Complex equilibrium dynamics in a simple OLG model of neoclassical growth with endogenous retirement age and public pensions

Transcription

1 MPRA Munich Prsonal RPEc Archiv Complx quilibrium namics in a simpl OLG mol of noclassical growh wih nognous rirmn ag an public pnsions Luciano Fani an Luca Gori Dparmn of Economics, Univrsi of Pisa, Dparmn of Economics, Univrsi of Pisa 6. Jul 2010 Onlin a hp://mpra.ub.uni-munchn./23694/ MPRA Papr No , pos 8. Jul :37 UTC

2 Complx quilibrium namics in a simpl OLG mol of noclassical growh wih nognous rirmn ag an public pnsions Luciano Fani an Luca Gori Dparmn of Economics, Univrsi of Pisa, Via Cosimo Riolfi, 10, I Pisa (PI), Ial Absrac W anals h sa-sa quilibrium namics of h convnional ovrlapping gnraions conom à la Diamon (1965) wih pa-as-ou-go public pnsions an scon prio of lif ivi bwn working an rirmn im in a proporion pnn on h iniviual halh saus (a rahr ralisic assumpion spciall in h currn worl wih high longvi). In conras o an conom wihou public halh spning which is alwas sabl wih monoonic rajcoris, an conom wih ax-financ halh car srvics (which in urn affc h iniviual halh saus an hnc h lngh of h rirmn im) ma xprinc complx quilibrium namics wih rminisic chaoic businss ccls an, in paricular, complica namical phnomna, such as mulipl bubblings ma occur whn crucial conomic paramrs chang. Inrsingl, i is shown ha incrasing h siz of PAYG pnsions, alhough iniiall ma riggr chaoic ccls, vnuall works for sabili. Kwors Halh; Ol-ag workrs; OLG mol; Prfc forsigh; Public PAYG pnsions JEL Classificaion C62; H55; I18; O41 arss: fani.luciano@gmail.com; l.: ; fax: Corrsponing auhor. arss: luca.gori@c.unipi.i; l.: ; fax:

3 1. Inroucion Th problms concrning h public provision of halh car srvics an pnsion bnfis for rir maur workrs ar currnl high on h poliical agnas in svral inusrialis counris. As is known, halh saus ma affc h iniviual conomic bhaviours hrough mulipl channls. On of hm is surl rprsn b h ffcs ha halh ma xr on boh h abili o work an h proucivi of work. 1 Th link bwn halh saus an labour proucivi has bn arl rcognis b h pionring Grossman (1972). Morovr, svral suis hav mpiricall foun ha halh saus has a significan ffc on labour forc of olr popl (.g. Chirikos, 1993; Curri an Marian, 1999; Campolii, 2002; Cai an Kalb, 2006; Disn al., 2006). Howvr, h rol pla b boh public halh spning an rirmn ag on h quilibrium namics of a noclassical growh mol has no so far bn pl invsiga. In his papr w aim o fill his gap in h framwork of h famous Diamon s (1965) mol xn wih h following hr simpl ralisic assumpions: (i) ol popl ma suppl labour for a fracion of hir im nowmn whn ol pning on hir halh saus, (ii) h iniviual halh saus, in urn, pns in non linar form b h siz of h public halh spning, an (iii) public PAYG pnsions xis o suppor h ol-ag unabl o work an hus rir. In orr o concnra on h ffcs of halh spning an pnsion bnfis on h lngh of h ag of rirmn w absrac from consiring aul morali an h uili ffc of halh pr s, an morovr w also nglc h rol of lisur. In paricular, w assum ha h lngh of h rirmn prio (convrsl, h working prio), which is of cours a fracion of h scon prio of lif span of popl, onl pns on h iniviual halh saus whn ol. This mans ha h a of rirmn is chosn nihr 1 Th ohr main channl hrough which halh ma affc conomic growh an ha has bn mor largl invsiga is ha bwn halh an longvi ras (.g. Chakrabor, 2004; la Croix an Ponhir, 2010; Lung an Wang, 2010). 1

