MPRA Munich Prsonal RPEc Archiv PAYG pnsions and conomic cycls Luciano Fani and Luca Gori Univrsiy of Pisa, Univrsiy of Pisa January 00 Onlin a hps://mpra.ub.uni-munchn.d/9984/ MPRA Papr No. 9984, posd 5 January 00 4:0 UTC
PAYG pnsions and conomic cycls Luciano Fani * and Luca Gori ** Dparmn of Economics, Univrsiy of Pisa, Via Cosimo Ridolfi, 0, I 564 Pisa (PI), Ialy Absrac This aricl analyss h dynamics of an ovrlapping gnraions conomy (Diamond, 965) wih pay-as-you-go financd public pnsions and myopic xpcaions. I is shown ha larg PAYG pnsions may riggr conomic flucuaions dpnding on h muual rlaionship bwn chnology and prfrnc paramrs. Our findings consiu a policy warning abou h siz of social scuriy and provid anohr xplanaion of h occurrnc of prsisn cycls. Kywords Myopic forsigh; PAYG pnsions; Sabiliy; OLG modl JEL Classificaion C6; H55; J6 W wish o hank sminar paricipans a h 0 h Annual Confrnc of h Associaion for Public Economic Thory (PET 009), hld on Jun 8 0, 009 a h Naional Univrsiy of Irland, Galway, Irland, for hlpful commns. Usual disclaimr applis. * E-mail addrss: lfani@c.unipi.i; l.: +39 050 6 369; fax: +39 050 6 384. ** Corrsponding auhor. E-mail addrss: luca.gori@c.unipi.i; l.: +39 050 6 ; fax: +39 050 6 384.
. Inroducion Social scuriy in many dvlopd counris is basd on pay-as-you-go (PAYG) public pnsions. Moivad by h hrif of aging on h viabiliy of h widsprad PAYG schms in h long-run, social scuriy rforms ar currnly high in h poliical agnda. Whil a growing body of conomic liraur daling wih h rlaionship bwn pnsions, friliy, longviy and conomic growh has bn dvlopd in h las dcads (s, amongs many ohrs, Zhang al., 00, 003; van Grozn al. 003; Pcchnino and Pollard, 005), lss anion has bn paid o h dynamical ffcs of h PAYG sysms in h ovrlapping gnraions (OLG) conx. Morovr, whil h ffcs of highr longviy as a hra for h susainabiliy of public pnsions has bn xnsivly dbad, is rol in drmining whhr and how PAYG pnsions can affc conomic sabiliy has no bn so far invsigad. In his papr w show ha if, as commonly raind, a ris in longviy can hran h balancing of h PAYG budg, i can favour, howvr, h sabiliy of h conomy. Th aim of his papr, hrfor, is o provid a sabiliy analysis of h convnional gnral quilibrium OLG conomy wih PAYG pnsions, showing h condiions undr which conomic cycls can occur and also how h lar ar affcd by h siz of h pnsion sysm as wll as by ohr conomic variabls. In paricular, w focus on h PAYG dfind-conribuion schm, which is h mos largly invsigad in liraur and also sms o b incrasingly implmnd in By passing, w no ha Fani and Gori (008) showd ha in many cass a ris in longviy may favour h viabiliy of h PAYG plan in h long run. In ordr o prsrv solvncy, a govrnmn ha wishs o rdisribu across gnraions wih a PAYG pnsions nds o adjus ihr conribuion ras (dfind-bnfi schm) or pnsion promiss (dfind-conribuion schm).
