The Fallacy of the Fiscal Theory of the Price Level *

Transcription

1 The Fallacy of he Fiscal Theory of he rice Level * Willem H. Buier ** rofessor of Inernaional Macroeconomics Universiy of Cambridge and Member, Moneary olicy Commiee, Bank of England December 999 * Willem H. Buier, 999 ** The views and opinions expressed are hose of he auhor only. They do no necessarily reflec he views and opinions of he Bank of England or of he oher members of he Moneary olicy Commiee. I would like o hank Charlie Bean, David Begg, Alec Chrysal, Jon Faus, Dale Henderson, Berhold Herrendorf, Greg Hess, Mervyn King, obu Kiyoaki, Erzo Lumer, David Miles, William erraudin, Chris issarides, Danny Quah, Anne Siber, Ralph Tryon, John Vickers, Mike Woodford, Sephen Wrigh and paricipans in seminars a he LSE, Birkbeck College, he Bank of England and he Federal Reserve Board, for helpful discussions and commens on earlier incarnaions of his paper. The responsibiliy for any errors is mine alone.

2 Absrac This paper argues ha a recen line of lieraure, he fiscal heory of he price level developed by Woodford, Cochrane, Sims and ohers, is based on a fallacy. The source of he fallacy is an economic misspecificaion. The proponens of he fiscal heory of he price level do no accep he fundamenal proposiion ha he governmen s ineremporal budge consrain is a consrain on he governmen s insrumens ha mus be saisfied for all admissible values of he economy-wide endogenous variables. Insead hey require i o be saisfied only in equilibrium. This economic misspecificaion has implicaions for he mahemaical or logical properies of he equilibria suppored by models purporing o demonsrae he properies of he fiscal approach. These include: overdeermined (inernally inconsisen) equilibria; anomalies like he apparen abiliy o price hings ha do no exis; he need for arbirary resricions on he exogenous and predeermined variables in he governmen s budge consrain; and anomalous behaviour of he equilibrium price sequences, including behaviour ha will ulimaely violae physical resource consrains. The issue is of more han academic ineres. olicy conclusions could be drawn from he fiscal heory of he price level ha would be harmful if hey influenced he acual behaviour of he fiscal and moneary auhoriies. The fiscal heory of he price level implies ha a governmen could exogenously fix is real spending, revenue and seigniorage plans, and ha he general price level would adjus he real value of is conracual nominal deb obligaions so as o ensure governmen solvency. When realiy dawns, he resul could be painful fiscal ighening, governmen defaul or unplanned recourse o he inflaion ax. JEL classificaions: E3, E4, E5, E52, E62, E63. Key words: Fiscal heory of he price level; Ricardian fiscal rules; Governmen budge consrain; price level indeerminacy. Auhor: Willem H. Buier, Bank of England, Threadneedle Sree, London WC2A 2AE, UK Tel. # Fax. # willem.buier@bankofengland.co.uk Web page: hp://

3 I. Inroducion I is no common for an enire scholarly lieraure o be based on a fallacy, ha is, on fauly reasoning; misleading or unsound argumen. The recenly revived fiscal heory of he price level is an example of a research programme ha is faally flawed, concepually and logically. 2 The source of he fallacy is an economic misspecificaion. The proponens of he fiscal heory of he price level do no accep he fundamenal proposiion ha he governmen s ineremporal budge consrain is a consrain on he governmen s insrumens ha mus be saisfied for all admissible values of he economy-wide endogenous variables. Insead hey require i o be saisfied only in equilibrium. This economic misspecificaion has implicaions for he mahemaical or logical properies of he equilibria suppored by models purporing o demonsrae he properies of he fiscal approach. These include: overdeermined (inernally inconsisen) equilibria; anomalies like he apparen abiliy o price hings ha do no exis; he need for arbirary resricions on he exogenous and predeermined variables in he governmen s budge consrain; and anomalous behaviour of he equilibrium price sequences, including behaviour ha may ulimaely violae physical resource consrains. The purpose of his paper is o pu an end o his fruiless line of enquiry by demonsraing he naure and origins of he fallacy. In he Concise Oxford Dicionary [995], he following definiions of a fallacy can be found. () a misaken belief, esp. based on unsound argumen; (2) fauly reasoning; misleading or unsound argumen; (3) (Logic) a flaw ha viiaes an argumen. 2 The seminal conribuion is Begg and Haque [984]. The recen revival includes Leeper [99], Leeper and Sims [994], Woodford [994, 995, 996, 998a,b], Auernheimer and Conreras [995] Sims [994, 997], Cochrane [996, 999a,b], Dupor [997], Loyo [997a,b],

4 The issue is of more han academic ineres. olicy conclusions could be drawn from he fiscal heory of he price level ha would be harmful if hey influenced he acual policy behaviour of he fiscal and moneary auhoriies. Specifically, a governmen migh infer from he fiscal heory of he price level, ha here is never any hrea of defaul, no maer how i ses is fiscal-financial-moneary programme. For insance, even when i fixes exogenously he real sequence of primary surpluses plus seigniorage, he general price level will always adjus he real value of is non-moneary nominal deb so as o ensure is solvency. The fiscal heory of he price level is based on he disincion beween wo kinds of budgeary rules for he governmen. Following Woodford (and wih apologies o he memory of David Ricardo) I shall refer o hese as Ricardian and non-ricardian fiscal rules. In wha follows, he governmen is o be inerpreed as he consolidaed general governmen and cenral bank. The governmen spends on goods and services (exhausive public spending), makes ransfer paymens, raises curren revenues and, eiher mees is deb obligaions (ineres paymens and repaymen of principal) or defauls. Is financial defici is financed eiher by issuing non-ineres-bearing fia money or ineres-bearing non-moneary liabiliies (bonds). A fiscal rule (a beer bu excessively long label would be a fiscal-financial-moneary programme) is a complee sequence of rules specifying public spending, axes ne of ransfers, money issuance (seigniorage) and bond issuance in each period and in each sae of naure. Exhausive public spending is no he issue in his lieraure. I is herefore helpful o keep he sequence of real public spending on goods and services consan hroughou and o focus on ne axes, money issuance and governmen borrowing. Lumer [997], Olivei [997]; criical evaluaions include Canzoneri, Cumby and Diba [998a,b], Buier [998], McCallum [998] and Clemens, Herrendorf and Valeninyi [998]. 2

