Fertility-related pensions and cyclical instability

Transcription

1 MPRA Munich Prsonal RPEc Archiv Friliy-rlad pnsions and cyclical insabiliy Luciano Fani and Luca Gori Univrsiy of Pisa, Dparmn of Economics, Univrsiy of Pisa, Dparmn of Economics 23. January 200 Onlin a hp://mpra.ub.uni-munchn.d/2022/ MPRA Papr No. 2022, posd 26. January 200 0:20 UTC

2 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 W show ha h inroducion of unfundd public pnsions in a Cobb-Douglas conomy wih ovrlapping gnraions and ndognous friliy may caus complx conomic cycls 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. Our rsuls provid a possibl xplanaion of h occurrnc of prsisn cycls in an ovrlapping gnraions conx and rprsn a policy warning abou h dramaic dsabilising ffcs of a friliy-rlad pnsion rform. Kywords Endognous friliy; Friliy-rlad pnsions; Myopic forsigh; OLG modl JEL Classificaion C62; H55; J4; J8; J26 * addrss: lfani@c.unipi.i; l.: ; fax: ** Corrsponding auhor. addrss: luca.gori@c.unipi.i; l.: ; fax:

3 . 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 bnfis 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). 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. As is known, cyclical bhaviour can occur in many-good OLG modls (Grandmon, 985) as wll as in h on-good Diamond-yp OLG conx (Farmr, 986; Richlin, 986), bu only whn

4 producion facors ar rlaivly complmn. 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 whn h inr-mporal lasiciy of subsiuion in h uiliy funcion is highr han uniy (i.., highr han in h cas of Cobb-Douglas prfrncs). 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 rlaiv wigh of h public pnsions in capial accumulaion is highr han in h cas of a pur PAYG schm. Friliy-rlad pnsions, hrfor, ac as an conomic d-sabilisr in ovrlapping gnraions conomis. 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 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 ] θ, () = w + 2

5 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 a im, n, wih a shar ω of h conribuion, and (ii) h avrag ra of friliy in h whol conomy a im, n, 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 an ovrlapping gnraions (OLG) 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 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 3

6 cos q pr child, wih 0 < q < bing h prcnag of child-raring cos on working incom. 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 ). 2 Assuming logarihmic prfrncs, h rprsnaiv individual nring h working priod a solvs h following problm: { 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: This child cos srucur is similar o ha adopd by, amongs many ohrs, Wiggr (999) and Boldrin and Jons (2002). 2 S Ecksin and Wolpin (985) and Galor and Wil (996). 4

7 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) 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 5

8 numbr of hir dscndan raiss (i.. 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), 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 L Y = AK, whr A > 0 is a scal paramr and 0 < < 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 Ak y =. Assuming oal dprciaion of capial a h nd of ach priod and normalising h pric of final oupu o uniy, profi maximisaion implis ha facor inpus ar paid hir marginal producs, ha is: r = Ak, (9) w ( ) Ak =. (0) 2.4. Equilibrium 6

9 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. () 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. 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 h individual dgr of hrifinss ( β ) as a ponial dsabilising paramr. A ris in dgr of parsimony, in fac, incrass boh h priva saving componn and h public pnsion componn and, hnc, is final ffc on capial accumulaion may b ambiguous. Blow w sudy how h dynamic pah of capial accumulaion volvs dpnding on whhr individuals hav ihr prfc or 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 7

10 + r w + + = = Ak ( ) + Ak +. (3) Combining Eqs. (9), (0), (2) and (3), h dynamic quilibrium squnc of capial can b wrin as k ( ) A ( )( β ω + φ) q β. (4) φ + θ + = k Sady-sa implis * k +, so ha: ( ) A ( )( β ω + φ) * q β k = 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 + + = = Ak ( ) Ak. (6) Combining Eqs. (9), (0), (2) and (6), h dynamic pah of capial accumulaion is now givn by: k + β β ω φ q( ) Ak + θ = k. (7) φ φ Th sady-sa is sill drmind by Eq. (5), s Michl and D La Croix (2000). Dspi Eq. (7) is a simpl firs ordr non-linar diffrnc quaion, h dynamics of capial may b highly non-linar and ndognous flucuaions may mrg. Th local sabiliy propris of 8