h rcn yars. 3 W conribu som findings o h xising liraur on OLG conomis and social scuriy. Firs, w find ha h siz of h PAYG dfind-conribuion schm may b rsponsibl of h xisnc of ndognous cycls, adding anohr possibl xplanaion o h cyclical bhaviour in OLG modls. 4 Scond, w show ha counris wih a high capial shar (.g., Ialy, Japan and Spain) may suppor largr PAYG sysms han counris wih low capial shars (.g., Grmany and h US) wihou dsabilising h conomy. 5 Third, pnsionrs in counris wih a rlaivly high individual dgr of hrifinss as wll as wih long-livd agns can rciv a mor gnrous pnsion arrangmn han pnsionrs in counris whr individuals ar rlaivly impain and adul moraliy is much largr. 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). On h on hand, howvr, a low valu of h lasiciy of subsiuion in producion is rquird (in paricular, a valu wll blow uniy) in ordr o mak cycls possibl. 6 On h ohr hand, wih myopic forsigh, h sady sa quilibrium may b oscillaory and xhibi drminisic complx cycls (Michl and d la Croix, 000, d la Croix and Michl, 00; Fani and Spaaro, 008), bu only 3 As nod by Wagnr (003), svral counris rcnly swichd o a dfind-conribuion schm (.g., Swdn), or chos h fixd-conribuion opion whn nwly dsigning hir pnsion sysm in h la 990s, such as Lavia and Poland, or a las movd ino ha dircion (.g., Ialy and Grmany). 4 Wagnr (003) invsigas a PAYG dfind-rplacmn-raio schm (i.., pnsions as a givn fracion of wags during working ag) wih prfc forsigh, and shows ha changs in h rplacmn ra may xpos h conomy o priodic or unsabl dynamics, bcaus diffrn from a dfind-conribuion sysm h dfind-bnfi sysm is characrizd by a scond ordr diffrnc quaion which is inrinsically mor pron o gnra unsabl dynamics. Diffrn from Wagnr (003), howvr, in his papr w focusd on a dfind-conribuion PAYG sysm wih myopic forsighd individuals also o invsiga h ffcs of longviy on h sabiliy of h conomy. 5 For rcn simas of h capial shar in svral counris, s,.g., Jons (003) and Rodriguz and Orga (006). 6 Richlin (986) discussd h Lonif cas (wih no subsiuabiliy), whil Farmr (986) showd ha cycls occur only whn chnologis xhibi lowr facor subsiuabiliy han h Cobb-Douglas funcion and discussd h CES xampl. 3
whn h inr-mporal lasiciy of subsiuion in h uiliy funcion is highr han uniy (i.., highr han h inr-mporal lasiciy of subsiuion in h Cobb-Douglas uiliy funcion). Thrfor, alhough i is wll known ha OLG conomis wih myopic xpcaions and lasiciy of subsiuion in h producion and in h uiliy funcions rlaivly low and high, rspcivly, may show cyclical (and vn chaoic) dynamics, in his papr w show ha h siz of PAYG pnsions may play a crucial rol in drmining h sabiliy of h conomy vn in a doubl Cobb-Douglas conx and, in paricular, larg PAYG sysms may caus prsisn cycls. This is, o h bs of our knowldg, a novl xampl of h possibiliy of cyclical bhaviour in OLG modls. Th rmaindr of h papr is organisd. In Scion w prsn h modl. In Scion 3 w analys h dynamics of h conomy showing ha h siz h PAYG sysm may riggr conomic flucuaions showing also an xampl of chaoic dynamics. Scion 4 concluds.. Th modl.. Individuals Considr a wo-priod ovrlapping gnraions conomy (Diamond, 965) wih saionary populaion and idnical individuals. 7 In h firs priod of lif (working priod) young individuals blonging o gnraion ( N ) ar ndowd wih on uni of im supplid inlasically on h labour mark, whil rciving wag incom a h compiiv ra w. This incom is usd o consum, o sav and o suppor marial consumpion of h ldrly (hrough a public PAYG-basd pnsion sysm). In h scond priod of lif (rirmn priod) old-ag individuals ar rird and 7 Assuming a consan ra of populaion growh dos no alr any of h subsaniv conclusions of h modl and, hnc, i is no includd hr. 4
