Should central banks provide reserves via repos or outright bond purchases?

Transcription

1 Should enral banks provide reserves via repos or ourigh bond purhases? Johen Shanz 1 and David Miles 2 Bank of England This draf: 5 Augus Absra: In he wake of he finanial risis banks are likely o wish o hold far more highly liquid asses han before. Some of hose liquid asses are likely o be held in he form of reserves a he enral bank. We ask wheher he enral bank should provide hese reserves by purhasing nominal, fixed rae governmen bonds ourigh, or by repo ing hem in for a limied period. The key differene beween hese opions is ha wih repos, he privae seor reains he prie risk assoiaed wih bonds, whereas his risk ress wih he enral bank if i purhases hese bonds ourigh. There is a signifian, praial poliy issue for enral banks here: should hose enral banks mos noably he Fed and he Bank of England who buil up a large sok of bonds during he QE operaions, whih were finaned by reaing reserves for ommerial banks, expe o sell hose bonds in due ourse or oninue o hold a high proporion of hem for a long period sine he demand for reserves will be permanenly higher? We develop and alibrae a simple LG model in whih risk averse households hold money and bonds o insure agains risk. We find ha he omposiion of he enral bank's asses should depend on how fisal poliy is ondued; bu in general i has only a small impa on welfare. Key words: Liquidiy provision; Cenral bank balane shee JEL lassifiaion: E52, E58 1 Bank of England 2 The Moneary Poliy Commiee, Bank of England and Imperial College, London.

2 1 Inroduion The finanial risis, during whih he inerbank markes in many ounries beame wholly dysfunional, is likely o mean ha banks in fuure will wan o hold far more highly liquid asses han hey did. Some of hose liquid asses are likely o be held in he form of reserves a he enral bank. Assuming ha he enral bank aommodaes hanges in he demand for is reserves, we invesigae wha asses i should hold o bak hese reserves. Speifially, we ask wheher he enral bank should bak is liabiliies by purhasing nominal, fixed-rae governmen bonds ourigh, or by repo-ing hem in for a limied period. There is a signifian, praial poliy issue for enral banks here: should hose enral banks mos noably he Fed and he Bank of England who buil up a large sok of bonds during he QE operaions, whih were finaned by reaing reserves for ommerial banks, expe o sell hose bonds in due ourse or oninue o hold a high proporion of hem for a long period sine he demand for reserves will be permanenly higher? The key differene beween hese opions is ha wih repos, he privae seor reains he prie risk assoiaed wih fixed-rae governmen bonds, whereas his risk ress wih he enral bank if i purhases hese bonds ourigh. We absra from finanial inermediaion by banks and invesigae direly how households alloae heir wealh beween enral bank reserves money and fixed-rae nominal governmen bonds. Households need money for ransaion purposes. Bu for savings purposes, hey migh prefer o hold some governmen bonds. This ould be he ase if he yield on bonds is higher han ha of money, or if hanges in pries of bonds are posiively orrelaed wih onsumer prie inflaion, suh ha bonds help households o insure agains unexpeed inflaion. In his siuaion welfare migh be higher if he enral bank offered repos raher han purhased bonds ourigh. A repo failiy would allow an individual household o separae he problem of hoosing an opimal porfolio for savings purposes from ha of hoosing an opimal porfolio for ransaion purposes. There is a separae reason for why repos may be he preferred opion for providing reserves. Changing offi ial ineres raes affes households deisions no only onem- 1

3 poraneously bu also in fuure periods. When ineres raes are se, fuure realisaions of shoks are unknown, suh ha i is unerain wheher he level of oday s ineres raes is opimal given fuure realisaions of shoks. We show ha when reserves are provided via repos, moneary poliy affes onemporaneous deisions o a greaer exen, and fuure deisions o a lesser exen, han when reserves are provided via ourigh purhases of bonds. The inuiion is ha he value of households porfolio oday depends on oday s ineres raes only when households own bonds: he value of money oday is independen of oday s ineres rae. We deermine he welfare-maximising linear ineres rae rule for eah regime for providing money for ransaion purposes repo and ourigh bond purhases and ompare welfare ahieved in eah regime, given ha he enral bank follows he respeive opimal ineres rae rule. We find ha wheher reserves are provided via repos or ourigh purhases has only a small impa on welfare. Repos are generally opimal, unless he enral bank is unable o onrol he ineres rae preisely. When ineres onrol is poor, he marke prie of bonds beome very volaile. If households own bonds, his inreases he volailiy of he prie level beause households rely on selling heir bonds o finane onsumpion when old. They dislike his unerainy beause hey are risk-averse. In onras, he enral bank holds any bonds i purhases ourigh unil mauriy and an herefore look hrough variaions in heir marke prie. We also find ha he enral bank s welfare-maximising linear ineres rae rule only differs somewha beween he wo regimes for providing money for ransaion purposes. ur framework is an LG model in whih moneary poliy affes real variables no beause of friions in produ or labor markes, bu beause i leads o inergeneraional ransfers of wealh. We assume ha fisal poliy does no offse hese inergeneraional ransfers. 1 An aggregae produiviy shok reaes risk ha anno be raded in finanial markes. Households have o inves in money whih an be hough of as a shor-erm bond and long-erm bonds o parially insure agains risk. These bonds earn differen reurns in equilibrium. This is beause households are risk averse, and shor-erm and 1 See Serk and Tenreyro 2013 for a reen paper in whih moneary poliy works hrough similar hannels. 2

