Price Level Determinacy and Monetary Policy under a Balanced Budget Requirement

Price Level Determinacy and Monetary Policy under a Balanced-Budget Requirement StephanieSchmitt-Grohe´ Mart´nUribe (cid:3) BoardofGovernorsoftheFederal ReserveSystem April1997 Abstract Thispaperanalyzestheimplicationsofabalancedbudgetfiscal policyrulefor thedeterminacyof thepricelevelinacash–in–advanceeconomyunderthreealternativemonetarypolicyregimes.Itshows that in such stylized modelswith flexible prices and a period–by–periodbalanced budgetrequirement thepricelevelisdeterminateunderamoneygrowthratepegandisindeterminateunderapurenominal interestratepeg.Underafeedbackrulewherebythenominalinterestrateissetasanincreasingfunction of the inflation rate the price level is determinate for intermediate values of the inflation elasticity of thefeedbackruleandisindeterminateforbothverylowandveryhighvaluesoftheinflationelasticity. Finally, regardless of the particular monetary policy specification, a rational expectations equilibrium consistentwiththeoptimalquantityofmoneymaynotexist. JELClassification Numbers: E63,E52,E31 (cid:3) WewouldliketothankDaleHenderson,EricLeeper,andseminarparticipantsattheBoardofGovernors,IndianaUniversity, andMichiganStateUniversityforcomments. Theopinionsexpressedinthispaperarethoseoftheauthorsanddonotnecessarily reflectviewsoftheBoardofGovernorsoftheFederalReserveSystemorothermembersofitsstaff. Addressforcorrespondence: BoardofGovernorsoftheFederalReserveSystem,WashingtonDC20551;sgrohe@frb.govoruribem@frb.gov.

1 Introduction In the past decade, the idea of imposing fiscal discipline through a balanced-budget rule has gained considerable importance in the economic policy debate. This isreflected perhaps most clearly in the proposed balanced budget amendment thatwaspassed bytheUnitedStatesHouseofRepresentatives onJanuary 26, 1995butfailedpassageintheSenate. Yet,littlelighthasbeenshedontheconsequences ofbalancedbudget rules forbusiness cycle fluctuations beyond thebasic Keynesian insight that balanced budget rules amplify business cycles by requiring tax increases and expenditure cuts during recessions and the reverse during booms. Even less theoretical work has been devoted to understanding the implications of balanced budget rules for nominal stability, and in particular, to understanding the restrictions that such a fiscal policy rule mayimposeonmonetarypolicy, ifnominalstability istobepreserved. This paper fills this gap by providing an analysis of the implications of a balanced budget requirement for the determinacy of the price level under alternative monetary policy regimes. The type of balancedbudget rule we focus on is one in which each period the primary surplus—the difference between taxes and government expenditures—is required to be equal to the interest payments on the outstanding public debt. We combine this balanced budget rule with three simple monetary policy specifications: a nominal interest rate peg, a money growth rate peg, and a feedback rule whereby the nominal interest rate is set as anincreasing function oftheinflation rate. Weconduct theanalysis withinthecash-in-advance modelwith cashandcreditgoodsdeveloped byLucasandStokey(1987). Wefindthatunderthebalanced-budget rulethepricelevelisindeterminatewhenthemonetaryauthority follows an interest rate peg and is determinate when the monetary authority follows a money growth rate peg. Theseresultsarenotnecessaryconsequencesofthemonetarypolicyspecificationsalone. Forexample, Auernheimer and Contreras (1990), Sims (1994), and Woodford (1994), find that if the primary surplus is setexogenously,thenaninterestratepegdeliversauniquepricelevel. Thiscomparisonhighlightsthatgiven the monetary policy regime the adoption of a balanced-budget rule may have important consequences for nominalstability. If the balanced budget rule is combined withthe feedback rule, the price level isdeterminate when the nominal interest rate is moderately sensitive to the inflation rate, and is indeterminate when the nominal interest rate is either very responsive orlittle responsive to the inflation rate. Again, this result isdriven by the particular fiscal regime we analyze; for the same feedback rule we consider Leeper (1991) shows that whentheprimarysurplusisexogenous—afiscalpolicytowhichherefersasactive—thepricelevelisnot indeterminate regardlessofhowsensitivethefeedbackruleis.1 Leeper also showsthat ifthe primary surplus is increasing in and sensitive enough tothe stock ofpublic debt — a fiscal policy to which he refers as passive — the price level is indeterminate for relatively insensitive feedback rules and is determinate otherwise.2 Leeper's passive fiscal policy is similar to our balanced-budget rule because under both policies taxes are an increasing function of the stock of public 1Leeperstudieslocalequilibriabycharacterizingsolutionstoalinearapproximationoftheequilibriumconditionsnearasteady state.Bycontrast,weperformaglobalanalysischaracterizingsolutionstotheexactequilibriumconditions. 2See Woodford (1995) and Canzoneri and Diba (1996) for further analysis of the relationship between the monetary-fiscal regimeandpriceleveldetermination. 1

debt. The reason why in our model, unlike in Leeper's, highly sensitive monetary feedback rules render theequilibrium pricelevelindeterminate isthatinourmodelthenominal interest rateaffectstheconsumption/leisure, or cash/credit, margin. In Leeper's model this effect is not present because in his endowment money-in-the-utility-function model with a separable single-period utility function, the marginal utility of consumption isindependent ofthenominalinterestrateinequilibrium. In practice, balanced-budget proposals typically allow fiscal authorities to run secondary surpluses. Therefore,ourbenchmarkdefinitionofthebalanced-budgetrule,althoughanalyticallyconvenient,isclearly unrealisticsinceitforcesthegovernmenttorunazerosecondarysurplusonaperiod-by-period basis. However, it turns out that our main results are not driven by this particular specification of the balanced-budget rule. Specifically, we show that in the case in which either the real or the nominal secondary surplus is positiveandexogenous thepricelevelremainsindeterminate underaninterestratepeg. The balanced-budget requirement has implications for optimal monetary policy. We find that under a strict balanced-budget rule that eliminates budget surpluses as well as budget deficits the optimal quantity of money advocated byMilton Friedman —a monetary policy consistent witha zero nominal interest rate — cannot be attained under any of the three monetary regimes considered because in this case no rational expectations equilibrium exists. However,ifthebalanced budget requirement allowsforpositivesecondary surpluses anequilibrium consistent withtheoptimalquantityofmoneymayormaynotexist. Forexample, if the balanced budget rule is combined with a money growth rate peg and is implemented in such a way that the public debt converges to zero, or if the balanced budget rule is combined with an interest rate peg and is implemented in such a way that the real secondary surplus is at least in one period strictly positive, anequilibrium consistent withtheoptimalquantityofmoneyexists. In the next section we describe the formal model and the fiscal policy regime. In sections 3, 4, and 5 we analyze the implications of the balanced-budget rule for price level determinacy when the monetary authority follows, respectively, an interest rate peg, a money growth rate peg, and a feedback rule linking thenominalinterestratetoinflation. Section6concludes. 2 A Cash-in-Advance Economy Households Inthissection, wepresentamodelofacash-in-advance economyinwhichpublicandprivateconsumption arecashgoodsandleisureisacreditgood.3 Theeconomyisassumedtobepopulatedbyaninfinitenumber of identical households with log-linear single-period utility functions defined over consumption, c t , and leisure, 1 (cid:0) h t ,whoseektomaximizetheirlifetimeutility E 0 1 t= X 0 (cid:12) t [l n ( c t ) + (cid:18) l n ( 1 (cid:0) h t ) ]; (cid:18) > 0 (1) 3ThepresentationofthemodelfollowsWoodford(1994). 2