11 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 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 < < 3 hold. Thn θ < θ <, and: (.) if 0 < θ < θ, h dynamics of capial is monoonic and convrgn o (.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. * k ; (2) L 3 < < 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 < hold. Thn θ > θ >, and h dynamics of capial is monoonic and convrgn < o * k for any 0 < θ <, 3 Th (local) sabiliy propris of an conomy wih prfc forsigh is brifly prsnd in Appndix A. Diffrn from h cas wih myopic xpcaion, wih raional xpcaions h conomy dos no xhibi any inrsing dynamical faur. 9

12 whr θ = θ 2 φ θ = θ(, β, φ, ω) : =, (8) 2 ( β, φ, ω) ( ) β ω + φ ( + ) φ + = θ ( ) β ω + φ, : =, (9) 2 [ ], / 2 < = ( β, φ, ω) : = β ω + φ φ( β ω + φ) <, (20) β ω [ ], / < 3 3 = 3( β, φ, ω) : = 2β ω + 3φ φ( 8β ω + 9φ ) 3 <. (2) 2β ω Proof. Diffrniaing Eq. (7) wih rspc o k and using Eq. (5) givs: + * = θ k ( ) 2 β ω + φ. (22) φ Monoonic and non-monoonic dynamics + > From Eq. (22), h condiion * 0 k < implis ( ) 2 β ω + φ > < θ 0 θ θ, (23) 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 < < ( < < ). Morovr, θ < if and only if < and > 2, whr is dfind by [ ] 2 2. Sinc / 2 β ω < < and > 2 for any Eq. (20) and = ( β, φ, ω) : = β ω + φ + φ( β ω + φ) β, φ and 0 < ω, hn h cas > 2 can b ruld ou. 0

13 + Now, * < givs k θ ( ) 2 β ω + φ φ < θ >. (24) φ β ω + φ Thrfor, in h cas of monoonic dynamics h conomy always convrgs o h saionary sa + irrspciv of h siz of h pnsion sysm, i.. 0 < * < k for any 0 < θ <. Non-monoonic dynamics: sabiliy analysis + > Th condiion * k < implis: ( ) 2 β ω + φ > < θ θ θ, (25) 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 < ( θ > ) for any 0 < < 3 ( 3 < < ). Morovr, θ < if and only if < 3 and > 4, whr is dfind by Eq. (2), ( β, φ, ω) : = 2β ω + 3φ + φ( 8β ω 9φ ) 3 [ ] = 4 and 3 <. Sinc 2β ω 4 + /3 < 3 < and > 4 for any β, φ and 0 < ω, hn h cas > 4 can b ruld ou. Thrfor, + (i) if 0 < < 3 hn θ < θ < and (.) 0 < * < k for any 0 < θ < θ, (.2) + < * < k 0 + for any θ < θ < θ, (.3) * = k if and only if θ = θ, and (.4) + * < k for any θ < θ <. This provs poin ();

14 + (ii) if 3 < < hn θ <, θ > and (2.) 0 < * < k for any 0 < θ < θ, and (2.2) + < * < k 0 for any θ < θ <. This provs poin (2); + (iii) if < < hn θ > θ > and 0 < * < 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 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. 2

15 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 wih FR-PAYG pnsions is highr han wih pur PAYG pnsions. A ris in h disribuiv capial shar ( ) 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, and a rducion in h as for childrn incrass h risk of cyclical insabiliy. Proof. Firs, in h cas of pur PAYG pnsions ( = 0 3 ω ) Eq. (8) bcoms θ = θ( ) : = 2 ( ) 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 ( ) < ( ( ) > ( 0 θ ) for any 0 < < / 2 ( / 2 < < ). Thrfor, wih FR-PAYG pnsions < ω ) h widh of h paramric rgion in h spac ( θ ), 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 ( β, φ, ω) Scond, in h cas of pur PAYG pnsions ( = 0, can b smallr han uniy vn whn / 2 < <. ω ) Eq. (9) bcoms θ = θ ( ) = θ ( ) + : (i.., h flip bifurcaion valu of h conribuion ra is indpndn of boh h subjciv discoun facor and as for childrn), so ha θ ( ) < ( ( ) > θ ) for any 0 < < / 3 ( /3 < < ). Thrfor, wih FR-PAYG pnsions ( 0 < ω ) h widh of h paramric rgion in h spac

