The Simple Analytics of Helicopter Money: Why It Works Always

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1 Vol. 8, Augus 21, 2014 hp://dx.doi.org/ /economics-ejournal.ja The Simple Analyics of Helicoper Money: Why I Works Always Willem H. Buier Absrac The auhor proides a rigorous analysis of Milon Friedman s parable of he helicoper drop of money a permanen/irreersible increase in he nominal sock of fia base money rae which respecs he ineremporal budge consrain of he consolidaed Cenral Bank and Treasury he Sae. Examples are a emporary fiscal simulus funded permanenly hrough an increase in he sock of base money and permanen QE an irreersible, moneized open marke purchase by he Cenral Bank of non-moneary soereign deb. Three condiions mus be saisfied for helicoper money always o boos aggregae demand. Firs, here mus be benefis from holding fia base money oher han is pecuniary rae of reurn. Second, fia base money is irredeemable iewed as an asse by he holder bu no as a liabiliy by he issuer. Third, he price of money is posiie. Gien hese hree condiions, here always exiss een in a permanen liquidiy rap a combined moneary and fiscal policy acion ha booss priae demand in principle wihou limi. Deflaion, lowflaion and secular sagnaion are herefore unnecessary. They are policy choices. JEL E2 E4 E5 E6 H6 Keywords Helicoper money; liquidiy rap; seigniorage; secular sagnaion; cenral bank; quaniaie easing Auhors Willem H. Buier, Ciigroup Global Markes Inc., 388 Greenwich Sree, New York, NY 10013, USA, Ciaion Willem H. Buier (2014). The Simple Analyics of Helicoper Money: Why I Works Always. Economics: The Open-Access, Open-Assessmen E-Journal, Vol. 8, hp://dx.doi.org/ /economicsejournal.ja Receied May 21, 2014 Published as Economics Discussion Paper June 13, 2014 Reised Augus 6, 2014 Acceped Augus 17, 2014 Published Augus 21, 2014 Auhor(s) Licensed under he Creaie Commons License - Aribuion 3.0

2 1 Inrocion Le us suppose now ha one day a helicoper flies oer his communiy and drops an addiional $1000 in bills from he sky,... Le us suppose furher ha eeryone is coninced ha his is a unique een which will neer be repeaed, (Friedman 1969, pp 4 5). This paper aims o proide a rigorous analysis of Milon Friedman s famous parable of he helicoper drop of money (Friedman 1948, 1969). A helicoper drop of money is a permanen/irreersible increase in he nominal sock of fia base money wih a zero nominal ineres rae, which respecs he ineremporal budge consrain of he consolidaed Cenral Bank and fiscal auhoriy/treasury henceforh he Sae. An example would be a emporary fiscal simulus (say a oneoff ransfer paymen o households, as in Friedman s example), funded permanenly hrough an increase in he sock of base money. I could also be a permanen increase in he sock of base money hrough an irreersible open marke purchase by he Cenral Bank of non-moneary soereign deb held by he public ha is, QE. The reason is ha QE, iewed as an irreersible or permanen purchase of non-moneary financial asses by he Cenral Bank funded hrough an irreersible or permanen increase in he sock of base money, relaxes he ineremporal budge consrain of he Sae. Consequenly, here will hae o be some combinaion of curren and fuure ax cus or curren and fuure increases in public spending o ensure ha he ineremporal budge consrain of he Sae remains saisfied. QE relaxes he ineremporal budge consrain of he consolidaed Cenral Bank and Treasury eiher if nominal ineres raes are posiie or because fia base money is irredeemable. In our simple model, QE is he irreersible purchase by he Cenral Bank of soereign deb funded hrough irreersible base money issuance. The same resuls would hold, howeer, if he Cenral Bank purchased priae securiies ourigh insead of soereign deb, or expanded is balance shee hrough collaeralized lending. There are hree condiions ha mus be saisfied for helicoper money as defined here o always boos aggregae demand. Firs, here mus be benefis from holding fia base money oher han is pecuniary rae of reurn. Only hen will base money be willingly held despie being dominaed as a sore of alue by nonmoneary asses wih a posiie risk-free nominal ineres rae. This means ha in a cashless economy, like he Woodford-Gali (Woodford 2003, Gali 2008) worlds in 1