4 volunaril b maur workrs nor in a manaor wa b govrnmns, whil bing objcivl rmin b h halh saus of maur iniviuals: if halh is low, hn maur workrs ar auhoris o rir an hus h ar nil o a pnsion bnfi (alrnaivl, i ma b assum ha for h prio of sicknss, in which ol iniviuals canno work, h rciv a paas-ou-go halh insuranc bonus). Th main fining of h prsn papr is h following: whn h convnional Diamon s mol is xn wih h hr assumpions abov mnion, h quilibrium namics ma b oscillaor. Morovr, boh rgular an chaoic businss ccls sm o b h rul rahr han h xcpion for his conom. In paricular, i is worh noing ha rminisic chaoic moions mrg in a simpl on-imnsional mol wih Cobb-Douglas uili an proucion funcions, which a rahr rar occurrnc in h OLG conx wih proucion. In fac, so far h liraur has shown ha complx oucoms picall occur ihr in highr imnsion mols for low valus of h lasici of subsiuion in proucion, in paricular for valus wll blow uni (i.., in h cas of h Lonif chnolog), 2 or abanoning h raional xpcaion paraigm an hn assuming mopic xpcaions, whr rminisic chaos can mrg a high valus of h inr-mporal lasici of subsiuion (.g., Michl an la Croix, 2000; la Croix an Michl, 2002; Chn al. 2008). Morovr, i is noworh ha h quilibrium namics of h prsn mol shows a rahr high complxi i.., muliplici of bubbling phnomna. Th bubbling phnomnon shows ha h rgular bifurcaion parn rvrss islf, unrgos prio halving via flip bifurcaions, an vnuall rurns o a uniqu sa sa for largr paramr valus. 3 2 For insanc, Richlin (1986) iscusss h Lonif cas, an in Farmr s (1986) xampl for h CES chnolog, nognous ccls occur onl whn h proucion funcion xhibi lowr facor subsiuabili han h Cobb-Douglas funcion. 3 Pionring iscussions abou his namical faur ar Bir an Bounis (1984) an Son (1993). 2

5 In aiion o h faur of muliplici of bubblings, for which a coninuum of rgulariis an irrgulariis in h conomic bhaviour appar for h mos par of h fasibl polic changs (w us h conribuion ra o h pnsion ssm as h bifurcaion paramr), i is inrsing o no ha rlaivl high valus of h conribuion ra vnuall n o smooh conomic flucuaions, hus proprl working for h global sabili of h conomic ssm. This wofol rol of such a polic paramr is rmarkabl from an conomic poin of viw: on h on han, i ma concur o xplain h obsrv irrgular businss flucuaions showing ha an nognous rminisic origin of conomic ccls ma b complmnar o h sochasic origin which is a h cor of h ral businss ccl hor bu, on h ohr han, an mab mor imporan, h polic variabl ma vn b us o conrol an poniall o supprss unsirabl businss flucuaions. Thus, w ma conclu ha h quilibrium namics of his simpl conom ma mbo wo unispuabl slis facs: h xisnc of boh irrgular businss ccls an h slfvin prsisnc of h conomic ssm. Th papr procs as follows. Scion 2 prsns h mol. In Scion 3 h namics of h conom is anals an iscuss. In Scion 4 h complxi of h namic bhaviour of h conom is shown. Scion 5 iscusss h rsuls an conclus. 2. Th conom 2.1. Iniviuals Consir a gnral quilibrium ovrlapping gnraions (OLG) clos conom popula b a coninuum of inical iniviuals of masur on. Lif is ivi ino ouh (firs prio) an olag (scon prio). In ach prio, iniviuals ar now wih on uni of im. Whn oung, agns of gnraions work an inlasicall offr hir whol im nowmn on h labour mark, whil rciving wag incom a h compiiv ra 3 w. This incom is us