5 Like every agen in a muli-period economy, he governmen faces an ineremporal budge consrain or solvency consrain. A Ricardian fiscal rule requires ha he governmen s solvency consrain holds for all admissible sequences of he endogenous variables. Tha is, he governmen s ineremporal budge consrain holds idenically, no jus in equilibrium. A non- Ricardian fiscal rule requires he governmen s solvency consrain o hold only for equilibrium sequences of he endogenous variables. For exposiional simpliciy, his paper models a world wihou uncerainy. Wih a Ricardian fiscal rule, here is never any defaul on he public deb: conracual deb obligaions are me. 3 Loosely speaking, wih a Ricardian fiscal rule, here has o remain one degree of freedom in he fiscal-financial-moneary programme. The governmen canno, for insance, given is inheried deb obligaions, specify exogenous sequences for real public spending, real axes ne of ransfers and real moneary issuance, and sill expec, excep by chance, o mee is conracual obligaions in full. Eiher public spending, or axes or seigniorage would have o be residually deermined during a leas one period (as a funcion of he economy-wide endogenous variables in he model and of he governmen s oher policy insrumens) o ensure ha he ineremporal budge consrain is saisfied for all admissible sequences of he economy-wide endogenous variables. A non-ricardian fiscal rule permis wha, from a Ricardian perspecive, would be overdeermined fiscal-financial-moneary programmes. The solvency consrain is required o hold only in equilibrium, no for all admissible sequences of he endogenous variables. An 3 Wih uncerainy, a sric Ricardian fiscal rule, ha is, one which rules ou governmen defaul wih cerainy, is only possible if here is a complee se of coningen spo and fuures markes. In an incomplee markes world, defaul is in general a possibiliy. Specifying an appropriae ineremporal budge consrain (or sequence of budge consrains) in a world wih incomplee markes is beyond he scope of his paper. 3

6 example of a non-ricardian fiscal rule would be one which fixes exogenously he enire sequence of real primary surpluses 4 plus real seigniorage. The Ricardian view implies ha if such a non-ricardian fiscal rule is adoped, he governmen will no, excep in very special circumsances, be able o mee is conracual deb obligaions exacly: here will eiher be defaul on he public deb, or he governmen could be super-solven : afer all conracual obligaions are me, here are surplus resources lef. From a Ricardian perspecive, he over-deermined non-ricardian fiscal rule herefore requires he addiion of anoher endogenous variable o he model: he defaul discoun facor on he noional or conracual value of he governmen s non-moneary deb insrumens. The non-ricardian view mainains ha despie he prima-facie overdeermined fiscalfinancial-moneary programme, here will be no discoun (or premium) on he noional or conracual value of he governmen s non-moneary deb insrumens. Insead he equilibrium general price level adjuss he real value of he governmen s nominal conracual deb obligaions o a level ha is consisen wih he governmen solvency consrain - he requiremen ha he aggregae real value of all he governmen s ousanding conracual deb obligaions be equal o he presen discouned value of fuure real primary surpluses plus seigniorage. I views his as an alernaive heory of price deerminaion o he Ricardian one, a fiscal heory of he price level. There are wo ways of refuing he fiscal heory of he price level. The firs is based on a- priori economic consideraions. I consider i o be axiomaic ha only hose models of a marke economy are well-posed, in which, if defaul is ruled ou, budge consrains (including he governmen budge consrain), mus be saisfied for all admissible values of he economy-wide 4 The governmen s primary surplus is is financial surplus, excluding ne ineres receips. 4

7 endogenous variables. I does no maer wheher he governmen (or he privae agens) are small (price-aking) or large (monopolisic or monopsonisic). I does no maer wheher he governmen opimises (or wha i opimises), saisfices or acs according o ad-hoc decision rules. According o his Ricardian posulae abou he proper specificaion of budge consrains, a non-ricardian fiscal rule ha rules ou defaul, is ill-posed. Any model ha incorporaes a non- Ricardian fiscal rule, ye assumes ha all conracual deb obligaions are me, does no make economic sense. 5 The second way o refue he fiscal heory of he price level applies even if one does no accep he a-priori asserion or posulae ha budge consrains mus be saisfied always, no only in equilibrium, and ha consequenly a non-ricardian fiscal rule only makes sense if we explicily inroduce an endogenous defaul discoun facor on he public deb. This second approach involves he demonsraion of a number of mahemaical (or logical) and concepual anomalies ha characerise equilibria purpored o be suppored by non-ricardian fiscal rules wihou defaul. The paper shows ha i is no rue ha he general price level can mimic, generically, he role of a defaul discoun facor on he public deb, even if we only consider he governmen s budge ideniy and solvency consrain. Because he general price level canno be negaive, an arbirary resricion on he predeermined socks of governmen deb insrumens and on he exogenous sequences of public spending and revenues (including seigniorage), is necessary for he general price level o mimic he role of a defaul discoun facor on he public deb, even when we consider only he governmen s accouns. 5 oe ha here is no disagreemen abou he fac, ha, if privae defaul is o be ruled ou, households and businesses mus saisfy heir budge consrains for all admissible sequences of 5