4 long-erm finanial asses allow hem o respond o risks in differen ways. Variaions in he enral bank s holdings of governmen bonds an depending on how fisal poliy is se affe a household s wealh, and herefore is real deisions. Beause a larger ransaion friion learly favours reserve provision via repos being able o separae he problem of hoosing an opimal porfolio for savings purposes from ha of hoosing an opimal porfolio for ransaion purposes only maers wih srily posiive ransaion friions we fous in our numerial resuls on he ase in whih hese ransaion friions are zero. Even in his ase,] we find ha providing reserves via repos is generally preferred, alhough he reserve provisioning regime has relaively lile influene on welfare. Geing he reserves provisioning regime wrong resuls in a welfare loss of abou 0.03% of GDP per year in our base alibraion, equivalen o 1.5% of GDP when umulaed and disouned a a real rae of 2%. Holding he nominal poliy ineres rae onsan insead of allowing i o vary opimally wih produiviy leads o a welfare loss five imes as large, even when he reserve provisioning regime is hosen opimally. In he onex of our model, he problem of hoosing he bes way of providing reserves o he privae seor is equivalen o ha of hoosing he opimal mauriy of governmen deb. A siuaion in whih he enral bank provides reserves via ourigh purhases of governmen bonds is equivalen o one in whih he deb managemen offi e redues he mauriy of he bonds i issues. There is a well-esablished lieraure on he impa of governmen deb mauriy and he enral bank balane shee. Muh of he lieraure finds ha swihes in he omposiion of asses by he enral bank are irrelevan. For example, Wallae 1981 showed ha he pah of he governmen s sok of liabiliies ha is he omposiion of is porfolio for a given fisal poliy is irrelevan in a model wih omplee markes. In his ase open marke operaions ondued by a enral bank ha purhases governmen bonds of any mauriy in exhange for oher liabiliies wih differen mauriies have no impa on real ouomes. Chamley and Polemarhakis 1984 showed ha he neuraliy resul ould also hold in a world wih inomplee markes bu only so long as he enral bank purhased real asses. Sargen and Smih 1987 showed ha a neuraliy, or ineffeiveness, 3

5 resul for governmen finanial poliies whih inlude he kind of enral bank purhases of asses we onsider in his paper ould hold in a world in whih governmen issued fia urreny is dominaed in rae of reurn. Bu as in Chamley and Polemarhakis, he resul holds for open marke operaions where physial apial is exhanged for urreny and where here are simulaneously aleraions in lump sum axes and ransfers. In Woodford and Eggerson 2003 an infiniely lived, represenaive household maximizes uiliy in a world wih omplee markes and faes no limi on borrowing agains fuure inome. I is lear ha wih hese assumpions enral bank purhases - whih are essenially swaps of asses wih he represenaive agen an do nohing beause ha single represenaive agen owns he balane shee of he enral bank and suh swaps do no hange is opporuniy se. Bu heir resuls do no apply in more realisi seings, in whih households are no homogenous and markes are no omplee, and in whih fisal poliy anno be se o offse he disribuional impa of moneary poliy. ur model allows for all hose feaures whih mean ha hese neuraliy resuls do no hold. So he quesion as o he bes way for he enral bank o fund is liabiliies largely reserves held by ommerial banks is no made irrelevan by assumpion. In he firs par of his paper we desribe he model. Seion 2 desribes he model in non-ehnial erms; Seion 3 presens he formal sruure. Seion 4 onains he resuls. We onlude in Seion 5. 2 Model verview The model has he same sruure as Miles and Shanz Following Wallae 1981, Sargen and Smih 1987 and ohers, we model he impa of enral bank purhases of governmen bonds wihin he framework of an overlapping generaions model. There are wo asses in he eonomy: money, and governmen bonds. Households live for wo periods. When bonds are issued hey also have a mauriy of wo periods. Eah generaion is born wihou an endowmen bu he abiliy o ransform heir own labor ino a perishable onsumpion good. Produion uses labor only; here is no apial 4

6 aumulaion. The produion ehnology has sohasi produiviy. Eah young household deides how muh labor o supply o produe he onsumpion good, and how muh of his o onsume. They sell he remainder o old households, in exhange for money, and deide how many governmen bonds o buy. Beause he onsumpion good perishes unless onsumed, young households an only ransfer wealh o when hey are old by holding money or governmen bonds. Neiher young nor old households an borrow. Money is remuneraed a he poliy rae se by he enral bank. Money ould be hough of as Treasury bills, or bills issued by he enral bank. Bu we ould jus as well hink ha here are 100% reserve baked ommerial banks ha are inermediaries beween households, who hold deposis, and he enral bank, whih pays ineres on reserves. Eiher way, we an hink of money in his model as deposis reserves of he enral bank ha are is liabiliy and whih pay a rae of ineres equal o he enral bank s poliy rae. All money is ineres bearing as long as he enral bank ses a nonzero ineres rae. We make his assumpion beause in developed eonomies non-ineres bearing noes and oins are very muh smaller han ineres bearing aouns whih an be easily used o finane ransaions. The Treasury issues bonds a a disoun; bonds do no pay oupons. The amoun issued is onsan in eah period. Bonds have a mauriy of wo periods a issuane. This fisal rule is a very simple one whih keeps he fae value of deb onsan he marke value of governmen deb depends on real shoks o produiviy. Taxes, whih are lump sum, are varied o saisfy he fisal rule. We absra from redi risk of governmen bonds. Households pay sae-dependen nominal lump sum axes o he Treasury. The governmen is able o levy differen axes on he young and old. ld households have simple deisions o make: hey have no beques moive so simply liquidae all heir asses o finane purhasing he onsumpion good whih hey buy from young households. We assume old households do no supply labor. Wha disinguishes money and bonds in our model? Money in his model is he only 5

7 asse ha an be used o buy goods. Bonds mus be sold for money o purhase he onsumpion good. The enral bank akes deposis, whih we ould hink of as reserves ha a ommerial banking seor holds agains deposis held by households. Reserves money pay a 1 period ineres rae se by he enral bank. The enral bank will use a poliy rule o se he rae, whih will be some funion of he sae variables in he model. The wo finanial asses money and bonds differ beause money has a known nominal value 1 period ahead one plus he ineres rae se by he enral bank oday while newly issued bonds have a value one period ahead whih depends on he ineres rae ha he enral bank will se in he nex period whih is no known oday. Bonds wih one period lef o mauriy are perfe subsiues for money beause boh asses have a know nominal value one period ahead. This means ha he prie of a one period bond is ied o he ineres rae se by he enral bank. The finanial asses in he model are dramaially simple in fa as simple as is possible while allowing longer daed governmen bonds o be imperfe subsiues for shorer daed finanial asses. The enral bank balane shee is sraighforward: i holds 1 period bonds and eiher 2 period bonds regime 1 or repos regime 2 as asses whih i aquires in exhange for issuing money is liabiliy. Repos are repurhase agreemens beween he enral bank and a household. In a repo, a household sells a 2 period bond o he enral bank agains enral bank reserves and onras o purhase i bak a period laer for he sales prie imes he enral bank s gross poliy rae. The enral bank remuneraes he reserves reaed by he poliy rae: so overall, he repo is osless for boh he enral bank and households. The repo only serves o replae bonds wih money in households porfolio during a period during whih households migh have o ransa using money. Any profis or losses made by he enral bank from is porfolio of asses and liabiliies is passed o he Treasury and axes are raised or lowered aordingly so as o ensure ha he Treasury an oninue o issue an unhanged quaniy of new bonds o replae hose ha maure. In eah period : 1. The sohasi produiviy of young households beomes known. This is an ag- 6

