LONG-TERM DEBT AND OPTIMAL POLICY IN THE FISCAL THEORY OF THE PRICE LEVEL

Transcription

1 Economerica, Vol. 69, No. 1 January, 001, LONG-TERM DEBT AND OPTIMAL POLICY IN THE FISCAL THEORY OF THE PRICE LEVEL BY JOHN H. COCHRANE 1 The fiscal heory says ha he price level is deermined by he raio of nominal deb o he presen value of real primary surpluses. I analyze long-erm deb and opimal policy in he fiscal heory. I find ha he mauriy srucure of he deb maers. For example, i deermines wheher news of fuure deficis implies curren inflaion or fuure inflaion. When long-erm deb is presen, he governmen can rade curren inflaion for fuure inflaion by deb operaions; his radeoff is no presen if he governmen rolls over shor-erm deb. The mauriy srucure of ousanding deb acs as a budge consrain deermining which periods price levels he governmen can affec by deb variaion alone. In addiion, deb policyhe expeced paern of fuure sae-coningen deb sales, repurchases and redempionsmaers crucially for he effecs of a deb operaion. I solve for opimal deb policies o minimize he variance of inflaion. I find cases in which long-erm deb helps o sabilize inflaion. I also find ha he opimal policy produces ime series ha are similar o U.S. surplus and deb ime series. To undersand he daa, I mus assume ha deb policy offses he inflaionary impac of cyclical surplus shocks, raher han causing price level disurbances by policy-induced shocks. Shifing he obecive from price level variance o inflaion variance, he opimal policy produces much less volaile inflaion a he cos of a uni roo in he price level; his is consisen wih he sabilizaion of U.S. inflaion afer he gold sandard was abandoned. KEYWORDS: Fiscal heory of he price level, governmen deb, price level, inflaion. 1. INTRODUCTION THE FISCAL THEORY STATES ha he price level is deermined by he raio of nominal deb o he presen value of real primary surpluses, nominal deb Ž 1. presen value of real surpluses. price level The fiscal heory is developed by Leeper Ž 1991., Sims Ž 1994, 1997., Woodford Ž 1995, 1997, 1998a, 1998b. and Dupor Ž 000. wih one-period deb, building on Sargen and Wallace Ž Cochrane Ž 1999, 000. reviews he fiscal heory, argues for is plausibiliy, and addresses many heoreical dispues. In his paper, I exend he fiscal heory o include long-erm deb. Wih long-erm deb, he nominal value of he deb on he lef-hand side of Ž. 1 is no fixed; i depends on nominal bond prices which in urn depend on expeced fuure price levels. To see why his fac migh maer, suppose ha here is bad 1 I hank he CRSP, Graduae School of Business, and he Naional Science Foundaion for research suppor, and I hank Andrea Eisfeld for research assisance. I hank Angel Serra, Michael Woodford, and an anonymous referee for unusually helpful commens. An early draf of his paper circulaed under he ile Mauriy Maers: Long Term Deb in he Fiscal Theory of he Price Level. 69

2 70 JOHN H. COCHRANE news abou fuure surpluses so he righ-hand side of Ž. 1 declines. If here is no long-erm deb, he nominal value of governmen deb is predeermined, so he price level mus rise o re-equilibrae Ž. 1. However, if long-erm bonds are ousanding, heir relaie price and hus he numeraor of he lef-hand side migh fall insead, leaving oday s price level unchanged. Lower bond prices oday correspond o expecaions of higher price levels in he fuure, so long-erm deb means ha bad news abou fuure surpluses can resul in fuure raher han curren inflaion. To analyze issues of his sor, I solve equaions like Ž. 1 for he price level, wih curren and expeced fuure surpluses and deb on he righ-hand side. I presen an exac soluion, bu i is algebraically complex. I also presen wo approximae soluions which are more convenien for many applicaions. Comparaie Saics I use he soluions o undersand he obvious comparaive saics exercises: Ž. i How does he price level reac o curren and fuure surpluses, holding deb consan? Ž ii. How does he price level reac o curren and fuure deb, holding surpluses consan? Answers o he firs quesion are paricularly useful in hinking abou evens such as currency crashes or he ends of hyperinflaions. Answers o he second quesion sugges ways in which governmen choices abou he quaniy and mauriy srucure of nominal deb can cause inflaion or offse he inflaionary impac of surplus shocks. They also allow us o hink abou open marke operaions, deliberae wiss in he mauriy srucure, and oher deb-managemen issues. In answer o he firs quesion, I find ha he effecs of surpluses on he price level depend on deb policy: Curren and expecaions of fuure sae-coningen deb sales and redempions maer as well as he mauriy srucure of ousanding deb. The effecs are ofen surprisingly differen han hose in he shor-erm deb case. For example, if he governmen pays off ousanding perpeuiies raher han roll over shor-erm deb, he price level a each dae is deermined by he surplus a ha dae raher han by he presen value of surpluses. In answer o he second quesion, I find ha he effecs of deb on he price level also depend on he mauriy srucure and on expecaions of fuure deb policy. For example, I find ha he governmen can rade inflaion oday for inflaion in he fuure, wih no change in surpluses, if and only if some long-erm deb is ousanding. Suppose ha he governmen sells some addiional deb, holding surpluses consan. If no long-erm deb is ousanding, he governmen faces a uni-elasic demand curve. Bonds are nominal claims o he same real resources, so bond prices fall one-for-one wih he number sold; real revenue from bond sales and he price level oday are unaffeced by he number sold. However, if here are long-erm bonds ousanding, selling exra deb dilues he exising long-erm bonds as claims o he fixed sream of fuure real resources. In his case, unexpeced deb sales can raise revenue oday and lower oday s price level, wih no change in curren or fuure surpluses, or in he oal marke

