Research Division Federal Reserve Bank of St. Louis Working Paper Series

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1 Research Dvson Federal Reserve Bank of St. Lous Workng Paper Seres Optmal Response to a Transtory Demographc Shock n Socal Securty Fnancng Juan C. Conesa and Carlos Garrga Workng Paper A September 2007 FEDERAL RESERVE BANK OF ST. LOUS Research Dvson P.O. Box 442 St. Lous, MO 6366 The vews expressed are those of the ndvdual authors and do not necessarly reflect offcal postons of the Federal Reserve Bank of St. Lous, the Federal Reserve System, or the Board of Governors. Federal Reserve Bank of St. Lous Workng Papers are prelmnary materals crculated to stmulate dscusson and crtcal comment. References n publcatons to Federal Reserve Bank of St. Lous Workng Papers (other than an acknowledgment that the wrter has had access to unpublshed materal) should be cleared wth the author or authors.

2 Ths verson: January 2007 Optmal Response to a Transtory Demographc Shock n Socal Securty Fnancng Juan C. Conesa Unverstat Autònoma de Barcelona Carlos Garrga Federal Reserve Bank of St. Lous ABSTRACT We examne the optmal polcy response to a transtory demographc shock that affects negatvely the fnancng of retrement pensons. n contrast to exstng lterature, we endogenously determne optmal polces rather than explorng mplcatons of exogenous parametrc polces. Our approach dentfes optmal strateges of the socal securty admnstraton to guarantee the fnancal sustanablty of exstng retrement pensons n a Pareto mprovng way. Hence, no cohort wll pay the cost of the demographc shock. We fnd that the optmal strategy s based n the followng ngredents: elmnaton of compulsory retrement, a change n the structure of labor ncome taxaton and a temporary ncrease n the level of government debt. Keywords: Optmal taxaton, Ramsey Polces, Constraned Optma J.E.L. codes: E62, H2 * We thank very useful comments by Georges de Menl, Perre Pesteau and two anonymous referees. Both authors acknowledge fnancal support from the Spansh Mnstero de Educacón y Cenca, SEJ Conesa also acknowledges support from the Barcelona Economcs Program of CREA, and Generaltat de Catalunya through 2005SGR The vews expressed heren do not necessarly reflect those of the Federal Reserve Bank of St. Lous nor those of the Federal Reserve System. The authors can be reached va e-mal at HTUjuancarlos.conesa@uab.esUTH, and HTUcarlos.garrga@stls.frb.orgUTH.

3 . ntroducton Fnancal sustanablty of the socal securty system s an mportant polcy concern due to the agng of the US populaton and n partcular of the baby boom. Accordng to estmates of the Socal Securty Admnstraton the dependency rato (measured as populaton 65 or older over populaton between 20 and 64) wll ncrease from ts present 2% to 27% n the year 2020, 37% n 2050 and 42% n 2080 under the scenaro they label as the medum populaton growth. Fgure.: Populaton 65+ / Populaton from SSA Low Growth Medum Growth Hgh Growth Year Under ths demographc scenaro the Socal Securty system, of a Pay-As-You- Go (PAYG) nature, wll face clear fnancal mbalances unless some reforms are 2

4 DPT n ntroduced. n ths paper we explore the optmal response to a transtory demographc shock.tpd contrast to exstng lterature we follow a smlar approach to that used n Conesa and Garrga (2007) and endogenously determne optmal polces rather than explorng mplcatons of exogenous parametrc polces. Our approach determnes the optmal strategy of the socal securty admnstraton to guarantee the fnancal sustanablty of current retrement pensons n the least dstortonary way. Moreover, no cohort wll have to pay the welfare cost of the demographc shock. Notce that we are only concerned about effcency consderatons n the fnancng of retrement pensons, rather than n the effcency of ther exstence n the 2 frst place. Ther exstence mght be justfed on dfferent grounds.tpd DPT We do not model why socal securty was mplemented n the frst place, nor why socal securty benefts are provded through a potentally neffcent tax system. We wll consder for our experments an unexpected transtory demographc shock, whch mght sound qute awkward snce these shocks are certanly qute predctable by lookng at Fgure. The reason why we do that s because f the demographc shock s predctable the fscal authorty should have reacted to t n advance. However, we beleve t s more nterestng to focus on what should be done from now on, rather than focusng on what should have been done. n that sense predctng t but not dong anythng about t s equvalent to the shock beng unexpected. However, the transtory nature of the shock consdered s a lmtaton of the analyss drven by computatonal tractablty. The quanttatve evaluaton of socal securty reforms has been wdely analyzed 3 n the lterature.tpd DPT Demographc consderatons play an mportant role n the socal securty debate, but there are few quanttatve studes of polcy responses to demographc shocks, and none to our knowledge from an optmal fscal polcy perspectve. n partcular, De Nard et al. (999) consders the economc consequences 3