8 gregae shok. oung households deide how muh labor o supply o produe he onsumpion good. The enral bank ses he ineres rae for he remuneraion of money from o ld households reeive ineres on heir money holdings from 1 o. If hey enered a repo wih he enral bank in 1, his repo is now unwound; old households hen sell hese bonds whih now have a remaining mauriy of one period o he enral bank in exhange for money. The enral bank has o aep all 1-period bonds sold by old households a he prie implied by is hoie for he poliy rae. ne an hink of hese ransaions as open marke operaions ondued by he enral bank o implemen a pariular deision over he poliy rae. ld households use heir money o purhase some of he young households newly-produed onsumpion good. ld households die. 3. The Treasury issues new 2-period bonds o young households and olles axes o balane he budge. 4. The enral bank eiher offers young households o purhase newly issued bonds ourigh regime 1 a he marke prie, or o repo hem in regime 2. oung households are assumed o need ash o exploi subsequen, unmodelled, rade opporuniies. Figure 1 is a shemai overview of he iming. ne-period bonds are perfe subsiues o money in our model beause heir prie is known a he sar of period i is deermined by he ineres rae prevailing from o + 1; he assumpion ha he enral bank purhases hese bonds is herefore inonsequenial. In onras, he enral bank s hoie beween 1 purhasing bonds wih a remaining mauriy of wo periods, and 2 repo-ing in wo-period bonds, maers beause i affes he reurn haraerisis of households porfolio. We do no expliily model why young households migh need o ransa using ash wih eah oher. Expliily modeling hese ransaions would ompliae he model wihou affeing he impa ha alering households porfolio omposiion has on heir 7

9 behavior. Bu one migh hink of he following jusifiaion, whih would leave he model s firs order ondiions unhanged. Imagine ha one households have sold a share of heir produion o he old, eah young household may, wih some probabiliy, find ha he good hey reained for onsumpion has perished. This risk is idiosynrai and an herefore be perfely insured. Insurane is implemened as follows. There is a single insurane ompany whih ompensaes in ash households who have los heir good a he end of eah period. This payou is finaned from onribuions by households whose good has no perished. Suppose households need o onsume before he end of he period. Then a young household whose good has perished needs o finane purhases of he onsumpion good while i is waiing for he insurane payou. If oher households aep only ash, he household muh prefers o own ash raher han bonds for he purpose of hese ransaions. A osless repo wih he enral bank allows he household o raise his ash wihou hanging he reurn disribuions of his savings. How does moneary poliy affe real variables in his onex? Variaions in nominal ineres raes affe he aions of boh urren and fuure generaions of households. When oday s ineres rae is raised, oday s young expe o earn a higher nominal reurn on heir porfolio. The assoiaed posiive wealh effe means ha he young work and save less in real erms. Their real onsumpion inreases. In addiion, if old households hold bonds in heir porfolio, heir nominal wealh falls. The prie level falls as well, and young households have less inenive o sell heir good. So young households work and save even less. This means ha varying he poliy rae has a sronger impa on households onemporaneous behaviour when bonds are held by households raher han by he enral bank. An inrease in oday s ineres rae also affes real variables in he nex period beause he wealh of omorrow s old inreases, raising he prie level omorrow, hereby enouraging omorrow s young o produe more. When bonds are owned by households, he impa of moneary poliy on oday s households is sronger, and he volailiy of he prie level omorrow is smaller beause households own less money, and he value of heir bonds omorrow does no depend on 8

10 oday s ineres rae. 3 Model: Deailed Speifiaion 3.1 Households ur noaion is as follows. We index individual households by j. Labor supply of a young household born in period is h j,. Using his labor inpu, he household produes y j, = ω h α j, real unis of he onsumpion good. ω is an aggregae produiviy shok, disribued independenly aross periods aording o a log-normal disribuion wih mean µ ω and sandard deviaion σ ω. A young household s real onsumpion in period is j,. Is nominal money holdings are m j,, and is holdings of bonds issued in and mauring in + 2 are denoed by g j,. The marke-learing prie of hese bonds is P g,+2, and ha of bonds wih a remaining mauriy of one period is P g,+1. The prie of he onsumpion good is P. The lump-sum ax is denoed τ if levied on he young, and τ if levied on he old. We do no indiae he dependene of he endogenous variables on he sae variables of our model bu i should be aken as read. Eah household s period uiliy is of he CRRA variey: u, h = 1 ρ 1 h ρ 1 σ 1 1 σ 1 and lifeime uiliy realling no-one an work when hey are old is: U j,, j,+1, h j, = u j,, h j, + βe [ u j,+1, 0 s ] 2 where he expeaion is aken over fuure saes given he household s informaion in period, summarized by he model s sae variables s. Eah household maximizes is lifeime uiliy over { h j,, m j,, g j,} subje o he budge onsrains P yj, j, = m j, + P,+2g g j, + τ 3 m j, 1 + i + P g +1,+2g j, τ +1 = P +1 j,+1 4 9