6 o consum ( 1 ) an o sav ( s ). Morovr, h govrnmn lvis wag incom axs on h c, oung s labour incom, a h consan ras 0 < τ < 1 an 0 < θ < 1, o sparal financs a a balanc bug halh car srvics an public PAYG pnsions, rspcivl. Thrfor, h bug consrain of a oung agn born a ras as ( τ θ ) c + s = w 1. (1.1) 1, Th uniar im nowmn of h ol-ag is ivi bwn working im ( + 1 ) an rirmn im ( ). For h fracion of im suppli on h labour mark, h rciv arnings qual o w 1, whr w + 1 is h wag iniviuals xpc o arn a im + 1, whil as rgars h rirmn im, h xpc o b nil o an amoun of public pnsions ( 1 ) p + 1 (s, amongs ohrs, Hu, 1979; Momoa, 2003). Th bug consrain a + 1 of an ol prson born a, hrfor, is ( 1 + r + 1) s + w ( ) p + 1 c, (1.2) 2, + 1 = ha is, consumpion whn ol c 2, + 1 is suppor b savings plus xpc inrss from o + 1, accru a h ra r + 1, h xpc wag incom for h working im whn ol,, an h xpc pnsion bnfi for h rirmn im, 1. In his papr w assum h cofficin + 1 is nognous an rmin b h iniviual halh saus whn ol, which, in urn, is assum o posiivl pn on h public halh spning h govrnmn provis a, h, i.. h halhir an ol iniviual, h largr h fracion of im vo o suppl labour. Th rlaionship bwn h working im in h ol-ag prio (i.. h ol-ag labour suppl) an halh xpniur is assum o b capur b h gnric non-crasing-boun funcion: Alrnaivl, his amouns o assuming ha h halh chnolog is a non-crasing boun funcion of halh xpniur an h working fracion of h scon prio of lif is proporional o h iniviual halh saus wih a proporionali cofficin qual o on. 4

7 whr ( 0) = 0 0, lim ( ) h h = 1 ( ) an 0 0 < 1 < = h, (2) Prfrncs of iniviuals of gnraion ovr oung-ag consumpion an ol-ag consumpion ar scrib b h following lifim logarihmic uili funcion: ( c ) + ( c ) U β, (3) = ln 1, ln 2, + 1 whr 0 < β < 1 is h gr of iniviual (im)painc o consum ovr h lif ccl. Th rprsnaiv iniviual wishs o choos how much o sav ou of hr isposabl incom in orr o maximis h lifim uili inx Eq. (3) subjc o Eqs. (1), whr acual an xpc facor prics, h cofficin + 1 an h xpc pnsion bnfi p + 1 ar akn as givn. Thrfor, h saving ra is: s ( 1 τ θ ) β w 1+ β 1 ( ) ( )( ) [ + 1w p + 1] 1+ β 1+ r + 1 =. (4) 2.2. Firms A im firms prouc a homognous goo, Y, b combining capial an labour, K an L, rspcivl, hrough h consan rurns o scal Cobb-Douglas chnolog Y = AK α 1 α L, whr A > 0 is a scal paramr an 0 < α < 1 h oupu lasici of capial. Labour suppl is ( ) L = L 1 +, whr L is h consan numbr of workrs in ach cohor (oung an ol); hn, w s L = 1 wihou loss of gnrali. Thrfor, oupu pr fficin workr ( ) as a funcion of capial pr fficin workr ( k ) is α = Ak, (5) whr / := Y L an k K / L :=. 5

8 Firms maximis profis 5 an prfc compiion guarans ha facor inpus ar pai hir marginal proucs, ha is α r = αak 1 1, (6.1) w α ( α ) Ak = 1. (6.2) 2.3. Govrnmn Th govrnmn sparal financs wih labour incom axs boh halh car srvics (.g., vaccins, hospials, scinific rsarch an so on) o amliora h iniviual halh saus, an unfun PAYG pnsions o ransfr rsourcs from h oung o h ol, a a balanc bug. As rgars halh car xpniur a, i is consrain b h following pr capia govrnmn bug: h = τ, (7.1) w h lf-han si bing h halh xpniur an h righ-han si h ax rcip (s Chakrabor, 2004). Th pr capia pnsion accouning rul follow a, insa, ras as: ( ) p = w ( 1+ ) 1 θ, (7.2) h lf-han si bing h pnsion xpniur an h righ-han si h ax rcips Gnral quilibrium Exploiing h on-prio forwar pnsion accouning rul Eq. (7.2) o subsiu ou for h pnsion xpniur ino Eq. (4) an using Eqs. (2), h saving ra can b rarrang as 5 Wihou loss of gnrali, w assum h pric of final oupu is normalis o uni an capial oall prcias a h n of ach prio. 6

