, whou an à pror clear ne effec on he workng lves. The exsng emprcal evdence on he relaonshps beween educaonal aanmen, enry age no he labor marke and reremen suggess posve relaonshps beween he frs wo, and a negave relaonshp beween educaonal level and he lengh of workng lves. (See Brugavn and Peracch, 2005). We beleve ha our resuls are farly robus o hs assumpon, because he predced reremen age for more sklled ndvduals hardly exceeds ha for less sklled ndvduals, excep for ype-1 ndvduals when hey are pad he mnmum penson. 5. Appendx A 5.1. Soluon o households problem Opmal educaon and reremen penson. If he penson benef depends on he labor ncome ha he ndvdual obaned when he/she was a worker, he frs order necessary condon comes from solvng he followng problem rep Φ q τ w h A w h A n, (1 +1 ) max NPV ( e ) + +. = w n, h A e 1 { e } R + 1 R +1

21 Socal secury, educaon, reremen and growh 29 Dfferenang NPV 1 (e ) wh respec o e, akng no accoun Eq. (2), equang o 0 and solvng ha equaon for e yelds he soluon for educaon expendure 36 rep 1 γ ss Φ q τ w (1+ λ n )(1 ), e 1, ( θ ) A γθ w 1 τ + +, R +1 R+1 where λ = (A +1 A ) / A. If he penson benef does no depend on he labor ncome ha he ndvdual obaned when he/she was a worker, however, he frs order necessary condon comes from solvng he followng problem w n, +1 h A +1 (1 +1 ) max NPV e = w h A + e. 2 ( ) n, { e } R+1 Dfferenang NPV 2 (e ) wh respec e o [agan, akng no accoun Eq. (2)], equang o 0 and solvng for he frs-order-necessary (and suffcen) condon for e yelds he soluon for educaon expendure 1 γ w n, +1 (1+ λ )(1 +1) e 2, ( θ ) = A γθ w n, +. R+1 The oher opmal decsons: consumpon, savngs and lesure. From Eqs. (3) and (4) we oban he neremporal budge consran w h A (1 c + w h A + + = e + c +. [A.1] R +1 R+1 R+1 R+1 ss +1 n, ) b o, +1 n, y, Maxmzng Eq. (1) wh respec o, c y,, co,+1 and +1, subjec o Eqs. (5), (6), (8) and (A.1), and usng Eq. (3), yelds he followng sysem of non-lnear equaons whch [along wh Eq. (A.1)] characerze he opmal neror, c y,, co,+1, +1 and sy,: 1 βr = +1, c c ξc y, o, +1 o, +1 b+1 L < U n, = w h A b, for <, where L N mn 0, f < +1 < or b +1 = b+1 b +1 α 1 b = +1, f N < U mn < and b +1 < b +1 q α 2 b +1 L N mn, f < +1 < and b +1 < b +1 Φ where h s gven by Eq. (2). 1 1