11 and m j,, gj, 0. The lef-hand side of 3 is he proeeds from selling he onsumpion good, and he righ-hand side is he young household s use of he proeeds: i holds some of i in money, uses some o purhase newly issued bonds, and pays some lump-sum axes. Noe ha he young do no buy 1 period bonds, whih are perfe subsiues for money. We assume ha he enral bank sands ready o swap one period bonds for money hese are open marke operaions required o esablish a pariular 1 period ineres rae. The lef-hand side of 4 is he nominal wealh of he old afer axes, wj,+1: his is he sum of remuneraed money holdings and he reeips from selling bonds now 1-period o he enral bank, minus ax paymens. Noie ha one ould equally wrie his problem as one of hoosing any oher hree of he period- deision variables { j,, h j,, m j,, gj,} ; i is learly opimal for he old households o spend heir enire nominal wealh on he onsumpion good in he absene of a beques moive. 3.2 Moneary poliy The enral bank s poliy insrumen is he nominal ineres rae a whih i remuneraes money Bank Rae. We ompare welfare under wo regimes of supplying money o households for ransaion purposes. In he firs, he enral bank repoes in wo-period governmen bonds agains money. Repos are unwound in he following period. in he seond, he enral bank purhases wo-period governmen bonds ourigh from he privae seor and holds hem unil mauriy. Beause we are ineresed in he deision of how bes o sruure he enral bank s asses in he long run raher han in using bond purhases as a onjunural insrumen, we assume ha he amoun of bonds purhased or repo-ed in is onsan. We make he inonsequenial assumpion ha he enral bank buys all bonds wih a remaining mauriy of one period from old households. The enral bank s asses herefore omprise all governmen bonds wih a remaining mauriy of one period, and eiher woperiod bonds purhased ourigh regime 1 or repo-ed in regime 2. Is liabiliies onsis of money whih we an hink of as issued direly o households or else as held as reserves by ommerial banks who issue deposis o households of exaly equivalen value. 10

12 We assume ha he poliy rule for he ineres rae is a linear funion of he produiviy shok, ω, he random innovaion o he level of Bank Rae a moneary poliy shok, ε, and old households money holdings a he sar of period, l 1 = m i 1, before hey onver heir bond holdings ino money. We show below ha he model an be wrien in erms of hese hree variables as sae variables. We also assume ha he poliy ineres rae is subje o a zero lower bound. Wihin his bound, he poliy rule akes he form i = a 1 + a 2 ω µ ω + a 3 l 1 µ l + ε 5 where a 1, a 2, and a 3 are salars. The amoun of bonds ha he enral bank eiher purhases or repos in, g CB, is limied by he Treasury s onsan issuane of bonds, γ. The enral bank anno issue any liabiliies oher han money g CB [0, γ]. 3.3 Fisal poliy Fisal poliy only involves seing he size of he nominal lump-sum axes o levy on households. There are no governmen expendiures oher han ransfers o he enral bank on mauriy of he bonds, and hose required o indemnify he enral bank for any losses i may make. Governmen revenues onsis only of axes levied on households and of profis ha he enral bank may make. We assume ha he governmen balanes is budge in eah period by levying an appropriae amoun of lump-sum axes. The amoun of bonds issued is assumed o be onsan in eah period. We se his quaniy a γ = 1 so ha a any ime here are bonds wih aggregae fae value of 2 ousanding. The oss of serviing he ousanding zero-oupon deb are booked on an arual basis. In eah period, he ax is hen equal o he nominal reurn ha ha period s old earned on heir porfolio: τ + τ = i 1 m 1 + P g,+1 P g 1,+1 γ g CB The governmen may be able o levy non-zero lump sum axes a differen raes on he 11

13 young and old alive a he same ime. Bu ax poliy may be more onsrained. We will onsider wo more onsrained senarios: eiher he young are axed τ = 0, or he old τ = 0. If he old are axed, eah generaion in equilibrium pays as ax he exa amoun hey earned on heir finanial asses during heir lifeime. This should be inerpreed as a fisal poliy ha does no aemp o redisribue wealh aross generaions. I may herefore no ome as a surprise ha he omposiion of he enral bank s asses has no impa under his ax rule: any impa of asse purhases on he reurn on households porfolio is neuralized by his ype of fisal poliy. We show his formally in our ompanion paper, Miles and Shanz In onras, if he young are axed, eah generaion in equilibrium pays as ax he amoun ha he previous generaion earned on is finanial asses during is lifeime. In his ase, enral bank asse purhases hange he omposiion of he youngs porfolio, ransfer wealh aross generaions, and has he poenial o affe heir real deisions. We herefore fous on a fisal poliy whih axes exlusively he young. 3.4 Equilibrium This seion haraerizes he equilibrium of he model. Le x j, { j,, j,, h j,, y j,, m j,, g j, denoe he veor of household j s hoies. A symmeri raional expeaions equilibrium, ha is, one where all agens wihin one generaion ake he same deisions, is a se of oningen plans { j,, j,, h j,, y j,, m j,, g j,}, pries, axes, a nominal ineres rae and bond purhases by he enral bank, { P, P g,+1, P g,+2, τ, τ, i, g CB}, and exogenous } 12

14 proesses {ω, ε }, saisfying a all daes, for all households j, and a all saes: U m j, U g j, = 0 6 = 0 7 U h j, = 0 8 P j, = m j, + P g,+2g j, + τ 9 P j, = m j, i 1 + P g,+1g j, 1 10 y j, = ω h α j, 11 1/P g,+1 = 1 + i 12 g = γ g CB 13 + = y 14 τ = m 1i 1 + P g,+1 P g 1,+1 g 15 i = a 1 + a 2 ω µ ω + a 3 l 1 µ l + ε 16 l 1 = m i 1 17 In equilibrium, all households of a given ohor make he same deisions. 6 is household j s firs-order ondiion wih respe o he young s money holdings; 7 he firs-order ondiion wih respe o bond holdings; and 8 he firs-order ondiion wih respe o labor supply. The firs-order ondiions for money and bonds an be expressed in he ypial Euler equaion form. Subsiuing he budge onsrains for a young household s onsumpion ino he lifeime uiliy yields = U 1 ω h α j, τ 1 σ [ 1 m +βe 1 σ j, 1 + i + P+1,+2g g j, τ + m j, + P,+2g g j, /P 1 ρ 1 hj, ρ 1 σ 1 /P +1 1 ρ1 σ 1 ] 18 The opimal soluion has firs order ondiions 13