9 s β w = 1+ β ( 1 τ θ ) ( h ) w + 1 ( 1+ β )( 1+ r + 1) θ w + 1 [ 1+ ( h )] ( 1+ β )( 1+ r + 1). (8) Givn h govrnmn bugs Eqs. (7), mark-claring in goos an capial marks is rmin as h quali bwn invsmns an savings K = +1 S, ha can also b xprss as: [ ( h )] s k =. (9) Now, combining Eqs. (8) an (9), assuming iniviuals ar prfc forsigh (i.. h fuur valus of boh h wag an inrs ra is xpc o pn on h fuur valu of h capial sock), an using Eqs. (6.2) an (7.1) o obain a rlaionship bwn h fracion of working im whn ol an capial pr fficin workr, i.. ( k ), h namic voluion of capial is simpl scrib b h following firs orr non-linar iffrnc quaion wih consan cofficins: ( k ) ( k ) I k +1 =, (10) L + M whr I : = βα ( 1 α )( 1 τ θ ), L : = α( 1+ β ) + θ ( 1 α ) an : = 1+ αβ + θ ( 1 α ) α consans an ( k ) = Ak from Eq. (5). M ar posiiv 3. Dnamics Analsis of h phas map Eq. (10) givs h following proposiions as rgars h xisnc an uniqunss of h sa sa, monoonic an oscillaor namics an h possibili of nognous flucuaions pning on h lngh of h rirmn im. Proposiion 1. (Exisnc an uniqunss of h sa sa). A uniqu non-rivial sa sa k > 0 of h namic ssm scrib b Eq. (10) xiss. 7

10 Proof. Dfin h righ-han si of Eq. (10) as J ( k). Sinc ( 0) = 0 0 o vrif ha J ( 0 ) = 0, hn i is sraighforwar. Now, fix poins of Eq. (10) ar rmin as inrior soluions o α k = J ( k), ha can also b rarrang as Z1 ( k) = Z2( k), whr Z ( ) = 1 1 k : k an Z ( k ) Thrfor, sinc Z 1 ( k) an ( k) Z 2 ar coninuous funcions an: IA := L + M ( k) 2. (i) Z ( 0) 0, ( k) = ( 1 α ) k 0 1 = α 1, > Z k for an > 0 k an lim Z ( k) = + k + 1, an an > 0 IA IA Z 2 = = > 0, whr : 0 0 L + M G = L + M > (ii) ( 0) ( 0) k sinc k ( k) > 0 an Z ( k) IA IA <, sinc E > G bcaus 0 0 < 1 < 1. E G G, ( ) MIA k ( k) Z, k = 2 [ L + M( k) ] ( ) k 0 for 2 < lim IA IA = = > 0 k +, whr : L + M E E = L + M >, Thn for an > 0 k w g ( k) Z ( k) Z 2 1 = onl onc a k. Q.E.D. Proposiion 2. (Non-monoonic bhaviour). If an ε > ε, (11) ( k ) > ˆ ε ( k ): = Q ε ε hn h law of moion in Eq. (10) is non-monoonic, whr, (12) 8

11 k k ε : = k ( k ) an ( k ) : = k ( k ) ( k ) ε ar, rspcivl, h lasici of h suppl labour whn ol an h lasici of GDP pr fficin workr wih rspc o h sock of capial pr fficin workr valua a k, an : < 1 L Q =. M Proof. Diffrniaing J ( k) wih rspc o k givs an valuaing i a J ( k ) IA L = k α Sinc sgn{ J ( k )} = sgn{ Lε M ( ε ε ) ( k )} [ ε M ( ε ε ) ( k )] 1 ( k ) [ L + M( k )] 2 k, hn ε sufficin o hav ( k ) < 0 J k. If ε ε < hn ( ) 0 k givs:. (13) ε > is ncssar an ( k ) > ˆ( k ) J k k > alwas hols. Q.E.D. Proposiion 3. (Enognous flucuaions). If (11) an (12) hol an (1) if ˆ( k ) ( k ) < ( k ) k ; (2) if ( k ) ( k ) <, hn h law of moion in Eq. (10) is non-monoonic an convrgn o =, hn a flip bifurcaion ma gnricall occur; 6 (3) if ( k ) ( k ) < ( k ) k ; (4) if ( k ) ( k ) <, hn h law of moion in Eq. (10) is non-monoonic an ivrgn from = a rvrs flip bifurcaion ma gnricall occur; (5) if ( k ) < ( k ) < 1 whr, hn h law of moion in Eq. (10) is non-monoonic an convrgn o ( k ) : 2L 1 α ( k ) + IA( ε ε ) P( k ) 2M ( k ) α = 1 k,, (14) 6 In h numrical xampl in h nx scion w show ha h flip bifurcaion is supr-criical an, hnc, araciv (i.., h bifurcaion poins ar smmrical an sabl). 9

