Price-level Determinacy, Lower Bounds on the Nominal Interest Rate, and Liquidity Traps

Board of Governors of the Federal Reserve System International Finance Discussion Papers Number 795 April 2004 PRICE-LEVEL DETERMINACY, LOWER BOUNDS ON THE NOMINAL INTEREST RATE, AND LIQUIDITY TRAPS Ragna Alstadheim and Dale W. Henderson NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors. Recent IFDPs are available on the Web at www.federalreserve.gov/pubs/ifdp.

PRICE-LEVEL DETERMINACY, LOWER BOUNDS ON THE NOMINAL INTEREST RATE, AND LIQUIDITY TRAPS Ragna Alstadheim and Dale W. Henderson* Abstract: We consider monetary-policy rules with inflation-rate targets and interest-rate or money-growth instruments using a flexible-price, perfect-foresight model. There is always a locally-unique target equilibrium. There may also be below-target equilibria (BTE) with inflation always below target and constant, asymptotically approaching or eventually reaching a below-target value, or oscillating. Liquidity traps are neither necessary nor sufficient for BTE which can arise if monetary policy keeps the interest rate above a lower bound. We construct monetary rules that preclude BTE when fiscal policy does not. Plausible fiscal policies preclude BTE for any monetary policy; those policies exclude surpluses and, possibly, balanced budgets. Keywords: price-level indeterminacy, multiple equilibria, zero bound, liquidity trap, monetary policy, monetary rule, fiscal policy, money demand *Alstadheim is an Assistant Professor at Howard University, and Henderson is a Special Adviser in the Division of International Finance at the Federal Reserve Board. For helpful comments, we thank David Bowman, Matthew Canzoneri, Behzad Diba, Refet Gurkaynak, Berthold Herrendorf, Lars Svensson, and participants in a session at the 2001 meetings of the Southern Economics Association; in seminars at Georgetown University, the Federal Reserve Board, and the European Central Bank; and in the Konstanz Seminar on Monetary Theory and Monetary Policy. Remaining errors are our own. Alstadheim thanks the Center for Monetary and Financial Research in Norway and the Federal Reserve Board for financial support and hospitality. The views in this paper are solely the responsibility of the authors and should not be interpreted as reflecting those of the Board of Governors of the Federal Reserve System or any other person associated with the Federal Reserve System. The email addresses of the authors are, respectively, ra7@georgetown.edu and dale.henderson@frb.gov.

1 Introduction and summary In this paper we discuss price-level determinacy when there is a lower bound on the (nominal) interest rate. The lower bound may arise either because of the behavior of (private) agents or because of monetary policy. For generality and relevance, our analysis is conducted in terms of the inflation rate instead of the price level.1 In our terminology, a model exhibits (inflation-rate) indeterminacy if it has multiple equilibria. Determinacy in flexible-price models is of both theoretical and practical interest. As regards theory, inflation-rate determinacy is a standard topic. Furthermore, in models with synchronized price contracts, agents must be able to determine what expectedinflationwouldbeunder priceflexibilityin order to set theircontractprices. As regards practice, the current situation in Japan, with deflation and zero shortterm interest rates, makes it more urgent to ascertain whether the possible existence of multiple equilibria is more than a theoretical curiosum. Recently, the possibility of indeterminacy has received much attention. Models with standard interest-rate rules or