Fujii, Takao; Hayashi, Fumiaki; Iri Author(s) Oguro, Kazumasa.

Transcription

1 TileDesigig a Opimal Public Pesio Fujii, Takao; Hayashi, Fumiaki; Ii Auho(s) Oguo, Kazumasa Ciaio Issue 3- Dae Type Techical Repo Tex Vesio publishe URL hp://hdl.hadle.e/86/54 Righ Hiosubashi Uivesiy Reposioy

2 Desigig a Opimal Public Pesio Sysem Takao Fujii Associae Pofesso, Kobe Uivesiy Fumiaki Hayashi Pos Gaduae Sudies, Kobe Uivesiy Ju Iiai Pofesso, Fukuyama Uivesiy Kazumasa Oguo Associae Pofesso, Hiosubashi Uivesiy This vesio: Jauay 7, 3 We would like o expess ou sicee gaiude fo he ivaluable advice eceived fom Pofesso Takashi Kamihigashi (Kobe Uivesiy) as well as Seio Ecoomis Ryo Ishida (Policy Reseach Isiue, Miisy of Fiace) duig he wiig of he fis daf of his pape.

3 Absac This pape uses a wo-peiod ovelappig geeaios model i ode o povide a heoeical desig fo a opimal public pesio sysem based o a paial equilibium aalysis. Household pefeeces oly deped o wo peiods cosumpio ad leisue ad is homogeeous of degee m wih espec o cosumpio i he wokig ad eied peiods. We pese chaaceisic feaues of a opimal public pesio sysem i his pape. Fis, diffeeces i he populaio gowh ae do o affec he elaive level of he opimal e lifeime bude ae of each geeaio. Secod, if m o m, he opimal public pesio sysem ca be expessed explicily. Thid, he diffeece bewee he make ime-pefeece ae ad social ime-pefeece ae povides a cucial isigh io he opimal bude ae of each geeaio. Keywods: Ovelappig geeaios model, public pesio, opimal bude ae JEL codes: D3, D6, D9, H, H6. Ioducio Alhough cosideable eseach exiss o public pesio plas, such as he sudies by Feldsei (995, 998) of he possibiliy fo social welfae impoveme based o a asiio pocess fom he pay-as-you-go sysem o he fuded sysem, mos wok i his aea heoeically ad empiically

4 examies he effeciveess ad faiess of exisig ad efomed plas. By coas, few sudies have bee caied ou o desigig a opimal pesio pla (bu see Oguo, 8). The pese pape hus uses a simplified ovelappig geeaios (OLG) model i ode o aalyze he chaaceisics of a opimal public pesio sysem ha maximizes he sum of idiec uiliies of each geeaio ude is ieempoal budge cosai. The peseed heoeical aalysis hus aims o bidge his gap i he cue body of kowledge o his opic. Lile eseach exiss specifically o opimal public pesio plas owig o aalyical difficulies. I paicula, he chaaceisics of he wo macoecoomic models applied o such eseach, amely he epeseaive household model ad OLG model, hide he aalysis of opimal axaio. Well-kow agumes i favo of he fome ae offeed by Bao (979, 999) ad Boh (99) o ax smoohig ad by Judd (999) egadig dyamic opimal axaio. Howeve, as he epeseaive household model does o usually ivolve geeaio aleaio, i is difficul o aalyze a public pesio pla (e.g., a pay-as-you-go sysem), which is a iegeeaioal icome disibuio policy. The OLG models poposed by Samuelso (958) ad Diamod (965) assume geeaio aleaio ad hus faciliae he cosideaio of public pesio plas. Howeve, ulike he epeseaive household model, he key ages i hese models icease by umbe of geeaios, fuhe complicaig he heoeical aalysis. Theefoe, mos eseach caied ou sice he semial 3

5 achievemes peseed by Auebach ad Kolikoff (987) eds o adop a muli-peiod OLG model fo he empiical aalysis i ode o adap he ifluece of cue pesio plas ad efoms o he uiliy of each geeaio. To avoid he summaio of idiec uiliies divegig o ifiiy i is ecessay fo he sequece of uiliies i his pape o be discoued usig he ime-pefeece ae. We desigae his ae he social ime-pefeece ae i ode o diffeeiae i fom he iees ae, which we em he make ime-pefeece ae. I he fis sep of desigig ou opimal pesio pla, we sugges hee simplificaios o he adiioal OLG model, which ca cope wih a ifiie umbe of households ad goods. Fis, we igoe he geeal equilibium ad isead adop a paial equilibium aalysis. Secod, alhough he pay-as-you-go pesio sysems i mos couies cosis of wo pas (e.g., he basic pesio ad he icome popoioal pesio), we ecogize oly icome popoioal paymes fo he sake of simpliciy. Thid, we use a ypical wo-peiod OLG model (e.g., wokig peiod ad eied peiod) ad assume ha each household s uiliy depeds o wo peiods of cosumpio ad labo. Thus, we assume ha he uiliy fucio is homogeeous of degee m wih espec o cosumpio i he wokig ad eied peiods. Assumig he fis simplificaio above allows us o se wage icome ad he iees ae as exogeous vaiables. By coas, wage icome ad he iees ae become a fucio of he pesio beefi ae ad pemium ae i he geeal equilibium model, edeig opimal pesio 4