3 which somehing called money seres as a numéraire bu eiher has no exisence as a sore of alue (currency, an accoun wih he Cenral Bank or e-money) or yields no non-pecuniary benefis, earns he same pecuniary rae of reurn as bonds and is no irredeemable, helicoper money is ineffecie. Second, fia base money is irredeemable: i is iew as an asse by he holder bu no as a liabiliy by he issuer. This is necessary for helicoper money o work een in a permanen liquidiy rap, wih risk-free nominal ineres raes a zero for all mauriies. Third, he price of money is posiie. The paper shows ha, when he Sae can issue unbacked, irredeemable fia money or base money wih a zero nominal ineres rae, which can be proced a zero marginal cos and is held in posiie amouns by households and oher priae agens despie he aailabiliy of risk-free securiies carrying a posiie nominal ineres rae, here always exiss a combined moneary and fiscal policy acion ha booss priae demand in principle wihou limi. Deflaion, inflaion below arge, lowflaion, subflaion and he deficien demand-drien ersion of secular sagnaion are herefore unnecessary. 1 They are policy choices. This effecieness resul holds when he economy is away from he zero lower bound (ZLB), a he ZLB for a limied ime period or a he ZLB foreer. The feaure of irredeemable base money ha is key for his paper is ha he accepance of paymen in base money by he goernmen o a priae agen consiues a final selemen beween ha priae agen (and any oher priae agen wih whom he exchanges ha base money) and he goernmen. I leaes he priae agen wihou any furher claim on he goernmen, now or in he fuure. The helicoper money drop effecieness issue is closely relaed o he quesion as o wheher Sae-issued fia money is ne wealh for he priae secor, despie being echnically an inside asse, where for eery credior ha holds he asse here is a debor who owes a claim of equal alue (see Painkin 1965, Gurley and 1 The erm lowflaion is, I beliee, e o Moghadam e al. (2014). The erm subflaion has been around he blogosphere for a while. I use i o refer o an inflaion rae below he arge leel or lower han is opimal. Secular sagnaion heories go back o Alin Hansen (1939). I refer here o he Keynesian arian, which holds ha here will be long-erm sagnaion of employmen and economic aciiy wihou goernmen demand-side inerenion. There also is a long-erm supply side arian, associaed e.g. wih Rober Gordon (2014), which focuses on falering innoaion and prociiy growh. Larry Summers (2013) marries he demand-side and supply-side secular sagnaion approaches by inoking a number of hyseresis mechanisms. For a formal model see Eggersson and Mehrora (2014). 2

4 Shaw 1960 and Pesek and Saing 1967), Weil (1991). The discussions in Hall (1983), Sockman (1983), King (1983), Fama (1983), Helpman (1983), Sargen and Wallace (1984), Sargen (1987) and Weil (1991) of ouside money, priae money and he paymen of ineres on money ask some of he same quesions as his paper, bu do no offer he same answer, because hey don address he irredeemabiliy of fia base money. Krugman (1998), Sims (2001, 2004), Buier (2003a, 2004) and Eggersson and Woodford (2003, 2006) all sress ha o boos demand in a liquidiy rap, base money increases should no be, or expeced o be, reersed. None of hese papers recognized ha een a permanen increase in he sock of base money will no hae an expansionary wealh effec in a permanen liquidiy rap unless money is irredeemable in he sense deeloped here; wihou his, here is no wealh effec or real balance effec from irreersible base money issuance in a permanen liquidiy rap. Ben Bernanke spen years liing down he moniker helicoper Ben which he acquired following a (non-echnical) discussion of helicoper money (Bernanke 2003). The issue has also been reisied by Buier (2003b, 2007) and, in an informal manner, by Turner (2013), by Reichlin e al. (2013). The paper shows ha, because of is irredeemabiliy, sae-issued fia money is indeed ne wealh o he priae secor, in a ery precise way: he iniial sock of base money plus he presen discouned alue of all fuure ne base money issuance is ne wealh, an ouside asse o he priae secor, een afer he ineremporal budge consrain of he Sae (which includes he Cenral Bank) has been consolidaed wih ha of he household secor. This irredeemabiliy of base money and he resuling asymmeric reamen of base money in he solency consrains of households and of he sae accouns for our base money expansion/qe effecieness a he zero lower bound (ZLB), when Eggersson and Woodford (2003) (henceforh EW) esablished he exisence of a self-fulfilling deflaionary rap a he ZLB and ineffecie base money issuance or QE. In mos of he EW paper, base money is reaed symmerically in he solency consrains of he Sae and he household secor. When, owards he end of he EW paper, a fiscal rule is inroced ha effeciely imposes asymmeric reamen of base money in he solency consrains of he Sae and he household secor idenical o wha we assume, QE effecieness a he ZLB is presen, een in he EW model. 3