where (cid:12) 2 ( 0 ; 1 ) denotesthesubjectivediscount factorand E t denotestheexpectation operatorconditional oninformation available inperiod t . Eachperiod t (cid:21) 0 isdividedintotwonon-overlapping markets. Inthe firstmarkethouseholdsusetheirnominalwealthatthebeginningoftheperiod, W t ,topaylump–sumtaxes, T t , to acquire money, M c t , and to purchase state–contingent claims, D t+ 1 , which cost E t r t+ 1 D t+ 1 dollars andpay D t+ 1 dollars inperiod t + 1 (i.e., r t+ 1 isthepriceofaone–period contingent claimdivided bythe probability of occurrence of that state). The household's budget constraint in the first market is then given by T t + M c t + E t f r t+ 1 D t+ 1 g (cid:20) W t (2.a) Inthesecond market, goods andlabor services are traded. Thehousehold purchases consumption goods at apriceof P t dollarsperunitusingthemoneybalancesitheldatthebeginningofthegoodsmarket. Further, thehousehold hasaccesstoalineartechnology thatenables ittoproduceoneunitoftheconsumption good per unit of labor input. The household sells these consumption goods at a price of P t dollars per unit and since it cannot use sales revenues to purchase consumption goods in the current goods market its nominal assetholdings atthebeginning ofperiod t + 1 are W t+ 1 = D t+ 1 + M c t (cid:0) P t c t + P t h t (2.b) Purchasesofgoodsaresubject toacash-in-advance constraint oftheform M c t (cid:21) P t c t : (2.c) The household chooses sequences for W t+ 1 , D t+ 1 , M c t , h t , and c t , given W 0 > 0 , so as to maximize (1) subjectto c t , M c t (cid:21) 0 , 0 (cid:20) h t (cid:20) 1 ,(2.a),(2.b),(2.c)andthefollowingborrowingconstraint thatpreventsit fromengaging inPonzischemes W t+ 1 (cid:21) (cid:0) q (cid:0) t+ 1 1 1 j = X 1 E t+ 1 f q t+ j + 1 P t+ j (cid:0) q t+ j T t+ j g (2.d) thepriceinperiod 0 ofonedollarinperiod t inaparticular stateoftheworld,whichisdefinedas q t (cid:17) r 1 r 2 (cid:1) (cid:1) (cid:1) r t with q 0 (cid:17) 1 : Theborrowinglimit(2.d)ensuresthatineverystateoftheworldprivatedebtisnotgreaterthantheamount an agent would be able torepay, which is equal to thepresent discounted value ofthe timeendowment net oftaxes. One can show that the set of sequences f c t ; h t ; M c t g satisfying the budget constraints (2.a)-(2.d) are equivalent to the set of sequences f c t ; h t ; M c t g satisfying the cash-in-advance constraint (2.c) and the fol- 3

lowingpresentvaluebudgetconstraint4 E 0 1 t= X 0 W [( q t 0 (cid:0) + q E t+ 0 1 1 t= X ) ( 0 M [q t+ c (cid:0) t 1 h P P t c t t t (cid:0) ) + q t q T t t p ] t (cid:21) c t ]: (3) Fromthe first–order conditions ofthehousehold's optimization problem, consumption andhours mustsatisfy P 1 t c t r t+ 1 = (cid:12) P t+ 1 1 c t+ 1 (4) 1 (cid:0) (cid:18) h t = (cid:12) E t P t+ P 1 t c t+ 1 : (5) Thefirstequation isastandard pricingequation foraone-step-ahead contingent claim,andequates theloss inutilityfrombuyingacontingent claimtodaywiththeexpectedgaininutilityrealizedfromconsumingits paysoff. Thesecondequationisalaborsupplyequationandsaysthatthedisutilityofworkinganextrahour inperiod t hastoequaltheutilityderivedfromspendingthewageonconsumptiongoodsinperiod t + 1 . A further requirement for optimality of the household's consumption, money demand, and labor plans is that theysatisfythepresentvaluebudget constraint, (3),withequality. The government We assume that the government issues only a riskless pure discount bond (i.e., a bond that pays off one dollar in the following period regardless of the state realized) which is denoted by B t . The government startsinperiodzerowithsomeoutstanding stockofdebt, B (cid:0) 1 . Itsperiod–by–period budget constraint is M t + B R t t = B (cid:0)t 1 + M (cid:0)t 1 + P t g (cid:0) T t (6) where M t denotes the money supply, g denotes constant real government purchases, and R t denotes the grossnominalinterest ratepaidontheriskless bond. Theinterestratemustsatisfy R t = E t 1 r t+ 1 : (7) Thegovernment, likehouseholds, issubjecttoacash-in-advance constraint onitspurchases ofgoods M g t (cid:21) P t g : The central element that distinguishes this paper from other studies of price level determination under alternative monetary policy regimes is the specification of fiscal policy. We assume that the government is subjecttoabalanced-budgetrequirementwherebytheprimarysurplusmustbeequaltotheinterestpayments 4SeeWoodford(1994). 4

ontheoutstanding publicdebt T t (cid:0) P t g = B (cid:0)t 1 = R (cid:0)t 1 ( R (cid:0)t 1 (cid:0) 1 ) : (8) This specification of the balanced budget rule implies that seignorage income cannot be used to finance currentspending ortopayinterestonthedebt. Combining(6)and(8)yields M t + B R t t = M (cid:0)t 1 + B R (cid:0)t (cid:0)t 1 1 ; (9) which says that total nominal government liabilities are constant over time; that is, under the balancedbudgetrule,changes inthestockofmoneyareimplementedexclusively throughopenmarketoperations. Throughout the paper, we maintain the same fiscal policy regime (i.e., a balanced budget requirement) but consider three alternative monetary policy regimes: (i) a pure interest rate peg, (ii) a money supply growth rate peg, and (iii) afeedback rule whereby the nominal interest rate isset as an increasing function of the inflation rate. Under policy regimes (i) and (iii), the central bank sets the nominal interest rate by fixing the price of the riskless one-period nominal bond and stands ready to exchange money for bonds in any quantities demanded. This means that M t and B t are endogenous. Under policy regime (ii), the governmentspecifiesadeterministic pathforthemoneysupply, sothat B t and R t areendogenous. Equilibrium Inequilibrium theproduct andmoneymarketclear, h t = c t + g (10) and M t = M c t + M g t = M c t + P t g ; where the last equality follows from the assumption that the government's cash–in–advance constraint is always binding. Because all agents are identical and government bonds are the only financial assets in positiveaggregatenetsupply, itmustbethecasethat D t+ 1 = B t : In any equilibrium, the nominal interest rate must be non-negative. When the nominal interest rate is positive, the household's cash-in-advance constraint holds with equality, M c t = P t c t , and when the nominal interestrateiszero( R t = 1 ),consumption isequaltotheunconstrained optimum, c t = ^c (cid:17) argmax [l n c t + (cid:18) l n ( 1 (cid:0) c t (cid:0) g ) ]: Therefore, inanyequilibrium c t = m i n ( ^c ; m t (cid:0) g ) (11) 5

