Sustaining Ramsey plans in overlapping generations economies

Transcription

1 June 202 Suaining Ramey plan in overlapping generaion economie Juan C. Conea Univeria Auònoma de Barcelona and Barcelona GSE Begoña Domínguez Univeriy of Queenland ABSTRACT Thi paper udie he opimal fical policy and deb rucure in an economy wih overlapping generaion. We obain ha wihou commimen here are muliple governmen deb rucure ha can uain he Ramey policy plan. The agen finie overlapping live and heerogeneiy generae ample degree of freedom in how o rerucure he opimal governmen deb. Thi mulipliciy allow implifying he mauriy rucure of he opimal deb. We how ha here i a lea one opimal deb rucure compoed of only one period bond ha enure imeconiency of he Ramey plan. * We han ueful commen by Tim Kehoe and paricipan a SAE-Zaragoza and SWM-Aucland. Conea acnowledge uppor hrough he prize CREA Academia for excellence in reearch funded by he Generalia de Caalunya gran 2009SGR350 from Generalia de Caalunya gran ECO from he Minierio de Educación and he uppor of he Barcelona GSE Reearch Newor. The auhor can be reached a JuanCarlo.Conea@uab.e and b.dominguez@uq.edu.au.

2 . nroducion n hi paper we characerize he deb rucure neceary o render he opimal fical policy ime conien in an overlapping generaion economy. The procedure of deb rerucuring ha received wo main criicim in he lieraure. The fir criicim i of a concepual naure. ay ha i i inconien o aume ha he fical auhoriy can commi o a ax equence while a he ame ime he procedure require a commimen o he choen deb rucure. For inance many wor uch a Klein and Río-Rull (2003) conider a one-period commimen o capial axe bu do no allow for deb-rerucuring a i would require long mauriy bond and herefore a long-erm commimen. The econd criicim i an empirical one. ay ha he complicaed deb rucure he andard procedure would precribe are no nearly cloe o he much imple deb rucure oberved in he daa. Our reul how ha hee criicim do no apply o a andard overlapping generaion eup. We how ha in hi environmen here i a mulipliciy of deb rucure ha could guaranee ime coniency of he Ramey policy. Among hoe here i one in which only one period bond are needed. We conclude ha he wo criicim do no apply o overlapping generaion environmen. One of he cenral queion in public finance i wheher opimal fical policie are imeconien. n a cloed economy wihou capial or money Luca and Soey (983) how ha he careful managemen of he rucure of governmen deb can mae he opimal policy imeconien. Thi mehod called deb rerucuring ha been exended o a grea variey of eing: open economie by Peron and Svenon (986) and Faig (99); economie wih endogenou governmen pending by Faig (994); moneary policy by Alvarez e al. (2004) Peron e al. (2006) and Albanei (2005); economie wih heerogenou agen by Baeo (999) and Albanei (2005); economie wih capial by Zhu (995) and Domínguez (2007) ec. n general he policy precripion delivered by he deb rerucuring procedure remain exremely complicaed. One of he difficulie i ha governmen are required o iue pecific amoun of differen ype of bond a each imaginable mauriy (from one period o infiniy). 23 Alernaive approache coni of direcly compuing he opimal ime-conien policie eiher he e of ubgame perfec equilibria a in Phelan and Sacchei (200) or he Marov-perfec equilibria a in Klein Kruell and Río- Rull (2008). Noe ha deb rerucuring involve chooing an opimal bond rucure ha mae he Ramey equilibrium Marov. 2 Typically he opimal amoun of deb iue a each mauriy depend on he allocaion a ha fuure dae and hould be coningen on any hoc occurring a ha dae. Then unle he raniion oward he eady ae i very hor and aben of hoc he opimal deb iuing problem can be coniderable. 2

