Fertility-related pensions and cyclical instability. Luciano Fanti * and Luca Gori **

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1 Friliy-rlad pnsions and cyclical insabiliy Luciano Fani * and Luca Gori ** Dparmn of Economics, Univrsiy of Pisa, Via Cosimo Ridolfi, 0, I 5624 Pisa (PI), Ialy. Absrac Using an ovrlapping gnraions Cobb-Douglas conomy wih ndognous friliy, w show ha h inroducion of a friliy rlad componn in unfundd public pnsions may dsabilis h conomy and caus chaoic flucuaions whn individuals ar shor-sighd. In paricular, h risk of cyclical insabiliy incrass wih boh h individual dgr of hrifinss and h rlaiv wigh of individual friliy in h pnsion sysm, whil bing rducd by a highr prfrnc for having childrn. I is illusrad for ralisic conomis ha if PAYG pnsions ar linkd o individual friliy, hn vn a small-sizd pnsion sysm brings h conomy ino h unsabl rgion wih chaoic flucuaions. Our rsuls idnify a novl possibl facor rsponsibl for h riggr of prsisn drminisic cycls in an ovrlapping gnraions conx, and also rprsn a policy warning abou h dramaic dsabilising ffcs of friliy-rlad pnsion rforms, which ar currnly high in h horical dba as wll as in h poliical agnda in svral dvlopd counris. Kywords Endognous friliy; Friliy-rlad pnsions; Myopic forsigh; OLG modl JEL Classificaion C62; H55; J4; J8; J26 W wish o hank Profssor Odd Galor for hlpful commns. Usual disclaimrs apply. * addrss: lfani@c.unipi.i; l.: ; fax: ** Corrsponding auhor. addrss: luca.gori@c.unipi.i; l.: ; fax:

2 . Inroducion Social scuriy is a pillar of h wlfar sa in svral dvlopd counris, and is ssnially basd on pay-as-you-go (PAYG) public pnsions, i.. currn workrs financ pnsions o currn pnsionrs. Th friliy crisis ha has affcd and indd sill affcs a lo of counris around h world (.g., Grmany, Ialy, Japan and Spain) is hraning h viabiliy of public pnsion budgs, as h numbr of young conribuors is sadily falling and h numbr of old bnficiaris is sadily rising (du o also h rducd adul moraliy). Moivad by h hrif of boh aging and blow-rplacmn friliy on h xisnc of h widsprad PAYG sysms, pnsion rforms ar currnly high on h poliical agndas of many govrnmns, spcially in Europ (s,.g., Bori al., 200, 2002; Blindr and Krugr, 2004). As a rmdy agains h ponial ngaiv ffcs of h friliy crisis on PAYG pnsions, i has bn suggsd, amongs ohr hings, o incniv familis o hav mor childrn in ordr o incras h raio of conomically aciv o oal populaion, for insanc hrough h public provision of child allowancs (van Grozn al, 2003; van Grozn and Mijdam, 2008). Morovr, linking h siz of h pnsion arrangmn rcivd whn by h old-agd o h numbr of childrn raisd whn young may b anohr inrsing insrumn ha migh b usd o promo h friliy rcovry as wll as for opimaliy purposs (s, Kolmar, 997; Abio al, 2004; Fng and Mir, 2005, 2009; Cigno and Wrding, 2007). As policy implicaions, alhough friliy-rlad pnsions wr alrady prsn in som pnsion sysms, many conomiss and policy makrs argud for a furhr xnsion, 2 ofn wih vry laborad proposals. 3 In h words of Cigno (2007, p. 39) xampls of his ar h majoraion d dur d assuranc pour nfans in h Frnch Rgim Gnral, and h Swdish xrapnsion for barn. In 986, h Grman govrnmn sard crdiing parns who wihdraw from h labour mark o look afr a child wih a noional pnsion conribuion, Kindrrzihungszin, originally s a 75 prcn of avrag arnings, for up o on yar. Lar, his noional

3 As rgards h imporan issu of drminisic businss cycls in compiiv conomy, h ida ha cyclical bhaviours can occur in OLG modls vn wih prfc forsigh is wll known in liraur (Grandmon, 985), in paricular in h noclassical growh OLG modl in boh h manygoods (Bnhabib and Nishimura, 979) and on-good conomis (Farmr, 986; Richlin, 986), bu his crucially rquirs ha producion facors mus b rlaivly complmn (i.. h lasiciy of capial-labour subsiuion mus b lowr han ha of h Cobb-Douglas chnology) and consumpion and lisur mus b gross subsiu. Morovr, wih myopic forsigh, h sady sa may b oscillaory and xhibi drminisic complx cycls (Michl and d la Croix, 2000, d la Croix and Michl, 2002; Fani and Spaaro, 2008), bu only providd ha h inr-mporal lasiciy of subsiuion in h uiliy funcion is highr han uniy (i.., highr han in h cas of Cobb-Douglas prfrncs). Whil a growing body of liraur on h rlaionship bwn pnsions, friliy, longviy and conomic growh has bn dvlopd in h las dcads (s, amongs many ohrs, Zhang al., 200, 2003; Pcchnino and Pollard, 2005), lss anion has bn paid o h dynamical ffcs of public PAYG pnsions in an conomy wih ovrlapping gnraions (OLG) and ndognous friliy, in paricular in h cas of friliy-rlad PAYG schms. conribuion was raisd o 00 prcn of avrag arnings, and xndd o hr yars. Sinc 996, howvr, h condiion ha h parn should acually giv up work in ordr o qualify for h bnfi has bn rmovd, and Kindrrzihungszin has bcom a friliy-rlad pnsion bnfi jus lik h Frnch and Swdish ons. 2 For insanc Sinn (2007, p. 0) argus ha h currn friliy rlad lmn in h pnsion formula for Grmany is oo low: Th pnsion sysm in Grmany provids som rlif for mohrs who rais an addiional child and work n yars afr h birh. Thy rciv, in rms of currn valu,,000 Euros as an addiional pnsion. This is clos o nohing. 3 Only for rfrnc o on of h mos auhoriaiv rform proposals, w ci Cigno al. (2003) ha claims ha an ffciv policy is o inroduc pnsion bnfis coningn on h oal (or ponial rahr han acual bcaus vn h childrn may wan o wihdraw from h labour mark for a crain im priod o rais childrn) arning capaciy of h pnsionr s own childrn (s also Cigno, 2007; Cigno and Wrding, 2007). 2