where m t = M t = P t : (12) Usingthedefinitions F ( m ) (cid:17) (cid:18) m = [1 (cid:0) m i n ( m ; ^c + g ) ]; and G ( m ) (cid:17) m = [m i n ( m ; ^c + g ) (cid:0) g ]; andcombining themarketclearingconditions withthefirstorderconditions (4)and(5)yields, G ( m t ) r t+ 1 = (cid:12) G ( m t+ 1 ) M t = M t+ 1 (13) and F ( m t ) = (cid:12) E t [G ( m t+ 1 ) M t = M t+ 1 ] : (14) Takingexpectedvaluesofbothsidesof(13)andsubstituting(7)and(14)impliesademandforrealbalances oftheform R t = G F ( ( m m t t ) ) : (15) Since G ( :) isstrictlydecreasing and F ( :) isstrictlyincreasingin m for m (cid:20) ^c + g , R t isstrictlydecreasing in m t for m (cid:20) ^c + g . Ontheotherhand,since G ( m ) = F ( m ) for m (cid:21) ^c + g , m (cid:21) ^c + g for R = 1 . From(13)wecanexpressthepresent valuedeflatoras q t = (cid:12) t M 0 = M t G G ( ( m m t 0 ) ) : (16) Substituting themarketclearing conditions into(3),whichinequilibrium mustholdwithequality, yields 1 t= X 0 E 0 ( q t (cid:0) q t+ 1 ) M t = ( M (cid:0) 1 + B (cid:0) 1 ) + 1 t= X 0 E 0 q t ( P t g (cid:0) T t ) ; (17) Given(6),(17)isequivalent to l i !t m 1 E 0 [q t ( M t + B t = R t ) + ( q t+ 1 (cid:0) q t ) M t ] = 0 : (18) Using(9)and(16)toeliminate M t + B t = R t and q t fromthisexpression yields l i !t m 1 E 0 (cid:12) t [G ( m t ) A 0 = M t + F ( m t ) (cid:0) G ( m t ) ] = 0 : (19) where A 0 ( (cid:17) M (cid:0) 1 + B (cid:0) 1 = R (cid:0) 1 ) denotesinitialnominalgovernmentliabilitiesandisaninitialconditionin period 0 . Weassumethat A 0 > 0 . A rational expectations equilibrium is a set of processes m t > 0 , M t > 0 , and R t (cid:21) 1 satisfying (14),(15), (19),andoneadditional equation specifying themonetarypolicyregime. Onecanthenuniquely 6

determineconsumption, c t ,from(11);labor, h t ,from(10);thepricelevel, P t ,from(12);thepricingkernel, r t+ 1 , from (13); the present value deflator, q t , from (16); the stock of public debt, B t = R t , from (9); and lumpsumtaxes, T t ,from(6). 3 Equilibrium under an Interest Rate Peg The class of monetary policy regimes to be considered in this section are interest-rate pegs defined by the choiceofaconstantnominalinterestrate R t = R (cid:21) 1 : Wefirstanalyzethecaseinwhichthenominalinterestrateissetatzero( R = 1 ). Thispolicycorrespondsto theoptimalquantityofmoneybecauseiteliminatesanyinefficienciesassociatedwithholdingmoney. Since for m (cid:21) ^c + g , F ( m ) = G ( m ) , it follows that when R t = 1 , any m t (cid:21) ^c + g solves (15). However, (19) will not be satisfied by any of these solutions. Since in this case F ( m ) = G ( m ) , (19) simply requires that l i m !t 1 E 0 (cid:12) t G ( m t ) = M t A 0 = 0 . Using(14)and(15)onecanexpress E 0 G ( m t ) = M t as ( (cid:12) R ) (cid:0) t G ( m 0 ) = M 0 so that for R = 1 l i m !t 1 E 0 (cid:12) t E 0 G ( m t ) = M t A 0 = G ( m 0 ) = M 0 A 0 6= 0 . Consequently, under a balanced budgetrulenorationalexpectations equilibrium existswhenthenominalinterestrateispeggedatzero,and therefore theoptimalquantity ofmoneycannotbebroughtabout. Next, we show that when the nominal interest rate is pegged at a positive value, then a rational expectations equilibrium exists and the associated resource allocation is unique while the price level is indeterminate. Let the nominal interest rate be pegged at some positive value R > 1 . Since G ( m ) = F ( m ) is monotonically decreasing in m ,strictlygreaterthanoneforall m < ^c + g ,andequaltoonefor m = ^c + g , itfollowsfrom(15)thatrealbalances areuniquely determined, thatis, m t = m 8 t (cid:21) 0 and g < m < ^c + g : If R > 1 ,thehousehold'scash-in-advanceconstraintisalwaysbindingandtherefore c t m (cid:0) g forall t . From market clearing in the product market it follows that h t = m . That is, under a pure interest rate peg with R > 1 therealresource allocation isuniqueandconsumption andhoursareconstant. Using E 0 [G ( m t ) A 0 = M t ] = ( (cid:12) R ) (cid:0) t G ( m 0 ) A 0 = M 0 and m t = m in(19)yields l i !t m 1 R (cid:0) t A 0 + (cid:12) t ( R (cid:0) 1 (cid:0) 1 ) M 0 = 0 (20) Thisequation issatisfiedforany M 0 . Therefore, inequilibrium theprice levelinperiod 0 isindeterminate, P 0 = M 0 = m . Thisrepresents the mainresult ofthis section, namely that abalanced budget rulecombined withapureinterestratepegleadstonominalindeterminacy. Perfect foresightequilibria Itisusefultocharacterize theevolution ofrealdebtandrealtaxesunderthemonetary-fiscal regimeconsidered in this section. Tosimplify matters, wewillrestrict the analysis toperfect foresight paths. Given M 0 , 7

theperfectforesight pathof M t isuniquely determined by(14)as M t = ( (cid:12) R ) t M 0 : Using this expression and the fact that both total nominal government liabilities and real balances are constant,realdebtcanbeexpressed as b t (cid:17) B t = P t = R m ( (cid:12) R ) (cid:0) t A 0 = M 0 (cid:0) R m : Accordingtothisexpression, thestockofrealdebtdependsontheinitialmoneysupply M 0 andistherefore not unique. The long–run level of real debt depends on the level of the nominal interest rate. If the monetary authority pegs the nominal interest rate at a value that exceeds the real interest rate ( R > (cid:12) (cid:0) 1 ), then l i m !t 1 b t = (cid:0) R m ,sotherealstockofdebtconverges toafinite(negative)value. Thatis,thegovernment ends up being a net lender to the public. Alternatively, if the monetary authority pegs the nominal interest rate at a value below the real interest rate ( 1 < R < (cid:12) (cid:0) 1 ), then l i m !t 1 b t = 1 , so the real stock of debt growswithoutbound.5 Thereason forthisexplosive behavior ofrealdebtisthatwhen (cid:12) R < 1 ,themoney supply falls at the rate 1 (cid:0) (cid:12) R and therefore seignorage income is negative. In turn, these losses must be financed with new debt because, according to the balanced budget rule, the government is only allowed to raise taxestocover government spending andinterest ontheoutstanding debt. However,private agents are willing to hold the ever increasing government debt because they face a path of real lump-sum taxes, (cid:28) t , whichisalsoincreasing overtime (cid:28) t = T t = P t = g + b (cid:0)t 1 = R ( R (cid:0) 1 ) = ( (cid:12) R ) : Allowingforfiscalsurpluses Inthissubsection weconsider balanced-budget rules that allowforpositive secondary surpluses. Thistype ofrulesareclearlymorerealisticthanourbaseline specification ofazerosecondary surplus.6 Supposethat the fiscal authority were to set exogenously the time path of the (non-negative) secondary surplus.7 For simplicityweassumethateithertherealsecondarysurplus, s t ,isexogenousandboundedaboveby s orthat thenominalsecondary surplus, S t = P t s t ,isexogenous andbounded aboveby S . Undereither assumption thefiscalpolicyruletakestheform T t (cid:0) P t g (cid:0) ( R (cid:0)t 1 (cid:0) 1 ) B (cid:0)t 1 = R (cid:0)t 1 = S t : 5Notethatalthoughtherealstockofdebtisincreasingovertime,itspresentdiscountedvalue( lim !t 1 (cid:12) t b t )iszero. 6Forexample,theproposedbalancedbudgetamendmentthatwaspassedinJanuary1995bytheU.S.HouseofRepresentatives allowsforpositivesecondarysurpluses. 7Therequirementofan,atleastsometimes,strictlypositivesecondarysurplusbyitselfisnotacompletedescriptionofthefiscal policyregime.Furtherrestrictionsonthewayinwhichthesecondarysurplusisbroughtaboutarenecessary. 8