8 When his arbirary resricion is saisfied, a furher concepual anomaly arises. The fiscal heory of he price level can deermine he price of money in a model wihou money, ha is, in a world where money does no exis as a physical commodiy or financial claim (privae or public, inside or ouside ) and does no funcion as a medium of exchange, means of paymen or sore of value. In such a world money only serves as a uni of accoun or numeraire, and here exiss a financial claim commiing he issuer o make a paymen whose value is fixed in erms of his pure numeraire money. The fiscal heory of he price level would appear o be able o deermine he price of phlogison if phlogison were used as he numeraire in a general equilibrium model and if some agen had issued phlogison-denominaed financial claims. 6 Mahemaical/logical anomalies include he following. In any finie-horizon economy (wih flexible nominal prices or wih nominal price rigidiies), a non-ricardian fiscal rule wihou an endogenous defaul discoun facor on he public deb leads o an overdeermined (inernally inconsisen) price level when he governmen follows a moneary rule 7. A finiehorizon economy wih a non-ricardian fiscal rule and nominal price rigidiies also is overdeermined when he governmen adops a nominal ineres rae rule. In an infinie horizon economy wih flexible nominal prices, here is an overdeermined price level under a non-ricardian moneary rule if he velociy of circulaion of money is consan (as in simple cash-in-advance models). When velociy is endogenous, here is no price level overdeerminacy, bu explosive or implosive behaviour of he equilibrium price sequence is possible, even wih consan fundamenals. In many moneary models, such implosive or he economy-wide endogenous variables. I is my mainained hypohesis ha he governmen is in no way differen. 6 hlogison is an imaginary elemen formerly believed o cause combusion. 6

9 explosive price level behaviour would imply he evenual violaion of economy-wide real resource consrains. When here are nominal price rigidiies, here is overdeerminacy in he infinie-horizon economy (as in he finie-horizon economy) under boh moneary rules and nominal ineres rae rules. By inroducing he governmen deb defaul discoun facor as an addiional endogenous variable, I presen a Ricardian resoluion of he fiscal fallacy. The over-deermined non- Ricardian fiscal rule implies, generically, a non-uniary equilibrium value for he defaul discoun facor. This Ricardian resoluion of he fiscal fallacy also desroys he fiscal heory of he price level. In an economy wih flexible nominal prices, money is now neural when he governmen follows a moneary rule and here is price level indeerminacy when he governmen follows a nominal ineres rae rule. II. The Model John Cochrane [999a] provides a published example of a model purporing o suppor he fiscal heory of he price level, whose inadequacy can be esablished simply by couning equaions and unknowns. According o Cochrane Fiscal price deerminaion is easies o see in a erminal period, or a period in which he governmen sells no new deb (Cochrane [999a, p. 327]). A key advanage of he finie-horizon, complee markes specificaion is ha he appropriae ineremporal budge consrains are self-eviden. The model ha follows differs from Cochrane s in wo ways. The firs is he addiion of an index-linked one-period governmen bond o he asse menu. This is no essenial for proving 7 A moneary rule in his paper is an exogenous sequence of posiive nominal money socks. The sequence is such as o suppor a non-negaive nominal ineres sequence as an equilibrium. An ineres rae rule is a non-negaive sequence of period nominal ineres raes. 7

10 he inernal inconsisency of he fiscal approach, bu i helps bring ou more clearly an anomaly in he fiscal approach ha should have given is proponens cause for concern, even wihou couning equaions and unknowns. The second difference from Cochrane s presenaion is ha I presen a fully specified general equilibrium model. Cochrane presens jus he governmen s single period budge ideniy and is ineremporal budge consrain. The res of he model (household behaviour, producion and equilibrium condiions), is omied. Couning equaions and unknowns in his governmen secor sub-model (and making he innocuous simplifying assumpion ha he real ineres rae sequence is exogenous), Cochrane concludes ha his sub-model alone deermines he equilibrium price level sequence. Had he wrien down he missing bis of he general equilibrium model, he would have concluded ha he model is overdeermined under a moneary rule, or whenever he price level is a predeermined variable. Time, indexed by, is measured in discree inervals of equal lengh, normalised o uniy. There are periods indexed by,. Iniial asse socks are inheried from period. There is no uncerainy, markes are complee and conracs are enforced coslessly. Defaul herefore does no occur. In Secion III, I will propose an inerpreaion of he fallacy involving defaul on public deb. For ha reason I will refer o he prices of public deb insrumens in he absence of defaul as noional prices and o he prices of public deb insrumens when here is defaul as effecive prices. II. Household Behaviour Households ac as price akers in all markes in which hey operae. They receive an exogenous perishable endowmen, y >, each period, consume c and pay real lump-sum axes 8

11 τ. 8 They have access o hree sores of value: non-ineres-bearing fia money; a nominal oneperiod bond wih a noional money price B in period, which eniles he buyer o a single conracual nominal coupon paymen Γ > in period +; and a real or index-linked one-period bond wih a noional money price b in period, which eniles he buyer o a single conracual real coupon paymen γ> in period +. The noional bond prices are he prices ha will prevail if here is no defaul, ha is, if he conracual paymens (Γorγ) are made wih cerainy. The effecive bond prices are he marke prices ha acually prevail, if here is (a risk of) parial or complee defaul. Unil furher noice in Secion III, defaul is ruled ou and he effecive prices herefore equal he noional prices. A richer menu of liabiliies (longer mauriies, coningen coupon paymens) could be included, bu would no add o he analysis. The quaniies of money, nominal bonds and real bonds ousanding a he end of period (and he beginning of period +) are denoed M, B and b, respecively. The money price of oupu in period is. We will only consider equilibria in which money is weakly dominaed as a sore of value, ha is equilibria supporing a non-negaive nominal ineres rae sequence. The moive for holding money is ha end-of-period real money balances are an argumen in he direc uiliy funcion. To keep he analysis as ransparen as possible, he period feliciy funcion is assumed o be iso-elasic and money is assumed o ener he period feliciy funcion in an addiively separable manner. All key proposiions in his paper would go hrough for more general funcional forms and for mos alernaive ways of inroducing money ino he model including money in he shopping funcion and money in he producion funcion. For he sric Clower [967] cash-in-advance models, here exiss no finie-horizon equilibrium wih a posiive price of 8 They ake he sequence of axes as given. 9