5 of dfferent alternatve fscal adjustment packages to solve the future socal securty mbalances assocated to the projected demographcs n the U.S. They fnd that all fscal adjustments mpose welfare losses on transtonal generatons. n partcular, polces that partally reduce retrement benefts (by taxng benefts, postponng retrement or taxng consumpton), or that gradually phase benefts out wthout compensaton yeld welfare gans for future generatons, but make most of the current generatons worseoff. They conclude that a sustanable socal securty reform requres reducng dstortons n labor/lesure, consumpton/savng choces and some transton polces to compensate current generatons (ssung government debt). Our approach allows for the endogenous determnaton of such polces n a way that nobody faces welfare losses. n other words, everybody wll be guaranteed the same level of welfare as n the benchmark economy wthout a demographc shock. However, for computatonal tractablty we wll substantally smplfy the nature of the demographc shocks relatve to De Nard et al. (999). Jeske (2003) also analyzes payroll adjustments to demographc shocks n an economy smlar to ours. He fnds that n contrast wth the benchmark economy not all cohorts are worse-off because of the arrval of the baby boomers. n partcular, the parents of the baby boomers gan about 0.5 percent of average lfetme consumpton; the baby boomers loose percent, the chldren of the boomers gan 2 percent, and the grandchldren loose more than 2 percent. The ntuton for ths result comes from movements n factor prces mpled by the demographc shock, and the mpled payroll taxes adjustment to balance the government perod budget constrant. n contrast to them, we do not analyze the dfferent mplcatons of exogenously specfed strateges to guarantee sustanablty, but rather we optmze over ths polcy response to demographc shocks followng the Ramsey approach. The quanttatve analyss of optmal fscal polcy n overlappng generatons economes was poneered 4

6 by Escolano (992) and has been recently consdered by Erosa and Gervas (2002) and Garrga (999). Conesa and Garrga (2007) used a smlar framework to analyze the desgn of socal securty reforms and therefore the focus was on effcency consderatons, abstractng from sustanablty ssues. Our man conclusons ndcate that the optmal strategy n absorbng a negatve demographc shock conssts of:. Changng the age structure of labor ncome taxaton. n partcular, labor ncome taxes of the young should be substantally decreased. 2. Elmnatng compulsory retrement and allowng cohorts older than 65 to supply labor n the market. 3. ncreasng the level of government debt durng the duraton of the demographc shock and then repayng t slowly. We fnd that the welfare gans wll be concentrated for generatons born n the dstant future after the demographc shock s over, whle t does mantan the benchmark welfare level for exstng cohorts and current newborns durng the shock. Therefore, no generaton s worse-off along the fscal adjustment process mpled by the demographc shock. Ths result contrasts wth the fndngs of De Nard et al. (999), and Jeske (2003), where ether current or future generatons suffer mportant welfare losses. More mportantly, we fnd that a sustanable socal securty reform does not necessarly requre reducng dstortons n consumpton/savng choce. t s suffcent wth a reducton n labor/lesure dstortons, and ssung government debt to compensate current generatons. n addton, we show that the welfare costs of dstortonary taxaton are quanttatvely mportant rght after the demographc shock, but are relatvely less mportant n the long run. 5

7 μ t s mportant to notce that the dstortonary mpact of the fnancng of pensons n our artfcal economy s rather assumed than endogenously determned. n our benchmark economy pensons wll be fnanced through lnear age-ndependent payroll taxes and ndvduals do not establsh a lnk between ther ndvdual contrbutons to the system and ther future pensons. Hence, all the welfare gans obtaned n our analyss are generated by the mnmzaton of dstortons and the redstrbuton of these addtonal resources. ndeed, t could not be otherwse snce t s the presence of dstortons that generates the possblty of Pareto mprovements. We also show that when the ncome from retrement pensons s not taxable, the government could use ths fact to replcate lump-sum taxaton, and achevng frst-best allocatons. Yet, snce we want to focus n an envronment where the government s restrcted to dstortonary taxaton, we only consder an envronment where the fscal treatment of retrement pensons s constraned to be the same as that of regular labor ncome. The rest of the paper s organzed as follows. Secton 2 descrbes the benchmark theoretcal framework used. Secton 3 explans how we parameterze our benchmark economy. Secton 4 descrbes the optmal fscal polcy problem usng the prmal approach. Secton 5 descrbes the experment we perform, the demographc shock, and analyzes the optmal response. Secton 6 concludes. All the references are n Secton The theoretcal envronment n the benchmark economy Households The economy s populated by a constant measure of households who lve for perods. These households are forced to retre n perod r. We denote by μ t, the measure of households of age n perod t. Preferences of a household born n perod t depend on 6

8 the stream of consumpton and lesure ths household wll enjoy. Thus, the utlty functon s gven by: t t (, ) = β ( t, +, t, + ) = Uc l uc l () Every perod each household owns one unt of tme that they can allocate to work or lesure. One unt of tme devoted to work by a household of age translates ε nto ε effcency unts of labor n the market, and these are constant over tme. Technology The Producton Possblty Fronter s gven by an aggregate producton functon Yt = F( Kt, Lt), where K t denotes the captal stock at perod t and Lt = μ, tεl, t s the aggregate labor endowment measured n effcency unts. We assume the functon F dsplays constant returns to scale, s monotoncally ncreasng, strctly concave and satsfes the nada condtons. The captal stock deprecates at a constant rate δ. = Government The government nfluences ths economy through the Socal Securty and the general budget. For smplcty we wll assume that ntally (before the demographc shock) these two programs operate wth dfferent budgets. Then, pensons ( t ) are fnanced p through a payroll tax ( τ t ) and the socal securty budget s balanced. On the other hand, c l the government collects consumpton taxes ( τ t ), labor ncome taxes ( τ t ), captal ncome taxes ( τ k t ) and ssues publc debt ( b t ) n order to fnance an exogenously gven stream of government consumpton ( g t ). trt p 7