liv on h procds of hir savings ( s ) plus h xpcd inrs accrud a h ra r + as wll as on h xpcd pnsion bnfi, p +. Morovr, w suppos young individuals surviv o h firs priod wih (consan) probabiliy 0 < π < (i.., π is h probabiliy of dying afr on priod only). Th xisnc of a prfc annuiy mark implis old survivors will bnfi no only from hir own pas saving plus inrs, bu also from h saving plus inrs of hos who hav dcasd. Each young born a im mus choos how much o sav ou of wag incom so as o maximis a homohic and sparabl (lifim) uiliy funcion ( U ) dfind ovr young-agd and old-agd consumpion, and c, +, rspcivly, subjc o h firs and scond priod of lif budg c, consrains. Assuming logarihmic prfrncs, h rprsnaiv individual born a im solvs h following consraind maximisaion programm: subjc o max π +, (P) { s } U ln( c, ) + β ln( c, ) c, c + s w + r π ( θ ) +, + s + p +, whr 0 < θ < is h payroll ax paid by h young conribuors o financ pnsion arrangmns o h currn old-agd and 0 < β < is h subjciv discoun facor, ha is, β rprsns h dgr of individual (im)painc o consum ovr h lif cycl. Th highr β is h mor individuals ar pain and prfr o smooh consumpion ovr h rirmn priod. Maximisaion of (P) hus givs h following saving funcion: s ( θ ) w + π p + ( + )( + r + ). ().. Firms 5
As rgards h producion scor, w assum ha firms ar idnical and ac compiivly on h mark. Th (aggrga) consan rurns o scal chnology is Y AK L, whr Y, K and L ar oupu, capial and h im- labour inpu rspcivly, A > 0 rprsns a scal N 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 ha capial fully dprcias 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, () w ( ) Ak. (3).3. Govrnmn Th govrnmn rdisribus bwn gnraions hrough an unfundd PAYG social scuriy schm. Thrfor, in vry priod h bnfi rcivd by currn pnsionrs is nirly financd by currn workrs whos working incom is axd away a h consan ra 0 < θ <. Thrfor, h (pr workr) govrnmn pnsion budg a rads as π p θ, (4) w h lf-hand sid bing h social scuriy xpndiur and h righ-hand sid h ax rcip. Now, insring h on-priod-forward pnsion accouning rul Eq. (4) ino Eq. (), h saving ra is s ( θ ) w + θ w + + r + +. (5) From Eq. (5) i can radily b sn ha savings ar dividd in wo componns: (i) h priva saving componn (h firs rm on h righ-hand sid of Eq. 5), which dpnds xclusivly on h 6
marginal willingnss o sav ou of wag incom (undr h hypohsis of Cobb-Douglas uiliy), and (ii) h public pnsion componn (h scond rm on h righ-hand sid of Eq. 5), which dpnds on boh h xpcd pnsion bnfi and xpcd inrs ra. Noic ha h lowr adul moraliy and h highr h individual subjciv discoun facor, h lowr (highr) h rlaiv wigh of h public pnsion componn (priva saving componn) in savings..4. Equilibrium Givn h govrnmn budg Eq. (4) and knowing ha N + N (i.. saionary populaion hypohsis), mark-claring condiion in goods and capial marks is k + s, (6) ha is, h sock of capial insalld a im + is drmind by h amoun of rsourcs savd in a im. Combining Eqs. (5) and (6), quilibrium implis: k + ( θ ) w + θ w + + r + +. (7) Dpnding on whhr individuals ar prfc or myopic forsighd h dynamics of capial dramaically changs. Blow w show ha whil h sam uniqu posiiv sady-sa is prsrvd in boh cass (s Michl and d La Croix, 000), (i) h dynamics of capial wih raional xpcaions is always monoonic and convrgn o h sady-sa, and (ii) h dynamics of capial wih myopic xpcaions may b non-monoonic and divrgn o h sady-sa vn wih Cobb-Douglas uiliy and producion funcions. I is worh noing ha h non-monoonic bhaviour of h conomy rlis on (a) h siz of h (fixd) payroll ax paid by h young o financ pnsions o h old, and (b) h muual rlaionship bwn chnology and prfrnc paramrs..4.. Prfc forsigh 7
Wih prfc forsigh, boh h xpcd inrs and wag ras dpnd on h fuur valu of h sock of capial pr young, ha is + r + Ak + w + ( ) Ak +, (8) Thrfor, xploiing Eqs. (), (3), (7) and (8), h dynamic quilibrium squnc of capial can b wrin as k ( θ ) ( ) A ( + ) + θ ( ). (9) + k Sady-sa implis * k +, so ha: β ( θ ) ( ) A ( + ) + θ ( ) * π k. (0).4.. Myopic forsigh Wih myopic forsigh, boh h xpcd inrs and wag ras dpnd on h currn valu of h sock of capial pr prson, ha is + r + Ak w + ( ) Ak. () Using (), (3), (7) and (), h dynamic pah of capial accumulaion is now givn by: k + ( θ )( ) and h sady sa is sill drmind by Eq. (0). A θ k k, () + + 3. Local sabiliy wih myopic forsigh 8
In his scion w sudy h dynamics of h conomy wih PAYG pnsions and myopic xpcaions. Firs, w sudy h cas wih gnric uiliy and producion funcions. Scond, w analys h local sabiliy propris of h doubl Cobb-Douglas conomy. 8 Assuming gnric uiliy and producion funcions (as wll as h xisnc and uniqunss of h sady sa quilibrium), h mark-claring condiion Eq. (7) can b wrin as Sw ( k ), r + ( k ), P[ w + ( k ) r ( k )], (3) priva saving public pnsion k+, + whr h saving funcion S dpnds on boh h priva saving and public pnsion componns. Th priva saving componn is drmind by h individual s willingnss o sav (which, in urn, dpnds on boh h currn wag and h xpcd inrs ra), whil h (PAYG) public pnsion funcion P dpnds on h xpcd valus of boh h wag and inrs ras. 9 Thrfor, oally diffrniaing Eq. (3) wih rspc o k yilds + + + / + + dk + + + + S w S r S P w P r + + +, (4) dk w r + P + + w r + + / whr S / w > 0 rprsns h marginal propnsiy o sav ou of wag incom, S / r + capurs h ffcs of h xpcd ra of inrs wihin h priva componn of oal savings (boh dscrib how aggrga saving varis along wih h priva marginal willingnss o sav), 8 Th (local) sabiliy propris of a doubl Cobb-Douglas conomy wih PAYG pnsions and prfc forsigh is brifly prsnd in Appndix A. Diffrn from h cas wih myopia, an conomy wih prfc forsighd individuals dos no xhibi any inrsing dynamical faur. 9 Wih myopic xpcaions individuals will xpc h fuur valus of h wag and h inrs ra o b xacly h sam han hos prvailing in h currn priod. Thrfor, boh priva savings and h public pnsion funcion P dpnd xclusivly on h lvl of h sock of capial pr prson insalld in h currn priod. 9