15 wih respe o money holdings, m j, : U m j, = u j,, h j, 1 P + βe [ u j,+1, 0 ] 1 + i = 0 19 P+1 where we denoe he marginal uiliy wih respe o onsumpion as u, h = 1 ρ 1 σ1 ρ 1 1 h ρ1 σ 20 wih respe o holdings of newly issued bonds, g j,: U g j, = u j,, h j, P g,+2 P [ + βe u j,+1, 0 P g +1,+2 P +1 ] = 0 21 The firs-order ondiions for money and bonds an herefore be wrien as u u j,, h j, = βe [1 + i P j,, h j, = βe [ P g +1,+2 P g,+2 P +1 u j,+1, 0 ] 22 P u P+1 j,+1, 0 ] i P /P+1 is he real gross reurn on money; P g +1,+2/P g,+2 P /P+1 is he real gross reurn on bonds. 9 is he budge onsrain of he young: he revenues from selling heir produion ne of own onsumpion equals heir nominal savings and ax paymens. 10 is he budge onsrain of he old: hey onsume heir enire savings. 11 is he produion funion. 12 says ha he gross reurn of bonds wih a remaining mauriy of one period mus equal ha on money: his is beause he nominal reurn of hese wo asses is known a. 13 and 14 are he marke learing ondiions for bonds and he onsumpion good, respeively. 13 saes ha per-person purhases of newly issued bonds mus equal he ne per-person supply of bonds: he differene beween issuane, γ, and he amoun of newly issued bonds ha he enral bank buys per person, g CB. 14 saes ha in equilibrium, he period per-person onsumpion of he old and he young mus equal per-person produion in, y. 14

16 15 is he fisal poliy rule: axes are se equal o he governmen seor s paymens o he household seor: his is a balaned budge onsrain. 16 is he moneary poliy rule. The budge onsrains 9 and 10 imply a ondiion for equilibrium in he marke for money: m i 1 + P g,+1g j, 1 = m + P g,+2g j, + τ The lef-hand side is he nominal wealh of he old, whih he old use o pay for he onsumpion good; he righ-hand side shows wha he young do wih he money earned. The ax rule implies ha he nominal wealh of he young pos axes, w, is onsan and so is sae-independen. Using 9, and 15, w = m + P g,+2g j, = P = P τ m 1i 1 + P g,+1 P g 1,+1 = m i 1 + P g,+1g 1 = m 1 + P g 1,+1g 1 = w 1 g 1 m 1 i 1 + P g,+1 P g 1,+1 g 1 We impose he iniial ondiion ha w 1 = w. Equaions 6-17 show ha he endogenous variables in period depend on hisory via m 1, i 1, P g 1,+1, and g 1. Beause we assume ha he enral bank s purhases of governmen bonds are onsan, g 1 = γ g CB. We also know ha when he governmen axes he young, heir pos-ax nominal wealh is onsan. So m 1 + P 1,+1g g 1 = m 1 + P g 1,+1 γ g CB = w suh ha P g 1,+1 is a linear funion of m 1. Finally, noie ha he balaned budge rule requires he ax o be equal o he nominal reurn on he previous period s young households savings. So we an rewrie he ax as τ = l 1 + γ g CB / 1 + i w 15

17 where l 1 = m i 1. This means ha he model, afer imposing symmery, an be wrien suh ha i only depends on l 1, ω, and ε. We hoose hese hree variables as he sae variables. The remaining seven endogenous variables are,, h, i, m, P and P g,+2. They are deermined by he following seven equaions: u u, h = βe [1 + i P P +1 u +1, 0 ] 24 [ 1/ 1 + i+1 P, h = βe P g u,+2 P+1 +1, 0 ] 25 1 h = ρ/ 1 ρ 26 m = w P g,+2 γ g CB 27 P = l 1 + γ g CB / 1 + i / 28 = ω h α 29 i = a 1 + a 2 ω µ ω + a 3 l 1 µ l + ε 30 4 Resuls I is useful o keep in mind ha he value of having nominal asses is ha hey an lif welfare above he deenralised soluion ha would arise if hese asses did no exis. If households had no aess o nominal asses, hey would be unable o save. In he unique deenralised equilibrium, young households would onsume heir enire produion, he old would sarve, and he real ineres rae would be 100%, beause he onsumpion good is perishable. I is well known how o improve welfare above wha is ahieved in he deenralised soluion. ne possibiliy is a sheme ha resembles unfunded soial seuriy: ransfers beween young and old households ha provide old households wih jus he righ amoun of nominal wealh ha allows hem o onsume preisely he firs-bes amoun a given marke pries. This does no rely on households having aess o nominal asses for savings purposes: insead, he firs bes is implemened by a simple ransfer beween onemporaneous generaions, where he size of he ransfer depends on produiviy. The firs bes an herefore be implemened while keeping he fisal budge in balane. 16

18 However, he axes and ransfers assoiaed wih he implemenaion of he firs bes soluion require fisal poliy o be flexible o a degree ha appears unrealisi. In praie, workers onribue a larger share o he Treasury s inome han pensioners, for example via axes on wages and salaries and on firms profis. We herefore onsrain axes o fall only on he working generaion young households levied in order o pay he reurns on money and bonds issued by he Treasury and he enral bank. This onsrain, in ombinaion wih he requiremen ha he fisal budge needs o be balaned in eah period, renders fisal poliy ompleely passive. Taxes do no lead o a disribuion of wealh ha ahieves he firs bes. In his siuaion, nominal asses - reserves and bonds - an help improve welfare beause hey allow households o ransfer wealh ino reiremen, and he reserve provisioning regime maers o he exen ha i affes he reurns ha households earn on hese asses. We firs haraerize he firs bes alloaion assuming ha he poliymaker an hoose households onsumpion and labour supply direly and show how i an be implemened via ransfers when fisal poliy is unonsrained Seion 4.1. The more ineresing ase is a moneary eonomy - ha is one in whih money an bonds are willingly held - in whih fisal poliy is onsrained o axing only he young o balane he budge in eah period Seion Firs bes alloaion and implemenaion when fisal poliy is only onsrained o ahieve a balaned budge We define welfare as he unondiional expeed uiliy of all households alive a some poin in ime: W = E s [ u, h + u, 0 ] where he expeaion is aken over all realizaions of he sae variables s = l 1, ω, ε, ha is, before he realisaion of he shoks is known. 2 To deermine he firs-bes alloaion wih respe o his definiion of welfare, we assume ha he soial planner hooses 2 This is a imeless measure of welfare; i is no ondiioned on l 1. When evaluaing he expeaion numerially, we firs alulae he disribuion of l 1 and hen draw from his disribuion. 17