9 gven by: Thus the Socal Securty and the government budget constrants are respectvely r p t wt l, t = pt = = r (2) τ με μ r c l p k t ct, + t ( t ) wt lt, + t rt at, + bt+ = gt+ ( + rt) bt = = = (3) τ μ τ τ με τ μ n response to the demographc shock, however, both budgets wll be ntegrated and we wll allow the government to transfer resources across budgets to fnance the retrement pensons. Market arrangements We assume there s a sngle representatve frm that operates the aggregate technology takng factor prces as gven. Households sell an endogenously chosen fracton of ther tme as labor ( l t, ) n exchange for a compettve wage of w t per effcency unt of labor. They rent ther assets ( a t, ) to frms or the government n exchange for a compettve factor prce ( r t ), and decde how much to consume and save out of ther dsposable ncome. The sequental budget constrant for a workng age household s gven by: ( ) ( )( ) ( ( ) ),,..., + τ c l p k t c, t + a+, t+ = τt τt wtεl, t + + τt rt a, t = r (4) Upon retrement households do not work and receve a penson n a lump-sum fashon. Ther budget constrant s: ( ) ( ) ( ( ) ),,..., c l k + τt c, t + a+, t+ = τt pt + + τt rt a, t = r (5) The alternatve nterpretaton of a mandatory retrement rule s to consder dfferent labor ncome tax rates for ndvduals of ages above and below r. n 8

10 partcular, a confscatory tax on labor ncome beyond age r s equvalent to compulsory retrement. Both formulatons yeld the same results. However, when we study the optmal polcy we prefer ths alternatve nterpretaton snce t consders compulsory retrement as just one more dstortonary tax that the fscal authorty can optmze over. n the benchmark economy a market equlbrum s a sequence of prces and allocatons such that: ) consumers maxmze utlty () subject to ther correspondng budget constrants (4) and (5), gven the equlbrum prces; ) frms maxmze profts gven prces; ) the government and the socal securty budgets are balanced, (2) and (3); and v) markets clear and feasblty. 3. Parameterzaton of the Benchmark Economy Demographcs We wll choose one perod n the model to be the equvalent of 5 years. Gven our choce of perod we assume households lve for 2 perods, so that the economcally actve lfe of a household starts at age 20 and we assume that households de wth certanty at age 80. n the benchmark economy households retre n perod 0 (equvalent to age 65 n years). Fnally, we assume that the mass of households n each perod s the same. All these assumptons mply that n the ntal Steady State the dependency rato s 0.33, rather than the 0.2 observed nowadays. The reason s that n our smple envronment there s no lfetme uncertanty. 9

11 Endowments The only endowment households have s ther effcency unts of labor at each perod. These are taken from the Hansen (993) estmates, convenently extrapolated to the 4 entre lfetme of households.tpd DPT Fgure 3.: Age-Profle of Effcency Unts of Labor from Hansen (993) Government We assume that n the benchmark economy the government runs two completely ndependent budgets. One s the socal securty budget that operates on a balanced budget. The payroll tax s taken from the data and s equal to 0.5%, whch s the Old- Age and Retrement nsurance, OAS (we exclude a fracton gong to dsablty nsurance, the OASD s 2.4%). Our assumptons about the demographcs together wth the balanced budget condton drectly determne the amount of the publc retrement penson. t wll be 3.5% of the average gross labor ncome. 0

12 The level of government consumpton s exogenously gven. t s fnanced through a consumpton tax, set equal to 5%, a margnal tax on captal ncome equal to 33% and a margnal tax on labor ncome net of socal securty contrbutons equal to 6%. We have estmated these effectve tax rates followng Mendoza, Razn and Tesar (994). The effectve dstorton of the consumpton-lesure margn s gven by l p c (- τ )(- τ )/(+ τ )=-0.3, yeldng an effectve tax of 30%. constrant. The government ssues publc debt n order to satsfy ts sequental budget Calbraton: Functonal Forms Households preferences are assumed to take the form: γ γ σ ( c ( l) ) β (6) σ = where β > 0 represents the dscount rate, γ (0,) denotes the share of consumpton on the utlty functon, and σ > 0 governs the concavty of the utlty functon. The mpled ntertemporal elastcty of substtuton n consumpton s equal to /(-(- σ ) γ ). Technology has constant returns to scale and takes the standard Cobb-Douglas form: Y = K L, where α represents the captal ncome share. α t t t Calbraton: Emprcal Targets We defne aggregate captal to be the level of Fxed Assets n the BEA statstcs. Therefore, our calbraton target wll be a rato K/Y=3 n yearly terms. Also, computng the rato of outstandng (federal, state and local) government debt to GDP we get the followng rato B/Y=0.5 n yearly terms. Deprecaton s also taken from the data, whch s a fracton of 2% of GDP. Another calbraton target s an average of /3 of the tme of households allocated to market actvtes. We wll choose a curvature parameter n

13 the utlty functon consstent wth a coeffcent of relatve rsk averson n consumpton of 2 (alternatvely a consumpton ntertemporal elastcty of substtuton of 0.5). Government consumpton wll be fxed to be 8.6% of output as n the data. Fnally, the captal ncome share s taken to be equal to 0.3, as measured n Golln (2002). Calbraton Results n order to calbrate our economy we proceed as follows. Frst, we fx the curvature parameter n the utlty functon to be σ = 4 and the captal share n the producton functon α = 0.3. Then the dscount factor β =.003 s chosen to match a wealth to output rato of 3.5,TPD 5DPT and the consumpton share γ = s chosen n order to match an average of /3 of tme devoted to workng n the market economy. The deprecaton rate s chosen so that n equlbrum deprecaton s 2% of output. Notce that σ = 4 and γ = together mply a consumpton ntertemporal elastcty of substtuton of 0.5 (CRRA of 2). Table 3. summarzes the parameters chosen and the emprcal targets that are more related to them. Table 3.: Calbraton Targets and Parameter Values Emprcal Targets A/Y ES Av.Hours wn/y Dep./Y Emprcal Values / Parameters β σ γ α δ Calbrated Values Usng the emprcal tax rates and rato of government consumpton to GDP, we derve from the government budget constrant an mpled equlbrum government debt of 50% of output. Ths fgure s consstent wth the average fgure n the data. Therefore, the captal/output rato s 3 as desred. 2