12 ( k ) : 2L 1 α ( k ) + IA( ε ε ) + P( k ) 2M ( k ) α = whr P( k ): = ( IA) ( ε ε ) 4IALε ( k ) α, (15) Proof. From Eq. (13) w fin ha ( ) 1 L k 1 α M If 2 ( ) ( ) > 0 2 J k k if an onl if 1 α [ ] ( k ) + L[ L( k ) + IAε ] 0 1 α 2 1 α ( k ) [ ( k )] + M 2L( k ) IA( ε ε ) IA ε ε L k hrough oscillaions. If 2 ( ) ( ) < 0. (16), hn (16) can nvr b vrifi an h sa sa is saionar 1 α IA ε ε, hn whn P ( k ) > 0 wo posiiv ral soluions J k if xis of (16), as givn b Eqs. (14) an (15) whr ( k ) > ( k ) > 0, such ha 1< ( k ) < 0 ( k ) < ( k ) ( k ), J k ( k ) = 1 if ( k ) = ( k ), J k ( k ) < 1 if ( k ) < ( k ) < ( k ), ( k ) = 1 ˆ < if ( k ) = ( k ) an 1 ( < J k k ) < 0 if ( k ) < ( k ) < 1. Q.E.D. J k In orr o lucia how h main sa-sa macroconomic variabls rac o a chang in h siz of h pnsion ssm, θ, as wll as o grasp h conomic inuiion bhin h rsuls, w now prform a snsiivi analsis b molling h rlaionship bwn h working fracion of h olag prio an halh xpniur wih h following rahr gnral funcion, following for insanc Blackburn an Cipriani (2002), Blackburn an Issa (2002),an la Croix an Ponhir (2010): 7 + δ h 1 = ( h ) =, (17) δ 1+ h 7 Alhough in h analss b Blackburn an Cipriani (2002), Blackburn an Issa (2002), Chakrabor (2004), la Croix an Ponhir (2010) an Lung an Wang (2010) h pnn variabl is h longvi ra rahr han h fracion of working (rirmn) prio in h ol-ag, h lin of rasoning o jusif his formulaion ma b h sam. 10

13 whr δ, > 0, 0 < 0 1, > 0, ( 0) = 0 0, ( ) δ 1 δ h ( 1 0 ) h = 1 0 > h > 0, δ ( 1+ h ) 2 lim ( ) = h h 1 1, ( h) < 0 hh if 1 > hh h for an < δ an ( ) 0 1 δ < δ 1 h h T : = > ( 1 δ ) if δ > 1. + Eq. (17) is abl o capur svral iffrn faurs of h abili o work whn ol of h pical agn as a funcion of h halh masur h : i ncompasss, in fac, (i) h sauraing funcion us in h numrical xampls b Chakrabor (2004) an Lung an Wang (2010) whn δ = = 1 an 0, whil also prsrving (iffrn from Chakrabor, 2004, an in accor wih 0 = Lung an Wang, 2010) a posiiv xognousl givn lvl of ol ag working prio rgarlss of whhr public halh spning xiss or no; (ii) h S-shap funcion whn δ > 1 (i.., hrshol ffcs of public halh invsmns xis) us in h numrical xampls b Blackburn an Cipriani (2002), Blackburn an Issa (2002), an la Croix an Ponhir (2010). Th rlaionship bwn halh saus an incom (ha, in his mol, can b consir as a prox of h public halh spning, h lar bing a consan fracion of incom), has bn shown o b S-shap b Ecob an Dav Smih (1999) 8 which argu ha hs inics of morbii, boh slf-rpor an masur, ar approximal linarl rla o h logarihm of incom, in all xcp vr high an low incoms (his mans ha incrasing incom is associa wih br halh, bu ha hr ar iminishing rurns a highr lvls of incom). (p. 693). Morovr, Srauss an Thomas (1998) conras an inx ha capurs nuriional saus an halh (h Bo Mass Inx) wih h logarihm of wag incom, fining an S-shap rlaionship for Brazil (s Srauss an Thomas, 1998, Figur 3, p. 774). In orr o clarif h inrsing namical propris of h mol, w now rsor o som numrical xrciss aking h following configuraion of paramrs (xclusivl chosn for illusraiv purposs): A = 22, α = (which is an avrag bwn h valus usuall rfrr o 8 S Ecob an Dav Smih (1999, Figurs 1 an 2, p ), ha plo som inics of morbii as a funcion of h logarihm of incom for Englan, Wals an Scolan in 1984 an