4 Th aim of his papr is o provid a dpr undrsanding of h sabiliy ffcs of public PAYG pnsions in a xbook OLG conomy (.g. Diamond, 965) whn friliy is ndognous and uiliy and producion funcions ar Cobb-Douglas. I is show ha whn individuals ar shor-sighd, h inroducion of a friliy-rlad componn in h pnsion formula may hav dramaic dsabilising ffcs and drminisic chaos appars vn for vry small-sizd PAYG schms. In such a cas, in fac, h ngaiv ffc of public pnsions on capial accumulaion is highr han in h cas of a pur PAYG schm, du o h rol playd by boh h prfrncs for childrn and prfrnc for fuur consumpion, which, on h conrary, do no play any rol in h absnc of friliy-rlad lmns in h pnsion formula. Friliy-rlad pnsions, hrfor, ac as an conomic dsabilisr in ovrlapping gnraions conomis. Th conribu of his papr o h liraur is wofold: (i) i invsigas h dynamical propris so far, o h bs of our knowldg, nglcd of an conomy wih a friliy-rlad public pnsion, nowihsanding h fac ha h lar is a vry dbad issu; (ii) i shows a novl drminan (i.. a friliy-rlad public pnsion) of drminisic conomic (rgular as wll as chaoic) cycls, which mrg vn wih boh Cobb-Douglas uiliy and producion funcions. Th policy implicaions of h papr s findings ar clar: o h xn ha dvlopd counris show, rahr plausibly, low prfrnc for childrn and high prfrnc for fuur consumpion (wih myopic forsigh), h ofn advocad rform for such counris of inroducing (or xnding) a friliy rlad lmn in h public pnsion formula may dsabilis h conomy and b rsponsibl of chaoic conomic flucuaions. Th rmaindr of h papr is organisd as follows. In scion 2 w dvlop h modl. In scion 3 h dynamical faurs ar analysd and discussd. Scion 4 concluds. 2. Th modl 2.. Govrnmn 3

5 Th govrnmn rdisribus across gnraions wih PAYG ransfrs from h young o h old ha ar parially or oally linkd o h numbr of childrn raisd whn young. A im, hrfor, currn workrs financ pnsions o currn pnsionrs, and h friliy-rlad pay-as-you-go (FR- PAYG hncforh) pnsion accouning rul in pr workr rms rads as p [( ω) n ω n ] = w + θ, () h lf-hand sid ( p ) bing h pnsion xpndiur and h righ-hand sid h ax rcips. In paricular, w is h wag arnd by h young workrs a im, 0 < θ < is h fixd conribuion ra and 0 ω is a wighing paramr of h diffrn disribuion ruls for oal conribuion o PAYG pnsions. In paricular, i masurs h imporanc of h individual numbr of childrn rlaiv o h avrag numbr of childrn in h PAYG sysm (s, for insanc, Kolmar, 997; Abio al., 2004; Fng and Mir, 2005, 2009; Fng and von Wizsäckr, 200). Th polar cass ω = 0 and ω = imply a pur PAYG schm and a PAYG schm oally linkd o individual friliy, rspcivly. Thrfor, Eq. () shows ha a im PAYG pnsions dpnd on (i) h individual ra of friliy, n, a im wih a shar ω of h conribuion, and (ii) h avrag ra of friliy in h whol conomy, n, a im wih a shar ω of h conribuion. Following Fng and Mir (2005, p. 34), w dfin h policy variabl ω h child facor Individuals Considr a gnral quilibrium OLG closd conomy populad by idnical individuals. Lif is dividd ino childhood and adulhood. In h formr priod ach individual dos no mak conomic dcisions. In h lar priod sh works and bars childrn whn young and sh is rird whn old. Only young individuals (of masur N ) join h workforc. Thy ar ndowd wih on uni of im supplid inlasically on h labour mark, whil rciving a uniary wag incom a h 4