14 DPT From Gven ths parameterzaton, socal securty annual payments n the benchmark economy amount to 7.35% of GDP and the socal securty mplct debt s equal to 28% of annual GDP. 4. The Government Problem: The Prmal Approach We use the prmal approach to optmal taxaton as frst proposed by Atknson and Stgltz (980). Ths approach s based on characterzng the set of allocatons that the government can mplement wth the gven polcy nstruments avalable. A benevolent fscal authorty chooses the optmal tax burden takng nto account the decson rules of all ndvduals n the economy, and the effect of ther decsons on market prces. Therefore, the government problem amounts to maxmzng the socal welfare functon over the set of mplementable allocatons together wth the status quo constrants.tpd 6 the optmal allocatons we can decentralze the economy fndng the prces and the tax polcy assocated to the optmal polcy. A key ngredent s the dervaton of the set of mplementable allocatons, effectvely t amounts to usng the consumer s Euler condton and labor supply condton to express equlbrum prces as functons of ndvdual allocatons, and then substtute these prces n the consumer s ntertemporal budget constrant. Then, any allocaton satsfyng the mplementablty condton satsfes by constructon the household s frst order optmalty condtons, wth prces and polces approprately defned from the allocaton. See Char and Kehoe (999) for a descrpton of ths approach. To llustrate ths procedure we derve the mplementablty constrant for a newborn ndvdual. Notce that n our case the fscal authorty has to consder 3

15 retrement pensons as gven, and that s gong to ntroduce a dfference wth Erosa and Gervas (2002), Garrga (999), or Conesa and Garrga (2007). We wll dstngush two cases: frst, retrement pensons are consdered as regular labor ncome and are treated as such from a fscal pont of vew; second, retrement pensons are not subject to taxaton. Both cases have dfferent tax polcy mplcatons. Retrement Pensons as Taxable Labor ncome For clarty of exposton we wll suppress the tme subscrpts. Consder the household maxmzaton problem for a newborn ndvdual facng equlbrum prces and ndvdual specfc tax rates on consumpton, labor ncome and captal ncome: max β uc (, l) = st.. ( + τ ) c + a ( τ ) wε l + ( + ( τ ) r) a, =,..., c l k + r ( + τ ) c + a ( τ )( wε l + p) + ( + ( τ ) r) a, =,..., c l k + r a = 0, a = 0, c 0, l (0,) + Notce two mportant features of ths formulaton. The frst one s that ndvduals of age r and older have a retrement penson, denoted by p, as part of ther labor ncome (and t s taxed at the same rate as regular labor ncome). Second, upon retrement ndvduals could stll supply labor n the market. Denotng by υ the Lagrange multpler of the correspondng budget constrant, the necessary and suffcent frst order condtons for an nteror optmum are gven by: c [ c] β uc υ ( ) τ = + (7) l [ l] ul ( ) w β = υ τ ε (8) k [ a+ ] υ = υ+ [ + ( τ ) r] (9) 4

16 together wth the ntertemporal budget constrant. Multplyng these condtons by the correspondng varable we get: c β cu c υ ( ) τ c = + (0) β lu = υ ( τ ) wε l () l l k υa+ = υ+ [ + ( τ ) r] a+ (2) Let p = p f = r,...,, and zero otherwse. Addng up (0) and () over all : c l β cu c + lu l = β υ + τ c τ wεl = = [ ] [( ) ( ) ] l = υ ( τ ) p = where the second equalty comes from usng (2). Fnally, usng (8) we get: p ε β [ cu c + lu ] l = β ul = = w or: p β cu c + u 0 l l + = = wε (3) Any feasble allocaton of consumpton and lesure satsfyng equaton (3) can be decentralzed as the optmal behavor of a consumer facng dstortonary taxes. These dstortonary taxes can be constructed by usng the consumer s optmalty condtons for the labor/lesure and the consumpton/savngs margns. n partcular, gven an allocaton and ts correspondng prces, constructed from the margnal product of labor and captal, we can back up the optmal tax on captal and labor ncome by usng the Euler and labor 5

17 supply condtons: + τ u u [ ( ) r] c k c = β c + τ c + + τ + (4) l ul τ = wε c u + τ (5) c Notce that n ths case the optmal polcy s not unquely determned. Labor and consumpton taxaton are equvalent n the sense that they determne the same dstortonary margn. Also, the taxaton of captal ncome s equvalent to taxng consumpton at dfferent tmes at dfferent rates. n practce, ths mples that one of the nstruments s redundant. For example, we could set consumpton taxes to zero (or to any other constant) and decentralze the allocaton usng only labor and captal ncome k l taxes by solvng a system of two equatons (4) and (5) n two unknowns τ, τ. Fnally, drectly usng the consumer s budget constrants we could construct the correspondng sequence of assets. That way we would have constructed an allocaton that solves the consumer s maxmzaton problem. The prmal approach of optmal taxaton smply requres maxmzng a socal welfare functon over the set of mplementable allocatons,.e. subject to the feasblty constrant, an mplementablty condton such as (3) for the newborn cohorts, and addtonal mplementablty constrants for each cohort alve at the begnnng of the reform. We wll also mpose that allocatons must provde at least as much utlty as n the ntal Steady State of our economy. The allocaton mpled by the optmal polcy can be decentralzed wth dstortonary taxes n the way we have just outlned. Non-taxable Retrement Pensons f pensons are not taxable, the maxmzaton problem of the households s gven by: 6