3 Thi paper how ha hi condiion i no longer a requiremen in an overlapping generaion environmen. Thi paper conider an economy wih overlapping generaion wih capial where a benevolen governmen chooe labor axe capial axe and iue deb in order o finance an exogenou governmen pending. A in Domínguez (2007) we aume ha here i an implemenaion lag in he axaion of capial. We fir udy an economy wih a complee e of deb inrumen: deb indexed o conumpion and o he afer-ax wage of all age and all poible mauriie. We hen characerize he Ramey policy plan. 4 Our main reul are a follow. Fir we demonrae ha deb rerucuring can olve he ime-inconiency problem and uain he Ramey oucome. Then we how ha here are muliple deb rucure ha can mae he opimal policy ime-conien. Thi mulipliciy originae in he overlapping generaion rucure and i i amplified by he heerogeneiy among bond holder. Among he muliple opimal deb rucure we find ha here i a lea one deb rucure compoed of only one period bond ha can uain he Ramey policy plan. Several paper have udied ime-coniency in an overlapping generaion environmen. Calvo and Obfeld (988) udy opimal fical policy when he planner preference uffer from ime-inconiency a in Sroz (956) due o an aymmeric reamen beween born and unborn generaion. They how ha hi problem can be olved wih dae and age-dependen lump-um axe and ranfer. Ambler (2002) focue on Marov-perfec equilibria o characerize ime-conien opimal fical policie in an overlapping generaion framewor. More recenly Slee (997) and Farhi e al. (20) udy he e of uainable equilibria for overlapping generaion economie wih linear and non-linear axe repecively. Mulipliciy in he opimal deb rucure ha been found in earlier paper. Peron and Svenon (986) udy ime-coniency of opimal fical policy wih inernaional borrowing. They find ha while oal governmen iue are unique foreign and domeic iue are no eparaely. Thi indeerminacy arie from having differen poible bond holder. 3 Domínguez (2005) find ha when he governmen doe no have acce o deb of all mauriie and aim a building a repuaion o uain he Ramey policy plan i i opimal o concenrae he claim in few mauriie. 4 We are no he fir o do o our conribuion i he characerizaion of deb rerucuring in hi ind of environmen. Under he aumpion of full commimen he opimal policy in differen overlapping generaion environmen ha been previouly udied in Eroa and Gervai (2002) Garriga (2003) and Conea and Garriga (2008). 3

4 Similar reul are found in boh Baeo (999) and Albanei (2005). Baeo (999) udie he effec of axaion on rediribuion in an economy populaed by renier and axpayer. Albanei (2005) udie opimal fical and moneary policie in an economy wih high and low produciviy houehold. A in Peron and Svenon (986) here are differen bond holder and he opimal deb rucure i no unique. 5 The indeerminacy found in he above paper provide ome degree of freedom bu doe no implify he opimal deb iuing problem in a clear way. Thi paper alo ha differen bond holder bu i i he finie live ogeher wih he eparaion in he diorionary co of axaion of each generaion ha brea he need for long mauriie in he opimal deb rucure. The re of he paper i organized a follow. Secion 2 preen he model. Secion 3 olve he Ramey problem. Secion 4 focue on he deb rucure required o deliver imeconiency. Secion 5 conclude. 2. The Model The economy i populaed by overlapping generaion of idenical houehold who live for period. A each period x a new generaion i born (referred a generaion x). There i no populaion growh. The preference of a houehold born in period x are defined by he dicouned ream of he inananeou flow of uiliy derived from conumpion c ix i and leiure nix i where i denoe he age of he houehold ha i i U u cix i nix i imax x ( ) () wih 0 u() i ricly increaing concave and wice coninuouly differeniable. We aume ha all individual reire a age J and herefore do no wor i.e. nix i = 0 for all i J. Noe alo ha in period 0 here i a populaion of iniial old generaion x0... of age i 0 x A Baeo (999) and Albanei (2005) poin ou he opimal deb rucure depend on he diribuion of bond and oher ae. Then he governmen mu be able o oberve and reac o change in privae mare and brea he indifference in he compoiion of he houehold porfolio. 4

5 The houehold ource of income are wage while woring capial ren and income from governmen bond. Toal income i ued o buy conumpion and new ae (capial b and governmen bond i ). The budge conrain of a houehold of age i in period i i b q b i n Rb i pi ci i pi i wi ni i r i p i pi (2) wih (i) he iniial condiion b 0 for houehold of new generaion x 0 a age i i i and ˆ b b i0 0 for iniially old generaion x 0 a age i ; and (ii) he i00 i00 i00 erminal condiion i bi 0 for all houehold a age i. 6 Noaion i uch ha w i i n he wage rae r i he renal rae of capial i he depreciaion rae i i he ax rae on labor income i i he ax rae on capial income p i i he price of a uni of conumpion of an individual of age i a and b R and q are he reurn and price for governmen bond. More pecifically governmen bond b i are allowed o have a very rich rucure in erm of mauriy and indexaion. We conider bond indexed o conumpion and o he afer ax-wage. We denoe by b ch i conumpion an agen of age h... a bond owned by a houehold of age i in period ha pay a uni of in period.... One uni of b co ch i ch q and nh pay p h. We denoe by b i a bond owned by a houehold of age i in period ha pay a uni of afer-ax wage of an agen of age h... J nh n q and pay ph h wh in period.... Thu one uni of b co. The marice b i b nh i R and q ore hee governmen bond b b reurn and price repecively o ha R b r( R ' b ) and qb i ( ' ) r q b i are oal i i b bond paymen and oal purchae co of new bond in period. Tha i Rb i J ch n nh ph bi ph h wh bi and h h J ch ch nh nh i i i h h. q b q b q b 6 The iniial and erminal condiion (i) and (ii) on governmen bond i.e. b 0 and b 0 require ha he houehold holding of all ype of governmen bond are zero a hoe iniial and erminal age. 5