3 LONG-TERM DEBT 71 value of deb. Of course, selling more deb oday wih consan surpluses always raises he price level laer, as fixed real resources mus pay off a larger nominal deb. This limied conrol of he iming of inflaion is a differen mechanism han ha sudied by Sargen and Wallace Ž In ha paper, here is a moneary fricion, deb is real, and he moneary auhoriy deermines when seignorage revenue will be earned. The mechanism works wih shor-erm deb. Here, here is no moneary fricion, deb is nominal, he reasury deermines he price level pah, all revenues are held fixed, and he mechanism only works if long-erm deb is presen. For mos of he comparaive saics, sae-coningen deb policywhen he deb is expeced o be repurchased, redeemed or rolled overis crucially imporan o he resuling price level and nominal ineres rae pah. Thus, quesions such as wha is he effec of an open marke operaion? or wha is he effec of a change in he mauriy srucure canno be answered wihou specifying he full dae- and sae-coningen change in deb policy, as well as any implici changes in curren and expeced fuure surpluses. As always in dynamic ineremporal models, one mus hink abou policy rules or saeconingen sequences, raher han hink abou decisions aken in isolaion. Opimal Policy Afer sudying he comparaive saics of deb and surplus movemens, I ask wha deb and surplus policies opimally smooh inflaion, paying paricular aenion o moivaions for long-erm deb. The hree elemens of he governmen s policy choice are he average mauriy srucure, he choice of saeconingen deb sales and redempions in response o fiscal shocks, and a limied conrol of he surplus. I add each elemen in urn and analyze he resuls in erms of he above comparaive saics. I sar by analyzing opimal fixed-deb policy, in which he governmen deermines only he seady sae level of deb and is mauriy srucure; i does no adus deb in response o surplus shocks, and i canno conrol he surplus. I find ha shor mauriy srucures are preferred when he presen value of he surplus varies by less han he surplus iself; while long mauriy srucures are preferred when surpluses build up following a shock so ha he presen value varies by more han he surplus iself. This finding is a naural resul of he comparaive saics: he price level responds o he presen value of surpluses wih a shor mauriy srucure, while he price level responds o he surplus a each dae wih a long mauriy srucure. I hen analyze opimal acie policy, in which he governmen can also change he amoun of deb and is mauriy srucure each period in response o surplus shocks. Now here is a second moivaion for long-erm deb. If long-erm deb is ousanding, he governmen can smooh inflaion by occasionally and unexpecedly devaluing long-erm bonds, rading a lower price level oday for a higher price level in he fuure. This acion can smooh inflaion afer a shock

4 7 JOHN H. COCHRANE has hi. I sudy a quaniaive example in which he opimal fixed-deb policy consiss of shor-erm deb, bu he opimal acive policy includes long-erm deb so ha he governmen can smooh inflaion by such ex-pos devaluaions. Finally, I add a limied conrol over he long-erm surplus in order o model beer he siuaion faced by he U.S. governmen and he fac ha deb sales do seem o come wih promises of increased long-run surpluses. This opimal policy analysis solves some empirical puzzles. A simpleminded applicaion of Ž. 1 and is comparaive-saic predicions for he effecs of surplus and deb shocks seems disasrous for he fiscal heory in U.S. daa. However, if we regard he U.S. governmen as solving such an opimal policy problem, adaping deb and fiscal policy o defend price level sabiliy in he face of cyclical surplus shocks raher han causing price level disurbances by exogenous surplus and deb movemens, we can explain many of he iniially puzzling feaures of he daa. For example, equaion Ž. 1 suggess ha he price level should move ogeher wih oal nominal deb. On he reasonable assumpion ha he presen value of he surplus is high when he surplus iself is high, i also suggess ha he price level should move inversely wih he surplus and ha he real value of he deb should move ogeher wih he surplus. Bu none of hese paerns is an even vaguely plausible descripion of U.S. daa. Figure 1 presens he primary Federal surplusconsumpion raio and CPI inflaion. If anyhing here is a sligh posiive correlaion beween surplus and inflaion a business cycle frequencies. Figure presens he surplusconsumpion raio ogeher wih he level and FIGURE 1.Federal primary surplusnondurable services consumpion, and CPI inflaion. Boh series are expressed as percenages. For all he empirical work in his aricle, I use daa from and presened in more deail in Cochrane Ž I consruced he value of he deb as he marke value of all ousanding reasury securiies, and inferred he surplus from he rae of reurn on governmen deb and he quaniies ousanding. Dividing by consumpion gives a more plausibly saionary series, and he heory adaps easily o his ransformaion by adding consumpion growh o he rae of reurn in he formulas.

5 LONG-TERM DEBT 73 FIGURE.Real value of he debconsumpion; difference of real valueconsumpion, and surplusconsumpion raio. All series are expressed as percenages. Vc is shifed down by 45 percenage poins o fi on he same graph. difference in oal real value of he deb. Comparing he wo figures, we can see ha here is lile correlaion beween he level of deb and he price level, inflaion, or he surplus, as deb moves much more slowly han any of he oher series. The surplus is nicely negaiely correlaed wih changes in deb. Unsurprisingly wih a consan price level, bu surprisingly in erms of Ž. 1, high surpluses pay down he deb. By conras, I find ha he opimal policies ha smooh inflaion in he face of cyclical surplus shocks produce ime series ha are similar o hese U.S. ime series in many dimensions. For example, he opimal policies generae a negaive correlaion beween surpluses and deb growh, as in he daa. A Few Commens on he Fiscal Theory A hear, he fiscal heory recognizes ha even apparenly unbacked fia money is, ogeher wih nominal deb, a residual claim on governmen surpluses, and values hem as such. For example, sock is valued by number of shares Ž. presen value of fuure earnings. price per share If Microsof sock became numeraire, uni of accoun, and medium of exchange, we would ry o undersand price level deerminaionhe rae of exchange beween goods and one share of Microsofvia his equaion. The fiscal heory values governmen-issued nominal deb in exacly he same way. ŽCochrane Ž 000. pursues he sock analogy in deph.. As his analogy makes clear, he fiscal heory needs no fricionsno money demand or heory of moneyo deermine he price level. The fiscal heory can