12 money unless one inroduces anoher closure rule o ensure ha money is acceped in exchange for goods and services in he las period of he model. The fiscal heory of he price level fails in (infinie-horizon) cash-in-advance models as well, however, as shown in Secion IV. The represenaive compeiive and ax-aking consumer maximises he following uiliy funcional, defined over non-negaive sequences of consumpion and end-of-period real money balances u L M j + = η η c + j + j= η c M j φ η + j + δ, M ; η, φ, δ > + j + j F HG I KJ O F QHG I K J () The single-period household budge ideniy is, for, B b M M + B ΓB + b γ b ( y τ c ) (2) The ineremporal budge consrain or solvency consrain is ha a he end of period, he household canno have posiive deb, ha is, B B b + b (3) Since uiliy is increasing in consumpion and real balances, (3) will hold wih equaliy. Iniial financial asse socks are predeermined, ha is, B b = B = b M = M > (4) The riskless one-period nominal ineres rae se in period is i, + and he riskless oneperiod real ineres rae in period is r, +. Simple arbirage ensures ha c h Γ γ + i. + ` = + r, + = = B + + b (5)

13 Le R, + j be he nominal discoun facor beween periods - and +j, ha is, R, + j j k = + i for j + k, + k for j = (6) Solving he household budge ideniy (2) forward recursively, using (5) yields ΓB + γ b R ( c + τ y ) + ( M M ), + j + j + j + j + j + j + j j= B b + R ( B + b ), d i (7) The household solvency consrain (3), holding wih equaliy, implies ha ΓB + γ b R, + j + j ( c + j + τ + j y+ j ) + ( M+ j M+ j ) j= d i (8) The household opimal consumpion programme is characerised by M = c F HG c + c L F M HG I r + K J η = ( +, )( + δ ) (9) + i φ i, +, + IO KJ Q η () M = c φ η () Equaion () is he familiar opimaliy condiion relaing he opimal money sock in period o opimal consumpion in ha period. The money-in-he direc uiliy funcion approach views money as a consumer durable yielding a flow of unspecified liquidiy services each period. In he las period,, money only has value because of he liquidiy services i

14 yields ha period. Effecively, real money balances in period become a perishable commodiy, as shown in equaion (), which does no involve any ineremporal relaive price. II2. The Governmen Governmen decision rules are exogenously given, subjec only o some basic feasibiliy condiions, including he governmen s ineremporal budge consrain or solvency consrain. The governmen s single-period budge ideniy is given in (2), is solvency consrain in (3). B b M M + B ΓB + b γ b ( g τ ) (2) B B b + b (3) M B b = M > = B = b The governmen s solvency consrain is analogous o ha of he privae secor: a he end of he las period () he governmen canno have a posiive sock of deb ousanding. If he governmen solvency consrain is o hold idenically, governmen spending, money issuance or axes (or some combinaion of hese hree financing modes) mus adjus in such a way as o ensure ha he governmen always reaches he end of period wih non-posiive non-moneary deb. In Secion III, I allow for breaches of conrac or defaul. In his case he governmen solvency consrain helps deermine he appropriae defaul discoun on he noional value (or defaul-risk-free value) of he public deb. Equaions (2) and (5) imply ΓB + γ b R ( τ g ) + ( M M ), + j + j + j + j + j + j j= B b + R ( B + b ), d i (4) 2

15 Equaions (3) and (4) imply he presen value budge consrain given in (5) ΓB + γ b R, + j + j ( τ + j g+ j ) + ( M+ j M+ j ) j= d i (5) In equilibrium, equaion (5) will hold wih equaliy, because he household solvency consrain will also have o be saisfied. I will consider wo moneary regimes, an (open-loop) moneary rule and an (open-loop) nominal ineres rae rule. The moneary rule specifies an exogenous posiive sequence for he nominal money sock, lm ; q = mm > ; r. (6) The nominal money sock sequences considered are resriced o hose supporing a nonnegaive nominal ineres rae sequence. The nominal ineres rae rule specifies an exogenous non-negaive sequence for he nominal ineres rae, ni ; s = ni ; s (7),, II2A. Ricardian fiscal-financial-moneary programmes The real governmen spending sequence is exogenous, 9 g = g g < y (8) 9 One should no use general equilibrium consideraions o impose a-priori resricions on he insrumen choices of he governmen. The governmen could, in principle, se g > y. In ha case no equilibrium exiss. 3

16 Wih he nominal money sock sequence and he real spending sequence exogenous, he sequence of axes becomes endogenous: any rule for axes ha permis he governmen s ineremporal budge consrain (5) o be saisfied is appropriae (and equivalen) for our purposes. The reason ha, given he nominal money sock sequence { M},, and given he real public spending sequence, { g},, he borrowing mix and ax policy (he sequences { B }, { b }, and{ τ }, ) do no maer for eiher real or nominal equilibrium as long as he governmen s solvency consrain is saisfied, is ha he model exhibis deb neuraliy or Ricardian equivalence. The represenaive agen assumpion and he lump-sum naure of ne axes generae he deb neuraliy. The inflaion ax is in general a nonlump-sum ax, so deb neuraliy only applies o changes in he sequences of debs and axes, holding consan he nominal money sock sequence or he nominal ineres rae sequence. Since any lump-sum ax rule ha saisfies he governmen solvency consrain is equivalen, I shall for concreeness assume ha axes are se o achieve a zero nominal nonmoneary deb from he end of period on, ha is, τ = g M M ΓB + + γ b (9) τ = g M M < Equaions (6), (8) and (9) define our Ricardian moneary rule. Equaions (7), (8) and (9) define our Ricardian nominal ineres rae rule. or given he nominal ineres rae sequence 4

17 Many oher Ricardian fiscal-financial-moneary programmes are possible, including programmes based on ad-hoc or opimising feedback rules for he governmen s insrumens. The simple exogenous or open-loop moneary rule and nominal ineres rae rule ha he paper will focus on can be described as being characerised by dominan or acive moneary policy and subordinae, accommodaing or passive fiscal policy. Exogenous sequences are specified in each period for eiher he nominal money sock or he nominal ineres rae, and fiscal policy (axes in our model) accommodaes so as o ensure ha he governmen s solvency consrain is always saisfied, hrough equaion (9). Anoher Ricardian fiscal rule would specify exogenous sequences for real public spending and real axes ne of ransfers, and have he governmen borrow or lend jus enough o keep he real sock of public deb consan a is iniial value, unil he final period when i would have o go o zero, ha is, ΓB + + γ b + g = g τ = τ, ΓB = B B, + γ b b + b (2a) (2b) Under his Ricardian fiscal rule, fiscal policy (axes, spending and borrowing) is dominan or acive and moneary policy is subordinae, accommodaing or passive. Seigniorage 5