Substituting this specification ofthe balanced budget rule in the government's sequential budget constraint (6),theevolution oftotalnominalgovernmentliabilities, M t + B t = R t ,becomes M t + B R t t = M (cid:0)t 1 + B R (cid:0)t (cid:0)t 1 1 (cid:0) S t = : : : = A 0 (cid:0) j X t = 0 S j : Usingthisexpression toeliminate M t + B t = R t inthetransversality condition (18)yields l i !t m 1 E 0 2 4 q t ( A 0 (cid:0) j X t = 0 S j ) + ( q t+ 1 (cid:0) q t ) M t 3 5 = 0 ; (21) whichtogether with(16)replaces equilibrium condition (19). Suppose that R = 1 . In this case E t q t+ j = q t = R (cid:0) j = 1 , thus E 0 ( q t+ 1 (cid:0) q t ) M t = 0 and E 0 q t = 1 . Consequently (21)simplifiesto A 0 = l i !t m 1 E 0 q t j X t = 0 S j ) = 0 (22) We first analyze the case that the real secondary surplus is exogenous and deterministic. From (15) it followsthatinanyequilibrium m t (cid:21) ^c + g ,whichimplies,by(16),that G ( m t ) = m t = ^c and q t = (cid:12) t P 0 = P t so that E 0 q t P t j = 0 P j s j = (cid:12) t P 0 P t j = 0 E 0 P j = P t s j = (cid:12) t P 0 P t j = 0 (cid:12) j (cid:0) t s j , where the last equality follows from P j = P t = (cid:12) j (cid:0) t q t = q j and from the fact that if R = 1 , E 0 q t = q j = 1 for any t (cid:21) j (cid:21) 0 . Thus (22) becomes A 0 = P 0 l i !t m 1 j X t = 0 (cid:12) j s j : Since P t j = 0 (cid:12) j s j ismonotonically increasing andbounded aboveby s = ( 1 (cid:0) (cid:12) ) ,itconverges tosomefinite positivevalue s . Thereforethetransversality condition willbesatisfiedonlyif P 0 = ( 1 (cid:0) (cid:12) ) A 0 = s ; that is, the transversality condition uniquely determines the equilibrium price level. However, the initial moneysupplyisindeterminate, any M 0 thatsatisfies M 0 (cid:21) ( ^c + g ) A 0 P 0 isanequilibrium. Fromthisresult it follows that under a balanced budget requirement in which the secondary real surplus is deterministic and strictly positive inatleast oneperiod, (i)anequilibrium consistent withthe optimal quantity ofmoney existsand(ii)theassociatedpricelevelisunique. Thereasonwhyanequilibriumconsistentwiththeoptimal quantity ofmoneyexistswhentherealsecondary surplus ispositiveanddoesnotexistwhenthesecondary surplusisexactlyzeroperiodbyperiodisthefollowing. Inequilibrium theratioofthegovernment's initial nominal liabilities to the initial price level must be equal to the present discounted value of real primary surpluses. If R = 1 , the real primary surplus is equal to the real secondary surplus because interest payments on the debt are zero. Therefore, if the real secondary deficit is positive in at least one period, the present discountedvalueofrealprimarysurplusesisalsopositive. Consequently, thereisauniquepricelevelwhich 9

makes the real value ofthe initial nominal government liabilities equal to this present discounted value. If, however, the secondary surplus is equal to zero period by period, the real primary surplus is also equal to zero. This implies that unless the initial government liabilities are equal to zero, there exists no price level which makes the real value of the government's initial nominal liabilities equal to the present discounted valueofrealprimarysurpluses. Assumenowthatthenominalsecondary surplusisexogenous anddeterministic, then(22)becomes A 0 = l i !t m 1 j X t = 0 S j Butthiswillcertainlynotbetrueforanyarbitrarypathofthenominalsecondarysurplus. Further,evenifthe sequence ofnominal secondary surpluses satisfied the abovecondition, theprice level wouldbeindeterminate. Therefore weconclude that under a balanced budget requirement a rational expectations equilibrium consistent with the optimal quantity of money may or may not exist, and even if it does exist the price level may or may not be determinate depending on the particular implementation of the balanced budget requirement. Now suppose that R > 1 . In this case, (12) implies that real balances are uniquely determined and constant over time, m t = m (cid:3) . With real balances uniquely determined, the real allocation is also uniquely determined. Using m t = m (cid:3) , E t q t+ 1 = q t = E t r t+ 1 = R (cid:0) 1 , and (16), one can express the last term on the left side of (21), E 0 ( q t+ 1 (cid:0) q t ) M t , as ( R (cid:0) 1 (cid:0) 1 ) (cid:12) t M 0 which converges to zero as t converges to infinity. Replacing E 0 q t by R (cid:0) t yields E 0 q t A 0 = R (cid:0) t A 0 , which implies that the first term on the left side of (21) converges tozeroas t converges toinfinity. Thus(21)becomes l i !t m 1 E 0 q t j X t = 0 S j = 0 : (23) Assuming that the real secondary surplus is exogenous and using the facts that q t = (cid:12) t M 0 = M t and E 0 M j = M t = ( (cid:12) R ) j (cid:0) t theleftsideof(23)canbewrittenas E 0 q t j X t = 0 S j = = (cid:20) = = = l i !t l i !t l i !t l i !t l i !t 0 ; m 1 m 1 m 1 m 1 m 1 E (cid:12) (cid:12) (cid:12) R 0 t t t (cid:0) (cid:12) M m M m M m t M t M 0 (cid:3) 0 (cid:3) s 0 (cid:3) M m 0 t t j = X t j = X j X s 0 (cid:3) t j = X E 0 E 0 t ( = 0 0 0 0 (cid:12) 1 P M M M M R (cid:0) s j j j s t j s t (cid:0) j ) ( (cid:12) 1 (cid:0) j t R (cid:12) ) R t+ 1 10

This shows that the transversality condition is satisfied for any value of M 0 and hence the initial money supplyisindeterminate. Theinitialpricelevel,beingequalto M 0 = m (cid:3) ,isthusalsoindeterminate. Alternatively, ifthenominalsecondary surplusisexogenous, theleftsideof(23)canbewrittenas l i !t m 1 E 0 q t j X t = 0 S j (cid:20) = = = l i !t l i !t l i !t 0 ; m 1 m 1 m 1 E S S 0 t t q t E R j X 0 (cid:0) t = 0 q t t S whereweusedthefactsthat S j isboundedandthat E 0 q t = R (cid:0) t . Thisimpliesthatthetransversality conditionwillbesatisfiedregardlessofthevalueof M 0 andhencetheinitialpricelevelisagainindeterminate. It follows that our earlier finding that if R > 1 the price level isindeterminate, also holds when the balanced budgetrequirementisimplementedthroughafiscalpolicyinwhichthesecondarysurplusisexogenousand positive. Relatedliterature Weclose thissection bycomparing ourresults tothose obtained under afiscalpolicy inwhichtheprimary surplus is exogenous. Specifically, as mentioned earlier, Auernheimer and Contreras (1994), Sims (1994), andWoodford(1994)findthatifthepathsofgovernmentpurchasesandtaxesareexogenous, thentheprice level is uniquely determined under an interest rate peg. To see this, use equations (7) and (16) and the fact that for R > 1 real balances are constant (by the liquidity preference equation (15)), to express the transversality condition (17)as 1 t= X 0 (cid:12) t M 0 ( 1 (cid:0) R (cid:0) 1 + E 0 ( (cid:28) t (cid:0) g t ) = m ) = M (cid:0) 1 + B (cid:0) 1 : When the process for the primary surplus, (cid:28) t (cid:0) g t , is exogenous, this equation implies that M 0 is only a function of the predetermined variables M (cid:0) 1 and B (cid:0) 1 and of the present discounted value of current and expected future primary surpluses. Consequently, M 0 is uniquely determined and thus so is the price level P 0 = M 0 = m . Comparing this result with the one obtained under a balanced budget rule, it follows that given the monetary policy regime, the adoption of a balanced budget rule has important consequences for price stability. Under the monetary policy considered here, a fiscal policy shift whereby a policy in which the realprimary surplus issetexogenously isreplaced byabalanced budget policy inwhich thesecondary surplusissetexogenously wouldresultinthelossofthenominalanchor. 11