6 74 JOHN H. COCHRANE describe a well-deermined price level for apparenly unbacked fia money in a compleely cashless economy, one in which us-mauring governmen bonds are unis of accoun bu no media of exchange. The sock analogy also suggess ha he fiscal heory s predicions for he price level will no be much affeced by he presence of moneary fricionsif some caegories of deb help o faciliae ransacions. The only poenial effecs are he small fiscal consequences of seignorage or ineres rae spreads on ransacions-faciliaing asses. The analogy also shows ha fiscal price level deerminaion is immune o financial innovaion and o privae noe issue. An agen can issue a claim o a share of sock, payable from his holdings, wih no diluion effec on he value of he underlying shares, even if he agen s claim rades a a discoun due o he risk ha he may defaul. In he same way, agens can creae and rade claims o governmen deb or banknoes wih no effec on a fiscally-deermined price level. The basic fiscal heory equaion Ž. 1 is, like he sock example equaion Ž., an equilibrium valuaion equaion, no a consrain. There is nohing ha forces Microsof Ž or Amazon.com!. o adus fuure earnings o mach curren valuaions, any more han calling Ž. 1 a governmen budge consrain forces he governmen o raise fuure axes in response o an off-equilibrium deflaion. Since he equaions apply us as well o an economy ha uses Microsof sock as numeraire and medium of exchange, he fiscal heory does no require ha one assume anyhing differen abou governmen and privae budge consrains. Iniially, he idea ha nominal deb and surpluses are policy insrumens may seem srange. Mos of he above-cied fiscal heory analyses include a moneary fricion, and a moneary policy Žconrol of an ineres rae or moneary aggregae. hus implicily deermines he evoluion of nominal deb. Wih no moneary fricion, however, nominal deb does become he nominal policy ool direcly. I is also unusual ha nominal deb and surpluses are separae policy insrumens. We are used o hinking of deb as evolving from a surplus decision. For example, wih perfec foresigh, he real value of one-period nominal deb B ha maures a evolves as 1 B1 1 B s, p r p 1 where pprice level, sprimary surplus, and rgross real ineres rae. Thus, nex period s deb seems o be deermined from las period s deb and his period s surplus. This analysis is correc for real deb, or if prices are deermined elsewhere Ž e.g. by M p y.. In a fiscal equilibrium, however, he sequences B, s 4 are chosen firs, and prices follow; he governmen does no ake he price sequence p 4 as fixed when deciding on B, s 4. For example, if he governmen conemplaes doubling B 1, i knows ha p will also double, us as Microsof knows ha is share price will halve if i does a spli. Thus, he governmen can happily conemplae a change in deb wih no change in

7 LONG-TERM DEBT 75 surpluses. The governmen can choose deb and surplus as separae policy insrumens, even in a compleely cashless economy, and no us in a limi as in Woodford Ž 1998a.. Excep for occasional currency reforms, changes in nominal deb wih no change in surpluses are unfamiliar policy pahs. Mos exra sales of nominal deb increase he real value of oal deb, and hus mus come wih an increase in expeced fuure surpluses, since he oal real value of deb always equals he presen value of fuure surpluses. ŽA simulaneous decrease in he real discoun rae is heoreically possible, bu unlikely in his conex.. Thus, our experience is largely composed of increases in deb ha accompany a decreased curren surplus and increased fuure surpluses, and, as we shall see, for good reasons: changes in deb wih no accompanying change in surpluses have dramaic effecs on he price level, and mos governmens do no wan o cause sharp flucuaions in he price level. However, he fac ha mos policy acions consis of simulaneous changes in wo levers should no cloud he fac ha he wo policy levers are nominal deb and real surpluses. We can analyze wha happens if each is moved wihou moving he oher, and hen we can beer undersand why opimal policy ypically consiss of coinciden movemens in boh levers. Since he models here are fricionless, sandard Modigliani-Miller heorems by which he mauriy srucure of he deb is irrelevan for real quaniies sill apply. I sudy he effecs of he mauriy srucure on he nominal price level; such effecs can occur even in a fricionless economy and desired nominal resuls Ž such as smoohing inflaion. can deermine opimal mauriy srucures. The issues in his paper are differen han hose sudied by mos of he lieraure on he mauriy srucure of governmen deb. Lucas and Sokey Ž 1983., Blanchard and Missale Ž 1994., and many ohers analyze ime-consisency and precommimen issues. I ignore hese imporan issues; I describe governmen policy by a sequence of sae-coningen choices of deb and of he surplus, and I presume ha he governmen can commi o carrying ou such a policy once chosen. Taxes are lump sum, so his analysis is differen from Missale s Ž obecive of smoohing real governmen revenues over he cycle wih disorionary axaion, or Calvo and Guidoi s Ž 199. mixure of disoring axes and ime-consisency issues. Boh issues are imporan consideraions for fuure research.. FISCAL THEORY WITH LONG-TERM DEBT.1. The Basic Equaions Le B denoe he face value of zero-coupon nominal bonds ousanding a he end of period ha come due in period. Le Q denoe he nominal price a ime of a bond ha maures a ime. Of course, Q Ž. 1 and B 0 for. Le p denoe he price level and le s denoe he real primary surplus, i.e. ax collecions less governmen purchases. The appendix summarizes noaion.

8 76 JOHN H. COCHRANE I model a fricionless economy in which no cash is held overnigh. The economy need no be cashless; ransacions may be faciliaed by moneyclaims o us-mauring governmen bondscreaed each morning and reired each nigh via repurchase agreemens raher han by direc exchange of mauring bonds, and any amoun of privae money, bonds, banknoes, checking accouns ec. may be creaed wih no effec on he formulas ha deermine he price level. Ignoring moneary fricions simplifies he algebra a grea deal wihou alering he firs-order predicions of he fiscal heory. I assume a risk-neural economy wih consan gross real ineres rae 1; his assumpion simplifies he formulas wih no grea loss of generaliy. The enire analysis flows from wo equivalen equilibrium condiions, derived below. The flow condiion says ha he real primary surplus s mus equal bond redempions plus ne repurchases, ž / Ž 3. E B B s, B 1 1 Ý 1 p 1 p while he presen alue condiion says ha he real value of ousanding deb equals he presen value of real surpluses, ž / Ž 4. E B E s. B 1 1 Ý 1 Ý p 1 p 0 As discussed below, he erms E Ž 1p. give real bond prices in erms of expeced fuure price levels. I use whichever form is more convenien for a given applicaion. I use discree ime for clariy, bu he model works us as well in coninuous ime. An equilibrium is a sequence of prices p 4, of surpluses s 4, and of deb of all mauriies BŽ., 1,,... 4 such ha equaion Ž. 3 or Ž. 4 holds a each dae and sae. We are ineresed in finding he price level for various specificaions of he deb and surplus policy choices. A soluion is he equilibrium price sequence for given deb and surplus sequences, i.e. an equaion wih p on he lef and oher 3 quaniies on he righ. Because prices muliply quaniies in Ž. 3 Ž. 4, soluions are no rivial o find. I describe governmen policy by he sae-coningen sequences of prices and deb, s, B Ž.4 1. I assume ha he governmen can commi o such a sequence once chosen. 3 Prices, surpluses, and deb are each random variables, so p 4 denoes a sequence of random variables, wih p in he ime- informaion se. Thus, he qualificaion each dae and sae. I limi aenion o posiive and finie values of he surplus and deb, 0EÝ 0 s and 0 Ý Q B, and o rule ou 00, 0B. 0