19 labour supply and onsumpion levels direly. He solves max { j,,h j,, j,} W for all households j, subje o he ondiion ha aggregae onsumpion mus no exeed aggregae produion: + y. Proposiion 1 saes he inuiive resul ha in he welfare maximizing alloaion, he enire produion of he perishable good is onsumed in eah period, and ha he effe of inreasing he onsumpion of he young, 31, and of inreasing heir labour supply, 32, on he sum of he uiliies of he young and he old mus equal zero. Proposiion 1 The firs-bes alloaion is given by he soluion o u, h u, 0 = 0 31 u h, h + u, 0 y h ω, h = 0 32 for all. Proof. The firs bes alloaion has he propery ha all young households a a given poin in ime produe and onsume he same, and all old households a a given poin in ime onsume he same, beause he uiliy funion is srily onave in boh onsumpion and leisure. We herefore resri aenion o alloaions ha have he following symmery properies: for all households i, j, i, = j, 33 h i, = h j, 34 i, = j, 35 18

20 We omi individual-speifi subsrips in he following. Sine = y, we an wrie he welfare maximizaion problem as max {,h} E [ u, h + u y ω, h, 0 ] 36 The firs-order ondiions of his problem are, for all, given by 31 and 32. We solve equaions 31 and 32 for he speifi uiliy and produions funions 1 and 11 in he annex; see Proposiion 3. Labour supply is onsan in he firs bes alloaion, while onsumpion and produion are proporional o he produiviy shok. Proposiion 2 saes ha he firs-bes alloaion an be implemened under a simple and inuiive ombinaion of ax and ineres rae rules. Equaion 37 saes ha he ineres rae is onsan, and equal o he inverse of he disoun faor. Equaion 38 says ha axes on he old are se suh ha he old reain exaly he amoun of wealh neessary o purhase he firs-bes amoun of he good,. an be alulaed from he equaions deermining he firs bes alloaion, 31 and 32 in Proposiion 1 independenly of axes; w is he pos-ax nominal wealh of he young, a onsan. Equaion 39 saes ha axes on he young are se o balane he governmen s budge in eah period. Proposiion 2 Le denoe he firs-bes onsumpion level of old households. The poliymaker an implemen he firs-bes alloaion by seing 1 + i = 1/β 37 τ = w 1 + i w u, 0 38 τ + τ = iw 39 The firs bes an be implemened even hough he governmen s budge is required o be balaned in eah period: Taxes on young households ensure ha he governmen s budge is balaned in eah period eq. 39. The nominal ineres rae is held onsan a he inverse of he disoun faor eq. 37. Seing he nominal ineres rae a his 19

21 level enourages a young household o save a he welfare maximizing level. The proof is in he annex. Wih a onsan ineres rae, here is no role for bonds separae from ha of money in our deenralized eonomy. Bond purhases by he enral bank are ompleely neural, and welfare is he same independenly of how he enral bank provides reserves for ransaion purposes. As disussed above, neuraliy resuls of his sor have been derived elsewhere in he lieraure, under he assumpion ha fisal poliy an offse he disribuional effes of moneary poliy. Bu here are good reasons o assume ha his assumpion is oo srong. The ax rule ha implemens he firs bes in our model is, effeively, a ax ha depends only on he age of he household. In he following seion, we onsider a ax rule ha is more resriive bu probably more realisi: where axes are levied only on workers he young. 4.2 Impa of enral bank asse purhases when he firs bes anno be implemened We now onsider he impa of enral bank asse purhases when he Treasury an ax only young households. This makes sense for praial purposes, as mos of he governmen s ax revenue resuls from axing workers raher han pensioners. 3 By varying he nominal ineres rae, moneary poliy an hen play a role in improving welfare by reaing ransfers beween generaions whih fisal poliy, onsrained o axing only young households, is unable o ahieve by iself. We urn o numerial opimisaion o solve for opimal poliy. Seion explains how we alibrae he model. Seion shows he enral bank s balane shee for he base alibraion. Seion presens our resuls for he opimal reserve provision regime, and opimal linear ineres rae rules in hese regimes. 3 We invesigaed he opposie assumpion, where only he old are axed, in Miles and Shanz We showed ha in his ase, even hough bonds are no perfe subsiues for money he erm premium is generally non-zero, enral bank asse purhases are neural. Their proof arries hrough o he model presened here. 20

22 4.2.1 Calibraion Table 1 summarizes he parameers of he model in he base alibraion. There is ineviably a ension beween waning he model o be simple so using wo period lives and realism. Two period lifes means periods mus be long. Tha srehes he naure of he moneary poliy deision unomforably, beause we wan he poliy rae o be se for one period. Bu for our purposes wha really maers is ha we have one asse a bond wih a life whih is signifianly longer han he period for whih he ineres rae se by he enral bank an be known wih some erainy. Correspondingly, he key haraerisi of money in our model is no is mauriy bu he absene of ineres rae risk. We should hink of a period as abou half an adul life so of he order of 25 years. We se he disoun faor, β, o 0.66, implying a disoun rae of 0.5 or 50%. Wih a 25 year period ha orresponds o a disoun rae of abou 2% a year. For he uiliy funion, we se he exponen ρ on onsumpion and leisure suh ha in equilibrium hours worked are abou a fifh of maximum labor supply, orresponding o he idea ha people on average work abou 8 hours for 220 days per year, ha is 8 220/ = 20% of heir oal ime. This is approximaely he ase for ρ =.85. For he CRRA risk aversion parameer, σ, we use a value of 4 for our base ase bu also presen resuls for lower risk aversion. We assume ha he produion funion is linear in labor. This is a naural assumpion in a model ha overs he long run bu, a he same ime, omis apial as an explii produion faor: wha we all labor inpu should bes be hough of as a omposie apial and labor inpu. Assuming onsan reurns o sale hen seems plausible. We assume ha he labour produiviy shok is log-normally disribued wih parameers 0, 0.2. This ranslaes ino a sandard deviaion of 20% and a mean of jus above 1. The sandard deviaion of labor produiviy relaive o is mean is SD [y/h] /E [y/h] = SD [ωh/h] /E [ωh/h] = 20%. This orresponds approximaely o he sandard deviaion of derended labor produiviy in he UK and he US sine 1855 over non-overlapping 25-year periods. Using daa sine 1889 provided by Carer e al 21