6 compiiv ra w. This incom is usd o consum, o sav, o bar childrn and o financ marial consumpion of h ldrly hrough h public pnsion schm Eq. (). Raising childrn is cosly, and h amoun of rsourcs ha parns nd o ak car of hm is givn by a monary cos q pr child, wih 0 < q < bing h prcnag of child-raring cos on working incom. 4 w Thrfor, h budg consrain facd by an individual of h young (child baring) gnraion a rads as: ( θ ) c + s + q w n = w, (2), i.. wag incom n of conribuions paid o ransfr rsourcs from work im o rirmn im is dividd ino marial consumpion whn young, c,, savings, s, and h cos of baring childrn, q w n. Old individuals ar rird and liv wih h amoun of rsourcs savd whn young plus h xpcd inrss accrud a h ra r + and h xpcd public pnsion bnfi p +. A im +, hrfor, h budg consrain of an old rird prson sard working a is: ( + r + ) s + w ( ω), + = + [ n ω n ] c2 θ +, (3) i.. marial consumpion whn old, c,, is h sum of priva savings plus h xpcd inrs and h xpcd public pnsion bnfi. 2 + Each adul individual of gnraion draws uiliy from young-agd consumpion ( c, ), old-agd consumpion ( c 2, + ) and h numbr of childrn sh wishs o rais ( n ). 5 Assuming logarihmic prfrncs, h rprsnaiv individual nring h working priod a solvs h following problm: 4 This child cos srucur is similar o ha adopd by, amongs many ohrs, Wiggr (999) and Boldrin and Jons (2002). 5 This ramn of h ra of friliy in h uiliy funcion is rahr usual, (s,.g., Ecksin and Wolpin, 985; Galor and Wil, 996; van Grozn al., 2003; van Grozn and Mijdam, 2008). 5

7 { c c n } U ( c, c, n ) =,, 2, ln( c, ) + β ln( c2, ) + ln( n ), + + max +, (4) φ, 2, subjc o Eqs. (2) and (3), whr 0 < β < is h subjciv discoun facor or, alrnaivly, h individual rlaiv dgr of hrifinss, and 0 < φ < capurs h parns as for childrn. Th firs ordr condiions for an inrior soluion ar givn by: c 2, + c, = + r β +, (5) c, n w + φ = q w ω θ. (6) + r + Eq. (5) quas h marginal ra of subsiuion bwn working priod consumpion and rirmn priod consumpion o hir rlaiv prics (i.. h xpcd inrs ra drmind on h capial mark). Eq. (6) quas h marginal ra of subsiuion bwn working priod consumpion and h numbr of childrn o h xpcd marginal cos of raising an xra child. This cos is givn by h diffrnc bwn h monary cos of baring an addiional child and h prsn valu of h xpcd pnsion bnfi wighd by h child facor. Th highr h child facor, h lowr h xpcd n marginal cos of raising an xra child. If ω = 0 (pur PAYG pnsions), h cos of child raring is only drmind as a shar of h working incom. In conras, if 0 < ω (FR-PAYG pnsions), a posiiv inr-gnraional ffc xiss ha causs a rducion in h gross monary cos of childrn du o h highr bnfi rcivd by ach pnsionr, i.. individuals wan o subsiu young-agd consumpion wih childrn. Now, combining Eqs. (5) and (6) wih h individual lifim budg consrain givs h dmand for childrn and h saving ra, rspcivly: n = ( θ ) ( + β + φ) q w [( + β ) ω + ] φ w w φ θ + r + +, (7) s = ( θ ) ( + β + φ) q w [( + β ) ω + φ] w w θ + r β q w w + ( β ω + φ) θ + r (8) 6

8 Eq. (7) drmins h individual numbr of childrn in a parial quilibrium conx. A ris in h child facor causs a posiiv inr-gnraional ffc ha rducs h marginal cos of child baring and hus incrass friliy ( n / ω > 0 ). Eq. (8), insad, drmins h saving ra in a parial quilibrium conx. I rvals ha h child facor plays a wofold counrbalancing rol: (a) i rducs h saving ra bcaus individuals will xpc a highr pnsion bnfi as long as h numbr of hir dscndan raiss (i.. h ngaiv ffc givn by h xpcd public pnsion componn h scond rm in squar bracks of Eq. 8 incrass, whil kping h priva saving componn unaffcd h firs rm in squar bracks of Eq. 8), 6 and (b) i incrass h saving ra sinc a highr child facor maks mor convnin o subsiu young-agd consumpion wih childrn a im (i.. rducs h dnominaor of Eq. 8). Howvr, h final (parial quilibrium) ffc of a ris in h child facor on savings is ngaiv ( s / ω < 0 ), ha is h posiiv saving-ffc (b) is always dominad by h ngaiv saving-ffc (a) Firms Firms ar idnical and ac compiivly on h mark. Aggrga producion a im ( Y ) aks plac by combining capial ( K ) and labour ( L = N in quilibrium) according o h consan rurns o scal Cobb-Douglas chnology α α L Y = AK, whr A > 0 is a scal paramr and 0 < α < is h oupu lasiciy of capial. Dfining k := K / N and y := Y / N as capial and oupu pr workr, rspcivly, h innsiv form producion funcion may b wrin as α Ak y =. Assuming oal dprciaion of capial a h nd of ach priod and normalising h 6 W dnod h scond rm in squar bracks of Eq. (8) as h xpcd public pnsion componn and h firs rm in squar bracks of Eq. 8 as h priva saving componn only for xposiory purposs. 7 Th proof is no prsnd for conomy of spac, bu i is of cours availabl on rqus. 7