money-growth rules and a locally unique steadystatetargetequilibrium(TE)fortheinflationratemayhaveadditionalequilibria. To be more precise, there may be multiple below-target equilibria (BTE), paths along whichtheinflationrateisalwaysbelowtargetandisconstantoreitherasymptotically approaches or eventually reaches a below-target value.2 Fiscal policy may preclude BTE; in particular, a balanced-budget fiscal policy precludes BTE in which the interest rate is always at a zero lower bound.3 We illustrate, modify, and extend recent analysis using a perfect-foresight, superneutral model with flexible prices which may have a liquidity trap. We adopt what we regard as the conventional definition of a liquidity trap: a liquidity trap is a region of the money-demand function in which bonds and money are perfect substitutes so that open-market operations in bonds cannot lower the interest rate any further.4 In our model, a liquidity trap may arise at a zero or at a strictly positive interest rate. 1Ofcourse,inflation-ratedeterminacyandprice-leveldeterminacyarelinked. Iftheinflationrate (definedasthepercentagechangeinthepricelevelbetweentodayandyesterday)isdeterminedand yesterday’s price level is known, then today’s price level is determined. 2A 1999 version of Woodford (2003) cited by Benhabib, Schmitt-Grohe, and Uribe (2001a) containsanexplanationofthepossibleexistenceofBTEwithbothinterest-raterulesandmoney-supply rules. In Woodford (2003) and Benhabib, Schmitt-Grohe, and Uribe (2001b) the inflation rate may asymptoticallyapproachabelow-targetsteadystate;inSchmitt-GroheandUribe(2000),Benhabib, Schmitt-Grohe, and Uribe (2001a) and Eggertson and Woodford (2003) it may eventually reach a below-target steady state. 3Woodford (2001a) and Benhabib, Schmitt-Grohe, and Uribe (2001a) consider fiscal policy. Schmitt-Grohe and Uribe (2000) analyze balanced-budget fiscal policy. 4McCallum(2001)referstoa‘liquiditytrapsituation’asasituationinwhich‘the(usual)interest rate instrument is immobilized’. Svensson (2000) refers to a liquidity trap as a situation with a binding zero lower bound on the nominal interest rate. Krugman (1998) refers to a liquidity trap as a situation where ‘monetary policy loses its grip because the nominal interest rate is essentially 1

It is useful to summarize what we do. We present accessible derivations of the central results regarding indeterminacy given the existence of a lower bound on the interest rate.5 We distinguish clearly between a lower bound that arises because of monetary policy and one that arises because agents are in a liquidity trap. It turns out that a liquidity trap is neither necessary nor sufficient for BTE. We also consider monetary-policy rules that preclude BTE. The monetary policy rules used in deriving indeterminacy results are monotonic in the inflation rate. We presenttwokindsofmonetary-policyrulesthatmayprecludeBTE. First,elaborating on an observation in Benhabib, Schmitt-Grohe, and Uribe (2001b), we demonstrate that monetary-policy rules that are not monotonic in the inflation rate may preclude BTE. Second, we present a rule under which the interest rate responds to expected future inflation as well as to current inflation. This rule is asymmetric: the interest rate responds more strongly to expected future inflation if the current inflation rate is below the target rate. In addition, we show how conclusions about determinacy under alternative monetary rules depend on fiscal policy. For simplicity, we characterize fiscal policy by the growth rate of total nominal government debt.6 There is always a growth rate of debt high enough to preclude BTE no matter what the monetary policy because