14 vlop counris, i.. α = 0. 36, s.g., Kho an Prri, 2002, an hos usuall us for vloping counris, i.. α = 0. 50, s Puru Univrsi s Global Tra Analsis Projc 2005 aabas GTAP), β = (s Žamac, 2007), τ = 0. 06, = 0 0, = , δ = 50, whil also assuming = 1 wihou loss of gnrali, givn h purl chnical (an no conomicall inrprabl) naur of such a paramr. No ha h choic of h paramr δ = 50 9 amouns o assum ha halh invsmns hav a mor inns ffc in rucing h ol ag sicknss sa (ha is, nhancing h ol ag halh saus) whn a crain hrshol lvl of public halh xpniur is achiv, whil bcoming scarcl ffciv whn h abili o work (h goo halh sa) is clos o is sauraing valu No ha if ( h) appar in ha cas. wr concav, i.. δ 1, ε > ε canno hol an, hnc, non-monoonic rajcoris can nvr 10 As an xampl, w ma hink abou h xisnc of hrshol ffcs in h accumulaion of knowlg rquir for nw mical avancs an iscovris in h ramn of isass (.g. vaccins) o b ffciv: h public halh xpniur o financ nw rsarch projcs ma b high an apparnl uslss unil a crain gr of knowlg is achiv. Bon such a hrshol, howvr, a jump ffc xiss ha allows o riggr an bring o ligh h bnficial ffcs of h nw iscovris, o mak hm fficin, usabl an opraiv across populaion an vnuall ransform ino highr halh saus for ol ag iniviuals. 12

15 Figur 1. Th phas map Eq. (10) wih h corrsponing sa sa whn θ varis. Tabl 1 illusras Figur 1 as rgars h ffcs of a ris in θ on boh h voluion an sabili of h sa sa (o his purpos Tabl 1 rpors h valus of h firs rivaiv of Eq. 10 wih rspc o k valua a h sa sa, naml J k ( k ) ). Morovr, w also rpor h valus of ohr macroconomic variabls of inrs a h sa sa, such as h pr capia halh xpniur, h, h lngh of h im spn working whn ol, α ( ) ( 1 ) Y = A k +, h raio of pr capia halh spning o pr capia GDP, ( 1 ) of h pr capia pnsion xpniur o pr capia GDP, Y p, h lvl of pr capia GDP,. h /Y, an h raio Tabl 1. Macroconomic variabls a h sa sa whn θ varis. θ k

16 ( k ) J k h Y h /Y ( 1 ) p Y From Tabl 1 h following rsuls hol. Rsul 1. Th sa sa sock of capial pr fficin workr, k, h pr capia halh xpniur, h, h lngh of h working im whn ol,, an h pr capia GDP, Y, monoonicall ruc whn θ raiss. Rsul 2. Th raio of h pr capia halh xpniur o pr capia GDP a h sa sa, ( 1 ) h /Y, an h raio of h capia pnsion xpniur o pr capia GDP, monoonicall incras whn θ raiss. Y p Th conomic inuiion bhin Rsuls 1 an 2 is simpl. Firs, w rcall ha h lngh of h rirmn prio ( 1 ) (convrsl, h working prio, ) pns onl on h halh saus whn ol. Incrasing PAYG pnsions causs an xpc ngaiv ffc on capial accumulaion an wags, an his in urn rucs halh xpniur. Alhough h rucion in halh spning is 14