6 The houehold opimizaion problem i o chooe conumpion labor capial and bond o maximize i welfare () ubjec o i budge conrain (2) for i given iniial and erminal condiion on ae. The following equaion ummarize he neceary condiion for a oluion o he houehold problem for all 0 : 7 [ c ] u p for all i... i ci i n i ni i i i [ n ] u p w for all i... J [ i ] pi p i i r for all i... ch ch [ b ] q p for all h... and i h nh nh n i h h h [ b ] q p w for all h... J and. From he fir hree equaion we obain he following condiion: n ni ci i i u u w (3) uci u ci i r. A repreenaive compeiive firm produce he final good uing he echnology J f n... n wih i and f increaing concave and coninuouly differeniable. i2 Taing facor price a given he firm chooe capial and labor o maximize profi o ha [ ] r f n... n J i i ni J [ n ] w f n... n for all i... J. (4) The governmen collec ax revenue in order o finance an exogenouly given ream of governmen conumpion denoed by 0 g and iue governmen bond. We aume ha governmen conumpion i unproducive and no valued by houehold. Tax revenue are n colleced hrough axaion on labor income i and on capial income i. Boh ax rae are aumed o be age dependen. Thi i aumed for clariy and analogou reul are obained in he alernaive cae. Moreover i can be alo juified on he ground of capuring he non- 7 The Lagrangre muliplier for he budge conrain (2) i normalized o pix i for generaion x. x 6

7 lineariy of he ax yem a in Gervai (202). We aume ha g 0 i large enough o ha diorionary axaion i required in equilibrium. The governmen equenial budge conrain i given by J b q b r b n i i i wi ni i r i g. (5) i i2 i i i2 i To finalize he model we wrie down he reource conrain a 3. The Ramey Problem p ci g f n... nj. (6) i We now urn our aenion o he problem of a benevolen governmen. n hi ecion we aume ha here i a commimen echnology ha allow he governmen a dae 0 o bind he choice of all fuure governmen. We alo aume ha all governmen commi o honor deb paymen and ha for all iniial old generaion i given. i0 0 The governmen a dae 0 i benevolen and a uch maximize p x i uc ( ix i nix i ) (7) x imax x where 0 i he inergeneraional dicoun facor. We ue he Primal Approach o ha we map he equilibrium condiion ino an implemenabiliy condiion enuring ha a feaible allocaion choen by he fical auhoriy can be decenralized a a compeiive equilibrium wih he ax rucure provided in he previou ecion. Adding up he budge conrain of a given houehold over ime and plugging in i firorder condiion ogeher wih he iniial and erminal condiion on i ae we obain he following implemenabiliy condiion: 0 for x 0 W for x x... (8) x i ucix i cix i unix i nix i imax x x0 0 7

8 for all new generaion x 0 and iniial old generaion x0.... For old generaion of age i 0 x heir iniial wealh i J ch nh. (9) W u f u b u b i00 ci00 i00 0 i00 ch i00 nh i00 h 0 h 0 The Ramey problem can be now formulaed a he choice of houehold conumpion labor and capial ha maximize ocial welfare (7) ubjec o he reource conrain (6) for all 0 and he implemenabiliy condiion (8) for all generaion x for given iniial condiion on ae and iniial capial ax rae a dae Denoing and he muliplier on he reource conrain (6) and implemenabiliy condiion (8) repecively we obain he following Lagrangian: i L u( ci ni ) 0 i 0 f n... nj ci g 0 i J 0 0 ch i uci ci vni ni i 0 u ci00 i 00 f 0 i 00 uch bi 00 unh bi 0 0 i i0 2 h 0 h 0 and fir order condiion: nh 0 i 0 0 ci 0 ni 0 0 i ci i ci i0 cci i00 i 0 nci m0 ci0 i02 i02 c u R u b u b i 0 0 ci 0 i ni i ni i0 cni i00 i0 i02 i02 n u R u b u b f ni 0 0 nni m0 ni0 ni 0 0 i f wih Rci uci ucci ci unci ni Rni ucni ci uni unni ni and 0 0 ci0 i0 u cci00 i 00 f 0 i 00 for i i u f u f for i i ni0 i0 cni00 i00 0 i00 i0 ci00 i00 ni00 i00 0 i0 2 8