18 max β uc (, l) = st.. ( + τ ) c + a ( τ ) wε l + ( + ( τ ) r) a, =,..., c l k + r ( + τ ) c + a ( τ ) wε l + p+ ( + ( τ ) r) a, =,..., c l k + r a = 0, a = 0, c 0, l (0,) + Consequently, through the same procedure used as before we can obtan the expresson: c l β cu c + lu l = β υ + τ c τ wεl = = [ ] [( ) ( ) ] = = υ p Substtutng for the Lagrange multpler we get: p β uc 0 c c + lu l = = + τ (4) Notce that n ths case the mplementablty constrant does nclude a tax term n c t, τ. Ths dd not happen before, n expresson (3). Hence, t s always possble to choose a partcular taxaton of consumpton such that the mplementablty constrant s always satsfed. The reason s that now the fscal authorty could tax consumpton at a hgh level, but stll compensate the consumer through other taxes. n the prevous case ths strategy was not avalable snce t was mpossble to tax away the retrement pensons and compensate the consumers wthout ntroducng addtonal dstortons n the system. Another way to llustrate ths smple ntuton s by smply lookng at the ntertemporal budget constrant of the household: c l ( + τ ) c ( τ) wεl p = + R R R (5) = = = k where R =, R = [ + ( τ s ) rs]. s= 2 7

19 c Let τ c = τ,.e. we mpose the same taxaton of consumpton at each pont n tme of the lfetme of an ndvdual. Then we could rewrte (5) as: c τ wε l p l = + c c = R = + τ R + τ = R c Clearly, one could choose any desred level of taxaton of τ, and stll ntroduce no dstorton n the consumpton-lesure margn by choosng τ l l c = τ = τ. Effectvely c τ would act as a lump-sum tax. Therefore, under ths new scenaro the planner could decentralze a frst best allocaton by strategcally settng consumpton taxes to replcate lump-sum taxaton. Notce that ths strategy cannot be replcated for the case when retrement pensons are taxable as regular labor ncome, snce the equvalent of (5) would be: c τ wε l τ p l l = + c c = R = + τ R = + τ R (6) and hence the fscal authorty s forced to ntroduce a dstortonary wedge n the consumpton-lesure margn when tryng to mplement lump-sum taxaton as before. We are nterested n dstortonary tax responses to demographc shocks. Consequently we wll focus n the scenaro where the fscal treatment of retrement pensons has to be the same as the one of regular labor ncome. However, we wll compare the outcomes, n terms of welfare, wth the ones that could be obtaned f the government could mplement lump-sum taxaton. The Ramsey Problem We assume that n perod t= the economy s n a steady state wth a PAYG socal securty system, and no demographc shock or government nterventon has been antcpated by any of the agents n the economy. The expected utlty for each 8

20 generaton assocated to remanng n the benchmark economy s gven by s j U ( ˆ, ˆ j = β u cs ls),where s= j cˆ, l ˆ are steady state allocatons of generaton s. s s At the begnnng of perod 2, the demographc shock s known and then n response to t the optmal polcy from then on s announced and mplemented. We wll requre that the fscal authorty guarantees to everybody at least the level of utlty of the benchmark economy, so that the resultng polcy reform consttutes a Pareto mprovement. Ths partcpaton constrant wll ensure that the optmal response to a demographc shock generates no welfare losses (nether for the ntally alve nor the unborn). Notce that we are mposng a very strong partcpaton constrant, snce we requre that nobody s worst off relatve to a benchmark n whch actual fscal polces would have been sustanable forever (.e. the ntal Steady State). Alternatvely, we could have postulated dfferent arbtrary polcy responses to the demographc shock generatng welfare losses for some generatons, and then mprove upon those. Clearly, our specfcaton mposes stronger welfare requrements and s ndependent of any arbtrary non-optmal polcy we mght have chosen nstead. Besdes, the man concluson n the lterature s that no matter what polcy you choose somebody wll have to pay the cost of the demographc shock. We show ths s not necessarly the case. The government objectve functon s a utltaran welfare functon of all future newborn ndvduals, where the relatve weght that the government places between present and future generatons s captured by the geometrc dscount factor λ (0,), t t and Uc (, l ) represents the lfetme utlty of a generaton born n perod t. 7 Condtonal on our choce of weghts placed on dfferent generatonstpd DPT, the Ramsey allocaton s the one that solves the followng maxmzaton problem: 9

21 2 max t λ Uc ( t, l t ) t= 2 (7) st.. μ c + K ( δ) K + G F( K, μ ε l ), t 2 t, t, t+ t t t t, t, = = p β ct, + uc + u,,, 0, 2 t l l t t t = = wt+ ε (8) p u c u + u l + = ( + ( ) r ) a + p, = 2,..., (9) s c,2 k β ss, + 2 css, 2 lss, 2 ss, 2 τ c 2,2 s= ws + 2ε + τ2 s β ss, + 2 ss, + 2 s= uc (, l ) U, = 2,..., (20) t t Uc (, l) U, t 2 (2) Constrant (7) s the standard perod resource constrant. Constrant (8) s the mplementablty constrant for each generaton born after the reform s mplemented, and s exactly the one derved n (3). Ths equaton reveals that the government faces a trade off when determnng the optmal labor ncome tax of the older generatons. A hgher labor ncome tax s an effectve lump-sum tax on socal securty transfers, but t also reduces the ncentves of the older generatons to supply labor n the market. The optmal polcy wll have to balance these opposte forces. Constrant (9) represents the mplementablty constrants for those generatons alve at the begnnng of the reform, k where τ s the benchmark tax on captal ncome whch s taken as gven and a,2 are the ntal asset holdngs of generaton. Notce that takng τ k as gven s not an nnocuous assumpton, snce that way we avod confscatory taxaton of the ntal wealth. Fnally, constrants (20) and (2) guarantee that the polcy chosen makes everybody at least as well off as n the benchmark economy. n partcular, gven that the 20