9 LONG-TERM DEBT 77.. Deriaion To derive Ž.Ž. 3 4, sar wih he accouning ideniy ha he primary surplus equals purchases less sales of bonds, Ý Ž 5. B Q B B p s To express bond prices in erms of fuure price levels, denoe equilibrium marginal uiliy by už c., and condiional expecaion by E so už c. p p ž už c. p / ž p / Ž 6. Q E E. The righ-hand equaliy simplifies noaion wih he assumpion of a consan real discoun facor E už c. už c. 1 and by denoing expecaion E wih respec o a risk-neural se of probabiliies. The laer sep us simplifies noaion, avoiding a marginal uiliy in every formula. The model is fricionless, so changes in he price level sequence do no affec equilibrium consumpion or he real ineres rae. Subsiuing he one-period bond price Ž. 6 in Ž. 5 and dividing by p, we obain Ž. 3. To derive Ž. 4 noe ha Ž. 3 can be wrien as Ž 1 E 1L. s, where 1 Ý ž / B Ž p Ž 1 Ieraing forward on., or applying E 1L 1 o boh sides, ogeher wih he equilibrium condiion lim E T 0, we obain Ž. 4 and vice versa. T T 3. SOLUTIONS IN SPECIAL CASES, AND SURPLUS COMPARATIVE STATICS For several specificaions of deb policyhe pah of B4 we can easily derive soluions. These soluions also allow us o address he comparaive saics quesion, how does he price level reac o changes in curren and expeced fuure surpluses, holding deb consan? 3.1. One Period Deb Suppose ha he governmen only issues one period deb, rolled over every period. This is he sandard case analyzed in he fiscal heory, for example Woodford Ž All erms B oher han B 1 1 are zero. Then, he presen value condiion, Ž. 4, specializes o a soluion direcly, B 1 Ž 7. p. E Ý s 0

10 78 JOHN H. COCHRANE Wih one period deb, fuure surpluses affec he price level oday. The price level oday responds only o he presen value of surpluses. While his case is familiar o fiscal-heory readers, i is no generally rue ha he presen value condiion is also a soluion, as we see in he remaining cases. 3.. No New Deb Suppose insead ha a full mauriy srucure is ousanding a ime 0, and he governmen neiher issues new deb nor repurchases ousanding deb before i maures. For example, he governmen could pay off a perpeuiy. In his case, deb due a is consan over ime, B Ž. B Ž. B Ž The flow condi- ion Ž. 3 is now also a soluion, B 1 Ž 8. p. s Now, prices are deermined by bonds ha fall due a each dae divided by ha dae s surplus. Shocks o fuure deficis have no influence a all on he curren price level. Insead, long-erm bond prices, reflecing fuure inflaion, enirely absorb he shocks o he presen value of surpluses. To see his fac, apply Ž. 8 a ; a shock o expeced s changes expeced 1p and hus changes bond prices Q E Ž pp.. Since i is so much simpler, his mauriy srucure should prove more useful han rolled over shor-erm deb in many heoreical applicaions of he fiscal heory k-period Deb As an inermediae example, suppose ha each period he governmen issues BŽ k. k-period discoun bonds each period, and hen les hem maure. Wih his deb policy, BŽ k. B Ž k. B Ž k. 1 k1. The flow condi- ion Ž. 3 hen becomes ž / k B k 1 k E B Ž k. s. p p This is a k-period difference equaion, wih soluion B B k 1 p. k k E Ý s E Ý s 0 k 0 k The price level is sill deermined by a sor of presen value, bu only every kh erm maers! For example, if he governmen issues 5 year deb, hen expecaions of surpluses in years 5, 10, 15, ec. maer o oday s Ž. 0 price level, bu surpluses in years 4, 6 ec. do no maer. As k1 we recover he one period deb soluion Ž. 7 in which all fuure deficis maer. As k, we recover he case Ž. 8 in which only oday s surplus maers o oday s price level.

11 LONG-TERM DEBT Geomeric Mauriy Srucure A geomeric paern gives a racable way o analyze a rich mauriy srucure. Suppose ha he amoun of deb ousanding a he beginning of Ž end of 1. ha will maure a declines a a rae : 9 B B. 1 1 Equivalenly, he fracion of deb ha maures a dae, sold a dae, follows a geomeric paern, B B A Ž 1. ; 1. B 1 Ž. If he level of deb grows a a consan rae B1 B, hen his specifica- ion also implies ha deb declines geomerically wih mauriy a any given dae, B 1 B1 B. However, he laer conclusion is no he case for arbirary movemens in deb over ime. A specificaion in which deb always falls geomerically wih mauriy does no lead o a simple price soluion, since he governmen mus do a lo of buying and selling of deb a all mauriies o mainain i. To derive a soluion for his deb policy, plug Ž. 9 ino he presen value condiion Ž. 4, and plug Ž 10. ino he flow condiion Ž. 3. Adding he firs and Ž 1. imes he second equaions and solving for p we obain he soluion, B 1 Ž 11. p. s 1 E Ý s 0 This example also ness he one-period deb case and he no-change-in-deb case as varies from 0 o SOLUTIONS The above analysis gave some special cases of soluionsprice on he lef and oher variables on he righbu leaves one hungry for more general soluions, ha apply for arbirary deb policies. Here, I presen an exac soluion, and hen wo approximae soluions ha are convenien in some siuaions Exac Soluion To find a soluion for prices in erms of deb and surplus, I sar wih eiher he flow Ž. 3 or presen value Ž. 4 condiions and recursively subsiue he same equaions for fuure values of prices p. Afer some ugly algebra ha I