9 pric of final oupu o uniy, profi maximisaion implis ha facor inpus ar paid hir marginal producs, ha is: α r = αak, (9) w α ( α ) Ak =. (0) 2.4. Equilibrium Givn h govrnmn budg Eq. () and knowing ha populaion volvs according o N = n + N, mark-claring in goods and capial marks is xprssd (in pr workr rms) as n k = + s. () From Eq. (), and rcalling h analysis of Eqs. (7) and (8) in scion 2.2, w obsrv ha h xisnc of a friliy-rlad componn in h PAYG sysm ( 0 < ω ) ngaivly affcs capial accumulaion pr workr sinc, on h on sid, i incrass h friliy ra and, on h ohr sid, i dcrass h saving ra. Mor in dails, using Eqs. (7) and (8) o subsiu ou for n and s, rspcivly, quilibrium implis: k + β = q w φ β ω + φ w θ φ + r + +. (2) Eq. (2) shows ha h quilibrium sock of capial a + is drmind as h diffrnc bwn h priva saving componn and h xpcd public pnsion componn a, boh dividd by h as or childrn. Th formr (h firs addndum on h righ-hand sid of Eq. 2) xclusivly dpnds on h willingnss o sav ou of wag incom givn h assumpion of Cobb-Douglas prfrncs. Th lar (h scond addndum on h righ-hand sid of Eq. 2) dpnds on h xpcd valus of boh h wag and inrs ras. 8

10 Th xisnc of a friliy-rlad componn in h PAYG sysm ( 0 < ω ) has wo imporan ffcs on capial accumulaion: firs, i maks h crowding ou ffc of public pnsions on priva savings much srongr han h cas of pur PAYG pnsions ( ω = 0 ); scond, i maks boh h individual dgr of hrifinss ( β ) and parns as for childrn (φ ) as ponial dsabilising paramrs. In fac, a ris boh in dgr of parsimony and lov for childrn incrass h posiiv priva saving componn and h ngaiv public pnsion componn and, hnc, is final ffc on capial accumulaion may b ambiguous. As known, i is usual in h dynamical analyss of OLG modls (s.g., d la Croix and Michl, 2002) o invsiga how h pah of capial accumulaion volvs dpnding on whhr individuals hav ihr prfc or myopic xpcaions abou facor prics. Howvr, bfor saring ou wih h analysis of h dynamics of h modl, som clarificaions abou h assumpion ha an individual can choos h siz of hr pnsion by choosing h numbr of childrn sh will hav, ar in ordr. W no ha in hs modls individuals ar assumd o b aomisic and hus hy do no ak ino accoun h ra of friliy of h ohr popl, as is clar from h assumpions implici in h pnsion formula Eq. (). This mans ha h individuals ar unabl o coordina hir friliy dcisions. Ohrwis, h individuals should b, broadly spaking, ulra-raional, which would b an unusual assumpion in liraur and, according o Cigno (995, p. 7), clarly unralisic. Morovr, h rlaxaion of h aomisic individual s assumpion (ha is, individuals ar abl o coordina hir choics as rgards hir dscndans) would man ha h pur PAYG schm would always b, by consrucion, qual o a FR schm (s, Cigno, 995, p. 7). As a consqunc of h aomisic assumpion on which his class of modls is groundd, w may conjcur ha h myopic forsigh assumpion may b rahr naural, in ha if h coordinaion bwn individuals a h currn im is lacking, hn i sms o b a rahr srong hypohsis o imagin ha fuur friliy bhaviour will b prfcly fors, and, hnc, i dos no only 9

11 rprsn a spcial cas. Howvr, for h sak of complnss, blow w sudy h dynamics of h conomy in h cass of boh prfc and myopic xpcaions Prfc forsigh Wih prfc forsigh, h xpcd inrs and wag ras dpnd on h fuur valu of h pr workr sock of capial, ha is + r w + + = = αak ( α ) α + Ak α +. (3) Combining Eqs. (9), (0), (2) and (3), h dynamic quilibrium squnc of capial can b wrin as k ( α ) A ( α )( β ω + φ) q β α. (4) α φ + θ α + = k Sady-sa implis * k + = k = k, so ha: α( α ) A ( α )( β ω + φ) α * q β k = α φ θ. (5) Myopic forsigh Wih myopic forsigh, h xpcd inrs and wag ras dpnd on h currn valu of h pr workr sock of capial, ha is + r w + + = = αak ( α ) α Ak α. (6) 0

12 Combining Eqs. (9), (0), (2) and (6), h dynamic pah of capial accumulaion is now givn by: k + β β ω φ α q( α ) Ak α + θ = k. (7) φ φ α Whil h sady-sa is sill drmind by Eq. (5), h dynamics of myopic and prfc forsigh ar vry diffrn, as a simpl comparison bwn Eqs. (4) and (7) rvals (s also Michl and d la Croix, 2000). Dspi Eq. (7) is a simpl firs ordr non-linar diffrnc quaion, h dynamics of capial gnrad by such an quaion may b highly non-linar and, in paricular, ndognous flucuaions may mrg. Th local sabiliy propris of a doubl Cobb-Douglas conomy wih ndognous friliy, FR-PAYG pnsions and myopic xpcaions ar analysd in h nx scion Local sabiliy wih myopic xpcaions In his scion w wish o invsiga h drminisic dynamics dfind by Eq. (7) nar h sady sa and assss h prsnc of possibl local ndognous drminisic flucuaions. From Eqs. (5) and (7), h following proposiion holds: Proposiion. In a doubl Cobb-Douglas OLG conomy wih ndognous friliy, FR-PAYG pnsions and shor-sighd individuals, h dynamics of capial is h following. () L 0 < α < α3 hold. Thn θ < θ <, and: (.) if 0 < θ < θ, h dynamics of capial is monoonic and convrgn o * k ; 8 Th (local) sabiliy propris of an conomy wih prfc forsigh is brifly prsnd in Appndix. Diffrn from h cas of myopic xpcaion, in h cas of raional xpcaions (apar from h criicism discussd abov) h conomy dos no xhibi any inrsing dynamical faur.