17 rahr small an h raio bwn halh spning o GDP slighl incrass, 11 h worsning of h halh saus, ha is h rucion in h abili o work, is rahr srong: for xampl, whil in h absnc of public pnsions maur workrs spn almos 89 pr cn of hir scon prio of lif working, whn h siz of h pnsion ssm is assum in lin wih ha of svral Europan counris (i.. a paroll ax ra abou h 15 pr cn of wag incom), h working fracion of h scon prio of lif rops o abou 66 pr cn, an whn h conribuion ra furhr raiss o abou 25 pr cn of wag incom, as pric b man conomiss for h nar fuur, h im spn working in h ol-ag shrinks o almos 47 pr cn of h whol im nowmn. Thus, h highr is h siz of h PAYG ssm, h lowr pr capia GDP. This ngaiv ffc on h (noclassical) conomic growh is u o a wofol channl: no onl h wll-known crowing ou ffc of PAYG pnsions on savings an capial accumulaion proprl works, bu also a ngaiv ffc on h labour suppl of maur workrs xiss bcaus of an inirc crowing ou ffc of PAYG pnsions on halh spning. This mchanism, hrfor, rsmbls rahr o a vicious circl: h highr h siz of h PAYG ssm, h largr h numbr of pnsionrs (i.. h numbr of maur iniviuals wih ba halh, unabl o work, raiss). 4. Chaoic ccls unr prfc forsigh an mulipl bubbling phnomna In h prvious scion w anals h quilibrium namical propris of h phas map Eq. (10) an illusra (i) h bhaviour of h uniqu quilibrium, (ii) h swich o insabili an h rswich o sabili wih rspc o changs in h conribuion ra o h pnsion ssm. In his scion w invsiga wih numrical simulaions h local an global namics of h mol using h sam paramr θ as h k bifurcaion paramr. 11 No ha h raio of pr capia halh spning an pnsion xpniur o pr capia GDP ar rahr ralisic, passing from 1.7 pr cn o 2.2 pr cn an from 0 pr cn o 13.7 pr cn, rspcivl, whn h paroll ax ra raiss from zro up o

18 I is shown ha h non-linar namics scrib b h im map Eq. (10) ma b chaoic, as h following xampl rvals. Th paramr s is h sam as in h prvious scion. Morovr, w us k 1 as h iniial valu of h sock of capial. 0 = Figur 2. Bifurcaion iagram for θ (an nlarg viw for 0 < θ < 0. 6 an 1 < k < 3. 7 ). Figur 2 clarl rvals a noworh namical phnomnon: h prsnc of a wofol bubbling wih a prsisn wo-prio ccl bwn h wo bubbls an ovrall a significan inclusion of all rajcoris in a boun rgion, maning ha h conomic ssm is rsilin. In orr o br unrsan h apparanc of mulipl bubblings, i is usful o xamin h picur of h im map Eq. (10), Figur 1, oghr wih h corrsponing bifurcaion iagram, Figur 2, whn θ changs, showing how a prio-oubling casca iniias an hn rvrss. Th map has a bimoal shap wih an incrasing concav branch for larg capial socks. Th ffc of incrasing h polic variabl θ is boh o ransla h phas map vricall ownwars an flan ou h incrasing branch of i for larg capial socks. Thrfor whn θ is coninuousl incras 16