5 The paper also demonsraes ha fia base money issuance is effecie in boosing household demand regardless of wheher here is Ricardian equialence (deb neuraliy). Finally, he effecieness of helicoper money requires ha here is a rae-ofreurn dominaed (excep a he ZLB) sore of alue ha is willingly held by he priae secor and ha is irredeemable. Base money mus be rae-of-reurndominaed (equialenly, base money mus yield non-pecuniary benefis o he holder) if helicoper money is o hae wealh effecs away from he ZLB or if he economy is a he ZLB emporarily. Irredeemabiliy of base money is required for helicoper money o hae wealh effecs een if he economy is a he ZLB foreer. In a cashless economy, where money exiss only as a numéraire, he wealh effec of helicoper money drops canno exis eiher a or away from he ZLB and i is no possible do discuss he opic of helicoper money. In he Woodford (2003) cashless world, where here is a securiy issued by he goernmen called money which seres as he numéraire, yields he same pecuniary rae of reurn as nonmoneary securiies and yields no oher (non-pecuniary) benefis, here can be no effecie helicoper money drops away from he ZLB or if he economy is a he ZLB emporarily. If Woodford s money were irredeemable (his specificaion of he solency consrain of he Sae suggess i is no) here could be effecieness of helicoper money drops if he economy were a he ZLB foreer. 2 The model All imporan aspecs of how helicoper money drops work and wha makes helicoper money unique can be esablished wihou he need for a complee dynamic general (dis)equilibrium model. All ha is needed is a complee specificaion of he choice process of he household secor in a moneary economy, he period budge ideniy and solency consrain of he consolidaed general goernmen/treasury and Cenral Bank he Sae and he no-arbirage condiions equaing (in principle risk-adjused) reurns on all non-moneary sores of alue and consraining he insananeous nominal ineres rae o be nonnegaie. 4

6 I shall show ha, as long as he price of money is posiie, he issuance of fia base money can boos household consumpion demand by any amoun, gien he inheried socks of financial and real asses, gien curren and fuure wages and prices, and gien curren and fuure alues of public spending on goods and serices. Wheher such helicoper money drops change asse prices and ineres raes, goods prices, wages and/or oupu and employmen depends on he specificaion of he res of he model of he economy including, in more general models, he behaior of he financial secor and of non-financial businesses in driing inesmen demand, procion and labor demand, he res of he supply side of he economy and he res of he world, if he economy is open. The poin of his paper is o show ha, whaeer he equilibrium configuraion we sar from, helicoper money drops will boos household demand and mus disurb ha equilibrium. Wha gies ulimaely, in a fully ariculaed dynamic general equilibrium model nominal prices and wages, employmen or oupu, is no our concern here. The model of household behaior I use is as sripped-down and simple as I can make i wihou raising concerns ha he key resuls will no carry oer o more general and inricae models. The coninuous-ime Yaari-Blanchard ersion of he OLG model is used o characerize household behaior (see Yaari 1965, Blanchard 1985, Buier 1988 and Weil 1989). This model wih is easy aggregaion and is closed-form aggregae consumpion funcion includes he conenional (infinielied) represenaie agen model as a special case (when he birh rae is zero). Wih a posiie birh rae, here is no Ricardian equialence or deb neuraliy in he Yaari-Blanchard model. Wih a zero birh rae here is Ricardian equialence. This permis me o show ha helicoper money drops boos household demand regardless of wheher here is Ricardian equialence or no. Apar from he uncerain lifeime ha characerized households in he Yaari-Blanchard model (which plays no role eiher in Ricardian equialence or he effecieness of helicoper money drops), he model has no uncerainy. To sae on noaion I consider a closed economy. 5