9 a dae 0 and zero oherwie. The muliplier upercrip 0 indicae ha all choice are made a ime 0. A uual in hi lieraure we aume ha an opimal oluion exi. Uing hi oluion (he Ramey allocaion) he Ramey policy plan can be obained from he opimaliy condiion (3) and (4). We alo find he andard reul of irrelevance of he deb rucure under he aumpion of perfec commimen among governmen. n oher word he oal value of new bond iuance in each period i uniquely deermined by he Ramey allocaion bu he diribuion of hoe bond acro mauriie and indexaion i undeermined and irrelevan. 4. Opimal Fical Policy wihou Commimen From hi ecion on we aume ha fuure governmen can reconider heir policy plan bu commi o honor deb paymen. A aed earlier we alo aume an implemenaion lag in he capial axe. Tha i he governmen a dae can chooe he policy plan for all dae bu he iniial capial ax rae are inheried from he policy plan choen by he governmen a i dae. We laer dicu he condiion under which hi aumpion can be dipened. Now we urn o he ime-inconiency problem. Can he curren governmen chooe a deb rucure uch ha fuure governmen have no incenive o deviae from he coninuaion of he Ramey policy plan? Propoiion. f he equence J i i i i 0 c n and l J i i i i allocaion and policy plan hen he governmen can chooe a deb rucure a mare price ch nh q q and l J i i i i h h J 0 are he Ramey ch nh bi 0 bi 0 h h uch ha he coninuaion equence J i 0 c n 2 i i i i of he ame allocaion and policy are a oluion for he governmen problem when i i reconidered a dae. Thi opimal deb rucure i b b c u u u u ( bc) u u u u ( bn) ci 0 ci 0 ci nni ni nci i i i 0 i00 i i i ci i2 i02 ucci unni unci u cni ni 0 ni 0 ni cci cni ci i bi i0bi 00 i i ni ni i2 i02 ucci unni unci u cni J 9

10 where unni ci unci ni ucni ci ucci ni for ; for ; ci u and cci unni unci u cni ni ucci unni unci u cni 0 oherwie. 0 oherwie. By inducion he ame i rue for all laer period. Proof. See Appendix. Propoiion guaranee ha under he deb rucure ( ) plan i ime-conien. The opimal deb rucure ( ) bc bc bn he Ramey policy bn i characerized by hree erm: (i) he iniial deb rucure exiing in period 0 and mauring in period (weighed by he raio of he diorionary co of axaion on he bond holder generaion of governmen 0 relaive o ha of governmen ); (ii) he relaive difference in he co of diorionary axaion of generaion i for governmen 0 and ha for governmen 0 i i ime a funcion of he variable he bond i indexed o and i elaiciy of ineremporal ubiuion; 89 and (iii) a funcion of he effec of he iniial capial wealh value on he fir order condiion of he governmen a dae. The fir erm i an auoregreive erm i.e. how much he governmen hould iue of one ype of bond depend on how much here wa already iued of ha bond in he previou period. The econd erm balance he incenive o renege on he Ramey problem derived by change in he diorionary co of axaion for each curren and fuure generaion. The hird erm cancel ou he imeinconiency problem derived by he incenive o erode he iniial value of capial wealh. A can be een he mauriy of he deb i deermined by he fir wo erm; he hird one i only preen in he iniial period. One feaure ha i imporan o noice i ha he opimal deb rucure impoe no rericion on he ownerhip of he bond and capial bu hi ownerhip maer for he pecific deb rucure. Then in order o iue he righ amoun of each bond we need o now who will 8 The muliplier on he implemenabiliy condiion (8) x i inerpreed a he diorionary co of axaion for generaion x. However i hould be noed ha he value of hi muliplier i alo affeced by change in he compoiion of governmen bond. 9 Faig (994) provide a more deailed characerizaion of hi econd erm. 0

11 end up owning he bond who ha capial and how much hey have. Thi i capured by he muliplier and i 0 i0 and by and in he yem bc ( bn). ci ni n wha follow for clariy we aume ha here are no iniial deb claim exiing a zero. 0 Analyzing ( ) bc bn furher we find ha he opimal deb rucure i no unique. Propoiion 2. There are muliple deb rucure ch nh bi 0 bi 0 and herefore mae he Ramey policy plan ime-conien. Proof. See Appendix. h h J i 0 2 ha olve ( ) bc bn Thi mulipliciy originae in he overlapping generaion rucure. A we have overlapping generaion over an infiniy of period we coun x muliplier on heir x repecive implemenabiliy condiion (6) bu only curren bond holder which reul in muliple opimal deb rucure. Thi conra wih one muliplier for one implemenabiliy condiion in he infiniely-lived repreenaive agen cae which in general reul in a unique deb rucure. 2 The mulipliciy of opimal deb rucure in he OLG framewor i magnified by he exience of muliple bond holder which happen a long a we have 3. n hi cae we coun wih furher degree of freedom ince our precribed deb rucure bc ( bn) impoe no rericion on he ownerhip of bond and capial. Tha i for any fuure period he iniial value of wealh of he iniial old generaion i predeermined bu no he pecific porfolio ha add up o ha wealh. The reuling mulipliciy and degree of freedom are very large and here are infinie opimal deb rucure ha enure ime-coniency. Some of hoe have a imple mauriy rucure a he nex propoiion how: 0 To be conien wih he model pecificaion we alo aume ha privae claim are no required. Alernaively he model can be eaily modified o allow for privae claim. Mulipliciy can alo arie in he infiniely-lived repreenaive agen cae if here are addiional conrain on he implemenable allocaion ha inroduce exra muliplier in he governmen problem. For example impoing ha capial axe canno be above 00% if binding generae mulipliciy. Thi i illuraed in Domínguez (2007). 2 Our reul generalize o age-independen axe. The inuiion for hi connec wih foonoe 0. Addiional conrain in order o guaranee axe ha are independen of age inroduce new muliplier in he governmen problem a period which are no pinned down by he deb rucure and allow for furher degree of freedom.