22 government objectve functon does not nclude the ntal s generatons Equaton (20) wll be bndng. Ths formulaton mposes some restrctons, snce t rules out steady-state "golden-rule" equlbra. Also, the ntal generatons alve at the begnnng of the reform are not part of the objectve functon, and only appear as a polcy constrant. An equvalent formulaton would nclude the ntal s generatons n the objectve functon wth a specfc weght λ s, where the weght s chosen to guarantee that the status quo condtons for each generaton are satsfed. The polcy maker dscounts the future at the exponental rate λ. The Pareto mprovng nature of the reform mples that the rate λ has to be bg enough to satsfy the partcpaton constrants of all future generatons. n partcular, f λ were too low then the long run captal stock would be too low and then future generatons would be worse-off than n the benchmark economy. That restrcts the range of admssble values for λ. Of course, wthn a certan range there s some dscrectonalty n the choce of ths parameter, mplyng a dfferent allocaton of welfare gans across future generatons. n order to mpose some dscplne we choose λ so that the level of debt n the fnal steady state s equal to that of the benchmark economy, so that all debt ssued along the transton s fully pad back before reachng the new steady state. Our choce of the planner dscount factor, the parameter λ = 0.957, mples the full repayment of the level of debt ssued n response to the demographc shock. That does not mean that the rato of debt to output wll be the same n the fnal steady state, snce output does change. 2

23 Further Constrants n the Set of Tax nstruments We wll mpose addtonal restrctons n the set of fscal nstruments avalable to the fscal authorty. Ths can be done by usng the consumer s frst order condtons n order to rewrte fscal nstruments n terms of allocatons, and then mposng addtonal constrants on the Ramsey allocatons. n partcular, the regme we wll nvestgate s one n whch captal ncome taxes are left unchanged relatve to the benchmark. Then, reformulatng ths constrant n terms of allocatons we need to mpose: u u u c, t c2, t c, t k = =... = = β + ( τ )( fkt, + δ), t 2 (22) c2, t+ c3, t+ c, t+ u u u We ntroduce ths constrant snce we want to analyze an envronment n whch the reforms nvolve only changng the nature of labor ncome taxaton, so that welfare gans are accrued only because of the change n the nature of the fnancng of retrement pensons rather than a more comprehensve reform nvolvng also changes n the nature of captal ncome taxaton. Moreover, as Conesa and Garrga (2007) shows, the addtonal welfare gan of reformng captal ncome taxaton s very small. Wth such a constrant the only nstruments avalable to the fscal authorty wll be the taxaton of labor ncome and government debt. A Transtory Demographc Shock n our experment we ntroduce an unexpected transtory demographc shock, capturng the dea that an ncrease n the dependency rato s gong to break down the sustanablty of the socal securty system we had n the ntal Steady State for our benchmark economy. 22

24 The reason why we want to model t as an unexpected shock s that we want to nvestgate the optmal response from now on, nstead of focusng on what we should have done n advance to an expected shock. Snce ntroducng realstc demographc projectons would mply havng to change substantally the demographc structure of our framework, we wll choose a very smple strategy. We wll smply ncrease the measure of retrng ndvduals for three consecutve perods. Notce that the demographc shock s transtory, n the sense that for three perods (equvalent to 5 years) we wll face rasng dependency ratos, and then for another three perods the dependency rato falls untl reachng ts orgnal level and stayng there forever. We chose ths specfcaton of the demographc shock for computatonal convenence, snce otherwse the model would mply changes n the age structure over tme. The alternatve would have been an envronment where at some pont the fnal age permanently ncreases reflectng an ncrease n lfe expectancy. Ths rases some computatonal problems, especally f ndvduals could forecast the demographc evoluton and form expectatons about future paths of government acton. Hence, the benchmark economy would not be a Steady State anymore and the state of the economy at the benchmark date would be fully drven by arbtrarly chosen expectatons. 23

25 Fgure 4.: Evoluton of the Dependency Rato for Smulated Demographc Shock Year We have arbtrarly chosen to label the ntal Steady State n perod as the year 2000, and the demographc shock wll be observed and fully predctable at the begnnng of perod 2 (the year 2005). Hence the results that follow mply that the polcy response from 2005 on s publcly announced and mplemented at the begnnng of Notce that both ndvduals and the government are assumed to be surprsed by the demographc shock. The government learns that gven the demographc evoluton the system s not sustanable and then mplements a polcy that ratonalzes the fnancng of pensons. Not only the government wll optmally respond to the demographc shock guaranteeng the fnancal sustanablty of pensons n a Pareto mprovng way, but moreover the government wll permanently change the fnancng scheme of pensons, hence generatng long run welfare gans relatve to the benchmark economy. Our exercse s slent about the reasons why any collectve decson process 24