7 2.1 The household secor We consider he household and goernmen secors of a simple closed economy. The holding of inrinsically worhless fia base money is moiaed hrough a money-in-he-direc uiliy funcion approach, bu alernaie approaches o making money essenial (cash-in-adance, legal resricions, money-in-he ransacions-funcion or money-in-he procion funcion, say) would work also. For exposiory simpliciy, here is only priae capial. The helicoper money we discuss could, howeer, be used equally well o fund goernmen inesmen programs as ax cus or ransfer paymens ha benefi households, or boos o curren exhausie public spending Indiial household behaior A each ime 0, a household born a ime s maximizes he following uiliy funcional: α θ( ) 1 α ms (,) max E e ln c ( s, ) d P ( ) { c( s, ), ms (, ), b( s, ), k( s, ); s, } (1) c( s, ), ms (, ) 0, θ > 0,0 < α < 1 where E is he condiional expecaion operaor a ime, θ > 0 is he pure rae of ime preference, c(,) s is consumpion a ime by a household born a ime s, ms (,), b(,) s and k(,) s are, respeciely, he socks of nominal base money, nominal risk-free consan marke alue bonds and real capial held a ime by a household born a ime s, and P( ) 0 is he general price leel a ime. 2 The cashless economy where money only seres as a numéraire is he special case of his model when α = 0. Each household faces a consan (age-independen) insananeous probabiliy 1 of deah, λ 0. The remaining expeced life ime λ is herefore also age- 2 If a uni of real capial is inerpreed as an ownership claim o a uni of capial (equiy), hen k can be negaie, zero or posiie. If i is inerpreed as a uni of physical capial iself, k has o be nonnegaie. 6

8 independen and consan. The randomness of he iming of one s demise in he only source of uncerainy in he model. I follows ha he objecie funcional in (1) can be re-wrien as: α ( θ+ λ)( ) 1 α ms (,) max e ln c ( s, ) d P ( ) (2) { c( s, ), ms (, ), b( s, ), k( s, ); } Households ac compeiiely in all markes in which hey operae, and asse markes are complee and efficien, wih free enry. In paricular, here exis acuarially fair annuiies markes ha offer a household an insananeous rae of reurn of λ on each uni of non-financial wealh i owns for as long as i lies, in exchange for he annuiy-issuing eniy claiming he enire sock of financial wealh owned by he household a he ime of is deah. The household has hree sores of alue: fia base money, which carries a zero nominal rae of ineres and is an irredeemable financial insrumen issued by he Sae (he consolidaed general goernmen and Cenral Bank, in his noe), nominal insananeous bonds wih an insananeous nominal ineres rae i and real capial yielding an insananeous gross real rae of reurn ρ. 3 Capial goods and consumpion goods consis of he same physical suff and can be coslessly and insananeously ransformed ino each oher. Capial depreciaes a he consan insananeous rae δ 0. The real wage earned a ime by a household born a ime s is denoed ws (,) and he real alue of he lump-sum ax paid o he Treasury (lump-sum ransfer paymen receied if negaie) a ime by a household born a ime s is τ (,) s. The nominal alue of he helicoper money drop receied a ime by a household born a ime s is d(,) s. This can be iewed as a lump-sum ransfer paymen from he Cenral Bank (which is par of our consolidaed Sae) o he household secor. Labor supply is inelasic and scaled o 1. Compeiion ensures ha pecuniary raes of reurn on bonds and capial are equalized. Wih money yielding posiie uiliy, here can be no equilibrium wih a 3 In Secions 3.5, 3.6 and 3.7 we inerpre bonds as bonds ne of loans. Bonds and loans are assumed o be perfec subsiues as sores of alue. 7

9 negaie nominal ineres rae. Le r () be he insananeous risk-free real ineres P () rae and π () = he insananeous rae of inflaion. I follows ha P () i () 0 ρ() δ() = r () = i () π() (4) The insananeous budge ideniy of a household born a ime s ha has suried ill period is: 4 ms (,) + b (,) s b (,) ( () ) (, ) ( ( ) ) (,) s ms k s + ρ δ + λ k s + i + λ + λ (,) P ( ) P ( ) P ( ) (5) d(,) s + ws (,) τ (,) s + c(,) s P ( ) The real alue of oal non-human wealh (or financial wealh) a ime of a household born a ime s is ms (,) + b(,) s a(,) s k(,) s + (6) P ( ) The flow budge ideniy (5) can, using (4) and (6) be wrien as: ms (,) d(,) s a (,) s ( r () + λ) a(,) s i () + ws (,) τ (,) s + c(,) s (7) P ( ) P ( ) The no-ponzi finance solency consrain for he household is ha he presen discouned alue of is erminal financial wealh be non-negaie in he limi as he ime horizon goes o infiniy: ( r( u) ) lim a( se, ) +λ 0 (3) 4 k(,) s The noaional conenion is ha k (,) s. 8