6 pla chaaceisics exemely difficul o aalyze. By cosideig he secod simplificaio, he aalysis of he opimal pesio pla becomes he deemiaio of he e lifeime bude ae of each geeaio, as pesio beefis ad pemiums ae a fixed popoio of lifeime wage icome ude he lifeime budge cosai of households. Theefoe, he poposed opimal pesio pla desig esembles he discussios of Bao (979, 999), Boh (99), ad Judd (999). Fially, he hid simplificaio claifies he social welfae fucio, as show i Lemmas ad i Secio. The followig pois summaize he chaaceisics of he opimal pesio pla poposed i his pape: () The diffeeces i populaio gowh aes do o affec he elaive level of he e lifeime bude ae of each geeaio. () If m o m, (-) The opimal pesio sysem ca be expessed explicily. (-) The opimal pesio pla deceases he e lifeime bude ae of he geeaio wih he highe gowh ae of lifeime wage icome. (-3) If he social ime-pefeece ae is sufficiely lage ha he make ime-pefeece ae, he a icease i he populaio gowh ae of some geeaios would educe he e lifeime bude ae of each geeaio ude a opimal pesio sysem. (-4) If he social ime-pefeece ae is lage (smalle) ha ha of he make, he e lifeime 5

7 bude ae of fuue geeaios would be close o % (icease he cue geeaio s e lifeime bude, while educig he e lifeime bude of fuue geeaios). (-5) If he social ime-pefeece ae is equal o ha of he make, he e lifeime bude ae of fuue geeaios ude a opimal pesio pla would covege o he same level (i.e., he smoohig of he e lifeime bude of each geeaio holds a a ceai poi i ime). (3) If m, wih he possible excepio of oe geeaio, he e lifeime bude of each geeaio would become eihe % o %. Poi () is cucial fo he pesio pla desig peseed heei. Alhough Japa s decliig bihae implies a iceasig pesio bude o fuue geeaios fom a coveioal viewpoi, () suggess ha whe compaig he e lifeime bude of oe geeaio o ha of aohe, he elaive level of he e bude should emai uaffeced by a chage i he populaio gowh ae ude a opimal pesio sysem. By coas, as descibed i (-3), he populaio gowh ae affecs he e lifeime bude ae fo all geeaios. Fuhe, (-) idicaes ha a egessive pla is desiable fo he e lifeime bude of each geeaio ude a opimal pesio sysem. This sudy s aalysis is based o he assumpio ha each geeaio behaves selfishly wihou displayig dyasic aluism. Bao (974) agues ha whe hee exis iheiace asfes amog geeaios ad iegeeaioal aluism, he uiliy of each geeaio is iflueced o oly by is cosumpio bu also by he descedas uiliy, meaig ha he OLG model is fudameally 6

8 equivale o he epeseaive household model. Howeve, Takayama, Aso, Miyaji, ad Kamiya (996) ad Hoioka () show ha iegeeaioal aluism is aely evide i he aalysis of iheiace disibuio. The emaide of his pape is ogaized as follows. Secio ioduces he model. Secio 3 defies he opimal pesio pla as he maximizaio of social welfae. Secio 4 aalyzes he elaioship bewee he fuded sysem ad he e lifeime bude ae of each geeaio i ode o demosae ha he agume fo he opimal pla ude he pay-as-you-go sysem ca be fully applied o ha ude he fuded sysem. Secio 5 cocludes.. Cosume Choices ad Goveme Budge. Cosume Decisios Each geeaio lives fo wo peiods (i.e., wokig peiod ad eied peiod). The uiliy fo he geeaio bo i peiod is as follows: 3 u( c,, c,, h ), u : R R () Hee, ad c, epese he cosumpio a peiod ad a +, especively, while h c, epeses leisue. I addiio, we assume ha he uiliy fucio saisfies Assumpio : Assumpio : u is coiuous i he domai. Fo ieio pois, u is a sicly iceasig ad 7

9 sicly quasi-cocave fucio, ad homogeeous of degee m wih cosumpio levels (,, c, c ). Excep fo he assumpio of homogeeous of degee m wih he cosumpios above, he uiliy fucio is saisfied wih he omal desiable chaaceisics. We also assume m. If m, he cocave fucio ( c, c, h, ) i he Cobb Douglas fom ca be icluded i he same uiliy fucio class. The symbol epeses he pesio pemium ae fo wage icome ( w ) ad epeses he pesio beefi ae, expessig he pesio o be eceived i muliples of he pemium ae ( ). The uppe limi of leisue povided duig he wokig peiod is assumed o be. Fuhe, by epeseig he iees ae as, he budge cosai of geeaio will be he followig: c, s ( )w ( h ); c, ( )s w ( h ), whee s is he savig. These ca be cosolidaed io a sigle budge as: c, c, ( ) w h ( )w () Hee, epeses he e lifeime bude ae of geeaio as follows: This ca be ewie as ( )( ). Ude he codiio ha, w, ad ae give, each geeaio maximizes is uiliy hough he selecios of c,, c,, ad h. The, if wie 8