BTE paths would violate the transversality condition. A balanced-budget fiscal policy (a zero growth rate of debt) is not expansionary enough if the interest rate is positive at least part of the time either because of a positive lower bound or, for example, because of foreseen variation in productivity. Within a range, the combination of a small deficit with a standard interest-rate or money-supply rule guarantees that the TE is the unique equilibrium because the deficit precludes BTE. Inthe next sectionwe layour model anddiscuss twospecificmoney-demandfunctions. Section 3 is a presentationof some results regarding the existence of BTE with interest-rate rules. We discuss indeterminacy under money-growth rules in section 4. In section 5, we present monetary-policy rules that assure determinacy. Section 6 is a discussion of some implications of fiscal policy for determinacy. Concluding remarks are provided in section 7. zero[and]thequantityofmoneybecomesirrelevantbecausemoneyandbondsareessentiallyperfect substitutes’. OurdefinitionisthesameasKrugman’sexceptthat,likeSargent(1987)amongothers, we explicitly allow for a liquidity trap at a positive interest rate. 5Since our model has flexible prices and exhibits superneutrality, a lower bound can give rise to only nominal indeterminacy. However, using a model with sticky prices, Benhabib, Schmitt-Grohe, and Uribe (2001b) show that a lower bound can give rise not only to nominal indeterminacy but also to real indeterminacy. 6The characterization of fiscal policy in Eggertson and Woodford (2003) is more general and includes ours as a special case. 2

2 The model 2.1 Agents Our model economy is populated by a continuum of agents each of which acts simultaneously as a consumer and a producer. For simplicity, we assume that the product market is perfectly competitive, that prices are flexible, and that agents have perfect foresight. The problem of each agent is to find the ∞ Max βt U(C )+V(m ) [1/(2ρ )]Y 2 t t t t Ct,Bt,Mt,Yt − t=0 X © ª (1) U (C ) > 0, U (C ) < 0, m = M /P , V (m ) 0, V (m ) 0 0 t 00 t t t t 0 t 00 t ≥ ≤ subject to the following period budget and positivity constraints: P Y +M +I B = T +P C +M +B t t t 1 t 1 t 1 t t t t t − − − (2) C > 0,M > 0,P > 0,ρ > 0 t t t t t ∀ Periodutilityisincreasinginconsumption,weaklyincreasinginreal(money)balances m , and decreasing in output.7 ρ represents the level of productivity; as ρ increases t t t the production of a given level of output causes less disutility.8 The agent takes as given the money price of goods (P ) and the gross nominal interest rate (I ) earned t t on a bond held from period t to period t +1, and chooses holdings of two nominal financial assets, money (M ) and bonds (B ); consumption (C ); and output (Y ). t t t t According to the period budget constraint, nominal income from production in this period plus nominal money balances and bond holdings inclusive of interest from last periodmustequaltaxpaymentsT plusthesumofconsumptionandmoneyandbond t holdings for this period. In addition, each agent and, therefore, agents as a group are subject to a no-Ponzi-game condition: lim(M +B )Πk=t 1I 1 0, Πk= 1I 1 1 (3) t t t k=0− k− ≥ k=0− k− ≡ →∞ where M +B is their net (nominal) financial assets in period t. t t 7We consider specific functional forms for V(m ) in section 2.3 and for U(C ) in section 6.3. t t 8Increasesinρ canbeinterpretedeitherasincreasesinproductivityordecreasesinthedisutility t of labor if agents supply their desired amounts of labor either because they are consumer-producers or because they participate in a labor market that clears. Following Obstfeld and Rogoff (1996), let 1 φ l be the disutility associated with labor (l ) and let A l 2be the production function. Inverting − t t t t t the production function yields φ l = φ (y /A )2 = [1/(2ρ )]y2, ρ =A2/(2φ ) − t t − t t t − t t t t t 3