13 (.2) if θ < θ < θ, h dynamics of capial is oscillaory and convrgn o (.3) if θ = θ, a flip bifurcaion mrgs; (.4) if θ < θ <, h dynamics of capial is oscillaory and divrgn o * k. * k ; (2) L α 3 < α < α hold. Thn θ <, θ >, and: (2.) if 0 < θ < θ, h dynamics of capial is monoonic and convrgn o (2.2) if θ < θ <, h dynamics of capial is oscillaory and convrgn o * k ; * k. (3) L α < α hold. Thn θ > θ >, and h dynamics of capial is monoonic and convrgn < o * k for any 0 < < θ, whr θ = θ 2 α φ θ = θ( α, β, φ, ω) : =, (8) 2 ( α β, φ, ω) ( α ) β ω + φ ( + α ) φ + = θ ( α ) β ω + φ α α α, : =, (9) 2 [ ], / 2 < α = α( β, φ, ω) : = β ω + φ φ( β ω + φ) α <, (20) β ω [ ], / < α3 α 3 = α3( β, φ, ω) : = 2β ω + 3φ φ( 8β ω + 9φ ) 3 < α. (2) 2β ω Proof. Diffrniaing Eq. (7) wih rspc o k and using Eq. (5) givs: k k + * = α θ k = k ( α ) α 2 β ω + φ. (22) φ Monoonic and non-monoonic dynamics 2

14 k + > From Eq. (22), h condiion * 0 k = k k < implis ( α ) 2 β ω + φ > < α θ 0 θ θ, (23) α φ < > whr θ = θ (dfind by Eq. 8) rprsns h valu of h conribuion ra blow (byond) which h dynamics of capial is monoonic (non-monoonic). In paricular, θ < ( θ > ) for any 0 < α < α ( α < α < ). Morovr, θ < if and only if α < α and α > α 2, whr α is dfind by [ ] 2 2. Sinc / 2 β ω < α < and α > 2 for any Eq. (20) and α = α ( β, φ, ω) : = β ω + φ + φ( β ω + φ) β, φ and 0 < ω, hn h cas α > α 2 can b ruld ou. k+ Now, * < givs k = k k α θ ( α ) α 2 β ω + φ α φ < θ >. (24) φ α β ω + φ Thrfor, in h cas of monoonic dynamics h conomy always convrgs o h saionary sa k + irrspciv of h siz of h pnsion sysm, i.. 0 < * < k = k k for any 0 < θ <. Non-monoonic dynamics: sabiliy analysis k + > Th condiion * k = k k < implis: ( α ) 2 β ω + φ > < α θ θ θ, (25) α φ < > whr θ = θ > θ (dfind by Eq. 9) is h flip bifurcaion valu of h conribuion ra, i.. h hrshold valu of θ blow (byond) which h sady sa is sabl (unsabl). In paricular, θ < 3

15 ( θ > ) for any 0 < α < α3 ( α 3 < α < ). Morovr, θ < if and only if α < α3 and α > α 4, whr α is dfind by Eq. (2), α α ( β, φ, ω) : = 2β ω + 3φ + φ( 8β ω 9φ ) 3 [ ] = 4 and α 3 < α. Sinc 2β ω 4 + /3 < α 3 < α and α > 4 for any β, φ and 0 < ω, hn h cas α > α 4 can b ruld ou. Thrfor, k+ (i) if 0 < α < α3 hn θ < θ < and (.) 0 < * < k = k k for any 0 < θ < θ, (.2) k + < * < k = k k 0 k + for any θ < θ < θ, (.3) * = k = k k if and only if θ = θ, and (.4) k k + * < k = k for any θ < θ <. This provs poin (); k+ (ii) if α 3 < α < α hn θ <, θ > and (2.) 0 < * < k = k k for any 0 < θ < θ, and (2.2) k + < * < k = k k 0 for any θ < θ <. This provs poin (2); k+ (iii) if α < α < hn θ > θ > and 0 < * < k = k k for any 0 < θ <. This provs poin (3). Q.E.D. Proposiion can asily b inrprd as follows: h sock of capial insalld a im + is drmind as h saving ra dividd by h numbr of childrn a im (s Eqs. 7, 8 and ). Thrfor, h accumulaion of capial dpnds on diffrnc bwn h priva saving componn and h public pnsion componn, boh dividd by h as for childrn (s Eq. 2). Wih Cobb- Douglas uiliy, h priva saving componn xclusivly dpnds on h marginal willingnss o 4

16 sav ou of wag incom, and rflcs h posiiv ffc on capial accumulaion of a highr working incom following a ris in k. In conras, h public pnsion componn dpnds on boh h xpcd pnsion bnfi and h xpcd inrs ra, and rflcs h ngaiv (crowding ou) ffc on capial accumulaion following a ris in k. If h priva saving componn dominas (is dominad by) h public pnsion componn, h dynamics of capial is monoonic (nonmonoonic). Whn producion is rlaivly labour-orind and h conribuion ra is low nough, h priva saving componn dominas and hus h dynamics of h conomy is monoonic and h sady sa is always sabl, i.., h so-calld saddl nod bifurcaion can nvr occur. A ris in h conribuion ra incrass h rlaiv wigh of h public pnsion componn and a nonmonoonic unsabl dynamics mrgs in ha cas. In conras, whn producion is rlaivly capial-orind h dynamics is always monoonic irrspciv of h siz of h PAYG sysm. W now prform a snsiiviy analysis of h criical valus of h conribuion ra which discriminas bwn monoonic and non-monoonic dynamics (s Eq. 8), as wll as bwn non-monoonic sabl and unsabl dynamics (s Eq. 9) in h cass of boh FR-PAYG pnsions ( 0 < ω ) and pur PAYG ( ω = 0 ) pnsions. Analysis of Eqs. (8) and (9) givs h following proposiion: Proposiion 2. Th risk of cyclical insabiliy in an conomy wih FR-PAYG pnsions is highr han wih pur PAYG pnsions. A ris in h disribuiv capial shar (α ) monoonically rducs h risk of cyclical insabiliy irrspciv of h pnsion schm. Morovr, whil wih pur PAYG pnsions a chang in h individual dgr of hrifinss ( β ), and/or in h as for childrn (φ ) is nural for sabiliy, wih FR-PAYG pnsions a ris in h child facor (ω ), and/or in h individual dgr of hrifinss as wll as a rducion in h as for childrn incrass h risk of cyclical insabiliy. 5