12 Propoiion 3. For an economy wih overlapping generaion here exi a lea one deb rucure compoed of only one-period bond ha aifiebc ( bn) and herefore enure ime-coniency. Proof. See Appendix. A explained above he degree of freedom in he deign of he opimal deb rucure in an overlapping generaion model are very large. We can ue hee free parameer o enure ha he co of diorionary axaion of all fuure and all bu one (he olde) curren generaion for governmen i idenical o ha for governmen 0. How can a governmen enure ha? The governmen hould imply iue no bond a mauriie longer han one period. f he one-period bond are correcly iued (a maller problem o olve) he curren and fuure ime-inconiency problem are cancelled ou Some Numerical Example n wha follow we preen ome numerical example. We fir ar wih a benchmar economy a calibraion of an iniial eady ae ha correpond o an economy wih imilar policy and aiic o hoe of he US economy and we hen perform a Ramey ax reform. We conider 6 overlapping generaion. Each period la abou 0 year. ndividual wor from age o 4. Their produciviy level are aen from Hanen (993) and afer recaling are e o e e2.22 e3.26 and e4.20. ndividual are reired compulorily a age 4 and 5. The producion funcion i J f n... n A ( en e n e n e n ). (2) We e A = 2 and he hare of capial i Capial depreciae a a rae conien wih an invemen o oupu raio of 22.50%. We chooe a raio of capial o annual oupu of 3. K n order o arge 3 he individual dicoun facor i e o and he ame Y number i choen for he governmen generaional facor. We conider he uiliy funcion: u c i i c i ( ni ) n. () 2

13 The parameer i e equal o 4. The weigh of conumpion on he uiliy funcion i calibraed o mach an average labor of 0.30 which yield = For hee number he ineremporal elaiciy of ubiuion in conumpion equal The policy parameer a he iniial eady ae are e a follow. Labor and capial ax rae are aen from Gervai (202). 3 There i no iniial governmen deb. Governmen pending i obained from he governmen budge conrain and amoun o of eady ae oupu. Finally iniial capial holding a each age are choen o guaranee ha heir period budge conrain a he iniial eady ae are aified. We now perform he Ramey ax reform. We aume ha governmen pending i fixed a he level found a he iniial eady ae and ha he iniial condiion for he ae variable are alo hoe reuling from he benchmar economy. Then he governmen a period 0 chooe he Ramey policy plan he one ha olve he problem in Secion 3. Thi policy plan implie a unique real value of new deb iuance in period 0. The compoiion of hee bond which i no deermined by he Ramey approach affec he incenive of he governmen in period. Given an arbirarily fixed diribuion of capial in period Table -4 provide differen example of opimal deb rucure compoed of only one-period bond. Thee deb rucure mae he governmen a chooe a opimal policy he coninuaion of he Ramey policy plan choen by he governmen a dae 0. (NSERT TABLES -4) From he equaion ( bc) ( bn) and he above able one can ee ha ome iue of all ype (indexaion) of bond are required bu only one-period mauriy. Moreover he opimal amoun o be iued of each ype depend on who will own ha bond. f he bond are concenraed in few hand he opimal one-period bond iuance i furher implified. 5. Concluion n hi paper we have inveigaed he ime-coniency of opimal fical policy in an overlapping generaion economy. More pecifically we have focued on wheher here i an opimal deb rucure ha can mae he Ramey policy plan ime-conien. 3 n n n n Thee ax rae are and

14 One of he difficulie in he policy applicaion of he deb-rerucuring mehod i ha i uually require a very rich and complicaed mauriy rucure. n our eup we have found ha here are muliple governmen deb rucure ha can uain he Ramey policy plan. Some of hem have a imple mauriy rucure. Wih an implemenaion lag in capial ax rae we have hown ha i i alway poible o uain he Ramey wih one-period governmen bond. n fuure reearch we will explore he role of a ocial ecuriy yem a a commimen device. Penion benefi rule are claim of houehold o he governmen which are uually defined by he hiory of wage and are alo indexed o he CP. Thi mae penion an ideal inrumen o influence fuure governmen a i cloely mirror he real deb required o olve ime-coniency. Moreover penion can poenially be a more effecive commimen device han bond a hey are no purchaed bu impoed on he worer and alo allow governmen o reac o change in privae mare. 4