To simplify exposition, in sections 2 - 6 we express the nominal interest rate, the real interest rate and inflation rate in gross terms and refer to them as ‘the interest rate’, ‘the real interest rate’, and ‘the inflation rate’ respectively. In the introductory and concluding sections, we refer to the net nominal interest rate as ‘the interest rate’ in order to facilitate comparison of our results to those of others. Three necessary conditions for an optimum are I t U (C ) = βU (C ) (bonds) (4) 0 t 0 t+1 Π t+1 1 U (C ) = V (m )+βU (C ) (money) (5) 0 t 0 t 0 t+1 Π t+1 C = ρ U (C ) (output) (6) t t 0 t where Π = Pt+1 is the (backward looking) gross inflation rate. A fourth necessary t+1 Pt condition (the transversality condition) is that (3) hold with equality lim(M +B )Πk=t 1I 1 = 0, (7) t t k=0− k− t →∞ Informally, since the marginal utility of consumption is always positive, it cannot be optimal for the present value of agents’ ‘end of horizon’ net financial assets to be strictly positive. The first order conditions have been written as equilibrium conditions: Y has been set equal to C as it must be in equilibrium since there is t t no government spending, and desired asset stocks have been set equal to actual asset stocks. We assume that productivity is constant (ρ = ρ) except in section 6.3. Under t this assumption, the four equilibrium conditions (4), (5), (6), and (7) reduce to the Fisher equation, the money market equilibrium condition, the output determination equation, and the transversality condition: I = RΠ (8) t t+1 U (C¯ ) 0 I = (9) t U (C¯ ) V (m ) 0 0 t − C¯ = ρU (C¯ ) (10) 0 lim(M +B )Πk=t 1I 1 = 0, (11) t t k=0− k− t →∞ where R 1 > 1 is the constant gross real interest rate and C¯ is the constant ≡ β flexible-price value of consumption. I 0 is ruled out by (8), given R,Π > 0. In t t+1 ¯ ≤ turn, (9) implies that U (C) > V (m ), so depending on the functional formof V (m ) 0 0 t 0 t there may be a lower bound on m that is greater than zero. Since U (C¯ ) > 0 and t 0 V (m ) 0, (9) implies I Ω 1, where Ω is the lower bound on I implied by 0 t t t ≥ ≥ ≥ V (m ). Since R > 1 and I Ω, (8) also implies Π Ω. 0 t t ≥ t ≥ R 4

2.2 Policy We assume that fiscal policy determines the total amount of nominal government bonds outstanding, D , through control of the budget deficit inclusive of interest payt ments. Monetary policy determines whether these bonds are held by the monetary authority as a match for the money supply, M , or directly by the public, B , through t t control of open-market operations. The consolidated government balance sheet implies that D = M +B . Most of this paper is devoted to the analysis of determinacy t t t under alternative monetary-policy rules which are specified below. Except in section 6, we assume that fiscal policy is conducted so that (11) holds for any path of the nominal interest rate.9 Fiscal policyandequation(11), therefore, maybe disregarded until that section. So that we can discuss indeterminacy in our stripped-down model, we assume that there is a target equilibrium (TE) with a target value for inflation (Π ) given ∗ by Π > Ω because of considerations not included in the model. Since the target ∗ R nominal interest rate (I ) must satisfy I = RΠ , it is given by I > Ω. Absent such ∗ ∗ ∗ ∗ considerations, the optimal values for Π and I would be Ω and Ω, respectively. Our t t R main focus is on the possible existence of below-target equilibria (BTE). We define BTE as weakly increasing or