17 Proof. Firs, in h cas of pur PAYG pnsions ( = 0 6 ω ) Eq. (8) bcoms θ = θ( α ) : = 2 α ( α ) 2 (i.., h valu of h conribuion ra which discriminas bwn monoonic and non-monoonic dynamics is indpndn of boh h subjciv discoun facor and as for childrn), so ha θ ( α ) < ( ( α ) > ( 0 θ ) for any 0 < α < / 2 ( / 2 < α < ). Thrfor, wih FR-PAYG pnsions < ω ) h widh of h paramric rgion in h spac ( θ ) α, whr non-monoonic dynamics ar possibl is largr han h corrsponding rgion wih pur PAYG pnsions ( ω = 0 ). This mans ha whn 0 < ω, h hrshold θ ( α β, φ, ω) Scond, in h cas of pur PAYG pnsions ( = 0, can b smallr han uniy vn whn / 2 < α <. ω ) Eq. (9) bcoms θ = θ ( α ) = θ ( α ) + α : (i.., α h flip bifurcaion valu of h conribuion ra is indpndn of boh h subjciv discoun facor and as for childrn), so ha θ ( α ) < ( ( α ) > θ ) for any 0 < α < / 3 ( /3 < α < ). Thrfor, wih FR-PAYG pnsions ( 0 < ω ) h widh of h paramric rgion in h spac ( α, θ ) whr non-monoonic unsabl dynamics ar possibl is largr han h corrsponding rgion wih pur PAYG pnsions ( ω = 0 ). This mans ha whn 0 < ω, h flip bifurcaion valu ( α β, φ ω) θ,, can b smallr han uniy vn whn /3 < α <. Morovr, from Eq. (9) w g: for any 0 ω, and for any 0 < ω. Q.E.D. θ = α θ = ω θ = β θ = φ φ( + 3α ) ( α ) ( β ω + φ) 3 > α( + α ) φβ ( α ) ( β ω + φ) < 2 2 α( + α ) φω ( α ) ( β ω + φ) < 2 2 α( + α ) βω ( α ) ( β ω + φ) > 2 2 0, (26) 0, (27) 0, (28) 0, (29)

18 Figurs and 2 illusra Proposiion 2 and compar h paramric rgions in h spac ( α, θ ) ha dscrib h (sabl) monoonic and (sabl and unsabl) non-monoonic dynamics in h cass of pur PAYG pnsions (Figur ) and FR-PAYG pnsions (Figur 2). I is clarly shown ha whil in a pur PAYG conx cyclical insabiliy ariss only whn α < / 3, in a FR-PAYG conx h cyclical unsabl rgion in h spac ( α, θ ) is largr bcaus of h dsabilising ffcs playd by h child facor, h individual dgr of hrifinss and h as for childrn. Figur. Cas = 0 α,. ω (pur PAYG pnsions). Sabiliy and insabiliy rgions in h spac ( θ ) 7

19 Figur 2. Cas 0 α,. < ω (FR-PAYG pnsions). Sabiliy and insabiliy rgions in h spac ( θ ) Tabl. Paramric rgions of cyclical insabiliy ( 0 < θ < ) undr diffrn PAYG sysms. Pur PAYG ( ω = 0 ) Mixd FR-PAYG ( 0 < ω < ) Pur FR-PAYG ( ω = ) 0 < /3 < α < α < α ( β, φ, ω) 0 < α < α3( β, φ,) 0 3 Tabl summariss, for hr diffrn public PAYG schms, h hrshold valus of h oupu lasiciy of capial blow which cyclically insabiliy may mrg dpnding on h siz of h pnsion sysm. Sinc α ( β φ,) > α ( β, φ, ω) / 3 3, 3 >, i is vidn ha prsisn cycls mor likly occurs whn h wigh of individual friliy in h PAYG sysm is high. Morovr, from Proposiion 2 w may driv h following rsuls as rgards h ffcs of h prfrnc paramrs 9 on h sabiliy of h conomy: 9 I is worh noing ha diffrn paramr valus may b, broadly spaking, corrlad wih a diffrn lvl of conomic dvlopmn. For insanc: (i) h so-calld mor and mor slfish lifsyl in dvlopd counris has bn raind a rason for a rducd lov for having childrn, so ha h as for childrn migh b lowr in dvlopd rahr han dvloping and undrdvlopd counris; (ii) i is wll known ha conomic growh coms hand in hand 8