15 Reference Albanei Sefania (2005). Opimal and Time-Conien Moneary and Fical Policy wih Heerogeneou Agen. Manucrip. Alvarez Fernando; Kehoe Paric J. and Pablo Neumeyer (2004). The Time Coniency of Moneary and Fical Policy. Economerica Ambler Seve (2002). Opimal Time-Conien Taxaion wih Overlapping Generaion. Manucrip. Baeo Marco (989). Opimal Fical Policy wih Heerogeneou Agen. Manucrip. Calvo Guillermo and Maurice Obfeld (988). Opimal Time-Conien Fical Policy wih Finie Lifeime. Economerica Domínguez Begoña (2005). Repuaion in a Model wih a Limied Deb Srucure. Review of Economic Dynamic Conea Juan Carlo and Carlo Garriga (2008). Opimal Fical Policy in he Deign of Social Securiy Reform. nernaional Economic Review Domínguez Begoña (2007). On he Time-Coniency of Opimal Capial Taxe. Journal of Moneary Economic Eroa André and Marin Gervai (2002). Opimal Taxaion in Life-Cycle Economie. Journal of Economic Theory Faig Miquel (99). Time Coniency Capial Mobiliy and Deb Rerucuring in a Small Open Economy. Scandinavian Journal of Economic Faig Miquel (994). Deb Rerucuring and he Time Coniency of Opimal Policie. Journal of Money Credi and Baning Farhi Emmanuel; Slee Chriopher; Werning ván and Sevin Yelein (20). Nonlinear Capial Taxaion Wihou Commimen. Manucrip. Garriga Carlo (2003). Opimal Fical Policy in Overlapping Generaion Model. Mimeo. Gervai Marin (202). On he Opimaliy of Age-Dependen Taxe and he Progreive U.S. Tax Syem. Journal of Economic Dynamic and Conrol Hanen G. D. (993). The Cyclical and Secular Behaviour of he Labour npu: Comparing Efficiency Uni and Hour Wored. Journal of Applied Economeric Klein Paul and Joé-Vícor Río-Rull (2003). Time-Conien Opimal Fical Policy nernaional Economic Review Klein Paul; Kruell Per and Joé-Vícor Río-Rull (2008). Time-Conien Public Policy. Review of Economic Sudie Luca Rober E. and Nancy Soey (983). Opimal Fical and Moneary Policy in an Economy wihou Capial. Journal of Moneary Economic Peron Toren and Lar E. O. Svenon (986). nernaional Borrowing and Time 5

16 Coniency of Fical Policy. Scandinavian Journal of Economic Peron Ma; Peron Toren and Lar E.O. Svenon (2006). Time Coniency of Fical Policy and Moneary Policy: A Soluion. Economerica Phelan Chriopher and Ennio Sacchei (200). Sequenial Equilibria in a Ramey Tax Model. Economerica Slee Chriopher (997). Recurive Mehod for Solving Credible Governmen Policy Problem. Mimeo KSM-MEDS Norhweern Univeriy. Sroz Rober H. (956). Myopia and nconiency in Dynamic Uiliy Maximizaion. Review of Economic Sudie Zhu Xiaodong (995). Endogenou Capial Uilizaion nveor Effor and Opimal Fical Policy. Journal of Moneary Economic

17 Appendix Proof of Propoiion We conider he governmen a dae 0 and a dae and how ha he governmen a dae 0 can deign he deb rucure o a o mae he governmen a dae chooe he dae 0 policy plan. The governmen plan a dae 0 for all i ummarized in he following equaion: i 0 0 ci 0 ni 0 ci i ci i0 cci i00 i0 nci i00 i02 i02 i 0 0 ci 0 ni 0 uni i Rni i u 0 cni bi 00 i 0 nni i00 ni i0 2 i f u R u b u b for i... ( 0) i i J u b f for i... J (2 0) (3 0) c g f n... n x i ci x i i x i ni x i i x i imax x x0 (4 0) 0 for x 0 u c u n. (5 x0) W for x... n he eup of he governmen problem a dae we ubiue he upercrip 0 by a upercrip in he Lagrange muliplier o noe ha he choice i made in period. We alo ae ino accoun he new iniial condiion on ae. Then he governmen plan a dae for all i ummarized in he following equaion: i ci ni ci i ci i cci i i nci i ci i 2 m2 u R u b u b for i... ( ) i ci ni ni i ni i cni i i nni i ni ni i 2 i 2 u R u b u b f for i... J (2 ) f (3 ) ci g f n... nj i x i cix i ix i nix i ix i imax x x (4 ) 0 for x u c u n. (5 x) W for x...0 where and are analogou o hoe in page 8. The iniial wealh in period equal ci ni J W u f u b u b. ch nh i ci i i ch i nh i h h The dicoun facor appear o he power of in he above equaion becaue of our normalizaion of he Lagrange muliplier in he houehold opimizaion problem. 7