4 Equilibrium under a Money Growth Rate Peg Under the monetary policy regime to be considered in this section the government sets a constant growth rate (cid:22) > 0 forthemoneysupply M t = (cid:22) t M 0 : (24) Giventhemoneygrowthratepeg,(14)reduces to F ( m t ) = (cid:12) (cid:22) E t G ( m t+ 1 ) (25) andthetransversality condition (19)becomes l i !t m 1 (cid:12) t E 0 h G ( m t ) (cid:22) (cid:0) t A 0 + ( F ( m t ) (cid:0) G ( m t ) ) M 0 i = 0 : (26) Given A 0 and M 0 ,arationalexpectations equilibrium isastochastic process f m t g satisfying (25)and(26). Steady stateequilibria Wefirstanalyze the existence of steady state equilibria, that is, non-stochastic equilibria with constant real balances. Clearly, asteady state equilibrium does notexist if (cid:22) (cid:20) (cid:12) because inthat case no constant value ofrealbalances satisfies (26). Thismeans thatthepolicy maker cannot bring about theefficient allocation, or optimal quantity of money, by setting the money growth rate to (cid:12) (cid:0) 1 .8 Combining this result with the oneobtained intheprevious section, weconclude thatunderabalanced budgetrequirement aninterest rate peg as well as a money growth rate peg equilibrium may fail to be consistent with the optimal quantity of money advocated by Milton Friedman. This result is a consequence of the particular fiscal policy we are analyzing. For example, Woodford (1994) shows that for a fiscal policy in which the real primary surplus is positive and exogenous a steady state equilibrium consistent with the optimal quantity of money exists underapureinterestratepeg. Healsoshowsthatforafiscalpolicyinwhichthestockofpublicdebtiszero andgovernment purchases areexogenous asteady stateequilibrium consistent withtheoptimal quantity of moneyexistsunderamoneygrowthratepeg. In whatfollows, weassume that the growth rate of the moneysupply exceeds the discount factor ( (cid:12) (cid:22) > ). Giventhisassumption, anyconstant valueof m satisfies(26). Sothequestion ofwhetherasteadystate equilibriumexistsandwhetheritisunique,reducestostudyingsolutionsto(25)inwhich m t = m t+ 1 = m (cid:3) forall t F ( m (cid:3) ) = (cid:12) (cid:22) G ( m (cid:3) ) : (27) For g < m < g + ^c , G ( m ) is monotonically decreasing, and F ( m ) is monotonically increasing. For 8If the balanced budget rule allows for surpluses, a steady–state equilibrium consistent with the optimal quantity of money may exist. To see this, note that if (cid:22) = (cid:12) , then (16) implies that in steady state q t = q +t 1 = 1 and (15) together with (25) impliesthat R t = 1 . From(18)itthenfollowsthat lim !t 1 ( M t + B t ) mustbeequaltozeroinasteady–stateequilibrium. As lim !t 1 M t = lim !t 1 (cid:12) t M 0 = 0 ,thisrequiresthat lim !t 1 B t = 0 aswell. Thiscouldcertainlybethecasebutitdoesnot needtobe:abalancedbudgetruledoesnotnecessarilyimplythatthestockofpublicdebtconvergestozero. 12

m (cid:21) ^c + g , F ( m ) = G ( m ) , and since (cid:12) = (cid:22) < 1 the left-hand side of (27) is greater than the right-hand side of(27) (figure 1). As m approaches g from above, the right-hand side of(27) becomes larger than the left-handsidebecause G ( m ) convergesto 1 while F ( m ) convergesto (cid:18) g = ( 1 (cid:0) g ) ;therefore, thereexistsa uniquesolution m (cid:3) to(27)satisfying g < m (cid:3) < g + ^c . Since m (cid:3) < g + ^c ,theconsumer's cash-in-advance constraint is binding in any steady state equilibrium and from (15) the nominal interest rate, R (cid:3) (cid:0) 1 , is positive and equal to (cid:22) = (cid:12) (cid:0) 1 > 0 . Because real balances are constant in steady state, the inflation rate is also constant and equal to the growth rate of the money supply, (cid:22) . Using the fact that total nominal government liabilities ( M t + B t = R t = A 0 ), real balances, and the nominal interest rate are constant, real debtcanbeexpressed as b t = (cid:22) (cid:0) t R (cid:3) m (cid:3) A 0 = M 0 (cid:0) R (cid:3) m (cid:3) : Hence, ifthemoneygrowthrateisnegative( (cid:22) < 1 ), thentherealstock ofdebtassociated withthesteadystate equilibrium grows without bounds. If, on the other hand, the money growth rate is positive ( (cid:22) (cid:21) 1 ), thenthelong-run realstockofdebtisfinite(andnegativefor (cid:22) > 1 ). Non–steady–stateequilibria Wemakethefollowingassumption regardingpreferences andthesizeofgovernment purchases F ( g ) < (cid:12) = (cid:22) i m n > f g G ( m ) : (A1) This assumption is satisfied for any value of (cid:22) for which private consumption exceeds government consumption inthesteady-state equilibrium. Thefollowing proposition showsthatif(A1)issatisfied, then m t isbounded aboveandbelow(awayfrom g )inanyequilibrium. Inparticular, equilibria inwhichtheprivate sectorendsupcompletely demonetized areruledout. Proposition 4.1 Suppose that preferences satisfy (A1) and that (cid:22) > (cid:12) , then there exists alower bound m , g < m < ^c + g ,suchthat m t (cid:21) m atalltimesinanyequilibrium, andthereexistsanupperbound m < 1 suchthat m t (cid:20) m atalltimesinanyequilibrium. Proof: Theproofofthisproposition, whichdrawsheavilyontheproofofproposition5inWoodford(1994), ispresented intheappendix Proposition 4.1 establishes that in any equilibrium there are at most bounded fluctuations. This result depends crucially on the balanced-budget requirement and the assumption of a positive initial stock of governmentdebt. Foranalternative fiscalpolicy, oneinwhichthestockofdebtandgovernment purchases are zero, Woodford (1994) proves that Proposition 4.1holds for non-negative money growth rates ( (cid:22) (cid:21) 1 ) but not for negative money growth rates ( (cid:12) < (cid:22) < 1 ) . In particular, he shows that for negative money growth rates speculative hyperdeflations are possible. The existence of the lower bound for real balances m , and hence the impossibility of speculative hyperinflations, in our setup is due to the fact that under our fiscal policy government purchases are positive, exogenous, and subject to a cash-in-advance constraint which prevents the price level from growing (in the long run) at a faster rate than the money supply. However, the reason why we can rule out speculative hyperinflations is not simply a consequence of this aspect of 13