19 from zro, h fix poin 12 sail rucs an h slop of h curv a h inrscion poin shrinks unil h valu minus on (s h curvs A an B whn θ incrass from 0 o 0.1 in Figur 1) whn a prio-oubling bifurcaion occurs (a θ = , s Figur 2): h fix poin bcoms unsabl an a sabl wo-ccl is born, follow, as usual, b a succssion of priooublings as θ is grauall rais (s h curv C whn θ = 0. 3 in Figur 1, an h bifurcaion iagram Figur 2 in h inrval < θ < ). In conras, if w consir h cas whn θ is larg, h quilibrium poin is sabl bcaus i lis in h rgion of h map whr h slop is onl slighl ngaiv or vn posiiv (s, for xampl, h curv D in Figur 1 whr θ = 0. 5 an h slop a h inrscion poin is ). As θ is grauall ruc, h slop of h phas map coninuousl crass unil i rachs h valu minus on whn a prio-oubling bifurcaion mrgs ( θ = ); again, similar o h cas iscuss abov, h fix poin bcoms unsabl, a sabl wo-ccl mrgs, an for furhr rucions in θ a succssion of prio-oublings rsuls. Howvr, i is as o obsrv ha a θ = a rvrsal of prio-oublings iniiall occurs, laing o a sabl wo-prio ccl in h inrval < θ < 0. 39, an his is follow a θ = b h apparanc of nw prio-oubling cascas. I is inrsing o no ha, alhough in h inrval < θ < h slop of h im map a h fix poin is mor ngaiv han ousi of ha inrval (as inica boh b in Figur 1 an valus in h hir row of Tabl 1), a simpl wo-prio ccl prvails in such an inrval, whil ousi of i chaoic bhaviour mrgs: his is u o h fac ha bifurcaions ar govrn b h inracion bwn h wo faurs of h bimoal map: h hump, which is largl rsponsibl for prio-oubling, an b h concavi as wll as h high abov h abscissa-axis of h incrasing branch for larg capial socks, which ar rsponsibl for prio-halving. Thrfor, i happns ha, 12 As is known, h fix poin scrib b Eq. (10). Th sabili of valu) lss han uni. k ma b graphicall foun a h inrscion of h 45 lin k = k +1 an h curv k is nsur if h slop of h im map a h inrscion poin is (in absolu 17

20 for making a hurisic xampl, whn θ crass from 0.5 o 0.3 (curv D an curv C in Figur 1, rspcivl), on h on han h slop a h fix poin is alwas mor ngaiv, so ha priooublings occur mor likl, bu, on h ohr han, boh h concavi an h high of h incrasing branch for larg capial socks n o b largr, so ha prio halving also occur. Thrfor in h ovrall, h formr ffc iniiall prvails an h lar on vnuall ominas (as Figur 2 shows whn θ crass from 0.5 o 0.4). Thrfor w ma sum up his procss, saing ha as θ raiss from zro (i) h phnomnon of prio-oubling, onc appar, mus ncssaril b follow, a las for larg valus of θ, b prio-halving, an (ii) h bubbling phnomnon ma appar vn mor han onc in h prsn mol. Noworh, in an cas h quilibrium namics rmains boun in an conomicall maningful rgion alhough i ma ralisicall ispla irrgular businss ccls. 5. Conclusions W analz h sanar OLG mol wih capial accumulaion an PAYG pnsions whn h rirmn prio is pnn on h sa of iniviual halh whn ol, which is, in urn, rmin b h provision of public halh car srvics. Our focus is on h quilibrium namics an our rsuls ar h following: (i) quilibrium namics iffr consirabl bwn conomis wih an wihou h link bwn rirmn prio an sa of halh; (ii) whn h link os xis an h rlaionship bwn halh spning an halh chnolog is S-shap, as is ofn assum in liraur, w show ha h posiiv quilibrium is uniqu bu i ma b prioic an unsabl promping nognous businss ccls. Morovr, w foun h xisnc of complica namical phnomna, such as mulipl bubblings, whn h conribuion o h PAYG ssm chang. Howvr, larg PAYG pnsions ar foun o work for sabili. 18

21 Th conclusion is ha h quilibrium namics in his rahr simpl conom, which is compll rminisic, ma rconcil h xisnc of boh irrgular businss ccls as wll as h slf-vin prsisnc of h conomic ssm. Sinc h rirmn ag as wll as isabili pnsion an halh spning problms currnl figur prominnl on boh h pnsion an public halh ssm rform agnas, hn our analsis inicas ha arssing hos issus in a horical mol migh b worhwhil as rgars h comprhnsion of namical propris of h conom. Rfrncs Bir, M., Bounis, T.C., Rmrging Fignbaum rs in namical ssms. Phsics Lrs A 104, Blackburn, K., Cipriani, G.P., A mol of longvi, frili an growh. Journal of Economic Dnamics an Conrol 26, Blackburn, K., Issa, H., Enognous lif xpcanc in a simpl mol of growh. Cnr for Growh an Businss Ccl Rsarch, School of Economic Suis, Univrsi of Manchsr, Working Papr no. 13, availabl a hp:// Chakrabor, S., Enognous lifim an conomic growh. Journal of Economic Thor 116, Cai L., Kalb, G Halh saus an labour forc paricipaion: vinc from Ausralia. Halh Economics 15,

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