decreasing paths for inflation and the interest rate along which they are always below Π and I respectively, and are either constant at, ∗ ∗ asymptotically approach, or eventually reach values represented by ΠBTE and IBTE, respectively, where I > IBTE = RΠBTE Ω. ∗ ≥ 2.3 Money demand and lower bounds on the interest rate Weconsidertwoparticularspecificationsofmoneydemand. Underbothspecifications thegrossnominalinterestratehasalowerbound(possiblyone). Equation(9)implies U (C¯ ) U (C¯ ) 0 0 I = = Ω 1 (12) t ¯ ¯ U (C) V (m ) ≥ U (C) lim V (m ) ≥ 0 0 t 0 0 t − −mt →∞ Of course, a lower bound of one for I implies a lower bound of zero for the net t nominal interest rate. The lower bound Ω may be unattainable or attainable. To model an attainable lower bound (ALB) for I , we assume that the utility of real balances is given by t V m 1(m m )2 for m m 0 t − 2 − t t ≤ V(m ) = , m > 0 (13) t  V m for m > m  0 t t 9This means that (except in section 6) fiscal policy is assumed to be Ricardian according to the  definition of Benhabib, Schmitt-Grohe, and Uribe (1998). A sufficient condition for (11) to hold is that the gross growth rate of total government debt is low enough. Since the nominal interest rate enters equation (11), the maximum permissible growth rate depends on monetary policy and the rest of the model. 5

As before, V 0 represents the minimummarginal utilityof real balances. Equation 0 ≥ (12) implies U 0 (C¯) for m m I = U 0 (C¯) − (m − mt) − V 0 t ≤ U 0 (C¯ ) = ΩALB, (14) t    U U (C¯ 0 ( ) C¯) V for m t > m    ≥ U 0 (C¯ ) − V 0 0 − 0     To model an unattainable lower bound (ULB), we assume that the utility of real balances is given by γ V(m ) = V m +γlnm γ > 0, V 0, m > (15) t 0 t t 0 ≥ t U (C¯ ) V 0 0 − where V represents the lower limit of the marginal utility of real balances. With this 0 functional form, equation (9) implies U (C¯ ) U (C¯ ) U (C¯ ) I = 0 > 0 = 0 = ΩULB (16) t U (C ¯ ) γ V U (C ¯ ) lim γ V U (C ¯ ) V 0 − mt − 0 0 −mt mt − 0 0 − 0 →∞ Money-demand functions with an ALB and an ULB are represented in figure 1.10 In both cases, Ω = 1 if and only if V = 0. With an ALB there is a liquidity trap, 0 as conventionally defined, at the lower bound. Purchases of bonds with money can not lower the interest rate. With an ULB there is never a liquidity trap. Purchases of bonds with money can always lower the interest rate, if only by an infinitessimal amount. From (16), (14), and I > 0 (from (4)) we know that there exists a minimum level for real money balances denoted by m: m m = max m+V U (C¯ ),0 with an 0 0 ≥ { − } ALB model, and m > m = γ with an ULB. In order for lim I = in the ALB model, we need m+ U 0 V (C¯) − V U 0 (C¯ ) 0. m → m ∞ 0 − ≥ 3 Interest-rate rules and BTE We begin by assuming that monetary policy takes the form of interest-rate rules and consider two examples. The general form of the interest-rate rules is Y Π t t I = g(I , , ) (17) t ∗ Y¯ Π ∗ where Π is the target-equilibrium (TE) inflation rate and Y¯ = C¯ is the flexible-price ∗ output level. With flexible prices, output is always at its flexible-price level , Y = Y¯ , t so fromnow on we omit Yt. We assume that the elasticity of the interest rate with re- Y¯ spect to the inflation rate evaluated at the target inflation rate is greater than one. If 10For figure 1, V =0.001, γ =1, m=2, and U (C¯)=1. 0 0 6