and S / P < 0 rflcs h (ngaiv) public pnsion ffc on S, i.., h disincniv o sav causd by public social scuriy. In paricular, during h working priod h young know ha h govrnmn will provid a bnfi o suppor marial consumpion whn old, and his crowds ou priva savings. As rgards h ffcs of h priva saving componn on h accumulaion of capial, w no ha wih a gnric uiliy funcion h individual propnsiy o sav dpnds of cours on h inrs ra. As a consqunc, a rducion in h xpcd ra of inrs du o a marginal incras in h lvl of h fuur sock of capial posiivly (ngaivly) affcs aggrga savings dpnding on whhr h subsiuion ffc is dominad by (dominas) h incom ffc, i.., S / r < 0 ( S / r + > 0 ). Th subsiuion ffc dscribs h advanags or disadvanags o + subsiu consumpion bwn youh and oldnss, whil h incom ffc capurs h incrasd or rducd rvnu from savings, ohr hings bing qual (a lowr ra of inrs, hrfor, maks lss profiabl o subsiu consumpion bundls ovr h lif cycl and dcrass h dirc rvnu from savings). From Eq. (4) w s ha a marginal incras in h sock of capial in priod ambiguously affcs h valu of h capial sock insalld in h subsqun priod. Thrfor, boh monoonic and oscillaory dynamics ar possibl. In paricular, h final ffc is hrfold. As rgards h priva saving componn hr xis (i) a posiiv ffc ha incrass h currn wag, and hus h marginal propnsiy o sav ou of wag incom and h capial sock will b insalld in h fuur, and (ii) an ambiguous ffc of h rducd xpcd ra of inrs. Thrfor, (ii.) if h subsiuion ffc dominas h incom ffc (i.., S / r + > 0 ), hn a rducd xpcd inrs ra by dcrasing h rlaiv wigh of h priva saving componn nds o rduc h valu of h sock of capial insalld in h subsqun priod; (ii.) if h subsiuion ffc is dominad by h incom ffc (i.., S / r + < 0 ), hn a rducd xpcd inrs ra by incrasing h rlaiv wigh of h priva saving componn raiss fuur capial. Morovr, as rgards h public pnsion componn, a marginal incras in h currn sock 0
of capial causs (iii) a ngaiv ffc du o h incrasd rlaiv wigh of h public pnsion funcion P in oal savings, ha rducs, in urn, h sock of capial insalld in h fuur. In fac, h incrasd xpcd fuur marginal produciviy of labour (h wag ra ffc) as wll as h rducd xpcd marginal produciviy of capial (h inrs ra ffc) nd o nhanc h xpcd pnsion bnfi will b rcivd by currn pnsionrs, whil also rducing h priva marginal willingnss o sav. If h subsiuion ffc dominas h incom ffc, hn h posiiv ffc of h incrasd priva marginal willingnss o sav on h accumulaion of capial is counrwighd by h rducd inrs ra will prvail on h fuur (h priva saving componn) as wll as by h ngaiv public pnsion ffc (i.., h rlaiv wigh of h public pnsion funcion is highr). If h subsiuion ffc is dominad by h incom ffc, hn h highr marginal willingnss o sav ou of wag incom du o a ris k is rinforcd by h ffc playd by h lowr xpcd ra of inrs so ha his forc is counrwighd only by h ngaiv public pnsion ffc. Hnc, whn h subsiuion ffc dominas h incom ffc, oscillaory dynamics ar mor likly o occur han whn h subsiuion ffc is dominad by h incom ffc. If boh subsiuion and incom ffcs xacly cancl ou (i.., S / r + 0 Cobb-Douglas uiliy), hn h final ffc of a ris in h sock of capial a on h lvl of h capial sock insalld a + xclusivly dpnds on wo counrbalancing forcs: h posiiv priva saving ffc (ha conribus o incras oal savings du o h highr willingnss o sav whn young) and h ngaiv public pnsion ffc (which, insad, crowds ou savings du o h highr bnfi rcivd during h rirmn priod). Dfiniivly, in h cas of Cobb-Douglas uiliy h dynamics of capial wih myopic xpcaions is monoonic (oscillaory) if h priva saving componn dominas (is dominad by) h public pnsion componn. Blow w ypify h dynamics of a doubl Cobb-Douglas OLG conomy and showing ha h inroducion of a PAYG schm whn individuals ar myopic forsighd may caus ihr (convrgn) monoonic dynamics or (convrgn or divrgn) oscillaory dynamics. I is worh