20 Rsul. To h xn ha friliy is low bcaus h prfrnc for childrn is low (.g. dvlopd counris), h inroducion of FR-PAYG pnsions ( 0 < ω ) gnras a highr risk of cyclical insabiliy han whn friliy is high bcaus h prfrnc for childrn is high (.g. undrdvlopd or dvloping counris). Rsul 2. To h xn ha h dgr of hrifinss is high bcaus h financial ducaion of individuals is high (.g. dvlopd counris), h inroducion of FR-PAYG pnsions ( 0 < ω ) gnras a highr risk of cyclical insabiliy han whn h dgr of hrifinss is low bcaus h financial ducaion of individuals is low (.g. undr-dvlopd or dvloping counris). Rsuls and 2 lad o a rahr paradoxical policy ffc. Firs, sinc h inroducion of FR-PAYG pnsions is ssnially advocad in conomis wih low friliy in ordr o ovrcom h susainabiliy issu of h widsprad public PAYG sysms, our rsuls imply ha in conomis whr h as for childrn is rlaivly low, h insabiliy risk, inducd by a pnsion rform ha links h bnfi rcivd whn old o h numbr of childrn raisd whn young, is high. This rsul holds bcaus a rducion in h as for childrn incrass friliy, rducs savings and his, in urn, incrass h ngaiv wigh of h public pnsion componn in capial accumulaion, whil kping h priva componn unaffcd (s Eq. 22) and hus conribus o dsabilis h conomy. Th causal chain of his rsul is h following: (i) blow-rplacmn friliy in dvlopd counris is on of h mos imporan causs for svral conomiss and policymakrs o suggs h inroducion of friliy-rlad pnsions; (ii) on of h rasons why friliy is oo low in indusrialisd counris is ha h prfrnc for having childrn is ( φ ) is oo low. Sinc friliywih h financial dvlopmn and ha h lar, oghr wih h corrsponding financial ducaion, works for a highr valuaion of fuur consumpion, so ha h subjciv discoun facor migh b highr in dvlopd rahr han dvloping and undrdvlopd counris. 9

21 rlad pnsions ar inroducd ssnially as a simulus for h friliy rcovry and, hnc, o kp public pnsion budg susainabl ovr im, hn h rahr surprising rsul shown in his papr is ha h dsabilisaion of h conomy inducd by FR-PAYG public pnsions bcoms a plausibl scnario for svral conomis, as shown in h numrical xampl in h nx scion. Scond, anohr paradoxical rsul can b drivd abou h ffc of h paramr ha dscribs h financial ducaion of individuals whn FR-PAYG pnsions xis. A ris in subjciv discoun facor, in fac, mans ha individuals wish o smooh consumpion ovr h rirmn priod and, hnc, sav mor whn young. This apparnly causs a sabilising ffc. Howvr, h analysis of h local sabiliy propris of h sady sa rvals ha β is nural on h priva saving componn whil incrass h wigh of h crowding-ou ffc of h public pnsion componn, and hus acs as a dsabilising dvic. Thrfor, in a counry whr h individual dgr of hrifinss is high bcaus h financial ducaion is high (.g. dvlopd counris which, unforunaly, ar hos mos plagud by undr-populaion and hn pron o considr FR pnsion rforms), h inroducion of a FR-PAYG schm may caus unsabl cycls and, as shown in h nx scion, vn chaoic moions. 3.. Chaoic dynamics: a numrical xprimn W ar now inrsd in showing h possibl mrgnc of drminisic chaos in h doubl Cobb-Douglas conomy wih FR-PAYG pnsions prsnd abov. W ak h following paramr s (only for illusraiv purposs): A = 0, α = (as is usually assumd in h conomic liraur), β = (s Žamac, 2007), φ = 0. 05, q = Th 20

22 valus of φ and q ar calibrad such ha h corrsponding friliy ra ar clos o h currn blow-rplacmn lvl obsrvd in svral dvlopd counris. 0 In Figurs 3-5 w dpic h bifurcaion diagrams for h paramr θ (which lis on h horizonal axis), wih rspc o hr diffrn valus of h child facor (ω ), rspcivly, ha is pur PAYG pnsions ( ω = 0 ), mixd FR-PAYG pnsions ( ω = ) and pur FR-PAYG pnsions ( ω = ). 2 Th vrical axis shows h limi poins of h quilibrium squnc of capial, * k. Whn h conribuion ra is rlaivly low a uniqu limi poin xiss. Whn h conribuion ra raiss a priod doubling bifurcaion mrgs. Largr PAYG pnsions imply ha priod doubling bifurcaions appar mor and mor rapidly, hus bringing h conomy ino h chaoic rgion. 3 Mor in dail, hs diagrams ar bs undrsood if w sar from h valu of θ = 0 (i.. an conomy wihou social scuriy) and hn mov owards highr valus of h conribuion ra. L us compar h polar cass of pur PAYG and pur FR-PAYG schms, rspcivly. Iniially, h quilibrium poin is sabl for boh schms. As h conribuion ra raiss, such a poin bcoms unsabl for θ = (rsp. θ = ). Th diagrams show ha h mrging 2- priod cycl is sabl. 4 0 Wih h paramr s usd abov, in fac, h long-run friliy ra is abou 0.72 (i...42 childrn for ach coupl) (whn θ = 0. 6 and ω = ), which is fairly clos o ha obsrvd in svral dvlopd counris. W us only such a graphical ool for a picorial viw of possibl chaoic dynamic bhaviours wihou mbarking in mor sophisicad analyss (.g. Lyapunov s xponns) for h dcion of chaos, givn h conomical rahr han mahmaical moivaion of h papr. 2 Numrical simulaions ar prformd by using k = as h iniial valu of h sock of capial. 3 For a dpr undrsanding of h priod-doubling rou o chaos s,.g., Dvany (2003). 4 As is known, his dpnds on h sabiliy of h fixd poins mrging from h scond ira of h diffrnc quaion Eq. (7), which migh b also analyically ascraind. For simpliciy, w limi us o graphically show such a sabiliy. 2