23 2006, ha of he US appears o be a lile lower. For he base alibraion, we assume ha he sandard deviaion of he poliy innovaion is zero. We also show resuls for posiive sandard deviaions, mainly o be able o ompue impulse response funions for ineres rae shoks. Nominal pos-ax wealh of he young, w, and he fae value of bonds issued in eah period, γ, are seleed suh ha he enral bank s asses bonds and is money liabiliies are approximaely equal in seady sae for a poliy rule under whih he enral bank purhases all newly issued bonds. In his ase, he enral bank owns bonds wih a marke value of γ P g,+1 + P g,+2. Normalising γ = 1, we se w = 1/β + 1/β 2 = The enral bank s balane shee Table 2 shows how he enral bank s balane shee depends on he realisaion of he produiviy shok in olumns for he base parameerisaion of he model, in whih i is opimal o provide reserves via repos and o lower he ineres rae when produiviy is high. The enral bank s asses onsis of repos and bonds. In he ase illusraed here, he enral bank does no purhase any newly issued ie, wo-period bonds ourigh. Insead, households purhase hese bonds and repo hem ou o he enral bank. The repo posiion is herefore equal o he marke value of newly issued bonds. Beause he ineres rae falls in he aggregae produiviy shok, he marke value of hese bonds inreases as produiviy rises. The posiion Bonds wih remaining mauriy of one period refles our assumpion ha he enral bank purhases all bonds wih a remaining mauriy of one period. This assumpion is inonsequenial beause he prie of hese bonds is erain one he enral bank has se oday s ineres rae, so hese bonds are equivalen o money. The enral bank s liabiliies onsis of money. Some has been reaed from remuneraing reserve aouns; his amoun is independen of onemporaneous produiviy beause reserves were remuneraed a he previous period s ineres rae. Money is also reaed when he enral bank purhases bonds ourigh or when i repos hem in. Money is absorbed when new bonds are issued and when young households are axed. 22

24 Beause axes are se o balane he budge, hey keep onsan he value of laims on he governmen money and bonds. ne-period bonds are exhanged agains money, so he enral bank s liabiliies remain onsan when he enral bank provides reserves via repos. While axes ensure ha he enral bank s liabiliies remain onsan, is asses vary wih produiviy beause he value of newly issued bonds depends on he prevailing ineres rae. When produiviy is low, ineres raes are high, and newly issued bonds are worh lile, so he enral bank s equiy is negaive. If i held bonds ourigh, i would also book a loss on is holdings of bonds wih a remaining mauriy of one period. The opposie is rue when produiviy is high Welfare and opimal ineres raes for differen regimes for providing reserves When he Treasury an levy sae-dependen axes on boh he old and he young, he ineres rae ha implemens he firs bes is onsan and equal o he inverse of he disoun faor, while axes vary wih produiviy see Proposiion 2. When he Treasury an only ax he young, he opimal ineres rae shows some sae dependene. This is beause he enral bank an ransfer wealh beween generaions hrough one-off variaions of he poliy rae. Moneary poliy aims o offse he fa ha axes are resried o fall on he young only. Bu moneary poliy anno fully subsiue for flexible axaion. In pariular, varying oday s ineres rae shifs nominal wealh beween young and old households nex period. Nex period s ransfers depend on oday s produiviy. This is no a feaure of he firs-bes rule, where nex period s ransfers only depend on nex period s realisaion of produiviy. In his sense, an aivis moneary poliy risks inroduing noise in he disribuion of wealh of fuure generaions. So i may no surprise ha he ineres rules ha are opimal for he parameerisaions we onsider respond only weakly o produiviy shoks. When ineres raes are no very volaile, money and bonds are relaively lose sub- 23

25 siues in equilibrium, and hanges in he enral bank s balane shee sruure maer lile for welfare. ur resuls sugges ha he umulaed disouned welfare gain from geing he reserves provisioning regime righ lies in he order of 1% of he level of aggregae produion. Seion desribes he impulse responses o produiviy and ineres rae shoks. Seion fouses on he ineraion of opimal reserve provisioning and assoiaed opimal ineres rae rule. Impulse responses Table 3 shows impulse responses for parameers ha are idenial o he base alibraion Table 1 bu allows for modes poliy innovaions σ ε = 10% o be able o rae he impa of an ineres rae shok. Wealh effes dominae subsiuion effes hroughou in his alibraion. This is also he ase in he oher alibraions ha we onsider. Panel 3.1 holds he ineres rae onsan in order o isolae he effes of a produiviy shok of +20%. Produion rises, bu by less han produiviy as young households respond o he posiive wealh of higher produiviy effe by onsuming more and by reduing heir labour supply. This redues he prie level, inreases expeed inflaion, and lowers he real ineres rae. oung households respond o he negaive wealh effe of a lower real ineres rae by inreasing heir savings raio. Panel 3.2 shows impulse responses o an ineres rae shok of +10pp. A one-off inrease in he nominal ineres rae inreases he nominal wealh of he old omorrow. The assoiaed inrease in omorrow s prie level is dampened: a higher pries, omorrow s young have a greaer inenive o sell heir good. n balane, he real ineres rae rises. The young oday respond o he assoiaed posiive wealh effe by onsuming more and saving less. If old households hold bonds, a higher ineres rae redues heir nominal wealh, lowers he prie level, and redues oday s young households inenive o sell heir good. Moneary poliy does no lead o a wealh ransfer beween onemporaneous generaions beause axes are se o keep he youngs pos-ax wealh onsan. 24