noing ha h occurrnc of mporary or prmann oscillaions dpnds on h gnrosiy of h unfundd social scuriy sysm as wll as on h muual rlaionship bwn boh chnology and prfrnc paramrs. In paricular, whn h producion chnology is rlaivly labour-orind, hn a oo larg PAYG pnsions (ha is, a rlaivly high conribuion ra) may dsabilis h conomy. Morovr, h highr h ra of longviy is and h mor individuals prfr o smooh consumpion ovr h rirmn priod (a highr subjciv discoun facor), h lowr givn h incrasd rlaiv imporanc of h priva componn in oal saving h risk of cyclical insabiliy associad wih h PAYG schm is. Analysis of Eqs. (0) and () givs h following proposiion: Proposiion. In a doubl Cobb-Douglas conomy wih public PAYG pnsions and myopic xpcaions h dynamics of capial is h following. () L 0 < < 4 hold. Thn θ < θ <, and (.) if 0 < θ < θ, h dynamics of capial is monoonic and convrgn o (.) 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 ; () L 4 < < hold. Thn θ <, θ >, and (.) if 0 < θ < θ, h dynamics of capial is monoonic and convrgn o (.) 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 <, whr θ ( β, π ) θ : +, (5), ( ) ( ) ( + ) ( + ) ( ) + θ (, β, π ) θ : θ, (6) + + ( β, π ): > 0, (7) ( 3 + ) + + 0π β + 9 4 4( β, π ): > 0. (8) Proof. S Appndix B. From Proposiion h following rmark can b drivd. Rmark. Th monoonic dynamics of capial in a doubl Cobb-Douglas PAYG-basd conomy wih myopic xpcaions is always convrgn o h saionary sa (i.., h so-calld saddl nod bifurcaion can nvr occur, or, in ohr words, h conomy may loos sabiliy only hrough oscillaions). Proposiion shows ha whn labour is rlaivly imporan in producion, larg PAYG sysms may imply oscillaory movmns and also riggr h occurrnc of prsisn cycls. In paricular, wo diffrn hrshold valus of h conribuion ra xis ha discriminas bwn (i) monoonic and non-monoonic rgions, and (ii) sabl and unsabl movmns xclusivly wihin 3
h non-monoonic rgion. If h payroll ax paid by h young conribuors is lowr (highr) han h hrshold θ, hn h public pnsion componn in oal savings is dominad by (dominas) h priva componn and hus h dynamics pah of capial accumulaion xhibis monoonic (oscillaory) movmns. Rising furhr h conribuion ra, howvr, may dsabilis h conomy (his is du o h fac ha h rlaiv wigh of h public pnsion componn in oal savings is highr. This happns whn h flip bifurcaion valu of h payroll ax θ is xcdd). Th following Figur illusras in h spac (, θ ) and for a givn valu of h ra of longviy and h individual subjciv discoun facor, h locus which discriminas bwn monoonic and oscillaory rgions ( θ ) as wll as h priod-doubling flip bifurcaion locus ( θ ), which, insad, discriminas bwn sabl and unsabl rgions. Th figur clarly shows ha h highr is h disribuiv capial shar h lowr is h risk of cyclical insabiliy. Morovr, a ris in ihr h individual subjciv discoun facor or lif xpcancy, or boh, shifs upward boh loci and hus nds o sabilis h conomy shrinking h siz of h cyclically unsabl rgion. 0 0 W rcall ha < / 4 3 and < / hold for any π, β ( 0, ). For h sak of brviy, w do no display hr h loci s shifs upward following a ris ihr in h individual subjciv discoun facor or in lif xpcancy. 4