10 Because he insananeous uiliy funcion is increasing in boh consumpion and he sock of real money balances, he solency consrain will bind: ( r( u) ) lim a( se, ) +λ = 0 (8) The erminal ne financial wealh whose presen discouned alue (NPV) mus be non-negaie includes he household s sock of base money. Noe ha in (8) base money is iewed as an asse by he holder (he household). The household may know ha base money is irredeemable ha when i owns/holds X amoun of base money, i has no claim on he issuer for anyhing oher han X amoun of base money. Base money in his model is fia base money: i is no backed by inrinsically aluable goods and serices a any fixed exchange rae). Like all fia money, i will only hae posiie alue if households beliee i o hae posiie alue. A leas in a flexible nominal price and wage economy, here will always be an equilibrium wih a zero price of money in eery period he barer equilibrium. This is no an issue will shall address in wha follows. I will resric he analysis o sricly posiie sequences of he general price leel. The opimaliy condiions of he household s choice problem imply he following decision rules for he household: ( ) c(,) s = (1 α) θ + λ j(,) s (9) j(,) s a(,) s + h(,) s (10) d(,) s ( r(u) + λ ) h(,) s = ws (,) τ (,) s e + d P ( ) (11) ms (,) α 1 = c(,) s P () 1 α i () i () 0 (12) The ne presen discouned alue of household afer-ax and afer helicoper money drops labor income, h(,) s, will be referred o as human wealh. A shorer life expecancy (a higher alue of λ ) raises he marginal propensiy o consume 9

11 ou of comprehensie wealh, or he sum of financial and human wealh j a + h. We assume in wha follows ha j > The case of saiaion in real base money balances The Cobb-Douglas insananeous uiliy funcion does no hae saiaion in real money balances for finie holdings of real money balances. There is a maerial issue wih he exisence of a liquidiy rap equilibrium or ZLB equilibrium when he demand for real money balances goes o infiniy as he nominal ineres rae goes o zero. 5 An infinie demand for real money balances can only be accommodaed by a zero price leel and/or an infinie sock of nominal money balances. In a Keynesian world (Old- or New-) he price leel is predeermined and canno drop o zero insananeously. Een in a model wih a perfecly flexible general price leel, a zero general price leel would hardly be an aracie or plausible equilibrium. Een he mos QE-enamored moneary auhoriy will hae rouble coming up wih an infinie sock of nominal base money. Wih a sicky general price leel, wha happens when i = 0 and he demand for money becomes unbounded, depends on he raioning mechanism imposed by he moneary auhoriies on would-be holders of base money when heir demand becomes unbounded (a he ZLB), and on he consequences of he raioning mechanism and he response of he priae agens o his mechanism for he equilibrium configuraion of prices and quaniies in a fully ariculaed model. I do no propose o go here in his paper. Insead I will consider a simple alernaie insananeous direc uiliy funcion ha has saiaion in real money balances a a finie leel of he sock of real money balances. The model has he exposiional adanage ha, when he economy is suck in an enring liquidiy rap (a he ZLB foreer), i exhibis effeciely he same behaior for aggregae consumpion as he Cobb-Douglas uiliy funcion model does away from he ZLB. The model wih saiaion a he ZLB shares wih he Cobb-Douglas model away from he ZLB he propery ha a permanen increase in he sock of base money always simulaes consumpion demand. 5 This issue is considered a lengh and in deph in Eggersson and Woodford (2003). I am indebed o an anonymous referee for poining ou he releance of he issue. 10