12 80 JOHN H. COCHRANE relegae o he Appendix, he resul can be expressed as B 1 Ž 1. p. E Ý W s 0, To define he W weighs, firs denoe he fracion of mauriy deb issued a ime by B B 1 Ž 13. A ; 1,,.... B 1 Then, he W are defined recursively by 14 W 1,,0 W A Ž 1.,,1 W A W A,, 1,1 W A 3 W A 3 W A 3,,3, 1,1 1 Ý W A W., k, k k0 To ge some sense of wha his means, wrie ou he firs wo erms of he general soluion, ž / B Ž 1. B Ž. 1 ½ ž / B 1 B 1 Ž. 5 Ž. B B Ž Ž 15. E s 1 s p B Ž B1 s. B 1 The weighs W, capure he effecs of deb policyhe curren and fuure mauriy srucure of he debon he relaion beween he price level and he sequence of surpluses. 4.. Approximae Soluion wih a Geomeric Baseline Fuure surpluses ener Ž 1. simply, hough wih complex coefficiens. Thus, we can easily characerize he effecs of surpluses on he price level for special cases of deb policy. Deb eners in a more complex and nonlinear manner, as seen in Ž 15.. Thus, o calculae he effecs of deb policy on he price level, as well as for he opimal policy quesions, I use an approximae soluion which is

13 LONG-TERM DEBT 81 much easier o manipulae. The approximae soluion is based on a firs-order Taylor expansion of he general soluion abou a simple baseline pah. The approximae soluion akes derivaives around a baseline pah s, B Ž 1., p 4 wih geomerically growing surplus and a geomeric mauriy srucure, 16 s s, 17 B B, 1 1 B 1 1 Ž 18. s, p 1 where. I denoe by x he proporional deviaion of each variable x from he baseline pah, p p s s B B 1 1 p ; s ; B 1. p s B 1 Wih his noaion, wo expressions for he approximae soluion are convenien, 1 1 ž Ý 1 /ž / 0 Ž 19. p B B s Ž 1. Es, where Ž 0. B B Ž.; 1 Ý 0 and, in lag operaor noaion, 1 Ž 1. Ž 1 1L 1 1L. Ž 1. p B 1 E s, Ž 1. Ž 1 1M 1 1L. 1 where M operaes on mauriy as L operaes on daes, M B BŽ 1..B 1 is a nominal deb aggregae ha I will use below. Keep in mind ha i is an aggregae of nominal, face values of he deb, no an aggregae of marke alues of deb, since i is unaffeced by variaion in he price level and hence bond prices. The approximaion uses he baseline price level o value ousanding deb raher han he acual price levels, and i uses he baseline mauriy srucure raher han he acual mauriy srucure o capure he rade-off beween curren and fuure price levels. I also linearizes he produc B 1 p.as usual, linearizing a produc gives he baseline value of each erm imes he

14 8 JOHN H. COCHRANE deviaion of he oher and ignores erms in which deviaions are muliplied by each oher. The surplus erms in equaion Ž 19. are comforingly similar o hose I derived above in equaion Ž 11. for a geomeric mauriy srucure. The version in 1 equaion 1 shows ha he price level is proporional o 1L of he presen value of he surplus. For 0, we recover he presen value, bu as 1, price becomes proporional o growh in he presen value of surpluses. Deriaion Taking derivaives of he presen value condiion Ž. 4 abou he baseline pah p, B Ž. 1, s, we obain an approximae version of he presen value condiion, B Ž. B p s s. 1 Ý ž 1 / Ý 0 p 0 Formula Ž. will obviously lead o a convenien represenaion if he baseline pah is geomeric. To ha end, I specify ha he baseline pah has a geomerically growing surplus, a geomeric mauriy srucure as in Ž 16. Ž 17., and ha he raio of deb o price grows geomerically, B 1 A A. p The baseline pah mus saisfy Ž. 4, which resrics is parameers, 3, A s 1. A 1 The firs equaion says ha he real value of he deb mus grow a he same rae as he surplus. The second equaion says ha he level of real deb mus equal he level of he presen value of fuure surpluses. Wih hese resricions, we have Ž 18.. The simples such pah feaures geomeric growh in p and B Ž., 4 p p, B B. p 1 B Ž. 1 However, he individual erms B and p need no grow geomerically, so long as heir raio does so. They may even be sochasic, and hey may share a common uni roo. The baseline pah mus saisfy 1 o keep he presen value of surpluses finie, and 1 o keep he presen value of he deb finie. I is no necessary ha 1, bu such mauriy srucures are unusual enough ha we may wan 1

15 LONG-TERM DEBT 83 o impose 1 in pracice. Firs, wih 1, he governmen sells some deb of every mauriy each period, and hen redeems i all when i maures. A 1, he governmen sells or purchases no deb, simply redeeming a sock ousanding a he iniial period. Wih 11, he governmen repurchases a lile bi of every mauriy deb each period, from an iniially ousanding sock. To Ž. 1 see his, wrie deb sales each period B B B Ž To say he leas, such a pah requires fundamenally new insiuions. The mos likely implemenaion are consols ha promise an increasing coupon. The limi 1 corresponds o he limi ha he coupons grow a he nominal ineres rae. Second, 11 and deb and price level ha grow over ime imply ha he face value increases wih mauriy, and herefore he oal face value is infinie. A a minimum, his will pose a srain for curren face-value based accouning pracices. The marke alue sill declines wih mauriy, which is why such parameers are allowed. To see his, noe ha wih geomeric deb growh as in 4, he face value of deb ousanding a each dae is 1 B 1 B Ž. B Ž., while he marke value of mauriy deb is 1 1 p p B B Ž.. Using he baseline pah 16 18, wih parameers 3, in, and aking condiional expecaions, we obain a linearized presen value condiion 5 1 Ý ž B1 E p / 1 Ý Es. 0 0 Since he weighs are geomeric, ieraing Ž 5. forward o solve for p is easy, and gives he approximae price soluion, Ž 19. Ž Approximae Soluion wih a Nongeomeric Baseline Pah A linearizaion abou a nongeomeric baseline pah capures some effecs ha are ignored by he linearizaion abou a geomeric pah, bu wihou he full complexiy of he general soluion. In his case, he coefficiens in he linearizaion are similar o he general soluion, and hus algebraically complex. However, hese coefficiens only need o be evaluaed once, and he approximae soluion is hen a convenien linear funcion of surplus and deb policy. Generalize Ž 17. o an arbirary baseline mauriy srucure, 6 B B. 1 1