10 wihou he ime idex, his becomes he followig: max u(c,c,h), subjec o c c ()wh ()w (3) We deoe he soluio of (3) as c (( ) w, ), c(( ) w, ), ad h(( ) w, ). Fuhe, i is obvious ha he followig codiio holds: c, (()w,) c,' (()w,),' c, (()w,) c,' (()w,) h (()w,) h ' (()w,) (4) This meas ha he fucioal fom is he same whe ime is also expessed as c, (()w,), c, (()w,), ad h (()w,). The papes of Oguo (8) ad Oguo, Nakakaumai, ad Takama (7) show ha whe m, he fucio h is o depede o w ad ha he fucios c, ad c, ae sepaable fom w. I he same mae, we also show ha a simila lemma also holds whe m. As show below, his simplifies he fomula of he social welfae fucio. Lemma : If he uiliy fucio saisfies Assumpio, he soluio h becomes a fucio of he iees ae ( ) aloe ad he soluios c, ad c, ae sepaable fom ( )w. I ohe wods, he followig holds: h (()w,) h (,) c, (()w,) ()w c,' (,) c, (()w,) ()w c, (,) (5) 9

11 Poof: I ode o solve he poblem of (3), we cosideed he followig poblem: max u(,,h), subjec o,,h h (6) Le us deoe soluios o he poblem (3) ad o (6) as (c *,c *,h * ) ad (ˆ, ˆ, h ˆ ), especively. As he pai (c * /()w,c * /()w,h * ) saisfies he budge i (6), he elaio u(ˆ, ˆ, h ˆ ) u(c * /()w,c * /()w,h * ) u(c *,c *,h * ) / ()w m holds. Covesely, as he pai (()w ˆ,()w ˆ, ˆ h ) saisfies he budge i (3), u(c *,c *,h * ) u(()w ˆ,()w ˆ, h ˆ ) u(ˆ, ˆ, h ˆ ) ()w m holds ad, heefoe, we ca obai u(c *,c *,h * ) u(()w ˆ,()w ˆ, ˆ h ). Sice he uiliy fucio is sicly covex, he soluio o maximizaio is uique. Theefoe, he followig holds: c * ()w ˆ, c * ()w ˆ, h * ˆ h As (6) does o iclude ()w as a paamee, we ca cofim he followig: c i (()w,) c * i ()w ˆ i ()wc i (,), i=, h(()w,) h * ˆ h h(,) The above esablishes he poof fo (5). Expessig uiliy as idiec uiliy ad usig Lemma, we obai he followig: V (,) u(c, (( )w,),c, (( )w,), h (( )w,)) ( )w m whee u(c (,),c (,), h(,)) (7)

12 Whe m, idiec uiliy is a sicly iceasig fucio of ad becomes a sicly cocave fucio. I addiio, i a eviome ha egads as fixed, i becomes a fucio of disposable icome ( ( ) w ) aloe.. Goveme Budge A ecoomy sas a peiod =, whe he wokig ad eied geeaios aleady exis. I peiod =, he goveme deemies icome asfe G o he eied geeaio, which is fuded by ax eveue ad he issuace of goveme bods ( B ). A he same ime, he goveme esablishes a pay-as-you-go pesio sysem wih ad (=,,, ). epeses he pesio pemium ae fo he wokig geeaio i peiod. is he pesio beefi ae ha he wokig geeaio eceives upo eiig expessed as a peceage of he pesio pemium paid by his geeaio. The amou of goveme bods issued is a edogeous vaiable ha is deemied i ode o saisfy he goveme budge cosai. By coas, a pesio pla wih ad (=,,, ) is assumed o be give. Thus, deoig he populaio of geeaio as L, he goveme budge cosai is he followig a peiod = : G w ( h )L B (8) This ca be cosideed o be he defiiio of B. Whe, he goveme budge cosai is: w ( h )L ( )B w ( h )L B (9)

13 Each B is edogeously deemied so ha equaios (8) ad (9) hold. By defiig such edogeous valuables, we ca use ad as he policy paamees. Assumpio : Ude he codiio ha he iees ae ( ), wage icome gowh ae ( g ), ad populaio gowh ae ( ) i peiod, as well as he iiial values ( w ad L ), ae give, wage icome i peiod ad he populaio of geeaio ae epeseed as follows: w w ( g j ), j L L ( j ). j I addiio, hee exiss a posiive eal umbe ( q ) such ha q, ad he followig codiio exiss: q( ) ( g )( ) fo =,, 3, Fis, his assumpio sigifies he followig. If ) ( g )( ) holds a ay give ime, he ( aioal icome gowh ae ( g ) is geae ha he iees ae ( ) o goveme bods. Thus, adopig a pay-as-you-go pesio sysem iceases he uiliy of he cue geeaio wihou deceasig ha of he fuue geeaio. I addiio, he discoued value of aioal icome i he fuue peiod will divege o ifiiy. I his case, i is uecessay o coside he bude o he pesio pla; by defiiio, his ype of ecoomy does o equie he caeful cosideaio of a opimal public pesio sysem.