16 (, 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 ( β, φ ω) θ,, can b smallr han uniy vn whn /3 < <. Morovr, from Eq. (9) w g: for any 0 ω, and for any 0 < ω. Q.E.D. θ = θ = ω θ = β θ = φ φ( + 3 ) ( ) ( β ω + φ) 3 > ( + ) φβ ( ) ( β ω + φ) < 2 2 ( + ) φω ( ) ( β ω + φ) < 2 2 ( + ) βω ( ) ( β ω + φ) > 2 2 0, (26) 0, (27) 0, (28) 0, (29) Figurs and 2 illusra Proposiion 2 and compar h paramric rgions in h spac (, 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 < / 3, in a FR-PAYG conx h cyclical unsabl rgion in h spac (, θ ) is largr bcaus of h dsabilising ffcs playd by h child facor, h individual dgr of hrifinss and h as for childrn. 4

17 Figur. Cas = 0,. ω (pur PAYG pnsions). Sabiliy and insabiliy rgions in h spac ( θ ) Figur 2. Cas 0,. < ω (FR-PAYG pnsions). Sabiliy and insabiliy rgions in h spac ( θ ) Tabl. Paramric insabiliy rgions ( 0 < θ < ) undr diffrn PAYG sysms. Pur PAYG ( ω = 0 ) Mixd FR-PAYG ( 0 < ω < ) Pur FR-PAYG ( ω = ) 5

18 0 < /3 < < < ( β, φ, ω) 0 < < 3( β, φ,) 0 3 Tabl summariss for hr diffrn PAYG schms h hrshold valus of h oupu lasiciy of capial blow which cyclically insabiliy may mrg. Sinc ( β φ,) > ( 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 on h sabiliy of h conomy: 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: 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 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 pnsion arrangmn rcivd whn old o h numbr of childrn chosn whn young is high. This rsul holds bcaus a rducion in h as for childrn incrass h rlaiv wigh of h public 6

19 pnsion componn in quilibrium and hus conribus o dsabilis h conomy, whil kping h priva saving componn unaffcd (s Eq. B in Appndix B). 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 posiiv 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 incrasing h rlaiv wigh 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 mrgnc of drminisic chaos in a doubl Cobb-Douglas conomy wih FR-PAYG pnsions. In Figurs 3-5 w dpic h bifurcaion diagrams for h paramr θ (on h horizonal axis), wih rspc o hr diffrn valus of h child facor (ω ). W ak h following paramr s (only for illusraiv purposs): A = 0, = 0. 25, β = 0. 60, φ = 0. 05, q = 0. 5 and k = (h iniial valu of h sock of capial). 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. As i is vidn, h chaoic bhaviour gnrad by FR-PAYG pnsions 7

20 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 of cours incrass h risk of cyclical insabiliy. In fac, whn PAYG pnsions ar fully linkd o individual friliy, a small-sizd pnsion sysm ( θ = ) dircly brings h conomy ino h chaoic rgion. Thrfor, alhough friliy-rlad pnsions ar ofn advocad as a possibl rmdy agains h pril of h fuur susainabiliy of unfundd public pnsions as wll as for opimaliy purposs (s Abio al., 2004), h ransiion from a pur PAYG sysm (Figur 3) o a PAYG sysm parially (Figur 4) or oally (Figur 5) linkd o individual friliy may asily opn h rou o drminisic chaos vn in prsnc of small-sizd public pnsions. Figur 3. Cas ω = 0 (pur PAYG schm). Bifurcaion diagram for θ ( θ = ). 8

21 Figur 4. Cas ω = (mixd FR-PAYG schm). Bifurcaion diagram for θ ( θ = ). Figur 5. Cas ω = (pur FR-PAYG schm). Bifurcaion diagram for θ ( θ = ). 4. Conclusions 9

22 W analysd h dynamics of an ovrlapping gnraions conomy wih ndognous friliy and friliy-rlad pay-as-you-go public pnsions whn individuals ar myopic forsighd. W showd ha a friliy-rlad pnsion rform dramaically incrass h risk of cyclical insabiliy gnrad by h PAYG sysm. Morovr, h xisnc of a friliy-rlad componn in h pnsion formula gnras wo counrinuiiv policy ffcs: a ris in h individual dgr of hrifinss and a rducion is as for childrn boh incras h ara of cyclical insabiliy. In fac, h capial accumulaion funcion is dividd ino wo componns: h priva saving componn and h public pnsion componn. Wha is imporan for sabiliy is h rlaiv siz of h lar componn. Wih FR-PAYG pnsions, h mor individuals wish o smooh consumpion ovr hir rirmn priod and h lss is h as for childrn, h highr is h rlaiv wigh of h public pnsion componn, i.. boh paramrs ac as conomic d-sabilisrs. Thrfor, a doubl Cobb-Douglas conomy wih FR-PAYG pnsions and myopic forsighd individuals conains in islf h possibiliy of drminisic complx cycls. Our rsuls hav a wofold inrpraion: (i) consiu a policy warning abou h risks of (cyclical) insabiliy causd by PAYG pnsion schms in prsnc of ralisic myopia of individuals, and (ii) hy rprsn an xplanaion of h occurrnc of prsisn cycls in conomis wih ndognous friliy. Appndix A 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 20 * k.