16 84 JOHN H. COCHRANE The counerpar o 5 no longer has a geomeric srucure, so finding a soluion requires more algebra. The soluion, derived in he Appendix, is 7 p Ý DEB 1Ý WEs. 0 0 Here, and B Ý B Ý 0 are naural generalizaions of heir counerpars wih a geomeric mauriy srucure. The W are given by W 1, 0 W A, 1 1 W A W A, Ý W A W ; k k k0 hese erms are he seady sae level of he general-soluion weighs W A are given by Ž 8. A 1 ;,. The hese erms are he seady sae level of he erms A in he general soluion. The D coefficiens are recursively generaed by Ž 9. D0 1, D1 1, DA1DA 1 D 0, D3A1DA D1A3D 0, k Ý D A D. k1 i1 ki i0 5. THE EFFECTS OF DEBT POLICY I use he approximae soluions o answer, wha are he effecs of deb changes on he price level, holding surpluses consan? Wih no change in surpluses, he approximae soluion abou a geomeric baseline pah, Ž 19., simplifies o Ž 30. p B B. 1

17 LONG-TERM DEBT 85 The firs B erm in Ž 30. means ha an increase in deb a dae 1, B 1, ha is repurchased a Ž so ha B does no also change. moves he price level p one for one. Wih one-period deb his effec is simple: more deb as a claim o he same fixed resources mus resul in a higher price level. The soluion shows ha more long-erm deb a ime 1 also raises he price level a ime, even hough he deb does no come due unil laer. The price level rises when he deb is repurchased, no when i maures. Working hrough he definiion of he deb aggregae B in Ž 0., if mauriy deb B 1 increases 1% and is hen repurchased a ime, he price level rises by percenage poins. Thus, he effec of increased deb on he price level is aenuaed for longer erm deb and as he mauriy srucure shorens. The second B erm in Ž 30. means ha an increase in deb a dae, B, can decrease he price level a ime, bu only if some long-erm deb is ousanding, i.e. if 0. If he governmen us rolls over shor-erm deb, his effec does no exis. New long-erm deb dilues ousanding long-erm deb as a claim o fixed fuure resources. The more long-erm deb is currenly ousanding, he less he diluion, and hence he more revenue he governmen can raise for each dollar of exra long-erm bond sales. In urn, he more real revenue raised, and used o redeem currenly mauring bonds, he greaer he impac on he price level. Only he aggregaes B ener his approximae soluion, so analysis using he approximae soluion will no disinguish changes in he deb aggregae B 1 brough abou by changes in deb of differen mauriies. The approximaion values changes in deb a he seady sae price level, as any firs-order approximaion mus. Thus, analysis using his approximaion will be silen abou he effecs of sae-coningen mauriy rearrangemens. Sudy of such policy will require a second-order approximaion or he exac soluion, and will no allow us o use simple linear ime-series ools. In mos cases he governmen does no sell long-erm deb and hen repurchase i one period laer. Raher, i sells addiional long-erm deb and hen les i maure. To calculae he effecs of such a policy, suppose ha a ime 0 he governmen sells an addiional 10 year bond and hen les ha bond maure. Normalizing o B Ž 10. 1, we have B Ž 10. 1, B Ž 10. 1,...,B Ž ŽSince he approximaion akes proporional deviaions from seady saes, a $1 increase in he quaniy ousanding is a larger proporional increase for longer mauriy bonds.. Using he definiion of B, Ž 0. and Ž 30., he resuling price pah is 0 p 10, 10 p 1, 1,,...9, 10 p 1. Figure 3 plos his price pah. A dae 0, we only have he second, negaive deb erm in 19 ; he price level is reduced if here is long-erm deb ousand-

18 86 JOHN H. COCHRANE FIGURE 3.Effec on he price level of an increase in 10 period deb a ime 0 ha is allowed o maure, saring in a seady sae wih a geomeric mauriy srucure. ing. A ime 10, we only have he firs, posiive erm in Ž 19., so he price level rises by 1.0 for any mauriy srucure. One more bond mus be redeemed from he same se of resources. In he inermediae daes, boh erms in Ž 19. are presen. Wih long erm deb, hey cancel so here is no inermediae effec on he price level. Wih shorer-erm deb, he price level increases all he way ou o period 10. The crucial quesion for he effecs of a deb sale is he paern of expeced fuure sales and repurchases. For example, he price level pah repored in 9 Figure 3 requires only B91, B8, B7,...,B0. This paern can be achieved us as well by selling an addiional one period bond and hen rolling over ha deb 10 periods before repaying i. All ha maers o he price pah is when he deb is expeced o be repaid. The mos imporan real-world deb operaion is an open marke operaion. In his model, an open marke operaion is exacly he same hing as a deb sale or repurchase. For example, o repurchase a bond, he governmen issues addiional us-mauring bonds, or equivalenly, money. The comparaive saics show ha he effecs of such an operaion on he price level and hence nominal ineres raes depend crucially on he mauriy srucure of ousanding deb, on simulaneous surplus movemens Žwheher he governmen spends addiional cash., and on expecaions of when and how he deb will be reiredwheher by raising fuure surpluses, or by compeing wih deb ha would be reired on a given day. A wide variey of resuls is possible by differen specificaions of hese componens of he policy change. Similarly, a revenue-neural shorening or lenghening of he mauriy srucure of he deb, as praciced by he Kennedy adminisraion and discussed in he early Clinon years Žsee Hall and Sargen Ž will have effecs on he price level and nominal ineres raes ha depend crucially on he paern of