our fiscal policy specification. Even if government purchases and the initial stock of debt were zero, as in Woodford,ourlog-linearpreferencespecificationwouldruleouttheexistenceofspeculativehyerinflations. If(A1)isreplaced bytheslightly strongerassumption that g < (cid:12) (cid:22) (cid:18) (cid:17) (cid:13) ; (A2) then one can show that the steady-state equilibrium is the unique rational expectations equilibrium. Assumption (A2) is still satisfied whenever private consumption exceeds government consumption in steady state.9 Some simple algebra shows that if (A2) holds, then m = ^c + g . Consider first the existence of perfect-foresight non-steady state equilibria. Assuming that m t (cid:20) ^c + g for all t , (25) can be solved for m t+ 1 asafunction of m t m t+ 1 = ( 1 + g (cid:13) m ) m t t (cid:0) (cid:13) (cid:17) f ( m t ) : (28) If (cid:22) > (cid:12) andif m t isbounded,thetransversalitycondition(26)isalwayssatisfied. Therefore,if(A2)holds, anysequence f m t g thatsatisfies(28)and m (cid:20) m t (cid:20) m constitutes aperfect foresightequilibrium. Proposition 4.2 If (cid:22) > (cid:12) and assumption (A2) is satisfied, then there exists a unique perfect foresight equilibrium m t = m (cid:3) forall t . Proof: Since the g ; (cid:12) , and (cid:22) that satisfy (A2) also satisfy (A1) it follows from our analysis of steady state equilibria that a steady state equilibrium m (cid:3) exists, that it is unique and that it satisfies g < m (cid:3) < ^c + g . Suppose there exists another perfect foresight equilibrium f m t g with m 0 < m (cid:3) . Let ~m t = m 2 t , t (cid:21) 0 ; From(28)itfollowsthat ~m t+ 1 = ( g 2 ~m t ) = ( ( 1 + (cid:13) ) ( g (cid:0) (cid:13) ) ~m t + (cid:13) 2 ) . (A2)and m 0 < m (cid:3) implythat f ~m t g isa monotonically decreasing sequence, andProposition 4.1implies thatitisbounded below by m . Therefore, the sequence f ~m t g must converge to some ~m , with m (cid:20) ~m < m (cid:3) . Being a limit, ~m must solve ( g 2 ~m ) = ( ( 1 + (cid:13) ) ( g (cid:0) (cid:13) ) ~m + (cid:13) 2 ) ~m = . Theonlynon-zerosolutiontothisequationis ~m = ( (cid:13) + g ) = ( 1 + (cid:13) ) = m (cid:3) , which is a contradiction. By a similar argument one can show that there does not exist a perfect foresight equilibrium with m 0 > m (cid:3) . Hencetheuniqueperfectforesight equilibrium is m t = m (cid:3) forall t Thisproposition canbeusedtoshowthattheonlyrational expectations equilibrium isthesteadystate. Proposition 4.3 If (cid:22) > (cid:12) andassumption (A2)issatisfied, thenthereexists aunique rational expectations equilibrium m t = m (cid:3) forall t . Proof: Supposethereexistsanotherrationalexpectationsequilibriumwith m t > m (cid:3) forsome t . Let f j ( m t ) m P t+ j (cid:17) , where f ( :) isdefined by(28). Itfollowsthat there exists an eveninteger J such that forall m P t+ s 2 [m ; m ] s (cid:20) J and g < m P t+ J + 1 < m . Note that m (cid:3) > m P t+ 1 and that P ( m t+ 1 (cid:20) m P t+ 1 ) > 0 because E t G ( m t+ 1 ) = G ( m P t+ 1 ) and G ( :) is strictly decreasing. Further, let m P t+ s (cid:0) 1 ) (cid:15) t+ s = P ( m t+ s (cid:21) m P t+ s jm t+ s (cid:0) 1 (cid:20) , if s is even, and let (cid:15) t+ s = P ( m t+ s (cid:20) m P t+ s jm t+ s (cid:0) 1 (cid:21) m P t+ s (cid:0) 1 ) , if s is odd. The probabilities (cid:15) t+ s arestrictly positiveforall s (cid:20) J . Toseethis,assumefirstthat m t+ s (cid:0) 1 (cid:20) m P t+ s (cid:0) 1 ,thensince F ( m ) is 9(A1)issatisfiedwhenever (cid:18) g < 1 (cid:12) (cid:0) = (cid:12) (cid:22) = (cid:22) .(A2)isstrongerthan(A1)because 1 (cid:12) (cid:0) = (cid:12) (cid:22) = (cid:22) > (cid:12) = (cid:22) . 14

strictly increasing, F ( m t+ s (cid:0) 1 ) (cid:20) F ( m P t+ s (cid:0) 1 ) ; using (25) this implies that E t+ s (cid:0) 1 G ( m t + s ) (cid:20) G ( m P t+ s ) and, since G(.) is strictly decreasing, this implies that (cid:15) t+ s > 0 . Similarly if m t+ s (cid:0) 1 (cid:21) m P t+ s (cid:0) 1 , then E t+ s (cid:0) 1 G ( m t+ s ) (cid:21) G ( m P t+ s ) andhence (cid:15) t+ s > 0 . Next,notethat P ( m t+ J (cid:21) m P t+ J ) = (cid:21) (cid:21) (cid:21) > (cid:15) t J + ( 1 (cid:0) (cid:15) t J + : : : J (cid:15) s = 2 Y 0 : P (cid:15) P t+ ( m t J + ( m P s t+ ) P t+ ( m J J (cid:0) ( m (cid:0) t+ 1 1 (cid:20) t+ (cid:20) (cid:20) 1 J m (cid:0) m m P t+ > 1 P t+ P t+ J J (cid:0) m (cid:0) ) 1 1 1 ) + P t+ ) J (cid:0) 1 ) Supposethat m t+ J (cid:21) m P t+ J ,then (cid:12) (cid:22) E t+ J G ( m t+ J + 1 ) = (cid:21) = F F (cid:12) (cid:22) ( ( G m m ( t+ P t+ m J J P t+ ) ) J + 1 ) Since g < m P t+ J + 1 < m and G ( :) ismonotone decreasing for m 2 ( g ; m ] ,thisimpliesthat P ( m t+ J + 1 < m jm t+ J (cid:21) m P t+ J ) > 0 : But by Proposition 4.1 this can never be true in any equilibrium. Therefore, in any equilibrium m P t+ J m t+ J < with probability one. But this is a contradiction because we just showed that if m t > m (cid:3) , then the probability that m t+ J (cid:21) m P t+ J is strictly positive. Hence, there does not exist a rational expectations equilibriumwith m t > m (cid:3) atany t . Similarly,onecanshowthatnorationalexpectationsequilibriumexists with m t < m (cid:3) atsome t ,andtherefore theuniquerational expectations equilibrium is m t = m (cid:3) forall t 5 Equilibrium under a Feedback Rule Themonetarypolicyregimeswehaveanalyzedthusfar—apureinterestratepegandapuremoneygrowth rate peg — share the characteristic of being insensitive to current economic conditions. By contrast, the monetarypolicyweconsiderinthissectionallowsforfeedbackfromendogenousvariables. Weassumethat the monetary authority responds to increases in inflation by raising the nominal interest rate. Specifically, themonetaryauthority isassumedtosetthenominalinterestrateaccording tothefollowingfeedbackrule R t = R + (cid:11) ( (cid:25) t (cid:0) (cid:12) R ) (cid:11) > 0 ; R (cid:21) 1 ; (29) 15