14 Nex, w is assumed o covege o a posiive value: lim w w. This assumpio is o cosideed o be a sic codiio, because he pice of goods i ou model is omalized o be uiy. Noe: The followig elaioship holds: if ad oly if /( ), fo =,, 3,. We mus coside he case /( ) whe. Based o he above cosideaios, he followig lemma is achieved. I his lemma, h is he demad fucio of leisue ( h (( )w,)). Lemma : Suppose ha he pesio pemium ae (, =,, 3, ) saisfies budges (8) ad (9), ad ha he No Pozi Game codiio holds as follows: B lim () ( ) The, if he goveme budge cosais (8) ad (9) hold, he followig also holds: I G () ( ) whee ad I w ( h )L (=,, 3, ). Covesely, if hee ae values (, =,, 3, ) ha saisfy (), he hee exis ad (=,, 3, ) such ha (8) ad (9) ae saisfied, ad he No Pozi Game codiio () holds. B 3

15 4 Poof: (Necessiy) By ewiig (9), we obai he followig: ( )w ( h )L B w ( h )L /( ) B /( ) The, eogaizig by wiig he oal wage icome of geeaio as I w ( h )L, we obai he followig: I I B I /( ) B /( ) Fom he above equaios wih =,, 3,, ad equaio (8), we obai he followig: ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( B I B I I B I B I I B I B I I B I B I I B I G () Addig he above esuls i: I ( ) G I ( ) B ( ) I addiio, he followig also holds: I ( ) w L ( ) w L ( g )( ) ( ) Theefoe, by Assumpio we kow ) )/( )( ( q g. This implies ha he above em coveges o zeo as eds o ifiiy. Fuhemoe, by usig he No Pozi Game codiio (), he goveme budge cosai ca be summaized as:

16 I ( ) G (Sufficiecy) The iiial icome asfe G is give ad we se (=,, 3, ) o saisfy equaio (). If, we ca saisfy /( ) by usig ad. If, we ca defie so as o saisfy ad /( ), by choosig a sufficiely small posiive value fo. Thus, we ca obai ad fo each peiod ha saisfies /( ), fom he give value of. I his case, we defie B as follows: B G I B ( )I ( ) I ( )B, =,, 3, The, we obai he followig: G I B I B B I I, =,, 3, Theefoe, he followig also holds: I ( ) G I ( ) B ( ) The above equaio saisfies he goveme budge, icludig bods, fo each peiod. The, because I /( ) holds if ad saisfies I /( ) G, he No Pozi Game codiio is saisfied as follows: 5

17 B lim ( ) This lemma does o ecessaily guaaee ha he se of, which saisfies (), is o-ull. Theefoe, we popose he followig hypohesis: Assumpio 3: I /( ) G holds. By coas, as show below, he seies I /( ) is fiie. 3. Maximizaio of Social Welfae By usig he idiec uiliy of each geeaio, we defie he social welfae fucio as follows: W(,,, ) (, ) V, whee L ad L ( j ) j, =,, 3, ( R) Hee, R epeses he social ime-pefeece ae. Applyig he esuls fom (7) fom Secio, we obai he followig:,,, ) ( ) W( w, m u(c (,),c (,), h(,)) I addiio, I w ( h (( )w,))l w ( h(,))l, =,, 3, allows us o egad I as a exogeous vaiable. Based o his seig, we ca coside he 6

18 followig poblem: max W(,,, ),,, subjec o I ( ) G which addesses a opimal public pesio sysem. By usig I /( ) (=,, 3, ), he above poblem is ewie as follows: max,,, ( ) w m subjec o G (3) Fo his poblem, we allow. Assumpio 4: Thee exiss he soluio * i (3). This assumpio equies R o be a appopiaely posiive value. We aalyze he case wih m ad m i Secio 3. ad he case wih m i Secios 3. ad The case wih m ad m Fis, V(,) / m( ) m w m holds. As lim V (,) /, hee ae o s saisfyig *. Theefoe, he soluio i (3) is a ieio soluio ad mus saisfy he followig by I he case of m, he uiliy value will be egaive because of he moooic ad homogeeous chaaceisics of he uiliy fucio. Theefoe, we mus keep ad m i mid. A uiliy fucio of m, fo example, () (is) / v C s, s /v 7 ca be used ofe i a pesio simulaio. Whe m, i holds ha ad m.

19 deoig he Lagage muliplie as : mw m, =,, 3, ( )w By cacelig ou ad, we obai he followig: 3 w m ( )w w ( )w, m * ( g * ) R This esuls i * * if ad oly if * * R ' /(m), =,, 3, ( g ) R /(m) (4) I geeal, a compaiso of wo pois i ime ad ' ( ' ), esuls i he followig: (')/(m) ' ( g j ) (5) j These esuls sigify he followig coceig a compaiso of geeaioal budes. Theoem Whe m ad m hold, he followig chaaceisics hold i a compaiso of opimal geeaioal budes: (i) Assume ha hee is o chage i he wage gowh ae. If R is lage ha ( is lage ha R ), he lage he value, a highe (lowe) e lifeime bude ae of he elaive fuue geeaio (geeaio ' ), i compaiso o he e lifeime bude ae of he pas geeaio 3 Aeio mus be paid o he followig equaliies as well: w, w g ( g )( ), R 8