23 For θ = (rsp. θ = 0. 96) a 4-priod cycl ariss. This priod-doubling procss coninus as θ dcrass. Evnually, his procss sops around θ = 0. 4 (whil in h pur PAYG cas i coninus unil h suprior limi of h conribuion ra). Indd, byond such a rahr low lvl of h conribuion ra, an aracing chaoic rgion no longr xiss and h conomy is, broadly spaking, disrupd. As is vidn by h comparison of Figurs 3-5, h chaoic bhaviour gnrad by FR-PAYG pnsions mor likly appars whn h wigh of childrn in drmining h siz of h pnsion arrangmn is high. In fac, h flip bifurcaion valu of h conribuion ra dramaically shrinks from θ = o θ = whn h social scuriy sysm shifs from a pur PAYG schm o a pur FR-PAYG schm. This mans ha h inroducion of friliy componns in h pnsion formula dramaically incrass h risk of cyclical insabiliy. Givn ha, as Liikann (2007, p. 4) claimd, h pnsion conribuions in Europ would ris from hir prsn lvl of around 6% of aggrga wags o around 28% by h yar Japan, which sars ou from a lowr bas, would nd up a approximaly h sam lvl his xampl, alhough only illusraiv, shows ha wih h currn siz of h mos par of h PAYG sysms (namly, an avrag conribuion ra in Europ around 6%) and, a foriori, wih h xpcd highr fuur conribuion ra, vn rahr small friliy-rlad lmns in h pnsion formula may dsabilis and riggr conomic chaoic flucuaions. Thrfor inroducing, in hos counris currnly plagud by blowrplacmn friliy, such as svral counris in Europ, ihr mixd or pur FR-PAYG pnsions vn wih valus of h conribuion ra wll blow h currn avrag valu of 6 pr cn may hav dramaic dsabilising ffcs. To sum up, alhough friliy-rlad pnsions ar ofn advocad as a possibl rmdy agains h pril of h fuur susainabiliy of h provision of unfundd public pnsions as wll as for opimaliy purposs (s Abio al., 2004, Cigno and Wrding, 2007), h ransiion from a pur PAYG sysm (Figur 3) o a PAYG sysm parially (Figur 4) or oally (Figur 5) linkd o 22

24 individual friliy may asily opn h rou o drminisic chaos vn in prsnc of small-sizd pnsion schms. Figur 3. Cas ω = 0 (pur PAYG). Bifurcaion diagram for θ ( θ = ). 23

25 Figur 4. Cas ω = (mixd FR-PAYG). Bifurcaion diagram for θ ( θ = ). Figur 5. Cas ω = (pur FR-PAYG). Bifurcaion diagram for θ ( θ = ). 4. Conclusions W analysd h dynamics of an ovrlapping gnraions doubl Cobb-Douglas conomy wih ndognous friliy and friliy-rlad pay-as-you-go public pnsions wih boh prfc and myopic xpcaions. 5 W showd ha a friliy-rlad pnsion rform dramaically incrass h risk of cyclical insabiliy gnrad by h PAYG sysm in h cas of myopic xpcaions. Morovr, h xisnc of a friliy-rlad componn in h pnsion formula implis ha a ris in h individual dgr of hrifinss and a rducion in h as for childrn boh incras h ara of cyclical 5 In paricular, w hav argud ha h cas of myopic forsigh is rahr plausibl for his class of modls in which individuals ar aomisic and unabl o coordina hir dcisions. 24

26 insabiliy, whil boh paramrs would no affc sabiliy in h radiional public pnsion sysm. This sms o b rahr paradoxical sinc friliy-rlad pnsion rforms ar proprly advocad in conomis plagud boh by rducd saving formaion and blow-rplacmn friliy ras, in which policis aiming a incrasing h lov for saving and childrn ar implmnd for. Thrfor, w may conclud ha h inroducion (or h xnsion) of friliy rlad lmns in h pnsion formula, as rcnly advocad by many conomiss and policy-makrs, ac as a srong conomic d-sabilisr. Morovr w showd ha an conomy wih FR-PAYG pnsions conains in islf h possibiliy of drminisic complx cycls. In paricular, a numrical illusraion has shown ha h dsabilisaion of h conomy du o FR-PAYG pnsions bcoms a plausibl scnario for svral conomis. Our rsuls hav a wofold inrpraion: (i) consiu a policy warning abou h risks of (cyclical) insabiliy causd by h inroducion of friliy-rlad lmns in PAYG pnsion schms in prsnc of ralisic myopia of individuals, and (ii) hy provid a furhr drminisic xplanaion of h occurrnc of prsisn cycls in conomis wih ndognous friliy. Finally, som cavas ar in ordr: sinc our rsuls prain o spcific uiliy and producion funcions and ohr modl assumpions, hy ar of cours naiv. Howvr, i is of valu o show ha h inroducion of friliy dpndan componns in pnsion schms may b dsabilising and gnra chaoic flucuaions in rahr ralisic conomis wih social scuriy. Appndix In his appndix w brifly show ha h dynamics of a Cobb-Douglas OLG conomy wih FR- PAYG pnsions and prfc forsigh canno b cyclical. Proposiion A.. Th dynamics of capial in a doubl Cobb-Douglas OLG conomy wih FR-PAYG pnsions and prfc forsighd individuals is always monoonic and convrgn o 25 * k.

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