19 LONG-TERM DEBT 87 expeced repaymens. If he governmen simply raises deb of mauriy and lowers ha of mauriy k, in a way ha he aggregae B is unaffeced Ž revenue-neural a baseline prices., and hen resores his paern every period as he deb maures Ži.e., sells some 1 mauriy deb nex period, buys some mauriy deb, ec.. hen here is no effec whasoever. However, if he governmen les he wis maure, hen he price level will rise when he deb maures and decline when he k deb maures; his expecaion will show up in ineres raes a he momen of he iniial wis Addiional Effecs wih a Nongeomeric Seady Sae The D coefficiens in Ž 7. measure he effec on he price pah of an expeced bond sale a dae 1, which will be repurchased a dae 0, 1 p k Dk. k B 1 Noe ha D 1, D Thus, despie he fac ha long-erm deb may be sold, here is no effec on prices pas period 0 when he deb is repurchased. D 1, so selling a lile more deb a period 1 and hen buying i back 0 a period 0 raises he period 0 price level. Since D 0, selling a lile 1 1 more deb a ime 1 can lower ime 1 prices, bu only if here is some long erm deb ousandingif 1 0. Ineresingly, wheher selling a lile exra period deb affecs prices immediaely depends on he presence of ousanding ime 1 deb Ž., no ime deb Ž. 1. The mauriy of he deb ha is sold does no maer; wha maers is when ha nominal deb will be repurchased, and compee wih oher deb for he fixed pool of resources. In general, he erms D, D3... are presen, so prices a can be affeced by all fuure expeced deb changes. These erms all specialize o zero wih he geomeric seady sae, in which case he price level a is only affeced by B 1 and B. To see he force of his effec, we need an example in which he mauriy srucure is far from geomeric. Suppose ha he seady sae mauriy srucure is 11, The governmen combines some shorerm deb wih some exremely long erm deb, for example a perpeuiy. Figure 4 plos he response of prices o an anicipaed deb sale a ime 0, which is hen repurchased a ime 1, for his case. All he ineresing dynamics before ime 0 would be absen wih a geomeric seady sae. 5.. Posponing InflaionThe Limis of Deb Policy As we have seen, addiional sales of long-erm deb can lower he price level oday while raising i in he fuure, when some long-erm deb is ousanding, even wih no change in surpluses. To wha exen can he governmen affec he price level oday hrough unexpeced bond sales? For example, can i compleely offse surplus shocks?

20 88 JOHN H. COCHRANE FIGURE 4.Price pah in response o an anicipaed deb sale a ime 1, which is hen repurchased a ime 0. The seady sae mauriy srucure is 11, 3 0.5, and he discoun facor is Ž. The presen value condiion 4 answers hese quesions direcly and exacly. Rewriing he condiion slighly, 1 31 E B E Ž s.. Ý 0 ž / Ý 1 p 0 We can read his equaion as budge consrain for achievable expeced inverse price levels. The mauriy srucure of ousanding deb B 1 gies he raes a which he goernmen can rade off he price leel oday for Žexpeced inerse. price leels in he fuure. The governmen can always raise fuure prices by selling more deb; he issue is wheher such sales affec oday s prices. Wih ousanding long-mauriy deb, erms B 0, 1 in Ž are presen, so ha raising fuure price levels Ž by selling more long-erm deb. can lower oday s price level. If only one-period bonds are ousanding, hese erms are absen so here is nohing he governmen can do wih deb policy o affec prices oday. Furhermore, here is a deb policya choice of B Ž i., B Ž i. 1...; i 1,,... 4 ha achiees any se of Ž expeced inerse. price pahs consisen wih he consrain Ž 31.. To verify his fac, we can consruc a policy ha works for a given price pah. I is no unique. Le he governmen adus is mauriy srucure once, deermining BŽ., and hen le he deb maure wih no furher purchases or sales. Fuure price levels are hen given by he soluion Ž. 8, and aking expecaions a ime, ž / ž / 1 s 1 E E. B p

21 Therefore, if he governmen ses E Ž s. B 1 E ž / LONG-TERM DEBT 89 p and les deb maure so ha B B Ž. 1, he desired pah of fuure price levels E Ž 1p.4 resuls. Equaion Ž 31. produces he price level a ime. The converse saemen is also rue. If here is no period deb ousanding a ime, hen here is no deb policyno choice of ŽB Ž i., B Ž i. 1...; i 1,,... by 4 which he goernmen can lower he price leel a ime in exchange for raising he price leel a ime. Can he governmen go so far as o aain a consan price level in he face of surplus shocks by appropriaely buying and selling bonds? The consrain Ž 31. shows ha his much is no possible, because deb a ime mus be in he ime informaion se. Take innovaions of equaion Ž 31., resuling in / 1 Ý Ý ž p E E s B E E. A consan price level implies Ž E E.Ž 1p. 1 0 for all. The righ side is zero and he lef side is no, so his canno be a soluion. This conclusion holds in coninuous ime versions of he model as well. Wih one period deb, we had B 1 E s. p Ý 0 Since deb was predeermined, he price level had o absorb any shocks o he presen value of fuure surpluses. Now we have equaion Ž 31.. Deb of each mauriy is sill predeermined, so revisions in he Ž expeced inverse. price level sequence mus absorb any surplus shocks. The governmen could aain a consan price level via deb policy alone if i issued sae-coningen nominal deb. For example, suppose ha he governmen issued sae-coningen deb a ime 0 and engaged in no furher deb sales or repurchases. Le BŽ. denoe he amoun of nominal deb ha comes due a dae in sae. Similarly, le sž. denoe he real surplus a ime in sae. The budge ideniy a each dae is hen simply pž. sž. BŽ.. In his case, he governmen can aain any sochasic process for prices, including a consan price level, by choosing he appropriae sae-coningen deb srucure. Though dynamic rading of long-erm deb allows a greaer array of sae-coningencies han does shor erm deb, i does no aain his complee-markes or sae-coningen limi. In his paper, I focus on non-sae-coningen nominal deb because ha is he nearly universal srucure of nominal governmen deb.