where (cid:25) t ( (cid:17) P t = P (cid:0)t 1 )denotesthegrossrateofinflation. Underthismonetarypolicyregimethequantityof moneyandbondsisendogenous, asisthecaseunderapureinterestratepegdiscussed insection three. In what follows we restrict the analysis to perfect foresight equilibria. Expressing (cid:25) t as M t = M (cid:0)t 1 m (cid:0)t 1 = m t andusing(15),thefeedback rulecanbewrittenas G F ( ( m m t t ) ) = R ( 1 (cid:0) (cid:11) (cid:12) ) + (cid:11) m m (cid:0)t t 1 M M t (cid:0)t 1 : (30) Using(14)toeliminate M t+ 1 = M t ,theaboveequation canbewrittenas G F ( ( m m 0 0 ) ) = R ( 1 (cid:0) (cid:11) (cid:12) ) + (cid:11) m m (cid:0) 0 1 M M (cid:0) 0 1 and (31) G F ( ( m m t+ t+ 1 1 ) ) = R ( 1 (cid:0) (cid:11) (cid:12) ) + (cid:11) (cid:12) m m t t+ 1 G F ( m ( m t+ t 1 ) ) ; all t (cid:21) 0 : (32) Inperfect foresight, thetransversality condition (19)becomes l i !t m 1 (cid:12) t [G ( m t ) ( A 0 = M t (cid:0) 1 ) + F ( m t ) ] = 0 : (33) A perfect foresight equilibrium consists ofa set ofsequences f m t ; M t g satisfying m t ; M t > 0 , (14), (31)- (33)given A 0 ; m (cid:0) 1 ; M (cid:0) 1 . Usingthedefinitionsof F ( :) and G ( :) equation (32)isasimplefirstorderlineardifference equation m i n [m t+ 1 ; ^c + g ] = a + b m i n [m t ; ^c + g ] (34) with a = ( 1 1 (cid:0) + (cid:11) (cid:18) (cid:12) R ) ( ( 1 1 + (cid:0) (cid:18) (cid:11) R (cid:12) g ) ) and b = 1 + (cid:18) R (cid:11) ( (cid:12) 1 (cid:0) (cid:11) (cid:12) ) : For R > 1 ,thisdifferenceequation hasauniqueconstant solution m (cid:3) = 1 1 + + (cid:18) R (cid:18) R g 2 ( g ; ^c + g ) : andfor R = 1 ithasacontinuum ofconstant solutions m (cid:3) (cid:21) ^c + g . Evaluating (30) at m t = m t+ 1 = m (cid:3) , it follows that both the steady-state gross money growth rate, M t = M (cid:0)t 1 , and the steady-state gross inflation rate are equal to (cid:12) R . Evaluating the feedback rule (29) at (cid:25) t = (cid:12) R impliesthat R isthesteady-statenominalinterestrate. Thefactthatthesteady-statemoneygrowth rateis (cid:12) R andimpliesthatthetransversalitycondition(33)willbesatisfiedif R > 1 butwillnotbesatisfied if R = 1 ;thatis,nosteadystateequilibriumexistsfor R = 1 . Accordingly,fortheremainderofthissection we assume that R > 1 . Note that the steady-state level of real balances is independent of (cid:11) and that it is identical tothelevelofrealbalances obtained underapureinterestratepeg. 16

Figure 2 plots the right and left hand sides of (34) as functions of m . The upper left panel displays the case 0 < b < 1 , which is equivalent to (cid:11) (cid:12) < 1 . For any m 0 2 ( g ; 1 ) , (34) implies a sequence of realbalances that converges monotonically tothesteady state m (cid:3) . Given m 0 ,equation (31) determines the nominal moneysupply intheinitial period, M 0 . Clearly, M 0 willbestrictly positive forany m 0 2 ( g ; 1 ) , if, in addition to (cid:11) (cid:12) < 1 , the condition R ( 1 (cid:0) (cid:11) (cid:12) ) < 1 is satisfied. So among the many possible perfect foresight equilibria there aresomeinwhichinitial real balances arearbitrarily large andtheinitial nominal interest rate is zero. If, on the other hand, R (cid:21) 1 = ( 1 (cid:0) (cid:11) (cid:12) ) , then M 0 will be strictly positive only if m 0 < 1 + 1 + g (cid:18) R (cid:18) R (cid:0) ( 1 (cid:0) ( 1 (cid:11) (cid:11) (cid:12) (cid:12) ) ) 2 ( m (cid:3) ; ^c + g ) . Thisimpliesthatinthiscase,thenominalinterestrateisalwayspositive along the perfect foresight path and real balances are always below the minimum level consistent with the efficient allocation ( m t < ^c + g ). Tosummarize, when the monetary authority follows a feedback rule for the nominal interest rate that is not very sensitive to current inflation ( (cid:11) (cid:12) < 1 ), both the price level and the real allocation are indeterminate. By comparison under a pure interest rate peg the price level is also indeterminate but the real allocation is unique. In this sense the feedback rule can make the indeterminacy problem worse. The lower left panel of figure 2 displays the case b 2 ( (cid:0) 1 ; 0 ) , that is, (cid:11) (cid:12) > ( 1 + (cid:18) R ) = ( (cid:18) R (cid:0) 1 ) and (cid:18) R > 1 . Inthiscasethereisacontinuumofperfectforesightequilibriaconvergingtothesteadystate m (cid:3) in anoscillatingfashion.10 Again,thepricelevelandtherealallocationareindeterminate. Interestingly, inthis case the indeterminacy occurs for a very `active' monetary policy. In models in which in equilibrium the marginal utility of consumption is constant or exogenous —as in the money-in-the-utility-function model withseparablesingle-periodpreferencesstudiedbyLeeper(1991)—thestandardresultisthatthehigherthe elasticity ofthefeedback rulethelesslikelyitisthattheequilibrium isindeterminate. The upper right panel of figure 2 shows the case b > 1 , which occurs when 1 < (cid:11) (cid:12) < 1 + 1 = ( (cid:18) R ) , and the lower right panel shows the case b < (cid:0) 1 which occurs when either (cid:11) (cid:12) > 1 + 1 = ( (cid:18) R ) and 1 (cid:18) R < or 1 + 1 = ( (cid:18) R ) < (cid:11) (cid:12) < ( 1 + 1 = ( (cid:18) R ) ) = ( 1 (cid:0) 1 = ( (cid:18) R ) ) and (cid:18) R > 1 . In these cases the only perfect foresight equilibrium isthesteady state, m t = m (cid:3) forall t ,because forany m 0 6= m (cid:3) thesequence ofreal balances implied by(34) either violates the lower bound g or is such that at some point the right hand side of (34) exceeds ^c + g . The determinacy of equilibrium when jb j > 1 demonstrates that under a balancedbudgetrequirementthemonetaryauthoritycanbringaboutnominalstabilitywithoutdirectlycontrollingthe monetaryaggregate. Figure3summarizestherelationbetweenpriceleveldeterminacyandtheelasticityofthefeedbackrule (cid:11) . Itshowswithopendotsthepairs( R ; (cid:11) (cid:12) )forwhichthepricelevelisdeterminate andwithsoliddotsthe pairsforwhichtheprice levelisindeterminate. Giventhesteady-state nominal interest rate, bothrelatively insensitive and very sensitive feedback rules generate indeterminacy of the perfect foresight equilibrium, andforintermediate degreesofsensitivity theequilibrium isunique. Atthesametime,giventhesensitivity of the feedback rule, the higher the nominal interest rate the more likely it is that the perfect foresight equilibrium isindeterminate. 10Theinitialrealbalances m 0 mustsatisfy m 0 (cid:21) m a x [g ; ( ^c + g (cid:0) a ) = b ] and m 0 < ( g (cid:0) a ) = b if ( g (cid:0) a ) = b < ^c + g . 17