20 (geeaio ), is desiable. (ii) Assume ha he iees ae is equal o he social ime-pefeece ae. A elaively low e lifeime bude ae of he geeaio wih he highe wage icome icease ae is desiable. (iii) The elaive level of he e lifeime bude ae of each geeaio is o iflueced by he populaio gowh ae. The sigificace of (iii) is ciical, as i asses ha a chage i he populaio gowh ae should o affec he elaive bude of he public pesio sysem compaed wih he e lifeime budes fo each geeaio. I addiio, (ii) agues ha he bude o he geeaio ha has a low wage icome icease ae should be iceased ad affims ha a egessive sysem fo pesio bude is desiable. Two iems eed o be oed coceig Theoem. Fis, i idicaes ha populaio gowh does o ifluece he elaive value of opimal bude aes. I ohe wods, i implies ha populaio gowh may ifluece he absolue value of he bude. Secod, Theoem idicaes ha (i) holds if hee exiss a opimal soluio; heefoe, i cao be deemied a his poi whehe he iequaliy R iself is cosise o o wih a cofiguaio of he poblem. 9

21 The elaive level of he e lifeime bude ae does o always claify he degee of ifluece fom he chage i he populaio gowh ae o explai wha he esuls of (i) eveal abou opimizaio. Le us hus pusue a explici fom of soluio i ode o claify hese facos. We kow ha (5) holds fo =,, 3,. The we obai he followig: * ( * ), - j ( g j ) R /(m) w w R /(m) Hee,. Usig he cosais of a opimized pesio pla, we defie α as follows: I his case, we have G ( * ) ( * ) ad, heefoe, we have he explici soluio of he opimal bude: * * j j G j j G A his poi, we assume ha values fo ad coside he exisece of ad. By seig (6), =,,, 3, (7) exis. Subsecios 3.. ad 3.. x (( ) /( R)) j /(m ), we ca obai: I i I i i i j j x i i j ( g j)( j) j x i j j i i Compaed wih he coespodig eies, we fid: i ( g j )( j )

22 j j x j ( g j )( j ) j j x ( g j ) j We assume fo each = i +, i +, / x ( g j ) j ( g ) Hee, followig: g epeses he aveage gowh ae of yeas of eal wages. I his case, we obai he i i Specifically, assume ha x is o lage ad ha he gowh ae of wage icome fo geeaio i + ad lae is high. The, a icease i elages i moe ha i elages, which booss ( G) /. Accodigly, he bude ae is compehesively educed. The followig heoem summaizes hese esuls: Theoem Whe m ad m hold, he followig chaaceisics hold fo he e lifeime bude ae of each geeaio. (iv) The opimal e lifeime bude aes of he public pesio sysem fo each geeaio ae expessed i equaios (6) ad (7). (v) Le x (( ) /( R)) /(m) ad g i be he aveage eal wage gowh ae fom peiod o geeaio i. If x g i j, j =,, 3, hold, he gowh ae icease fo geeaio i will educe

23 each e lifeime bude ae. Theoems ad allow us o pefom a elaive geeaioal compaiso of a opimal pesio sysem wih he chaaceisics of he e lifeime bude ae. Nex, we shall examie movemes of he opimal pesio sysem hough ime. 3.. Value of j j Hee, we coside he value of j j which appeas i equaios (6) ad (7). Accodig o Assumpio, we have j I I w L j I w L ( q q ) ( ) q ad hus we kow j j G is a fiie umbe. 3.. Values of ad As he soluio * saisfies budge (), j G lim( * ) j j j j * * is valid ad holds. Suppose. The igh-had side of he equaio is zeo. As he * lef-had side is a posiive value, his is a coadicio. Accodigly, mus be valid. I addiio, whe, he igh-had side becomes a ifiie value ad coadics

24 Assumpio. Thus, mus also be valid. Nex, we cofim he codiios ha validae I R /(m) j. To begi wih, j Now, puig x (( ) /( R)) /(m), a, a ( j ) /( ), =,, 3, we ca j ewie he above equaliy as follows: a I a x a x The igh-had side of his equaio is a powe seies. Accodig o Cauchy Hadamad s heoem, he covegece adius k of his powe seies ca be expessed as: 4 k limsup a / I addiio, he iequaliy a / ( ) ( ( ) ) / q holds fo a sufficiely lage. 5 Theefoe, k /q(). Two cases ca be disiguished as follows: (A) if x k, he. (B) if x k, he. The possibiliy of, which was excluded fom Theoem. This idicaes ha (A) above mus also be elimiaed. Fuhemoe, we have 4 See Yoshida (965, p. 5). 5 We ae gaeful o Ryo Ishida fo poiig ou ha he sufficiely lage esicio is esseial. 3

25 I ( a ( w / w ) x a x a x ) Assumig w coveges o w, ad usig he esuls fom (B), we have hee diffeig cases of paamees ha deemie he empoal value of a opimal pesio, (case ) If k x, he / (case ) If x,he / w( h) a j j (case 3) If x, he / By expessig he aveage value of peiod as fo he chage i he populaio gowh ae fom peiod o, we obai he followig: (( ) ( )) /, =,, 3, If, =,, coveges o k q, we obai he followig: 3..3 Value of * The bude aes fo each geeaio i (case ) o (case 3) above ae descibed below. is he make iees ae, which hus expesses he make ime-pefeece ae. Accodigly, x is he aio of he make o social ime-pefeece aes. Le us coside (case ) k x. Hee, he followig apply: 4