22 90 JOHN H. COCHRANE 6. OPTIMAL DEBT POLICY We have seen ha deb policy can affec he price level. Now, I search for policies ha opimally smooh inflaion. I proceed in hree sages: Firs, I find an opimal fixed-deb policy, i.e. an opimal seady sae mauriy srucure, given ha he governmen does no adus deb ex-pos in response o shocks. Then, I allow he governmen also o pursue acie deb policy, adusing he level of deb of various mauriies in order o offse surplus shocks. Finally, I allow he governmen o conrol par of he surplus as well. We can anicipae some of he qualiaive resuls. As we have seen, wih fixed deb, shorer mauriy srucures relae oday s price o many leads of he surplus, while long mauriy srucures relae oday s price o fewer leads of he surplus. Therefore, a shor mauriy srucure smoohs inflaion if surpluses have a large ransiory componen, while a long mauriy srucure will smooh inflaion when surpluses build following a shock. Long mauriy srucures also make acive deb policy possible, so ha he governmen can smooh a surplus shock as i happens by selling more long-erm deb. This fac weighs in favor of a long mauriy srucure, even when shor-erm deb is he opimal fixed-deb policy Saemen of he Problem Given a sochasic process for he surplus s 4, he governmen picks he parameers governing he seady sae mauriy srucure and a deb policy B 4 o minimize he variance of inflaion, Ž 3. minvarž p p., 1 given ha prices are generaed by he approximae soluion Ž 19.. I sae he obecive and consrains in erms of seady saes and deviaions abou he seady sae, since I use he approximae price soluion o solve he problems. In order o use he approximae soluion, I consrain he governmen s choice o a geomeric seady sae. A naural consrain se for he seady sae mauriy srucure is 01. However, as discussed above, soluions wih 1 1 are possible hough unusual given oday s insiuions. Thus, when he obecives poin o high values of, I will sudy soluions limied by 1 as well as soluions limied only by 1. Deb B mus be in he ime- informa- ion se and mus obey lim T B T T 0. Smoohing he volailiy of inflaion is a reasonable characerizaion of poswar cenral bank obecives. In his model, he level of inflaion is arbirary and so i is no ineresing o add i o he obecive. Modeling inflaion as he difference of proporional deviaions from he seady sae as in Ž 3. raher han he raio of price levels is an analyically convenien simplificaion. I also consider he obecive of minimizing variance of he price level, min varž p., which is a plausible characerizaion of moneary policy obecives in he prewar, gold-sandard regime. The mehods adap easily o oher obecives. For exam-

23 LONG-TERM DEBT 91 ple, one can minimize he variance of unexpeced inflaion min varž p E p. 1, moivaed by he Lucas Ž 197, world in which only unexpeced money has real effecs. Following a long radiion in moneary economics, for example Sargen and Wallace Ž 1975., I do no delay or complicae he analysis by usifying he price-smoohing obecive from welfare maximizaion in an economy wih specific fricions. 6.. Fixed-deb Policy I sar by analyzing fixed-deb policies: The governmen chooses only a geomeric seady sae mauriy srucure, governed by he parameer, in order o minimize he variance of inflaion given ha prices are generaed by he approximae soluion Ž 19.. I calculae resuls for an ARŽ. surplus process, s Ž. s s Figure 5 presens he opimal seady sae mauriy parameer as a funcion of he wo roos 1 and. The calculaion is deailed in he Appendix. For every saionary ARŽ.Ž 1 one roo equal o zero, he oher sricly less han one; his region is no shown in Figure 5 for clariy. he opimal mauriy is shor, 0. In hese cases he variance of he presen value of he surplus is smaller han he variance of he surplus, so shor-erm deb smoohs inflaion by making he price level equal o he smooher series. For he same reason, 0 is opimal for wo relaively small ARŽ. roos, as can be seen in he lower lef-hand corner of Figure 5. Two large posiive roos produce hump-shaped impulse response funcions ha coninue o rise afer an iniial shock, and for which he presen value varies by more han he series iself. In his case, he longes possible mauriy FIGURE 5.Opimal geomeric mauriy of passive deb policies ha minimizes he variance of Ž. inflaion, as a funcion of he wo roos of he AR surplus process

24 9 JOHN H. COCHRANE deb 1 minimizes he variance of he price leel. Long mauriies are also useful in his case o minimize he variance of inflaion, bu as Figure 5 shows, he opimal mauriy is inerior 01, and ineresingly is never much above 1. This case is no implausible, as many macroeconomic ime series have hump-shaped impulse-response paerns wih roos roughly hose of his region Acie Deb Policy Nex, I allow he governmen o adus deb of all mauriies, sill keeping he surplus process exogenous. As we have seen, his opion gives anoher moivaion for long-erm deb, since sae-coningen deb sales can pospone a shock o he price level if long-erm deb is ousanding. The problem now is o minimize varž p. or varž p p. 1 by choice of and B a each dae, given he surplus process, which I denoe Ý s Ž L.. 0 I solve his problem 4 by firs finding he opimal price process, for a given seady sae mauriy srucure. Wrie he price process as a funcion of surplus shocks as Ý p Ž L.. 0 I choose he coefficiens, subec o a consrain ha he price process mus be achievable by some deb policy Žchoice of B We can express ha consrain convenienly as follows. Wrie he linearized version of he presen value condiion Ž 5. as 1 33 EÝ p B1 EÝ s. 1 0 Since B is in he 1 informaion se, aking innovaions of Ž yields a relaion beween he responses of price and surplus o shocks ha does no involve deb, Ž 34. Ž 1. Ž 1. Ž.. Thus, I choose he weighs 4 o minimize he variance of he price level or inflaion rae subec o he consrain Ž 34.. This operaion is enough o fully characerize he opimal price process for given. Then, aking he variance of price level or inflaion, I find he opimal mauriy srucure. Finally, I solve Ž 33. for deb B 1 o characerize he deb policy ha suppors he opimal price process. 0 4 I hank Mike Woodford for suggesing his soluion sraegy.

25 LONG-TERM DEBT 93 Minimize he Variance of he Price Leel The obecive is Ý 4 0 min varž p. min subec o 34. A sraighforward Lagrangian minimizaion gives he opimum price level process, wih variance 1 Ž 1. 1 p, Ž 1. Ž 1L. Ž 1. p Ž.. Ž 1. The minimum variance occurs wih 0, and he resuling opimal price level process is Ž 35. p Ž 1.. The minimal-ariance price leel follows an i.i.d. process. Ineresingly, his is rue for any surplus process. Price variance is greaer, he greaer he response of he presen value of he surplus o is shocks, measured by. 5 Solving 33 for B, he deb policy supporing he opimal price process is 1 B1 1 E1 Ý s. 0 The deb process offses all he ime presen value of he surplus ha is known as of 1. The price level hen absorbs he shock o he presen value of surpluses only. Wih 0 deb policy can only affec he expeced price level bu canno offse shocks as hey come. Since varž p. varž E Ž p.. varž 1 p E p., deb policy aduss o se varže Ž p , by making he price level an i.i.d. process. 5 Solving 33, for B 1 and hence he resul., using 0 and subsiuing Ž 35., we have ž / Ý ž / 0 B1 1 1 E1 Ý s EE1 Ý s 0 0 Ž 1. Ž 1. E s 1