18 We now how ha here exi a deb rucure under which he equence J n J ci ni i i i i i i ha olve he governmen plan a dae 0 olve he plan evaluaed a dae. Fir ince hi equence olve (4 0) for all and (5 x0) for all x i alo olve (4 ) for all and (5 x ) for all x. Moreover given he one-period implemenaion lag he iniial capial ax rae of he governmen a dae equal he capial ax rae a dae choen by he governmen a dae 0. For he remaining condiion we mae ue of he deb rucure. Fir we require one deb inrumen a each mauriy o guaranee ha he ame allocaion ha olve (3 0) alo olve (3 ). For ha purpoe and wihou lo of generaliy we ue bond indexed o he c conumpion of a houehold of age i.e. bi Nex o olve he inra-period ci cj 0. condiion for all implied by ( ) we ue he remaining bond indexed o conumpion a each mauriy bi ci 0 i2. Then we guaranee ha he J conumpion-labor deciion for all implied by (2 ) in conjuncion wih ( ) are aified by uing he J bond indexed o he afer-ax wage a each mauriy bi ni 0 J i. Finally we are now lef wih implemenabiliy condiion (5 x) hoe of he iniial old generaion a dae i.e. x...0. Thi i olved by imply impoing ha he pah for he opimal deb J 0 h 0 h i0 2 ch nh inrumen bi bi i uch ha he diribuion of he iniial wealh mae he implemenabiliy condiion (5 x ) for he iniial old generaion x x...0 hold for he Ramey allocaion conidered a dae 0. We now find he deb rucure ha olve ime-coniency. Fir we guaranee ha he Ramey allocaion from he governmen a 0 olve he fir order condiion for capial (3 ) for all for he governmen a dae. For hi we find he deb rucure indexed o c conumpion of a houehold of age bi ha impoe 0 0 for all and hence 8

19 (3 ) hold for all. The fir order condiion for he conumpion of houehold of age a dae c for he governmen a dae 0 i 0 0 c 0 n 0 uc Rc i u 0 cc bi00 i u 0 nc bi00 i02 i02. The analogou condiion for he governmen a dae i c n uc Rc i ucc bi i unc bi i 2 m2. Subracing hi la equaion from he one of governmen 0 and impoing 0 we ge Rc iucc bi iunc bi iu 0 cc bi 0 iu 0 nc bi c n 0 c 0 n 0 0 i 2 m2 i02 i02 Then we obain he following opimal deb rucure indexed o conumpion of a houehold of age for all mauriie : c 0 c 0 unc u c unc n 0 n i bi i u 0 cc bi00 c n i bi i b 0 i00. i2 i02 ucc u cc ucc m2 i0 2 0 Thi deb rucure mae and in urn condiion (3 ) hold for all. Nex we find he deb rucure indexed o conumpion bi ci 0 i2 ha mae he Ramey allocaion from he governmen a 0 olve he inra-period ci cj condiion for all implied by ( ) from he governmen problem a dae. Wihou lo of generaliy we conider he ci c condiion for i The fir order condiion for he conumpion of houehold of age i a dae c i from he governmen a dae 0 i i 0 0 ci 0 ni 0 uci i Rci i u 0 cci bi 00 i u 0 nci bi 00 i02 i02. For he governmen a dae we have i ci ni uci i Rci i ucci bi i unci bi ci i 2 m2. 9

20 The RHS of hee equaion could be ubiued uing he fir order condiion for c. nead of ha we imply mae ue of he fac ha we have already guaraneed and ubrac he fir from he econd equaion o ge for all 0 i i Rci iucci bi iunci bi ci i u 0 cci bi 0 i u 0 nci bi ci ni 0 ci 0 ni 0 0 i 2 m2 i02 i02 Solving for he opimal bond rucure we ge b b c u n u u b b nci ci nci ci 0 ci 0 ni 0 ni i i i0 i00 i i i i i i i0 i00 ci i 2 i0 2 ucci u cci ucci m2 i0 2 ucci Thi deb rucure enure ha he Ramey allocaion from he governmen a 0 aifie he c c condiion for all implied by ( ). i j Finally we now guaranee ha he J conumpion-leiure deciion implied by ( ) (2 ) are aified for he governmen a dae. For governmen a dae 0 we have he following fir order condiion for labor:. i 0 0 ci 0 ni 0 uni i Rni i u 0 cni bi 00 i u 0 nni bi 00 fni i02 i02 For he governmen a period we have. i ci ni uni i Rni i ucni bi i unni bi ni fni i2 i2. 0 We could ubiue and in he RHS of hee equaion wih he correponding fir order condiion for conumpion. nead of ha we again mae ue of he implied 0. Then we ubrac he fir from he econd equaion o obain i i Rni iucnibi iunnibi ni iu 0 cnibi 0 iu 0 nnibi0 0 0 ci ni 0 ci 0 ni 0 0 i2 i2 i02 i02 Now olving for he opimal deb indexed o he afer-ax wage we obain u u u ni 0 ni 0 cni ni cni ci 0 ci ni i bi ib 0 i00 i i ci ni i bi ib 0 i00 i 2 i0 2 unni u nni unni i2 i02 unni. 20