12 Consider he case of an insananeous uiliy funcion which, unlike he Cobb- Douglas funcion used hus far, has saiaion in real money balances a a finie posiie leel of real money balances. We replace equaion (2) wih (2 ): ( θ λ)( ) max ln (, ) + ms (,) e c s + u P ( ) { c( s, ), ms (, ), b( s, ), k( s, ); } ms (,) ms (,) 1 ms (,) ms (,) η u ;0 ;, 0 P ( ) = η γ ηγ P ( ) 2 P ( ) > P ( ) γ 2 1 η ms (,) η = ; > 2 γ P ( ) γ 2 (2 ) The uiliy of real money balances increases in real money balances for 2 (,) 0 ms η P ( ) γ, reaches is maximum alue of 1 η ms (,) η a =, and is 2 γ P ( ) γ 2 1 η ms (,) η consan a for >. 2 γ P ( ) γ The firs-order condiions for a household opimum now imply: ( ) η 1 i ms (, ) = if i ( ) > 0 γ γ c(,) s η if i ( ) = 0 γ (12 ) For i ( ) > 0, household consumpion demand a ime is deermined from: 2 ( 2 r( u) θ+ λ) η ( r( u) + λ) c(,) s 1 ( i() ) e d + e d = j (,) s θ + λ γc(,) s (13) γ Equaion (13) defines indiial household consumpion a ime as an increasing funcion of comprehensie household wealh: 11

13 ( ) c(,) s = f j(,) s f' = 2 ( θ+ λγ ) c(,) s > 0 for c( s, ) > 0 γ θ λ [ 2 r( u) θ λ] c (,) s + ( + ) i() e d (9 ) This is hardly surprising, because boh consumpion and (unil saiaion ses in) real money balances are normal goods. From Engel aggregaion we know ha if we hae wo goods in he insananeous uiliy funcion, hey canno boh be inferior. Since, for i ( ) > 0,, real money balances and consumpion are posiiely relaed (see (12 )) consumpion demand and money demand are boh increasing in comprehensie wealh. So i suffices o show ha helicoper money can increase he comprehensie wealh of eery household o demonsrae is effecieness. This we do below. When i ( ) = 0, 0 we are in a permanen liquidiy rap and here is saiaion in real money balances a each insan. We assume ha real money balances remain finie. The household consumpion funcion for his case is gien by ( θ λ) c(,) s = + j(,) s (9 ) This is he same as he consumpion funcion deried in (9) from a Cobb- Douglas uiliy funcion wih α = 0. Noe, howeer, ha his is where he analogy wih he Cobb-Douglas funcion case ends: when α = 0 in he Cobb-Douglas model, he demand for real balances is, from equaion (12)zero we are in a cashless economy in which money will no be held if he nominal ineres rae is posiie, because money is rae-of-reurn dominaed as a sore of alue and does no yield any non-pecuniary benefis. When i = 0 base money is no longer rae-ofreurn dominaed and households will be indifferen beween holding money and non-moneary sores of alue. When i () = 0 households may end up holding real money balances in excess of η γ. To do so does, of course, use up comprehensie wealh wihou increasing insananeous uiliy oday. Wih he uiliy of consumpion increasing wihou bound in consumpion, would a uiliy maximizing household ake resources ou of 12

14 real money balances in excess of η γ and allocae hem o curren consumpion insead? If curren consumpion were he only opion i would, bu his household 1 has an expeced lifeime of raion λ, so i would wan o allocae more o fuure consumpion as well, since opimal consumpion oer ime is characerized, boh in he Cobb-Douglas model and in he model wih saiaion in real money balances for finie socks of real money, by c(,) s = c(,) s e ( r( u) θ ) So if faced wih rendan real money balances (a leel in excess of he saiaion leel), an opimizing household would wan o raise curren consumpion and consumpion in all fuure ime periods. To increase fuure consumpion oal comprehensie wealh has o be higher, bu he household will be indifferen beween holding ha wealh in he form of base money, bonds or real capial, as he nominal yield on all hese sores of alue is zero. In wha follows, I will, excep when I deal wih he permanen liquidiy rap case, work wih he Cobb-Douglas insananeous uiliy funcion. I permis a simple closed-form soluion - unlike he non-homoheic preferences ha generae insananeous uiliy funcions capable of procing saiaion for a finie sock of real money balances. When I consider he permanen liquidiy rap special case, in Secion 2.7, I will swich o he insananeous uiliy funcion wih saiaion, which, when i comes o aggregae consumpion behaior, in he special case under consideraion only requires one o se α = 0 in he aggregae consumpion funcion deried from he Cobb-Douglas insananeous uiliy funcion, bu wihou he implicaion ha he amoun of money held is zero Aggregaion We assume ha here is a consan and age-independen insananeous birh rae ( ) β 0. The size of he cohor born a ime is normalized o βe β λ. The size of he suriing cohor a ime which was born a ime s is herefore ( β λ) s λ( s) βe e. Toal populaion a ime is herefore gien, for β > 0 by 13