18 in a leas one period is endogenous and adjuss passively o ensure ha he governmen solvency consrain is saisfied for all admissible values of he economy-wide endogenous variables. oe ha, from a Ricardian perspecive, once we have specified (2a,b), i is self-eviden, from a consideraion of he governmen s ineremporal budge consrain alone, ha he sequence of nominal money socks, M, canno be exogenous, bu has o conain a leas one endogenous elemen. o general equilibrium consideraions are required o conclude ha moneary policy canno be se independenly once i is recognised ha he solvency consrain mus be saisfied for all admissible values of he economy-wide endogenous variables, including he general price level. II2B. on-ricardian fiscal-financial-moneary programmes Woodford, Sims, Leeper, Cochrane and oher proponens of he fiscal heory of he price level propose an alernaive fiscal-financial closure rule for dynamic general equilibrium models wih a governmen, called non-ricardian rules by Woodford. While here are by now a very large variey of non-ricardian rules, hey all suffer from he same faal flaw. The following very simple rule proposed in Woodford [995] is he mos convenien vehicle for making his clear. I is also a rule consisen wih Cochrane [999a]. where { s } Woodford proposes he following ax rule: = M τ s M (2), is an exogenously given real sequence of axes plus seigniorage. This is he policy in sage 2 of he Sargen and Wallace [98] unpleasan monearis arihmeic model. In our finie, horizon model, we could fix exogenously a mos - elemens of he nominal money sock sequence, {M, # # }. A leas one elemen would have o be 6

19 Our non-ricardian moneary rule will be defined by an exogenous sequence of real public spending, an exogenous sequence of s and an exogenous sricly posiive sequence of nominal money socks, ha is, by equaions (6), (8) and (2). Our non-ricardian nominal ineres rae rule will be defined by an exogenous sequence of real public spending, an exogenous sequence of s and an exogenous non-negaive sequence of nominal ineres raes, ha is, by equaions (7), (8) and (2). II3. Marke Clearing The goods marke clears each period, ha is, y = c + g For simpliciy, I assume in wha follows ha he real fundamenals are consan, ha is, y = y g = g Only non-negaive equilibrium price sequences are permissible. II4. Equilibrium Under he Ricardian Moneary Rule rovided he exogenous and sricly posiive nominal money sock sequence saisfies M+, he equilibrium nominal ineres rae sequence will be non-negaive. The M + δ equilibrium is characerised by equaion (6) and c = c = y g (22) deermined endogenously. Sargen and Wallace deermine a consan proporional growh rae of he nominal money sock. 7

20 r, + = δ (23) M + + i, + = ( + δ ) (24) = ( y g) L M + φ( + δ ) + ( + δ ) O Q η (25a) ΓB + γ b M = ( y g)φ η (25b) F I HG K J L F + M HG j M + j M j + j g + δ τ (26) j= + j I KJ O Q τ τ = g = g M M + + M M ΓB γ b < (27) M B b = M > = B = b (28) oe ha he governmen solvency consrain (26) has o hold in equilibrium (even if i did no have o hold as an ideniy), because of he household solvency consrain and he goods marke equilibrium condiion. 8

21 The key poin o noe is ha he moneary equilibrium condiions (25a,b) provide equaions ha uniquely 2 deermine he (non-negaive) equilibrium prices, =,,. Equaion (25b) alone deermines as a funcion of he nominal sock of money in he las period, M. The remaining - moneary equilibrium condiions given by (25a) deermine - down o, given he soluion for he price level in period and he exogenous values of he nominal money socks in periods o -. The ax rule given in (27) hen deermines he values of he lump-sum ax sequence. Given ha ax rule, he governmen s solvency consrain holds idenically. Anoher way of puing his is ha he governmen s solvency consrain (and he assumed exogeneiy of he real public spending sequence and he nominal money sock sequence) forces he ax sequence o become endogenous. The equilibrium real and nominal ineres rae sequences and he equilibrium consumpion sequence are also uniquely deermined. Under his Ricardian moneary rule, money is condiionally neural (see Buier [998]). Holding consan he iniial sock of nominal non-moneary deb, B, equal proporional changes in he sequence of nominal money socks (including he iniial nominal sock of money), B b { M},, and in he sequences of all endogenous nominal prices {,, }, leave he real equilibrium unchanged. If he iniial sock of non-moneary nominal deb is nonzero, he sequence of (endogenous) real lump-sum axes will change (according o (27)), because 2 This economy also has a non-moneary equilibrium, in which he value of money is zero in each period. As his is no relevan for he fiscal heory of he price level, his barer equilibrium is no considered here. 9

22 of he change in he real value of he iniial sock of nominal non-moneary governmen deb, B, when he iniial price level changes. The proof is by inspecion. Under he Ricardian moneary rule, money and he iniial sock of nominal non-moneary deb are joinly uncondiionally neural (see Buier [998]). Equal proporional changes in he sequence of nominal money socks (including he iniial nominal sock of money), { M},, in he iniial sock of nominal non-moneary deb, B, and in he sequences of B b all endogenous nominal prices {,, }, leave he real equilibrium unchanged. The (endogenous) sequence of real lump-sum axes will no change (again according o (27)). The proof is again by inspecion. I summarise his as roposiion. roposiion. Under he Ricardian moneary rule, money is neural. II5. Equilibrium Under he Ricardian ominal Ineres Rae Rule Wih an exogenous non-negaive nominal ineres rae sequence (and endogenous nominal money socks), he equilibrium is characerised by equaions (7), (22), (23), (24), (25a,b), (26), (27) and (28). I is helpful o rewrie, using (24), he wo moneary equilibrium condiions (25a,b) as (29a,b). M = ( y g) L M η φ( + i, + ) i, + O Q (29a) 2