26 , * :, *, lim * Sice x m ( ) /( R), x idicaes R. A his poi, he social ime-pefeece ae is lowe ha he make ime-pefeece ae ad he make may be moe sho-sighed ha sociey. I his case, he esuls show ha i is pefeable o place a lage bude o he cue geeaio ad educe he bude o fuue geeaios by offeig subsidies. Nex, le us coside (case ) x. This case implies he make ad social ime-pefeece aes coicide ad ha sociey uss he make. The bude ae * i i j ( j )( g j ) G /I ( ) ( j ) i i j ( ) * fo each geeaio is as follows: * j ( g j ) i i j ( j )( g j ) G /I ( ) ( j ) i i j ( ) Based o his, accodig o he cases g, g, o g, he followig * *, * *, * * mus hold especively. Fuhemoe, * j ( g j ) i i j * * coveges o a posiive value. ( j )( g j ) G /I ( ) ( j ) i i j ( ) 5

27 Fially, le us coside (case 3) x. This idicaes ha he social ime-pefeece ae is highe ha ha of he make ad ha sociey is moe sho-sighed ha he make. The bude ae * fo each geeaio is as follows. A his poi, * as ad i is hus pefeable o place he bude o fuue geeaios. Theoem 3 Whe m ad m hold, he followig chaaceisics hold fo he ime seies value of he opimal bude ae fo each geeaio. (vi) Whe he make ime-pefeece ae exceeds he social ime-pefeece ae, a egaive value is pefeable fo he e lifeime bude ae afe a ceai geeaio i ode o icease fuue cosumpio. (vii) Whe he make ad social ime-pefeece aes mach, he opimal bude aes will covege a a ceai value. (viii) Whe he make ime-pefeece ae falls below he social ime-pefeece ae, a ealy % bude ae is pefeable fo fuue geeaios. Of he above esuls, (vii) ca be cosideed o exed he ax smoohig heoems poposed by Bao (979, 999) ad Boh (99). Fis, hei eseach assumes R, which is equivale o he 6

28 assumpio i (vii) of x =. Secod, G is assiged o he iiial peiod i he cue model, which is compleely plausible i his discussio eve whe G is cosideed o be he oal public expediue of each peiod (discoued value). Fially, if he limi ae of bude j is a ( j G) w( h) a j posiive value ad ime is sufficiely lage, he value of lies i a small posiive ieval, * sigifyig bude smoohig. 3. The case wih m Uil his subsecio, he cosai poposed i equaio (3) was, bu hee we pesume a seies of fo. Fis, le us coside he case i which m. Hee, we defie he followig: b m w, =,, Le, =,,, 3, be bude aes saisfyig cosai (3). We also assume ad ' fo ad, especively. Thus, wihou ay loss of geealiy, we ca assume ' b ( ) m b ' ( ' ) m ( ) holds. A his ime, we assume a seies of sufficiely small posiive umbes fo, based o he followig: j j ' ' ' if j, (8) This opeaio is called ε-covesio. The, because 7

29 ' ' ' ' ' ' ',, j=,,, 3, saisfies he cosai i (3). Accodig o Taylo s heoem, some fom of exiss, esulig i he followig: m m / ) ( ) ゥ b ゥ ゥ ~ ~ ( W(,, ) W(,, )) m ゥ{ b ( Because we aleady have m he he elaio } b ' ( ' ) m b ' ( ' ) m b ( ) m b ( ' / ) m holds fo a sufficiely small. Theefoe, sice holds i his case, we also obai ~ ~ W (,, ) W(,, ) The soluio i (3), *, =,,, 3,, mus saisfy 6 : #{i * i } This idicaes ha, wih oe possible excepio, he opimal e lifeime bude aes ae eihe o, implyig ha he soluio is a ed-poi soluio. * Theoem 4 Whe m holds, geeaioal opimal bude aes, =,,,, wih he possible excepio of oe, ae eihe o. 3.3 The case wih m 6 Le A be a se. The symbol #A epeses he umbe of elemes i he se A. 8

30 Nex, we pesume a seies of fo. Defie b w, =,,, Below, we sudy his case fom wo pespecives: R ad R Whe R The elaio w L ( j ) j w (R ) ( g j ) j I /( ) R h holds. I ohe wods, b w /, =,,, is sicly moooic. Though moooic iceases (deceases), he esul diveges o ifiiy (coveges o zeo). I addiio, he fac ( )( g )/( ) q assumed i Assumpio leads us o (l )( g ) Accodigly,, =,, is sicly deceasig. A he same ime, lim. Lemma 3 Le ; =,,, be a seies of bude aes ha saisfy he budge G. If he elaio b w b ' 'w ',, ' ' holds fo wo disic peiods ad, he welfae will icease as he of -covesio is iceased uil eihe o ' holds. 9

31 Poof Assumig ; =,,, saisfies he budge G, ad ca be expessed by b w b ' 'w ' ',, ' Because b ', we defie k ; k =,,, accodig oε-covesio. These saisfy he budge i he same mae as peviously explaied. We also obai k ; k =,,, w (- ) ' w ' ( - ' ) - { w (- ) ' w ' ( - ')} w ( - ) ' w ' ( ' - ' ) ' (b - b ' ) Theefoe, welfae will icease as iceases uil eihe o ' holds. Accodig o Lemma 3, i ode o maximize welfae, he age of bude aes o be sudied should be limied o geeaios ha have a bude ae of eihe o, wih oe possible excepio. Accodig o Assumpio 3, fo ay aual umbe, G holds. The ses of aual umbes U ad Z ogehe wih umbe, ae defied below. Fis, we look a b ; =,,, ha is moooically iceasig: I N G I φ is self-evide. Le he fis eleme of Se I be 3. U ad Z ae defied as follows:

32 U {,,, }, Z {,, }. Nex, we examie b ; =,,, 3, ha is moooically deceasig: J N G As I φ, we deoe he las eleme of se J as. The las eleme exiss sice a sequece, s, =,,,, is moooic ad covegig o zeo ad sice s G. Z ad U ae defied as Z {,,..., }ad U {,,...}, especively. The followig he holds fo he U ad Z defied above: if U ad ' Z he b b ' (9) if U \ {}, he b b () Lemma 4 Usig he U, Z, ad obaied above, we defie ˆ j, if j Z, G h U \ {} h if, if j j U \ { } j This is he soluio fo equaio (3). Poof fo Lemma 4 Sep ( ˆ j ) j saisfies he budge cosai. 3

33 Based o he defiiios of U, Z, ad, j G j U j j U \ { } holds. Theefoe, he defiiio of ˆ esuls i: G j ˆ ^ j U \ {}, Accodigly, ( ˆ ) j saisfies he budge cosai. Sep Assume he soluio o welfae maximizaio is * ad =,,, 3,. If P { * }, he eihe P U o U P holds. Fom Lemma 3, we kow ha ( * j ) j has a value of o, wih oe possible excepio. We assume ha ad ' N exis ad ha U \ P ad ' P \ U ae valid. Hee, sice U ad ' Z, we obai he followig usig equaio (9): b ' b, * ', * ' Welfae ca be iceased by focusig o ad, ad ceaig k * ; k =,,, o k * ; k =,,, accodig oε-covesio. This esuls i a coadicio. Accodigly, he followig mus be valid: P U o U P Sep 3 P U holds. 3

34 If U = P he P U holds. The, i suffices fo us o examie he case U P ad U P. I is clea ha ˆ U \ {} ˆ G * U P () A ceai U exiss ad saisfies ˆ *. Whe ' P \ Uholds, he U ad ' Z, ad hus we obai he followig usig equaio (9): b ' b ad * ad * ' Accodig o Lemma 3, his coadics opimizaio. Sep 4 P = U Suppose P U. The case P U \ { } leads us o * G. This is also a coadicio. P The P. Fuhemoe, if P U \{ } ad U hold fo some, we obai * ad * ˆ. Whe cosideig, b b, * ad * is cofimed usig equaio (). Accodig o Lemma 3, his coadics opimizaio. These seps idicae * ˆ, =,,, Whe = R The followig holds: 33

35 w b; =,, 3,, whee b =/(-h). Le, =,,, be abiay bude aes saisfyig he cosai he followig esul:,, ) w ( ) b ( ) W ( b b b bg G. We obai This implies social welfae eaches a cosa level. Accodigly, egadless of he bude ae used, social welfae does o chage. Fially, we summaize he above esuls io oe heoem. Theoem 5 If m, he opimal e lifeime bude ae shows he followig chaaceisics. Whe he social ime-pefeece ae ad iees ae mach ( R ), ay esulig bude ae is opimal. Whe he social ime-pefeece ae ad iees ae diffe ( R ), i is desiable o saisfy he budge by assigig a bude ae of zeo o he geeaio ha has he highes cue icome value w / compaed wih is coibuio o welfae (whe he bude ae is zeo) ad a bude ae of % o he geeaio ha has he lowes such icome, while sequeially iceasig he ae. 4. Applicaio o a Fuded Sysem Uil his secio, we have focused o he pay-as-you-go appoach ha is used o maage he 34

36 Japaese pesio pla sysem. I his secio, we coside he cosequeces of ioducig a fuded sysem io he cue sucue. Fis, he household budge cosai is depiced as follows: c s ( ) w ( h ), c ( )s ( d ) w ( h ) Alhough pesio asses ae epeseed by w ( h ), his sysem oly allows a pesio o be eceived fom he picipal ad iees afe a ceai bude amou d w ( h ) has bee deduced. These vaious facos ca be summaized i oe budge usig he followig expessio: c c d w h d w Defie d /( ). The paamee is a subsaial e lifeime bude ae of he public pesio. Fo a give, w, ad, a geeaio maximizes is uiliy by selecig c, c, ad h. I ohe wods, by doppig he idex, he expessio becomes max u(c,c,h), subjec o c c w h w. Thus, his maximizaio poblem is simila o poblem (3) i he pay-as-you-go pesio sysem. The diffeece bewee he pay-as-you-go ad fuded sysems lies i ha of goveme budge. The goveme budge i fuded sysem ca be fiaced by wo agecies, amely goveme bods agecy ad pesio fuds agecy. The pocess of he fome woks as follows: 35