6 Conclusion In this paper we argue that a balanced-budget rule has important implications for the determinacy of the pricelevel. Usingastandard cash-in-advance frameworkwefindthatifthebalancedbudgetruleiscoupled withaninterestratepeg,thepricelevelisindeterminate, whereasifitiscoupled withamoneygrowthrate, the price levelis determinate. Wealso find that ifthe balanced budget rule iscoupled witha feedback rule wherebythenominalinterestrateisanincreasingfunctionoftheinflationrate,thedeterminacyoftheprice leveldependsontheresponsiveness ofthefeedbackrule. Forbothlittleandhighlysensitivefeedbackrules, thepricelevelisindeterminate, whereasformoderately responsive feedback rulesthepricelevelisunique. The results of this paper suggest that one largely ignored aspect of balanced-budget fiscal policy rules is their implications for price level determinacy. Foragiven monetary policy regime the price level can be determinate under some fiscal policy regime but indeterminate under a balanced budget requirement and viceversa. Therefore, anydiscussion onthemacroeconomic consequences ofbalanced budgetrulesshould takeintoaccount itsimplications fornominalstability. Our finding that for certain monetary policy specifications a balanced budget rule leads to nominal instability complements our earlier work on the macroeconomic consequences of balanced-budget rules, Schmitt-Grohe´ and Uribe (1996), where we show that in a standard neoclassical growth model without money such a fiscal policy may lead to real instability by allowing for equilibria in which expectations of futuretaxincreases canbeself-fulfilling. Thestudyoftheimplicationsofbalancedbudgetrulesfornominalstabilitypresentedinthispapercould beextended inseveraldirections. First,itwouldbeworthstudying howtheresultsaremodifiedinamodel augmented with nominal frictions such as sticky prices. Second, expanding the set of monetary policies might provide additional insights into the restrictions that the particular fiscal policy studied in this paper imposes on theconduct ofmonetary policy. Inthe context ofamodel withnominal rigidities thefamily of Taylorrules, that is, feedback rules whereby thenominal interest rate depends notonly oncurrent inflation butalsoontheoutputgap,areespecially interesting fromanempiricalpointofview. Lastly,theperiod-byperiod balanced-budget rule we consider is an obvious starting point, but there are certainly more realistic specifications, such as rules that require the budget to be balanced over non-overlapping intervals of more thanoneperiod. 18

Appendix ProofofProposition4.1 Wefirstprovetheexistenceofthelowerbound m . Define m (cid:17) i n f (cid:26) m (cid:21) g (cid:12) (cid:12) (cid:12) (cid:12) F ( m ) (cid:21) (cid:12) = (cid:22) m i n f 0 (cid:21) g G ( m 0 ) (cid:27) Note that m > g by (A1). Since m (cid:21) ^c + g , implies F ( m ) = G ( m ) > (cid:12) = (cid:22) G ( m ) ; one must have m < ^c + g . Suppose m t < m ,thiswouldrequirethat F ( m t ) < (cid:12) = (cid:22) m i n f 0 > g G ( m 0 ) (cid:20) (cid:12) = (cid:22) E t [G ( m t+ 1 ) ] contradicting (25). Therefore, m t (cid:21) m inanyequilibrium. Nextweshowtheexistence oftheupper bound m . Define m (cid:17) m a x ( (cid:12) = (cid:22) ^c m (cid:20) s m u p (cid:20) c^ + g G ( m ) ; ^c + g ) : Thefactthat G ( m ) iscontinuousonthecompactinterval [m ; ^c + g ] impliesthat m < 1 . Thedefinition of m furthermore impliesthatforall m intheinterval [m ; ^c + g ] , G ( m ) (cid:20) (cid:22) = (cid:12) m ^c (35) Observealsothatforall ^c + g < m (cid:20) m , G ( m ) = m 1 ^c (cid:20) m 1 ^c (cid:20) (cid:22) = (cid:12) m 1 ^c Thus(35)holds forall m intheinterval m (cid:20) m (cid:20) m . Nowsuppose that atsomedate m t (cid:21) m : Itfollows that m ^c t = = (cid:20) = (cid:20) = F (cid:12) (cid:22) (cid:12) (cid:22) P m ^c m ^c m ^c ( m P t P t ( m t (cid:0) + + ) t ( m ( m t+ P (cid:12) (cid:22) (cid:12) (cid:22) = t+ t+ 1 ( t P E (cid:12) E (cid:22) (cid:20) 1 > 1 m (cid:20) m t+ ( m t [m t [G ( m ) ] t t+ 1 m ) E [G ( m ) jm t t t + 1 + 1 m ) E [G ( m ) jm t t t + 1 + 1 1 (cid:12) ) m + P ( m > t t+ 1 ^c (cid:22) m (cid:12) > m ) + P ( m t t 1 + ^c (cid:22) 1 m ) > [E ( m jm t t t + 1 + 1 ^c 1 a x ( m (cid:0) m ; 0 ) ] t+ 1 ^c (cid:20) > m ) > 1 t+ m m E 1 ] + ] [m t m ) > E m jm t+ 1 [m t t+ ) (cid:0) m t+ 1 jm 1 ] > t+ m 1 ] > 1 ^c m ] 1 ^c where P t ( : : : ) denotestheprobabilityoftheeventinquestionconditionaluponinformationavailableattime 19

t . In deriving these equalities and inequalities we have used (25), (35), and the fact that m t+ 1 (cid:21) m with probability one. Thenforanyvalueof m t ,itfollowsthat E t [m a x ( m t+ 1 (cid:0) m ; 0 ) ] (cid:21) (cid:22) = (cid:12) m a x ( m t (cid:0) m ; 0 ) : Since G ( m t+ T ) (cid:21) ( 1 = ^c ) m a x ( m t+ T (cid:0) m ; 0 ) itfollowsthat E t G ( m t+ T ) (cid:21) ( 1 = ^c ) ( (cid:22) = (cid:12) ) T m a x [m t (cid:0) m ; 0 ] . Asaresult(26)canbewrittenas T l i ! m 1 (cid:12) T E t (cid:20) G ( m t+ T ) ( A (cid:22) t T (cid:0) M t ) + (cid:12) (cid:22) G ( m t+ T + 1 ) M t (cid:21) (cid:21) m a x [m t (cid:0) m ; 0 ] A ^c t : whichisviolated whenever m t > m . Hence,therecanbenoequilibrium inwhich m t > m 20

References Auernheimer, Leonardo and Benjamin Contreras, Control of the Interest Rate with a Government Budget Constraint: DeterminacyofthePriceLevelandOtherResults,mimeo,TexasA&MUniversity, 1990. Canzoneri, Matthew B.and Behzad Diba, Fiscal Constraints onCentral Bank Independence and Price Stability, mimeo,GeorgetownUniversity, July1996. Leeper, Eric,1991, Equilibria under `Active' and`Passive' Monetary andFiscalPolicies, Journal ofMonetaryEconomics27,129-147. Lucas, Robert E. Jr. and Nancy Stokey, 1987, Money and interest in a cash-in-advance economy, Econometrica55,491-514. Schmitt-Grohe´, Stephanie and Mart´n Uribe, Balanced-Budget Rules, Distortionary Taxes, and Aggregate Instability, working paper, The Board of Governors of the Federal Reserve System, Washington D.C., July1996, forthcoming JournalofPoliticalEconomy. Sims, Christopher, 1994, A simple model for the study of the determination of the price level and the interaction ofmonetaryandfiscalpolicy,EconomicTheory4,381-399. Woodford, Michael, 1994, Monetary policy and price level determinacy in a cash-in-advance economy, EconomicTheory4,345-380. Woodford, Michael, 1995, Price-level determinacy without control of the monetary aggregate, Carnegie- RochesterConference SeriesonPublicPolicy43,1-46. 21

Figure1 MoneyGrowthRatePeg ( (cid:22) > (cid:12) and(A2)) G F ( ( m m t t ) ) = = m 1 i (cid:0) n ( m m i m ; ^c t (cid:18) m n ( m t + t t ; g ^c ) + (cid:0) g g ) ‹ F(m ) t › b / m G(m ) t 0 g _m m* cˆ+g m t 22

Figure2 FeedbackRule: m i n [m t+ 1 ; ^c + g ] R = t = a + R b + m (cid:11) i ( n (cid:25) t [m (cid:0) t ; (cid:12) ^c R + ) g ] ——= m i n [m ; ^c + g ] -o-o-o= a + b m i n [m ; ^c + g ] 0<b<1 b>1 cˆ+g cˆ+g g g g m* cˆ+g g m* cˆ+g −1<b<0 −1>b cˆ+g cˆ+g g g g m* cˆ+g g m* cˆ+g 23

Figure3 Relationship betweenPriceLevelDeterminacyandvaluesof (cid:11) and R o pricelevelisdeterminate (cid:1) pricelevelisindeterminate a b 1 1 R 24