23 Proof. Diffrniaing Eq. (4) wih rspc o k and using Eq. (5) w find: + q β = φ + θ ( ) = ( )( ) ( ) * β ω + φ * k k A. (A) + Thrfor, 0 < * < k for any 0 < θ <. Q.E.D. Rfrncs Abio, G., Mahiu, G., Paxo, C., On h opimaliy of PAYG pnsion sysms in an ndognous friliy sing. Journal of Pnsion Economics and Financ 3, Blindr, A.S., Krugr, A.B., Wha dos h public know abou conomic policy, and how dos i know i? Brookings Paprs on Economic Aciviy 2004, Bori, T., Börsch-Supan, A., Tabllini, G., 200. Would you lik o shrink h wlfar sa? A survy of Europan ciizns. Economic Policy 6, Bori, T., Börsch-Supan, A., Tabllini, G., Pnsion rforms and h opinions of Europan ciizns. Amrican Economic Rviw 92, Boldrin, M., Jons, L.E., Moraliy, friliy and saving in a Malhusian conomy. Rviw of Economic Dynamics 5, d la Croix, D., and Michl, P., A Thory of Economic Growh. Dynamics and Policy in Ovrlapping Gnraions. Cambridg Univrsiy Prss, Cambridg. 2

24 Diamond, P., 965. Naional db in a noclassical growh modl, Amrican Economic Rviw 55, Ecksin, Z., Wolpin, K.I., 985. Endognous friliy and opimal populaion siz. Journal of Public Economics 27, Fani, L., Spaaro, L., Povry raps and inrgnraional ransfrs, Inrnaional Tax and Public Financ 5, Farmr, R.E., 986. Dficis and cycls, Journal of Economic Thory 40, Fng, R., Mir, V., Pnsions and friliy incnivs. Canadian Journal of Economics 38, Fng, R., Mir, V., Ar family allowancs and friliy-rlad pnsions prfc subsius? Inrnaional Tax and Public Financ 6, Fng, R., von Wizsäckr, J., 200. Mixing Bismarck and child pnsion sysms: an opimum axaion approach. Journal of Populaion Economics, forhcoming. Galor, O., Wil, D.N., 996. Th gndr gap, friliy, and growh. Amrican Economic Rviw 86, Grandmon, J.M., 985. On ndognous businss cycls, Economrica 53,

25 Grozn, B. van, Mijdam, L., Growing old and saying young: populaion policy in an aging closd conomy. Journal of Populaion Economics 2, Grozn, B. van, Lrs, T., Mijdam, L., Social scuriy and ndognous friliy: pnsions and child allowancs as Siams wins. Journal of Public Economics 87, Kolmar, M., 997. Inrgnraional rdisribuion in a small opn conomy wih ndognous friliy. Journal of Populaion Economics 0, Michl, P., d la Croix, D., Myopic and prfc forsigh in h OLG modl. Economics Lrs 67, Pcchnino, R.A., Pollard, P.S., Aging, myopia, and h pay-as-you-go public pnsion sysms of h G7: A brigh fuur?, Journal of Public Economic Thory 7, Richlin, P., 986. Equilibrium cycls in an ovrlapping gnraions conomy wih producion, Journal of Economic Thory 40, Wiggr, B.U., 999. Pay-as-you-go financd public pnsions in a modl of ndognous growh and friliy. Journal of Populaion Economics 2, Zhang, J., Zhang, J., L, R., 200. Moraliy dclin and long-run conomic growh, Journal of Public Economics 80, Zhang, J., Zhang, J., L, R., Rising longviy, ducaion, savings, and growh. Journal of Dvlopmn Economics 70,