23 M = ( y g)φ η (29b) The key poin o noe here is ha he moneary equilibrium condiions (29a,b) provide equaions ha uniquely 3 deermine he equilibrium real money socks, M /, =,,. The endogenous equilibrium nominal money sock sequence {M }, and he equilibrium price sequence { }, are indeerminae. The ax rule given in (27) hen deermines he values of he lump-sum ax sequence. If he iniial sock of nominal non-moneary deb, B, is non-zero, he firs erm in he equilibrium real ax sequence, τ, which depends on B, is also indeerminae. However, i coninues o be he case ha, given ha ax rule in (27), he governmen s solvency consrain holds idenically. The equilibrium real ineres rae sequence, he equilibrium inflaion rae sequence and he equilibrium consumpion sequence are also uniquely deermined. rice level indeerminacy under a Ricardian nominal ineres rae rule is a familiar resul. In a fricionless economy, wih flexible, marke-clearing nominal prices, an exogenous nominal ineres rae sequence does no provide a nominal anchor for he sysem. The reason is ha, despie is name, he shor nominal ineres rae is a real variable, he real pecuniary opporuniy cos of holding money balances. I summarise his as roposiion 2. roposiion 2. 3 See foonoe 2. 2

24 Under he Ricardian nominal ineres rae rule, all nominal equilibrium values are indeerminae in our flexible price model. Only real equilibrium values are (uniquely) deermined by he model. rice level indeerminacy under Ricardian moneary rules is a feaure of he class of flexible price level, general equilibrium models considered in his paper raher han a problem for moneary policy in he real world. More policy-relevan models would view he price level (and/or he money wage) in any given period as predeermined. Wih such Keynesian money wage or price rigidiies, price level indeerminacy is auomaically eliminaed (see Secion II8). II6. Equilibrium Under he on-ricardian Moneary Rule Subsiuing he rule given by (8) and (2), ha real ax revenue plus he real value of seigniorage is exogenously given, ino he governmen solvency consrain (26), we ge ΓB + γ b s + j g + δ (3) j= j F I HG K J The righ-hand side of (3) is exogenous. Everyhing on he lef-hand-side of (3) excep for is exogenous or predeermined. The governmen solvency consrain (3) under he non- Ricardian rule herefore deermines he equilibrium value of. This is he fiscal heory of he price level. Assume again ha he exogenous and sricly posiive nominal money sock sequence saisfies M+. The puaive equilibrium under he non-ricardian moneary rule is M + δ characerised by he equaions (6), (22), (23), (24), (25a,b), (28) and (3). 22

25 The key poin o noe is ha, jus as wih he Ricardian moneary rule, he moneary equilibrium condiions (25a,b) again provide equaions ha uniquely 4 deermine he equilibrium prices, =,,. Unforunaely for he fiscal heory of he price level, is also deermined by equaion (3). Excep for a se of parameer values of measure zero, he sysem is overdeermined. 5 There is no non-ricardian or fiscal heory of he price level under a moneary rule. I summarise his as roposiion 3. roposiion 3. Under he non-ricardian moneary rule, he price level sequence is overdeermined. Corollary There is no alernaive fiscal heory of he price level under a moneary rule. When here is a deerminae price level under a Ricardian rule, imposing a non-ricardian rule resuls in an inernally inconsisen model. oe ha roposiion 3 and is Corollary are logical proposiions, involving no more han a couning of equaions and unknowns. The relevance of he fiscal heory of he price level is herefore no an empirical issue. There is no inernally consisen fiscal heory of he price level. Of course here may well be empirical regulariies linking he general price level, anicipaed and unanicipaed inflaion, and measures of nominal public deb or deficis. Ricardian models like Sargen and Wallace s Unpleasan Monearis Arihmeic model (Sargen and Wallace [98]), 4 See foonoe 2. 23

26 which do no fall foul of he fallacy of he fiscal heory of he price level, offer one possible moivaion for such empirical regulariies. Adherens of he fiscal approach migh recognise he inconsisency of he fiscal approach and our exogenous moneary rule, bu conclude ha he incompaibiliy of he non-ricardian fiscal rule and our open-loop moneary policy only shows ha i is his paricular moneary rule ha has o be discarded, raher han he non-ricardian fiscal rule. 6 The argumen goes as follows: under a non-ricardian fiscal rule, fiscal policy is dominan and moneary policy mus be subordinae or accommodaing. Subordinae and accommodaing moneary policy means ha he nominal money sock sequence canno be specified exogenously. The nominal money sock in a leas one period mus be endogenous. 7 This would resolve he inconsisency. This argumen is flawed. I uses general equilibrium consideraions o resric he a-priori permissible insrumen choices of he governmen. If a governmen does no have o worry abou is solvency consrain being saisfied for all permissible values of he economy-wide endogenous variables (he cenral assumpion of he fiscal approach), i can cerainly choose boh sequences for is real primary surplus including seigniorage and is nominal money sock. As he following wo equaions, represening he non-ricardian moneary rule make clear, real axes ne of ransfers, τ, can adjus passively o allow he governmen o follow he non-ricardian fiscal rule and selec an exogenous non-negaive sequence for he nominal money sock. B Γ + F HG B + γ b ( + δ ) Γ + γ b + g s I KJ 5 Excep of course for cerain ( measure zero ) parameer configuraions. 6 I am indebed o John Vickers who, by playing devil s advocae mos effecively, induced me o address his issue. 7 Surprisingly, however, accommodaing or subordinae moneary policy would be consisen wih an enirely exogenous sequence of nominal ineres raes. 24