37 G B ( )B B d w ( h )L, =,,3, Fis, i peiod goveme bods B ae issued ad he amou of G is asfeed o he pesio agecy. Fom his poi owads, ew bods ae issued up o he amou B ad he pesio agecy uses he eus o hese fuds d w ( h )L as icome o pay he picipal ad iees fom he pevious bod issuace ( )B. O he ohe had, he pesio agecy s budge is show i he followig expessio, P A G w ( h )L ( ) w ( h )L A ( )A w ( h )L, =,, I peiod, we assume ha he goveme pays pesio beefis P ad fuds A usig G fom he goveme bod fiacig pogam ad he eveue deived fom pesio pemiums w( h ). I subseque peiods, he pesio agecy icus expediue i he fom of pesio beefis ad ivesme pofis, which i mus epay o he goveme, as well as asse eus ( d ) w ( h ) d w ( h ) A, which i pays fom he asse ivesmes ad pesio pemiums ( )A w ( h ). Combiig hese budges io a sigle udiffeeiaed sum, we have he followig: P A B w ( h )L 36

38 ( d ) w ( h )L A ( )B We ca ewie he expessio o show he e deb as P ˆ B w ( h )L ( )A w ( h )L B =,, ˆ B B A, =,,,, which esuls i ( d ) w ( h )L ( ) ˆ B w ( h )L ( ) ˆ B ( )w ( h - )L - ˆ B - w ( h )L B ˆ Defie I w ( h ) L, we aive a P ˆ B I I ˆ ˆ B I I B I I I Bˆ Bˆ ( ) ( ) ˆ I I I B ( ) ( ) ( ) ( ) Bˆ ( ) which fially esuls i I ( ) P I ( ) B ˆ ( ) This is he same equaio as obaied i he poof Theoem. Ad hus we kow ha he same esul as i Theoem holds i he fuded sysem. Accodigly, he agume pu fowad ealie ca also be applied o a fuded sysem. 5. Coclusio 37

39 This pape used a wo-peiod OLG model i ode o pu fowad a heoeical desig fo a opimal public pesio sysem based o a paial equilibium aalysis. Household pefeeces oly depeded o cosumpio ad leisue. The followig fidigs sugges ha a opimal public pesio sysem shaes simila chaaceisics if he uiliy fucio of households is homogeeous of degee m wih espec o cosumpio i he wokig ad eied peiods. Fis, diffeeces i he populaio gowh ae do o affec he elaive level of he opimal e lifeime bude ae of each geeaio. Secod, if m o m, he opimal public pesio sysem ca be expessed explicily. Thus, he diffeece bewee he make ime-pefeece ae ad social ime-pefeece ae povides a cucial isigh io he opimal bude ae of each geeaio. Oe limiaio of his sudy is ha we make assumpios based o ceai pefeeces, such as homogeeiy. The peseed fidigs would be moe wohwhile if i wee possible o show hese esuls moe geeally. We will cay ou his ask i subseque eseach. 38

40 Refeeces Auebach, A. J. ad L. J. Kolikoff (987) Dyamic Fiscal Policy, Cambidge: Cambidge Uivesiy Pess. Bao, R. J. (974) Ae Goveme Bods Ne Wealh?, Joual of Poliical Ecoomy, Vol. 8, No. 6, pp (979) O he Deemiaio of he Public Deb, Joual of Poliical Ecoomy, Vol. 87, pp (999) Noes o Opimal Deb Maageme, Joual of Applied Ecoomics, Vol., pp Boh, H. (99) Tax Smoohig wih Fiacial Isumes, Ameica Ecoomic Review, Vol. 8, pp Diamod, P. A. (965) Naioal Deb i A Neo-classical Gowh Model, Ameica Ecoomic Review, Vol. 55, pp Feldsei, M. (995) Would Pivaizig Social Secuiy Raise Ecoomic Welfae?, NBER Wokig Pape, No. 58. (998) The Effec of Pivaizig Social Secuiy o Ecoomic Welfae: Appedix o he Ioducio, i M. Feldsei ed., Pivaizig Social Secuiy, The Uivesiy of Chicago Pess, pp

41 Hoioka, C. Y. () Ae he Japaese Selfish, Aluisic, o Dyasic?, The Japaese Ecoomic Review, Vol. 53, No., pp Judd, K. L. (999) Opimal axaio ad spedig i geeal compeiive gowh models, Joual of Public Ecoomics, Vol. 7, pp. 6. Oguo K. (8) The Ie-geeaioal Gap i Social Secuiy: Smoohig he Ne Lifeime Bude of Each Geeaio as a Soluio, Quaely of Social Secuiy Reseach, Vol. xx, pp (i Japaese). Oguo, K., H. Nakakaumai, ad S. Takama (7) The Ie-geeaioal Gap i Social Secuiy: Smoohig he Ne Lifeime Bude of Each Geeaio as a Soluio Supplemeal Ioducio of Fudig Accou fo Log-em Cae Isuace, Miisy of Fiace, Policy Reseach Isiue, Discussio Pape Seies 7A-5 (i Japaese). Samuelso, P. A. (958) A Exac Cosumpio-Loa Model of Iees wih o wihou he Social Coivace of Moey, Joual of Poliical Ecoomy, Vol. 66, pp Takayama, N., Y. Aso, T. Miyaji, ad Y. Kamiya (996) Realiy of he Accumulaio of Household Asses ad he Successive Iheiace, i N. Takayama, C. Y. Hoioka, ad K. Oa, eds., The Savigs of Agig Sociey ad he Successive Iheiace, Nippyo Hyoo Sha Co., Ld. (i Japaese). Yoshida Y. (965) Theoy of Fucios, Iwaami Zesho (i Japaese). 4