26 would have resulted n such a dstortonary fnancng scheme n the frst place. ndeed, the demographc shock n our exercse trggers the government response, but there s no clear reason why the government should not reform the system n the frst place even n the absence of a demographc shock, purely for effcency consderatons. Ths s exactly what Conesa and Garrga (2007) does n an envronment where the government s not constraned to guarantee the pensons promsed n the past. 5. Dscusson of Results The optmal reform s obtaned by solvng the maxmzaton problem as stated n the prevous secton, wth the only dfference that we have ntroduced (22) as an addtonal constrant. We fnd that the optmal fnancng scheme mples dfferental labor ncome taxaton across age. Why would the government choose to tax dscrmnate? The crtcal nsght s that when ndvduals exhbt lfe cycle behavor labor productvty changes wth the household s age and the level of wealth also depends on age. As a result the response of consumpton, labor and savngs decsons to tax ncentves vares wth age as well. On the one hand, older cohorts are less lkely to substtute consumpton by savngs as ther remanng lfe span shortens. On the other hand, older households are more lkely to respond negatvely to an ncreasng labor ncome tax than younger cohorts born wth no assets, snce the elastcty of labor supply s ncreasng n wealth. Therefore, the optmal fscal polcy mples that the government fnds optmal to target these dfferental behavoral elastctes through tax dscrmnaton. 25

27 Fgure 5.: Evoluton of Average Taxes Captal Labor Consumpton Year Fgure 5. descrbes the evoluton of the average optmal taxes along the reform. We decentralze the resultng allocaton leavng consumpton taxes unchanged, even though t s possble to decentralze the same allocaton n alternatve ways. n partcular, we could set consumpton taxes to zero and ncrease labor ncome taxes so that they are consstent wth the optmal wedge chosen by the government. n dsplayng the results we arbtrarly label the year 2000 to be the Steady State of the benchmark economy and the reform s announced and mplemented the followng perod,.e. n Remember that a perod n the model s 5 years. Labor ncome taxes are substantally lowered the frst perod followng the reform (the combned mpact of labor ncome and payroll taxes was a 24.8% effectve tax on labor n the benchmark), but then they are ncreased to repay the ntal debt ssued and reach a new long run equlbrum around 22% on average. 26

28 Fgure 5.2 dsplays ts dstrbuton across age at dfferent ponts n tme. Fgure 5.2: Labor ncome Taxes across Dfferent Cohorts at Dfferent Tme 0.3 Fnal St.St Perod Perod Age The optmal labor ncome tax rate vares substantally across cohorts. n the fnal Steady State the optmal labor ncome tax schedule s concave and ncreasng as a functon of age, up to the pont at whch ndvduals start recevng a penson. Upon retrement the taxaton of labor ncome (remember that retrement pensons are taxed at the same rate as regular labor ncome) s hgher. Ths feature reflects the tenson between the ncentves for the fscal authorty to tax away the retrement pensons and the dstortons that ntroduces on labor supply. ntutvely, the fscal authorty ntroduces such labor ncome tax progressvty n order to undo the ntergeneratonal redstrbuton n favor of the older cohorts that the socal securty system s generatng. 27

29 As a result of ths new structure of labor ncome taxaton, ndvduals wll provde very lttle labor supply after age 65 and almost none n the last perod, as shown n Fgure 3.3. Fgure 5.3: Labor Supply across Dfferent Cohorts at Dfferent Tme ntal St.St. Perod Fnal St.St Age Notce that the shape of labor supply s not dramatcally changed wth the reform, except for the fact that ndvduals would stll provde some labor whle recevng a retrement penson. However, the amount of labor suppled by the oldest cohorts s qute small. The ntal tax cuts, together wth the ncreasng fnancal needs to fnance the retrement pensons, necessarly mply that government debt has to ncrease n the ntal perods followng the reform. Next, Fgure 5.4 dsplays the evoluton of government debt over GDP assocated 28

30 to the optmal reform. Fgure 5.4: Evoluton of Debt to GDP Rato Year n order to fnance retrement pensons debt would ncrease up to 77% of annual GDP (relatve to ts ntal 50%). Later on ths debt wll be progressvely repad. Overall, such a reform only generates welfare gans for those cohorts born once the demographc shock s over. However, the optmal response guarantees that the cohorts ntally alve and those born durng the shock enjoy the same level of utlty as n the benchmark economy. Notce that by constructon the ntal old were not ncluded n the objectve functon, and as a consequence the constrant to acheve at least the same utlty level as n the benchmark economy has to be necessarly bndng. Yet, ths was not the case for new generatons born durng the demographc shock snce they were ncluded n the objectve functon of the fscal authorty. Yet, the optmal polcy 29

31 response mples that the constrant wll be bndng, and only after the demographc shock s over wll newborn cohorts start enjoyng hgher welfare. The welfare gans accrung to newborns are plotted n Fgure 5.5. Fgure 5.5: Welfare Gans of Newborn Generatons.6.4 Planner Ramsey Year The optmal response assocated to the sustanable polcy contrasts wth the fndngs where polces are exogenously specfed as n De Nard et al. (999), where the ntal cohorts are worse-off, and Jeske (2003) where the baby boomers and the grandchldren of the baby boomers suffer welfare losses. n our economy the cost of the shock s dstrbuted over the cohorts ntally alve and those generatons born durng the shock. Remember that the latter do enter the government s objectve functon and hence the planner would be happy to allocate some welfare gans to these generatons f t 8 were possble.tpd DPT Notce that the welfare gans assocated to the reform just dscussed, labeled as 30