Figur. Sabiliy and insabiliy rgions in h spac (, θ ). Thrfor, counris wih a rlaivly high capial shar in producion as wll as wih pain individuals (ha ascrib a larg nough imporanc o h priva saving componn rahr hn o h public pnsion componn and hus prfr o smooh consumpion ovr h rirmn priod), may incras h siz of h conribuion ra paid by h young o fund h bnfis o rird popl wihou gnraing mporary or prmann oscillaions. In hs counris larg PAYG sysms do no vr consiu a pril for h conomic sabiliy. In conras, counris wih a rlaivly low disribuiv capial shar and wih a low dgr of hrifinss (i.., individuals ar impain and prfr o consum mor oday rahr han omorrow) ar much mor pron o conomic insabiliy whn h govrnmn riss h conribuion ra o financ PAYG pnsions. In h following proposiion w clarify h rol playd by boh h dgr of individual hrifinss and h ra of longviy on h sabiliy of h conomy. 5
Proposiion. L individuals b myopic forsighd. Thn a ris in ihr lif xpcancy or h dgr of individual hrifinss, or boh, nds o sabilis h conomy. Proof. Th proof can asily b drivd by diffrniaing Eqs. (5) and (6) wih rspc o π and β, ha is θ π β > ( ) 0, θ β π > ( ) 0, θ π ( + ) ( ) β > 0 and θ β ( + ) ( ) π > 0. Sinc h conomy is unsabl only hrough oscillaions, hn a ris in ihr lif xpcancy or in h dgr of individual hrifinss incrass h valu h conribuion ra byond which h dynamics of capial is displays oscillaory movmns as wll as h flip bifurcaion valu of h payroll ax. This causs a rducion in boh h widh of h oscillaory rgion and h widh of h cyclical unsabl rgion. Q.E.D. In Figurs and 3 w illusra Proposiion showing an xampl of h diffrn non-monoonic bhaviour whn adul moraliy is rlaivly high (Figur ) and low (Figur 3). As an xampl, w ak h following paramr valus for boh figurs: A 0, 0. 0, β 0. 50. Morovr, w choos π 0. 30 (Figur ) and π 0. 90 (Figur 3). This paramr ss gnra h following flip bifurcaion valus of h conribuion ra: θ 0. 43 (whn π 0. 30) and θ 0. 4 5437 (whn π 0.90 ). Thn w ak θ 0. 53. Th cobwb dpicd in Figur shows h non-monoonic divrgn dynamics of capial whn h ra of longviy is rlaivly low, whil h cobwb in Figur 3 displays h convrgn non-monoonic dynamics of capial whn h ra of longviy is rlaivly high. Th figurs clarly show ha a low adul moraliy acs as an conomic sabilisr. 6
Figur. Cas π 0. 30. A picorial viw of cyclical insabiliy ( k * 0., k 0. 0 08 ). Figur 3. Cas π 0. 90. A picorial viw of cyclical sabiliy ( k * 0. 393, k 0. 0 08 ). 7
This modl also shows drminisic chaos. Figur 4 dpics h bifurcaion diagram for h paramr θ (which lis on h horizonal axis). Th paramr valus ar h sam as in Figur. On h vrical axis w show h limi poins of h quilibrium squnc of capial. Whn h conribuion ra is rlaivly low (blow 0. 43 ) a uniqu limi poin xiss. Whn θ 0. 43 a priod doubling bifurcaion mrgs. Thn priod doubling bifurcaions appar mor rapidly afr θ 0.55. Finally, a furhr incras in h conribuion ra brings h conomy ino h chaoic rgion. This is an xampl of h complx dynamics gnrad by h larg PAYG sysms in an OLG Cobb-Douglas conomy. Figur 4. Bifurcaion diagram for θ. 4. Conclusions W analysd h dynamics of an OLG conomy wih PAYG (dfind-conribuion-ypd) pnsions and myopic forsighd individuals. 8