21 A een above he precribed deb indexed o conumpion depend on ha indexed o he aferax wage and vice vera. Solving hi yem of equaion he opimal deb rucure become ci 0 ci 0 ci nni ni nci i bi i b 0 i00 i i ci ci i2 i02 ucci unni unci u cni b b n ni 0 ni 0 i i i0 i00 i i i2 i02 u u u u u u u u ni cci cni ci i ni ucci unni unci u cni wih unni ci unci ni ucni ci ucci ni for ; for ; ci u and cci unni unci u cni ni ucci unni unci u cni 0 oherwie. 0 oherwie. Noe ha for age i > J uni ucni and unni mu be replaced by a zero. All in all we have found he deb rucure ( bc) ( bn) ha mae he allocaion and policy ha olve he Ramey allocaion and policy plan a dae idenical o he coninuaion of ha of he governmen a dae 0. We can replicae hi procedure for all laer dae. Therefore he Ramey policy plan a dae 0 can be made ime-conien hrough deb rerucuring. Proof of Propoiion 2 A noed in he proof of Propoiion we need o impoe he pah of he opimal deb rucure ( bc) ( bn) ono he implemenabiliy condiion (5 x ) of he iniial old generaion a dae i.e. x...0. Noe ha before ha we need o mae an aumpion abou he diribuion of bond. The precribed deb rucure doe no mae any rericion on he diribuion of bond and capial a long a hey are uch ha he oal value of iniial wealh i ha of he implemenabiliy condiion of he iniial old in period. Le u pic one diribuion of bond holding and capial. Afer impoing hi deb we have a yem of equaion and infinie unnown x. Thi yem i undeermined and herefore we have infinie x poible oluion x and infinie opimal deb rucure ha olve ( bc ) ( bn ) and reolve x he ime-inconiency problem. Since he choice of he diribuion of bond holding and capial among hoe conien wih he implemenabiliy condiion i alo arbirary he mulipliciy of oluion i amplified by he fac of having muliple bond holder. 2

22 Proof of Propoiion 3 Fir noe ha ince here are no iniial deb holding a 0 he mauriy of he opimal deb 0 rucure i governed olely by he difference i i. Recall ha we need o aify he deb rucure condiion bc bn and he implemenabiliy condiion of he iniial old. Le u ue he degree of freedom we have o e 0 for all x and aume an equal diribuion of bond acro bond holder. Saifying bc bn hi effecively implie b 0 and b 0 for all mauriie 2 and all iniial old generaion of age i 2... in ci ni i i period. We are now lef wih J ype of one period bond whoe oal value depend only on one muliplier and poible bond holder (he diribuion doe no depend on he muliplier). For 2 hi implie a deermined yem a we le adju o aify he (one) implemenabiliy condiion and ha muliplier give u he righ amoun for all he needed one-period bond. For 3 here are more han one deb holder and he diribuion of he bond can be ued o mee he iniial level of wealh. n fac for 3 we can chooe an arbirary diribuion for all bond excep for one and le hi one have a diribuion ha aifie one implemenabiliy condiion and le he muliplier adju o aify he econd condiion. Noe ha hi yem even wih only one period bond i undeermined and here are many poible deb rucure. A he number of generaion increae he number of implemenabiliy condiion ( ) o aify increae bu alo (and a a wealy higher rae) he number of bond whoe diribuion can be ued o mach he iniial wealh. Therefore for an OLG model wih any generaion we have a lea one period bond rucure ha guaranee ime-coniency. x x 22

23 Table. Example : Opimal One-Period Governmen Bond ndexed o /Owned by Age 2 Age 3 Age 4 Age 5 Age 6 Conumpion Conumpion Conumpion Conumpion Conumpion Conumpion Afer-Tax Wage Afer-Tax Wage Afer-Tax Wage Afer-Tax Wage Table 2. Example 2: Opimal One-Period Governmen Bond ndexed o /Owned by Age 2 Age 3 Age 4 Age 5 Age 6 Conumpion Conumpion Conumpion Conumpion Conumpion Conumpion Afer-Tax Wage Afer-Tax Wage Afer-Tax Wage Afer-Tax Wage

24 Table 3. Example 3: Opimal One-Period Governmen Bond ndexed o /Owned by Age 2 Age 3 Age 4 Age 5 Age 6 Conumpion Conumpion Conumpion Conumpion Conumpion Conumpion Afer-Tax Wage Afer-Tax Wage Afer-Tax Wage Afer-Tax Wage Table 4. Example 4: Opimal One-Period Governmen Bond ndexed o /Owned by Age 2 Age 3 Age 4 Age 5 Age 6 Conumpion Conumpion Conumpion Conumpion Conumpion Conumpion Afer-Tax Wage Afer-Tax Wage Afer-Tax Wage Afer-Tax Wage