15 λ βs ( β λ) βe e ds = e. For he case β = 0 we se he size of he populaion a = 0 o equal 1, so populaion size a ime is again ( β λ e ) λ = e. For any indiial household ariable x(,) s, we define he corresponding populaion aggregae X() as follows: λ βs X () = βe x( s, ) e ds if β > 0 λ = x(0, e ) if β = 0 We assume ha each household earns he same wage, pays he same axes and receied he same helicoper money drop, regardless of age: ws (,) = w () τ (,) s = τ () d(,) s = d() I follows ha each household, regardless of age, has he same human wealh: h(,) s = h () Finally, here are neiher olunary nor inolunary bequess in his model, so as (,s) = 0 (14) By brue-force aggregaion, if follows ha aggregae consumpion and money demand for he Cobb-Douglas model is deermined as follows: C () = (1 α)( θ + λ) J () (15) M() α 1 = P () 1 α i ( ) C () (16) M() D () A () r () A () i( ) + W() T() + C () (17) P () P () 14

16 D ( ) ( r( u) + β ) H ( ) = W ( ) T ( ) e + d P ( ) (18) M() + B () A () = K() + P () J() = A () + H() (19) is For fuure reference, he solency consrain of he aggregae household secor r( u) lim Ae ( ) = 0 or (20) M( ) + B ( ) r( u) lim K ( ) e + = 0 P ( ) Comparing he aggregae household financial wealh dynamics equaion (17), wih he indiial suriing household financial wealh dynamics equaion (7) shows ha he reurn on he annuiies, λ A is missing from he aggregae dynamics. This is as i should be, because λ A () is boh he exra reurns oer and aboe he risk-free rae earned by all suriing households a ime and he amoun of wealh paid o he annuiies sellers by he (esaes of he) fracion λ of he populaion ha dies a ime. Comparing he aggregae human wealh equaion (18) describing he human wealh of all generaions currenly alie bu no of hose ye o be born and he indiial suriing household s human wealh equaion (11), we noe ha if he households alie a ime were o discoun all fuure afer-ax labor income a he indiially appropriae, annuiy premium-augmened rae of reurn r + λ, hey would fail o allow for he fac ha he labor force o whom ha afer-ax labor income accrues includes he suriing members of generaions born afer ime. In he absence of he insiuion of inheried slaery, hose currenly alie canno claim he labor income of he fuure suriing members of generaions as ye unborn. Populaion and labor force grow a he proporional rae β λ, so he appropriae discoun rae applied o he fuure aggregae sreams of labor income is r + β. 15

17 2.2 The Sae The Sae whose budge ideniy and solency consrain we model is he consolidaed general goernmen (he Treasury in wha follows) and Cenral Bank. Le G denoe real public spending on goods and serices (exhausie public spending by he sae, curren and or capial). The Sae s budge ideniy and solency consrain are gien in equaion (21) and (22) respeciely. The implici assumpion ha base money can be creaed a zero marginal real resource cos (and indeed ha goernmen bonds can be issued a zero marginal M M real resource cos) is refleced in he absence of erms like µ () M (), µ () > 0 B B and µ () B (), µ () > 0 on he RHS of equaion (21). We also ignore any fixed cos of fia base money issuance, alhough any fixed cos could be buried in G(). For simpliciy we assume ha he Sae ges ax reenue only from he household secor and makes ransfer paymens (including helicoper money drops) only o he household secor. ( ) M () + B () B () D i () + G () T () + P () P () P () Because of he irredeemabiliy of base money, money is in no meaningful sense a liabiliy of he Sae. The solency consrain of he Sae herefore requires ha he presen discouned alue of is erminal ne non-moneary liabiliies be non-posiie, no ha he presen discouned alue of is erminal ne financial liabiliies be non-posiie. (21) B ( ) r( u) lim e 0 P ( ) (22) Equaion (22) is he naural way o formalize he familiar noion ha fia base money is an asse (wealh) o he holder (he owner households in his simple model) bu does no consiue in any meaningful sense a liabiliy o he issuer (he borrower he Sae or he Cenral Bank as an agen of he Sae). The owner of a $20 dollar Federal Resere Noe may find comfor in he fac ha This noe is legal ender for all debs, public and priae, bu she has no claim on he Federal Resere, now or eer, oher han for an amoun of Federal Resere Noes adding up o $20 in alue. UK currency noes worh X carry he proud inscripion 16