26 pimal reserve provisioning and ineres rae rules Table 4 shows he opimal reserve provisioning regime ol. 5 and he assoiaed ineres rae rule ols. 6-7, seady sae, and welfare ols for a variey of parameerisaions. We se he oeffi ien a 3 on he predeermined sae variable l 1 o zero: varying his parameer affeed welfare so lile ha we were ofen unable o find an opimum ha was robus o he differen ses of realisaions of he shoks. 4 The opimal reserve provisioning regime urns ou o be a orner soluion for all parameer ombinaions ha we invesigaed, wih he exepion of he ase in whih he ineres rae was, for all praial purposes, independen of he produiviy shok row 7. Repos appear o be generally preferable ol. 5, bu he welfare differenes are small. This is illusraed in olumns 12 and 13. In olumn 12, welfare is expressed in onsumpion equivalens relaive o firs bes. For example, a value of 98.9% means ha if, saring from he firs bes alloaion, he onsumpion levels of all households were redued o 98.9% of heir firs-bes levels, hen welfare would be jus as high as in he deenralised soluion. In olumn 13, we assume ha reserves are provided no by he opimal regime repos, in he base ase, bu by he worse mehod ourigh purhases, in he base ase. Column 13 shows by how many perenage poins he welfare measure of olumn 12 delines in his ase when moneary poliy is allowed o adjus opimally o he hanged provision of reserves. For example, he value 0.03 means ha he reduion in welfare under he wrong reserve provisioning regime is equal o 0.03% of aggregae onsumpion for as long as he regime is in plae. This number is relaively small. Disouned over all fuure periods he welfare gain of hoosing he reserve provisioning regime orrely is, if we disouned he fuure by a 2% real rae, equal o 0.03%/2% = 1.5% of annual GDP. This is he same order of magniude as in he oher alibraions. urigh purhases of all bonds are opimal when here is volailiy in he pries of bonds ha is unrelaed o hanges in produiviy: ha is, when innovaions o he enral bank s poliy rule have a high variane rows 5-6. This appears inuiive: If he enral bank holds hese bonds, he volailiy of bond pries has no impa on any 4 In mos ases, he algorihm found values of a 3 beween zero and 0.3 o be opimal. 25

27 variables in he model, in pariular no he prie level, beause axes do no hange when he marke prie of bonds hanges. In onras, if hey are held in privae seor, he marke value of he porfolio of he old is subje o shoks due o fluuaions in he ineres rae unrelaed o produiviy. Wih risk-averse households, welfare improves if hese random fluuaions are avoided. Rows 1-4 and 7-8 show ha when he enral bank an onrol he ineres rae preisely, repos are he opimal form of providing reserves. As explained before, his refles ha when bonds are owned by households, he desired impa of moneary poliy on oday s households is sronger, and he oss - in erms of volailiy of he prie level omorrow - is smaller beause households own less money, and he value of heir bonds omorrow does no depend on oday s ineres rae. Repos leave he prie risk assoiaed wih bond holdings wih households, and are herefore generally preferred. The same reasoning also suggess ha poliy should be more aivis when bonds are owned by households. This is indeed suggesed by he simulaions, whih show ha he response of ineres raes o produiviy shoks, a 2, has a larger absolue value when reserves are provided via repos. Rows 5-6 onsider senarios in whih ourigh purhases are preferred. For hese parameerisaions, he enral bank s ineres rae rule onains noise he sandard deviaion of he implemenaion error, σ ε, is posiive. If bonds are held by households, his inreases he volailiy of heir nominal wealh for reasons unrelaed o produiviy. If, in onras, he enral bank owns hese bonds, variaions in he marke value of he bonds have no impa on axes, nor on any oher nominal nor real variables in he model. Hene he resul ha if ineres raes are se wih some noise, reserves should be provided via ourigh purhases. For mos parameerisaions, i is opimal o lower ineres raes when produiviy is high. This means ha bonds are worh less when he prie level is high and he prie level is low, see Corollary 1 of Proposiion 2 and o raise raes when produiviy is low and he prie level is high. Bonds herefore do no provide insurane agains inflaion shoks. Bu households neverheless hold hem beause hey earn a small posiive erm 26

28 premium ol. 6. If he enral bank purhases bonds, his premium falls somewha no shown. This erm premium urns negaive in he parameerisaion in row 8, for whih i is opimal o raise he ineres rae when produiviy is high, beause bonds in his ase do provide insurane agains unexpeed inflaion. The erm premium is small beause ineres raes do no vary muh wih produiviy: for example, in he firs parameerisaion, he ineres rae falls by = 0.05, or 5pp, from is seady sae level of 48.4% in response o a one sandard deviaion inrease in produiviy. The erm premium ends o inrease wih he degree of risk aversion rows Conlusions We have developed a simple and highly sylized model of he eonomy o assess wheher shifs in he balane shee of he enral bank have a signifian impa on real variables. Given he reen finanial risis banks are likely o demand a onsiderably higher sok of liquid asses han before. Some of hose liquid asses are likely o be held in he form of reserves a he enral bank. Assuming ha he enral bank aommodaes hanges in he demand for is reserves, i would have o deide how o provide he reserves. We invesigae wo opions: ourigh purhases of nominal, fixed rae governmen bonds, versus repos, whih leave he prie risk of bonds wih he privae seor. Analyial soluions o ha model are no, in general, available. So we urn o simulaions of a alibraed version of his LG model. We find ha aross a fairly wide se of environmens wih differen rules for he seing of ineres raes and differen ways in whih fisal poliy is ondued he impa of hanging he way in whih he enral bank provides reserves o he privae seor working hrough a porfolio re-balaning hannel is weak or absen. This suggess ha enral banks ha bough large quaniies of bonds, finaned by reaing reserves, in heir moneary poliy operaions QE need no feel any urgeny abou selling hem should ommerial bank demand for reserves say high. This is beause our resuls sugges ha he benefis of he enral bank swihing from holding bonds agains reserves o providing reserves hrough repo are no likely o be large. If here is signif- 27