Π = Π , then I = I . In this and the following section, we assume that the interestt ∗ t ∗ rate rule is at least weakly increasing in the inflation rate and that the interest rate has a lower bound, eitherapreference-determinedlowerboundorapolicy-determined lower bound. A preference-determined lower bound might exist because of a liquidity trap. A policy-determined lower bound might exist, for example, because the monetary authority wants to keep money market funds economic.11 Under these assumptions, interest-rate rules are associated with two steady-state equilibria: a TE with Π = Π and a BTE, where Ω/R ΠBTE < Π and Ω IBTE < I . t ∗ ∗ ∗ ≤ ≤ 3.1 A preference-determined lower bound First, consider the case of an interest-rate rule under which the interest rate may go all the waytothepreference-determined lowerbound, ΩALB, associatedwithanALB money-demand function:12 Π λ I = max ΩALB,I t , λ > 1 (18) t ∗ Π " µ ∗¶ # The piecewise log-linear rule (18) is globally (weakly) increasing in the inflation rate, and the elasticity of the interest rate with respect to the inflation rate in the strictly increasing part is λ > 1.13 We refer to this case as the liquidity-trap case. The Fisher equation (8), the interest-rate rule (18), and I = RΠ imply a log- ∗ ∗ linear difference equation in Π when I > ΩALB: t t Π = I ∗ Πt λ = (Π )1 λ(Π )λ t+1 R Π ∗ − t ∗ (19) ¡ ¢ ˆ ˆ ˆ Π = (1 λ)Π +λΠ t+1 ∗ t − where variables with hats over them represent logarithms. The solution is Π ˆ = Π ˆ +(λ)k[Π ˆ Π ˆ ] (20) t+k ∗ t ∗ − One possible steady-state equilibrium is inflation equal to the target rate. If one could disregard the lower bound on the interest rate, this would be the only equilibrium. Deviationsfromtheinflationtargetwouldresultinexplosiveorimplosive paths of inflation since λ > 1. However, equation (20) applies only when it calls for an interest rate at or above ΩALB. The inflation rate cannot decline forever. If the inflation rate given by (20) 11For these funds to survive, there must be some spread between the market rates they earn and the deposit rates they pay as noted by Bernanke and Reinhart (2004). 12In this sense, the rule (18) is similar to the one used in Schmitt-Grohe and Uribe (2000) and in the appendix of Benhabib, Schmitt-Grohe, and Uribe (2001a). 13We use an interest-rate rule of the form (18) so that the inflation-rate term in the strictly increasing range is directly comparable to the inflation-rate term in the money-supply rule (28). 7

calls for an interest rate below ΩALB, the inflation rate is determined by the Fisher equation (8) together with I = ΩALB instead of by (20). That is, the inflation rate t will stop declining when it is equal to its lower bound: ΩALB Π = (21) t+1 R The difference equation reflecting the lower bound has the form shown in figure 2:14 ΩALB Π = max ,(Π )1 λ(Π )λ (22) t+1 ∗ − t R · ¸ The list of equilibria as indexed by Π is 0 1. Π = Π . Steady-state TE. 0 ∗ 2. Π ΩALB,Π . Π decreases to ΩALB in finite time. 0 ∈ h R ∗ i t R 3. Π = ΩALB. Steady-state BTE equilibrium. 0 R 4. Π 0, ΩALB . Equilibria with Π < Π = ΩALB, t > 0.15 0 ∈ h R i 0 t R Along anyBTE path, the inflation rate eventuallyreaches ΩALB sothat I reaches R t its liquidity-trap value ΩALB at which the levels of the nominal and real money supplies are indeterminate.16 Hence, the number of equilibria is even larger than indicated above. Let the liquidity trap be reached in period n at price level P , given n a particular initial P . There is an infinity of equilibria associated with each initial 0 P 0,Π P . Once the liquidity trap is reached, the set of possible paths for M , 0 ∗ 1 k k ∈ n h includ − es i all paths for which M k [m, , k n since agents are indifferent ≥ P k ∈ ∞i ≥ between money and bonds. There may be more equilibria than those listed above. Beginning on any initial inflation rate above Π the inflation rate follows follows a divergent path. Such ∗ divergent paths have been referred to as speculative hyperinflations, for example, by Obstfeld and Rogoff, who