32 Ramsey n Fgure 5.5, are much smaller than those assocated to the Frst Best allocaton, labeled as Planner. Remember the dscusson n Secton 4. By constructon we have prevented the fscal authorty from lump-sum taxng the retrement pensons. f we were to allow the fscal authorty to tax dfferently retrement pensons from regular labor ncome, the fscal authorty would choose to do so mposng on pensons taxes hgher than a 00% effectvely replcatng a system wth lump-sum taxes. Notce that the welfare gans from dong so (labeled as Planner ) would be much hgher, especally for the ntal generatons. Ths comparson ndcates that the welfare costs of havng to use dstortonary taxaton are very hgh, especally at the ntal perods of the reform. 6. Conclusons n ths paper we have provded an answer to a very smple and polcy relevant queston: what should be the optmal response to an unantcpated transtory demographc shock? n order to answer ths queston we use optmal fscal polcy to determne the optmal way to fnance some promsed level of retrement pensons through dstortonary taxaton. n our experment, the presence of a demographc shock renders the actual way of fnancng the socal securty system unsustanable and our approach endogenously determnes how to accommodate ths shock, at the same tme that the penson fnancng scheme s permanently changed n order to reduce dstortons. We fnd that the government can desgn a Pareto mprovng reform that exhbts szeable welfare gans n the dstant future, after the demographc shock s over. Ths shows that the pressure nduced by the demographc shock s substantal, snce the reducton of the exstng large dstortons only prevents welfare losses but does not 3

33 TP TP TP TP TP PT PT PT PT PT generate welfare gans untl further away n the future. Our approach explctly provdes quanttatve polcy prescrptons towards the polcy desgn of future and maybe unavodable socal securty reforms. The optmal response conssts of the elmnaton of compulsory retrement, decreasng labor ncome taxaton of the young and a temporary ncrease of government debt n order to accommodate the hgher fnancal needs generated by the ncrease n the dependency rato. The transtory nature of the demographc shock we analyze n our artfcal economy s assumed for computatonal convenence, whle Fgure clearly shows the permanent nature of the future demographc shock faced by the US populaton structure. 2 The basc reason mght be because of dynamc neffcences, see Damond (965) or Gale (973). Also, even n a dynamcally effcent economy, socal securty mght be sustaned because of poltcal economy consderatons, see Grossman and Helpman (998), Cooley and Soares (999) or Boldrn and Rustchn (2000). Also, socal securty mght be part of a more general socal contract, as n Boldrn and Montes (2005). 3 Feldsten and Lebman (200) summarzes the dscusson on transton to nvestment-based systems, analyzng the welfare effects and the rsks assocated to such systems. 4 n order to avod sample selecton bases we assume that the rate of decrease of effcency unts of labor after age 65 s the same as n the prevous perod. 5 Notce that n a fnte lfe framework there s no problem wth dscount factors larger than, and n fact emprcal estmates often take values as large. 32

34 TP TP TP PT PT PT 6 Throughout the paper we assume that the government can commt to ts polces gnorng tme consstency ssues. Clearly, ths s an mportant restrcton that affects the results. The analyss of a tme consstent reform goes beyond the scope of ths paper. 7 We are just dentfyng one Pareto mprovng reform, but t s clearly not unque. Placng dfferent weghts on generatons or the ntal old would generate a dfferent dstrbuton of welfare gans across agents. 8 That result shows how large s the pressure nduced by the demographc shock. Ths s specally mportant snce our demographc shock s much less severe than expected even under the most optmstc scenaro (compare Fgures and 3); and second, the level of dstortons present n our benchmark economy s very hgh. Hence, our exercse s based towards generatng large welfare gans. 33

35 7. References Atknson, A.B. and J. Stgltz (980), Lectures n Publc Economcs, McGraw- Hll, New York. Boldrn, M. and A. Montes (2005), The ntergeneratonal State: Educaton and Pensons, Revew of Economc Studes 72(3), Boldrn, M. and A. Rustchn (2000), Poltcal Equlbra wth Socal Securty, Revew of Economc Dynamcs 3(), Char, V.V. and P.J. Kehoe (999), Optmal Fscal and Monetary Polcy, n J.B. Taylor and M. Woodford, eds., Handbook of Macroeconomcs, Vol. C. Elsever Scence, North-Holland, Conesa, J.C. and C. Garrga (2006), Optmal Fscal Polcy n the Desgn of Socal Securty Reforms, mmeo. Cooley, T.F. and J. Soares (999), A Postve Theory of Socal Securty Based on Reputaton, Journal of Poltcal Economy 07(), De Nard, M., S. mrohoroglu and T.J. Sargent (999), Projected U.S. Demographcs and Socal Securty, Revew of Economc Dynamcs 2(3), Damond, P. (965), Natonal Debt n a Neoclasscal Growth Model, Amercan Economc Revew 55(5), Erosa, A. and M. Gervas (2002), Optmal Taxaton n Lfe-Cycle Economes, Journal of Economc Theory 05(2), Escolano, J. (992), Optmal Taxaton n Overlappng Generatons Models, mmeo. Feldsten, M. and J.B. Lebman (2002), Socal Securty, n Auerbach, A.J. and M. Feldsten (eds.) Handbook of Publc Economcs, vol. 4, Amsterdam, 34

36 London and New York: Elsever Scence, North-Holland. Gale, D. (973), Pure Exchange Equlbrum of Dynamc Economc Models, Journal of Economc Theory 6(), Garrga, C. (999), Optmal Fscal Polcy n Overlappng Generatons Models, mmeo. Golln, D. (2002), Gettng ncome Shares Rght, Journal of Poltcal Economy 0(2), Grossman, Gene M. and E. Helpman (998), ntegeneratonal Redstrbuton wth Short-Lved Governments, Economc Journal 08(450), Hansen, G.D. (993), The Cyclcal and Secular Behavour of the Labour nput: Comparng Effcency Unts and Hours Worked, Journal of Appled Econometrcs 8(), Jeske, K. (2003), Penson Systems and Aggregate Shocks, Federal Reserve Bank of Atlanta Economc Revew 88(), 5-3. Mendoza, E., A. Razn and L.L. Tesar (994), Effectve Tax Rates n Macroeconomcs: Cross-Country Estmates of Tax Rates on Factor ncomes and Consumpton, Journal of Monetary Economcs 34(3),