27 M M = ( s τ ) Of course, he behaviour of he public deb under his rule will, in general, be inconsisen wih governmen solvency. There may exis a unique sequence of values for he general price level ha makes his non-ricardian moneary rule consisen wih governmen solvency. However, ha sequence of general price level values does no, generically, saisfy he remaining equilibrium condiions. To proscribe he exogenous moneary rule because of is (general) equilibrium implicaions would compound he original concepual error of he fiscal heory of he price level, which was o impose governmen solvency only in equilibrium. Two wrongs do no make a righ. assive moneary policy cerainly can be a logical implicaion of a paricular parameerisaion of spending, ax and borrowing policies in a Ricardian framework. An example was given in Secion II2A and equaions (2a,b). The logical necessiy for moneary policy o be subordinaed o fiscal policy in ha example follows direcly and exclusively from he recogniion of he fac ha he governmen s solvency consrain has o hold idenically. I did no involve general equilibrium consideraions. II7. Equilibrium Under he on-ricardian ominal Ineres Rae Rule Wih an exogenous non-negaive nominal ineres rae sequence (and endogenous nominal money socks), he equilibrium is characerised by equaions (7), (22), (23), (24), (25a,b), (28) and (3). 25

28 The moneary equilibrium condiions (25a,b) provide equaions ha uniquely 8 deermine he equilibrium real money socks, M /, =,,. The equilibrium real ineres rae, inflaion and consumpion sequences are also uniquely deermined. Under a non-ricardian nominal ineres rae rule, he price level indeerminacy characerisic of he Ricardian nominal ineres rae rule can be resolved, provided a key condiion, explained below, is saisfied. oe ha he governmen solvency consrain (3) deermines. Given equaion (24) deermines all fuure price levels 2 o. Equaions (25a,b) hen uniquely deermine he endogenous nominal money sock sequence. The key cavea is ha, since he general price level canno be negaive, a necessary condiion for he governmen solvency consrain under he non-ricardian rule (equaion (3)) o make sense is ha condiion (3) be saisfied. sgn m r = R S T B sgn s + j g γ b (3) + δ j= j F I HG K J This says ha he iniial sock of non-moneary nominal public deb mus be posiive (negaive) if he excess of he presen discouned value of fuure primary governmen surpluses (including seigniorage revenues) over he value of he iniial sock of index-linked governmen deb is posiive (negaive). 9 Everyhing on eiher side of equaion (3) is exogenous or predeermined. There is no reason why arbirary configuraions of B, b, δ, g and { s }, would always saisfy (3), alhough hey may, foruiously, do so. U V W 8 See foonoe 2. 9 For insance, if all public deb is index-linked, ( B = ), condiion (3) is no saisfied and (3) canno deermine he iniial price level. In an open economy version of his model, he same would be rue if all public deb were denominaed in foreign currency. 26

29 The mos ha he fiscal heory of he price level herefore could aspire o, when he arbirary resricion (3) is saisfied, is o be a way of removing he price level indeerminacy characerisic of equilibria under a Ricardian nominal ineres rae rule, when nominal prices are flexible. As Secion II makes clear, acceping his inerpreaion would resul in a major concepual anomaly. I herefore rejec i. oe ha he fiscal or non-ricardian heory of he price level is never an alernaive o a moneary or Ricardian heory of he price level. When he moneary or Ricardian heory gives a deerminae equilibrium price level sequence (as i does under our Ricardian moneary rule), he imposiion of a non-ricardian fiscal rule leads o an overdeermined sysem. Generically 2, ha is, wihou imposing he arbirary resricion (3), an admissible (nonnegaive) iniial value for he general price level can herefore no be deermined from he governmen s ineremporal budge consrain, even under a nominal ineres rae rule. 2 The need for arbirary resricions such as (3) is inheren in he fiscal approach. Consider, for insance, an alernaive non-ricardian fiscal rule under which he governmen fixes is sequences of public spending and axes (plus seigniorage) in nominal erms raher han in real erms, e.g. 2 I am using he word generic or generical in he sandard logical sense of of, applicable o, or referring o all he members of a genus, class, group or kind; general. 2 There are oher, and more robus, ways of removing he nominal indeerminacy. Consider, for insance, he equilibrium configuraion under he Ricardian nominal ineres rule, given in equaions (7), (22), (23), (24), (25a,b), (26), (27) and (28). A deerminae price level sequence exiss, for insance, if he auhoriies fix jus one of he elemens in he sequence of nominal money socks { M},. This way of removing he nominal indeerminacy is no subjec o arbirary resricions such as (3). Finally, consider a world wih a more Keynesian deerminaion of nominal prices. ominal wage or price sickiness means ha he price level in any given period is predeermined and updaed hrough a dynamic money wage or price level adjusmen mechanism like he hillips curve. In such a world, a Ricardian nominal ineres rule suppors a deerminae equilibrium price level sequence, while a non-ricardian nominal ineres rule would resul in an overdeermined price level, even if (3) were saisfied (see Secion II.8). 27

30 g = G s = S The governmen s solvency consrain (assumed o hold wih equaliy) can be wrien as ΓB + b R S G γ, d i (32) + j + j + j j= Consider he case where he governmen fixes he sequence of shor nominal ineres raes. The R, + j are herefore exogenous. Again his non-ricardian rule canno generically resolve he price level indeerminacy characerisic of equilibria under a Ricardian nominal ineres rae rule. The reason is, again, ha equaion (32) canno be generically solved for a non-negaive iniial value of he price level. I can deermine a non-negaive iniial value of he price level if and only if he arbirary resricion in (33) is saisfied. d i Γ (33) sgn{ b } = sgn{ R S G B }, + j + j + j j= Thus, if here is no index-linked deb, his example of a non-ricardian nominal ineres rae rule would no be able o produce a deerminae price level. This is he counerpar, when he governmen fixes he nominal sequence of primary surpluses (plus seigniorage), of he resul ha here can be no fiscal heory of he price level, when he sequence of real primary surpluses (plus seigniorage) is given exogenously, unless here is a non-zero sock of nominal public deb (see equaion (3)). Arbirary resricions on he 28