18 promise o pay he bearer he sum of X bu his merely means ha he Bank of England will pay ou he face alue of any genuine Bank of England noe no maer how old. The promise o pay sands good for all ime bu simply means ha he Bank will always be willing o exchange one (old, faded) 10 Bank of England noe for one (new, crisp) 10 Bank of England noe (or een for wo 5 Bank of England noes). Because i promises only money in exchange for money, his promise o pay is, in fac, a saemen of he irredeemable naure of Bank of England noes. The asymmeric reamen of base money in he solency consrains of he households and he Sae is he key assumpion underlying our effecieness proposiions for base money expansions/qe een a he ZLB. I represens a deparure from he earlier lieraure, which specified he solency consrain of he sae in erms of he non-posiiiy of he NPV of he erminal deb boh moneary and non-moneary, of he Sae (see e.g. Leeper 1991). I beliee ha he irredeemabiliy propery of fia currency ha i is an asse o he holder bu no a liabiliy of he issuer exends also o he oher componen of base money (commercial bank reseres held wih he Cenral Bank), bu he simple heoreical model does no depend on his and does no make his disincion. Unil furher noice, we assume, alhough unlike wih he household secor, here is no opimizing jusificaion for i, ha he Sae saisfies is solency consrain wih sric equaliy. The case of he sae as NPV credior o he priae secor, een in he long run, is considered briefly in Secion 2.5. Equaion (21) implies ha M() + B () D ( ) M( ) r( u) T ( ) G( ) i( ) e d P ( ) + P ( ) P ( ) M( ) + B ( ) + lim e P ( ) r( u) (23) Because of he irredeemabiliy of base money (equaion (22)), assumed o hold wih sric equaliy, he ineremporal budge consrain of he Sae is 17

19 M() + B () D ( ) M( ) r( u) T ( ) G( ) i( ) e d P ( ) + P ( ) P ( ) M( ) + lim e P ( ) r( u) (24) Subsiuing he ineremporal budge consrain of he Sae ino he aggregae consumpion funcion (15), using (18) and (19), and rearranging yields, when i ( ) > 0, : 6 β ( ) ( ( ) ) ( ) ( ( ) ( )e ) r u + β K + W G e d D ( ) ( r ( u ) ) ( ) C( ) (1 )( ) T ( ) e + β β = α θ + λ 1 e d P ( ) i ( u ) i( ) M ( ) e d 1 + P () i ( u ) + lim M( e ) (25) ( ) ( ) ( ) M ( ) ( ) r u M r u i e d + lim e ( ) ( ) 6 P P Noe ha. 1 i( u) i( u) im ( ) ( e ) = d+ lim Me ( ) P () 18

20 From inegraion by pars i follows ha 7 ( ) ( ) ( ) ( ) i u i u im e d+ lim Me ( ) i ( u ) = M ( ) e d + M () (26) I follows ha (25) can also be wrien as: β ( ) ( ( ) ) ( ) ( ( ) ( )e ) r u β + K + W G e d D ( ) ( r ( u ) ) ( ) C( ) (1 )( ) T ( ) e + β β = α θ + λ 1 e d P ( ) 1 i ( u ) + M () + M ( ) e d P () (27) 7 If insead of haing a zero nominal ineres rae, fia base money carried he possibly ime-arying M nominal ineres rae i (), equaion (26) would become M ( ) ( ) ( ( ) ( )) ( ) i u i u i i Me d+ lim Me ( ) M i ( u ) = ( M ( ) i ( ) ) e d + M ( ) wih obious modificaions required in he ineremporal budge consrains of households and he Sae., 19

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