have discussed ways of precluding them.17 Throughout this paper we assume that paths with ever increasing inflation are precluded. For the sake of comparison, we briefly consider the case of an interest-rate peg in which λ = 0 and the difference equation in figure 2 is a horizontal line. Suppose 14For figure 2, Π =1.025, ΩALB =1.001, 1/R=0.975, and λ=2. ∗ 15TheinitialinflationrateisnotconstrainedbythelowerboundΩALB/R. Thereasonisthatthe constraintontheinflationrateisimpliedbytheFisherequationincombinationwiththeinterestrate given by the interest-rate rule. The Fisher equation has no implications for the initial price level or theinflationrate P0 =Π , but it hasimplicationsfortheinflationrate P1. Hence, Π ΩALB/R. P 1 0 P0 1 ≥ 16Otherwise,the−nominalmoneysupplyisdeterminedbythenominalinterestrateandthemoneydemand function, given the unique inflation rate associated with the TE. 17See Obstfeld and Rogoff (1983) and Obstfeld and Rogoff (1986). 8

it is announced that I will equal I in period t and all future periods. The Fisher t ∗ equation (8), implies that I would be associated with Π from period t + 1 on. ∗ ∗ However, this interest-rate rule would not pin down the initial inflation rate. There would be a continuum of equilibria, indexed by the initial inflation rate Π 0,Π . t ∗ ∈ h i However, if the monetary authorities specify the initial level of the money supply in addition to the interest-rate peg, the initial inflation rate is determinate, since there is a unique level of real balances associated with I = I . t ∗ 3.2 A policy-determined lower bound Now consider interest-rate rules designed to keep the gross nominal interest rate from falling below a policy-determined lower bound, Λ, that may be above Ω. The policy-determined lower bound may be attainable, ΛALB, or unattainable, ΛULB. For example, with the interest-rate rule Π λ I = max ΛALB,I t , ΛALB Ω, λ > 1 (23) t ∗ Π ≥ " µ ∗¶ # thepolicy-determinedlowerboundisattainable. Thisruleandthedifferenceequation in inflation that it implies are identical to the ones considered in the last subsection except that the lower bounds on the interest rate and inflation are determined by policy, not by preferences. The rule is implementable with both ALB money demand (ΛALB ΩALB) and with ULB money demand (ΛALB > ΩULB). The entire list of ≥ possible equilibria is given by items 1 through 4 in the last subsection except that ΛALB replaces ΩALB everywhere. In contrast to the liquidity trap case, when Π t reaches ΛALB along a BTE path, real balances and nominal balances are uniquely R determined.18 It is useful to consider a rule that is very similar to the continuously differentiable rules used in the seminal papers on the existence of BTE: I = I ΛULB Π t I∗− λ Λ I U ∗ LB +ΛULB, Π > 0 (24) t ∗ t − Π µ ∗¶ ¡ ¢ I = RΠ > ΛULB Ω 1, λ > 1 ∗ ∗ ≥ ≥ 18Consider the family of interest-rate rules given by I t =max  Υ,(I ∗ − Υ) Π Π t − Υ Υ / / R R λR[Π∗ I∗ − − Υ Υ /R] +Υ  , I ∗ >Υ ≥ max ΩALB,ΛALB,1 , λ>1 µ ∗− ¶ ¡ ¢   With this family, the strictly increasing part of the difference equation for Π begins at the point t Υ,Υ on the 45 line in (Π ,Π ) space. If Υ=1, the implied difference equation is qualitatively R R ◦ t+1 t identical to the one plotted in Figure 2.4 of Woodford (2003). ¡ ¢ 9