W showd ha counris wih a rlaivly capial-orind chnology and a rlaivly high dgr of parsimony (such as Ialy) may suppor largr public pnsion sysms wihou prils for h conomic sabiliy han counris whr chnology is rlaivly labour-orind and individuals hav a low dgr of parsimony (such as h US). Our rsuls hav a wofold inrpraion: (i) consiu a policy warning abou h risks of (cyclical) insabiliy causd by h widsprad and gnrous PAYG pnsion schms in prsnc of ralisic myopia of individuals, and (ii) hy rprsn a furhr xplanaion of conomic cycls in OLG conomis. Appndix A W brifly show hr ha wih prfc forsigh h dynamics of a PAYG-basd Cobb-Douglas OLG conomy canno b cyclical. Proposiion A.. In a Cobb-Douglas PAYG-basd conomy wih prfc forsighd individuals h dynamics of capial is always monoonic and convrgn o * k irrspcivly of h siz of h PAYG sysm. Proof. Diffrniaing (9) wih rspc o k and using (0) w find: + ( θ ) ( ) ( ) ( ) ( ) * + + θ. * k k A + Thrfor, 0 < * < k for any 0 < θ <. Q.E.D. Appndix B 9
Proof of Proposiion. Diffrniaing Eq. () in Scion.4.. wih rspc o k and valuaing i a h sady-sa givs: ( θ )( ) A * ( ) θ + + * k k k + Using Eq. (0) in Scion.4.., Eq. (B) bcoms θ + + * k k ( ). (B). (B) Monoonic and non-monoonic dynamics + > Th condiion * 0 k < implis ( ) 0 θ > < 0 θ θ, (B3) + < > whr θ θ (dfind by Eq. 5 in Scion 3.) rprsns h valu of h conribuion ra blow (byond) which h dynamics of capial is monoonic (oscillaory). In paricular, θ < ( θ > ) for any 0 < < ( < < ). θ < implis < <, whr + ( β, π ): < 0, (B4) + + ( β, π ): > 0. (B5) Sinc 0 i can b auomaically ruld ou. Noic also ha 0 < < / for any β and π. < + Now, * < givs k
( ) θ < θ > + + ( ). (B6) Thrfor, in h cas of monoonic bhaviour, h dynamics of capial is always convrgn o h + saionary sa, ha is, 0 < * <. k Sabiliy and insabiliy analysis in h cas of non-monoonic dynamics + > Th condiion * k < implis ( ) θ > < θ θ, (B7) + < > whr θ θ > θ (dfind by Eq. 6 in Scion 3.) rprsns h flip bifurcaion valu of h conribuion ra, ha is, h valu of θ blow (byond) which h quilibrium wih oscillaory dynamics is sabl (unsabl). In paricular, θ < ( θ > ) for any 0 < < 4 ( 4 < < ), wih 4 <. θ < implis 3 < < 4, whr ( 3 + ) + 0 + 9 3 3( β, π ): < 0, (B8) ( 3 + ) + + 0π β + 9 4 4( β, π ): > 0. (B9) Sinc 0 i can b auomaically ruld ou. Noic also ha 0 < </ 4 3 for any β and π. 3 < + Thrfor, if 0 < < 4 hn θ < θ <, and 0 < * < k for any 0 < θ < θ, + < * k < 0 for any θ < θ. < + for any θ < θ < θ, * k + if and only if θ θ, and * < k
If 4 < < hn θ <, θ >, and 0 + < * < k for any 0 < θ < θ and + < * k < 0 for any θ < θ. < + If < < hn θ > θ >, and 0 < * < k for any 0 < θ <. Q.E.D. Rfrncs d la Croix, D., and Michl, P., 00. A Thory of Economic Growh. Dynamics and Policy in Ovrlapping Gnraions. Cambridg Univrsiy Prss, Cambridg. Diamond, P., 965. Naional db in a noclassical growh modl, Amrican Economic Rviw 55, 6 50. Fani, L. Gori, L., 008. Longviy and PAYG pnsion sysms susainabiliy, Economics Bullin 0, 8. Fani, L., Spaaro, L., 008. Povry raps and inrgnraional ransfrs, Inrnaional Tax and Public Financ 5, 693 7. Farmr, R.E., 986. Dficis and cycls, Journal of Economic Thory 40, 77 86. Grandmon, J.M., 985. On ndognous businss cycls, Economrica 53, 995 045. Grozn, B. van, Lrs, T., Mijdam, L., 003, Social scuriy and ndognous friliy: Pnsions and child allowancs as Siams wins, Journal of Public Economics 87, 33 5.
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