31 predeermined and exogenous variables in he governmen solvency consrain are always required o suppor a non-negaive equilibrium price level sequence. II.8 on-ricardian Fiscal Rules in he resence of ominal rice Rigidiies The case for rejecing he non-ricardian fiscal rule is srenghened when i can be shown ha i leads o inernal inconsisencies under more general moneary policy rules, such as nominal ineres rae rules, even when (3) is saisfied. I his sub-secion I show ha any model in which he price level is predeermined will resul in an overdeermined equilibrium when a non-ricardian fiscal rule is imposed, regardless of he specificaion of he moneary policy rule. I will illusrae his wih a simple Keynesian modificaion of he earlier model. Assume ha, insead of a flexible price level and marke clearing in he goods marke, we have a predeermined price level and demand-deermined oupu. Acual oupu, y, is endogenous and demand-deermined. I can differ from he exogenous level of capaciy oupu, y. The price level is updaed hrough a simple acceleraionis hillips curve. 22 Any old- Keynesian or new-keynesian model of nominal price or wage rigidiies would have he same implicaions for he fiscal heory of he price level as our simple acceleraionis hillips curve. The Keynesian equilibrium condiions are (34), (35) and (36): y = c + g (34) + = ϕ( y y) + ϕ > (35) 22 The simple acceleraionis hillips curve has he inflaion rae predeermined, and no jus he price level. In period, boh and are predeermined. 29

32 = > = > (36) The remaining equilibrium condiions are given by (37) o (43). Equaion (4) is obained by subsiuing he household firs-order condiions (9), () and () ino he household solvency consrain (8) for period. oe ha (43) imposes he non-ricardian fiscal rule. M = c F HG c + c L F M HG I r + K J η = ( +, )( + δ ) (37) + i φ i, +, + IO KJ Q η (38) M = c φ η (39) M + ΓB + c L M + γ b + R ( y τ ), + j + j + j j= η 2 j η η j= F F HG L M I HG + K J + φ δ F HG I KJ OI J Q K η + i + j, 2+ j I i j 2 j J + HG + K J ( + φ ) +, + δ F O Q (4) + + i, + = ( + r, + ) (4) R, + j = j k = + r + k, + k + = j = j (42) ΓB γ, + j + j j= + b R s g (43) 3

33 Under a nominal ineres rae rule, equaions (34) o (43) provide 6+ equaions o deermine 6 unknowns ( values for each of y, c and M, =,..., ; - values for r,+, =,..., -; values for R, + j, j =,..., and + values for, =,...,. Wih he price level in period predeermined, i canno do he job of mimicking a defaul discoun facor on he public deb in equaion (43). The puaive equilibrium under he non-ricardian nominal ineres rule is always overdeermined. Overdeerminacy also characerises he sysem under he non-ricardian moneary rule. This overdeerminacy under boh he non-ricardian moneary rule and he non-ricardian fiscal rule, remains when he economy has an infinie horizon. I summarise he discussion in he following proposiion. roposiion 4. on-ricardian fiscal-financial-moneary rules never provide an alernaive heory of he price level o he convenional Ricardian rules. When Ricardian rules suppor a deerminae equilibrium price level sequence (under he Ricardian moneary rule in he flexible price model and under he Ricardian moneary rule and he Ricardian nominal ineres rule in he model wih nominal price rigidiy), imposing he non-ricardian rule resuls in overdeerminacy. When Ricardian rules resul in price level indeerminacy (as wih he Ricardian nominal ineres rae rule and flexible nominal prices), non-ricardian rules do no generically suppor a non-negaive iniial price level. An arbirary resricion on he predeermined public deb socks and he exogenous public spending and revenue sequences mus be saisfied. 3

34 II9. Wha Wen Wrong (): Incomplee Model Specificaion The source of Cochrane s error is easily locaed. Cochrane presens jus he governmen s single period budge ideniy and he governmen solvency consrain. The single-period governmen budge ideniy can be rewrien, for, as B Γ + F HG B + γ b ( + δ ) Γ + γ b + g s (44) I KJ This updaes he socks of public deb each period. 23 The governmen s solvency consrain is updaed each period, as in (45). ΓB F j H G I j= K J + γ b s + j g + δ (45) The iniial socks of nominal and real public deb, B and b, and heir coupon paymens Γ and γare given. 24 In Cochrane s example, as in he flexible price model of his paper, he sequence of equilibrium real raes of reurn is exogenous and consan. The fiscal heory of he price level hen assumes ha he enire sequence of real primary surpluses (including real seigniorage), s g, =,...,, is also exogenously given. I follows ha he iniial price level,, is deermined from (3) (or from (45) for = ) alone. Given, he res of he equilibrium price sequence, 2 o, is deermined from equaions (44) and (45) applied in each successive period = 2,...,. The equilibrium money price sequence is deermined from he governmen s ineremporal budge consrain alone. 23 The choice beween issuing index-linked or nominal bonds is reaed as exogenous. 24 Cochrane does no include index-linked deb and assumes ha Γ =. ohing essenial depends on his. 32

35 Cochrane hen offers he following reflecion: The reader may be uncomforable ha he res of he economy is no specified - where are preferences, echnology and shocks? The answer is ha a wide specificaion of models includes equaions such as...{(44) and (45)}...; hose equaions will deermine he price level no maer wha he res of he economy looks like, so we don have o spell hem ou. (Cochrane [999a, p. 328]) 25. This is wrong for wo reasons. Firs, even if we resric ourselves o a consideraion of he governmen secor alone (equaions (44), (45) and he iniial deb socks, B and b ), here is nohing o guaranee ha condiion (3) is saisfied. Only a non-negaive equilibrium price sequence makes economic sense. Second, assume (3) is saisfied. A unique, non-negaive value of he iniial price level,, saisfies (45) in period. Under ha condiion, he iniial value of he general price level can indeed mimic he operaion of a public deb defaul discoun facor in period, as regards he governmen solvency consrain. The problem hen becomes ha, while he iniial price level may indeed do he job of a period public deb defaul discoun facor as regards he governmen s solvency consrain, i is in fac likely o be occupied elsewhere. When he res of he economy is spelled ou, i is obvious ha he equilibrium price level sequence is overdeermined under a non-ricardian moneary rule even when he price level is flexible. When he price level is predeermined, he equilibrium is overdeermined boh under a non-ricardian moneary rule and under a non-ricardian nominal ineres rae rule. This rejecion 25 The equaion number references in Cochrane s quoe have been changed o refer o he equivalen equaions in his paper. 33