This rule has a policy-determined lower bound that is unattainable.19 It is implementable with both ALB and ULB money-demand functions. The interest rate rises with inflation, and the response is greater the higher is inflation.20 As before, the elasticity of the interest rate with respect to the inflation rate at Π is λ > 1. ∗ Withtherule(24), theremustbetwosteady-stateequilibriuminflationrates: one is Π and the other is ΠBTE which is below Π and above ΛULB/R but which may ∗ ∗ or may not involve deflation. Combining the rule (24) with the Fisher equation (8) yields a difference equation in inflation of the form plotted in figure 3:21 λR Π = 1 I ΛULB Π t I∗− ΛULB +ΛULB (25) t+1 ∗ R − Π " µ ∗¶ # ¡ ¢ This equation is a convex function that has a lower bound of ΛULB Ω and that R ≥ R crosses the 45 degree line from below at Π = Π = Π where its slope is greater ◦ t+1 t ∗ than one: dΠ t+1 = λ > 1, (26) dΠ t ¯Πt=Π ∗ ¯ that is, Π ∗ is an unstable steady state¯ equilibrium. In addition its slope approaches ¯ zero as Π approaches 0 from above and rises continuously with Π : t t dΠ t+1 = λ Π t I∗ λ − R Λ− 1 > 0 (27) dΠ Π t µ ∗¶ Therefore, itmustintersectthe45 lineasecondtimeatapointbelowΠ represented ◦ ∗ by ΠBTE where its slope is less than one.22 That is, ΠBTE is a stable equilibriumwith deflation (ΛULB/R < ΠBTE < 1), stable prices (ΛULB/R < ΠBTE = 1), or inflation (1 < ΠBTE). There is a continuum of equilibria, indexed by Π . Each Π is associated with one 0 0 of the two possible steady-state inflation rates. 19The continuous-time policy rule used in Benhabib, Schmitt-Grohe, and Uribe (2001b) and the main text of Benhabib, Schmitt-Grohe, and Uribe (2001a) has a policy-determined unattainable lower bound. The discrete-time analogue of this rule is RA I = I ΛULB exp (Π Π ) +ΛULB, A>1 t ∗ − I ΛULB t − ∗ · ∗− ¸ ¡ ¢ We employ the rule (24) because it is directly comparable to the other interest-rate rules and the money-supplyrules(28)usedinthispaper. Rule(24)isalsousedbyEvansandHonkapohja(2003), who assume that ΛULB =1. 20That is dI t =λR Π t I λ ∗ I − ∗ Λ− 1 >0, d2I t = λI ∗ 1 Π t − 1 dI t >0 dΠ Π dΠ 2 I Λ − Π dΠ t µ ∗¶ t µ ∗− ¶µ ∗¶ t 21For figure 3, Π =1.025, 1/R=0.975, ΛULB =1.02, and λ=2. ∗ 22If λ = 1 there would be a unique steady state at Π = Π , but there would be multiple initial t ∗ equilibrium inflation rates because all paths that start below the target rate would approach it. 10

1. Π = Π : Steady-state TE. 0 ∗ 2. Π ΠBTE,Π : Non-steady-state BTE; Π ΠBTE from above 0 ∗ t ∈ h i → 3. Π = ΠBTE : Steady-state BTE. 0 4. Π 0,ΠBTE : Non-steady-state BTE; Π ΠBTE from below.23 0 t ∈ → ­ ® 4 Indeterminacy under money-growth rules 4.1 Indeterminacy with ULB money demand When the money-demand function has an ULB, money-growth rules are consistent with the existence of both a TE and BTE in which real money balances are forever increasing and the interest rate is approaching its lower bound.24 Consider the money-growth rule: M /M Π τ t t 1 t − − = , τ > 1 (28) Π Π − ∗ µ ∗¶ If τ = 0, money grows at the constant target gross growth rate of money which is equal to Π . In that case, since M and P are given, M is determined by Π . If ∗ 1 1 0 ∗ τ = 0, there is one M associated w − ith each − P .25 In either case, there will be a range 0 0 6 of equilibria, each with its own level of initial real balances. The Fisher equation (8) and the money market equilibrium condition (9) with the functional form for V(m ) in equation (15) imply an expression for the inflation rate t in terms of real balances: ¯ 1 U (C) 0 Π = (29) t R U (C ¯ ) ( γ +V ) µ ¶ 0 − mt 1 0 − Furthermore, the money-growth rule can be rewritten as m = m (Π ) (1+τ)Π 1+τ (30) t t 1 t − ∗ − Combining (30) and (29), yields a difference equation in real balances, (1+τ) 1 U (C¯ ) − m = m (Π )1+τ 0 (31) t t − 1 ∗ " µ R ¶ U 0 (C¯ ) − mt γ 1 − V 0# − 23Π can be lower than ΩULB , see footnote 15. 0 R 24This possibility is pointed out, for example, in Woodford (1994), Woodford (2003), Christiano and Rostagno (2001), and Benhabib, Schmitt-Grohe, and Uribe (2001a)). We begin with the case of ULB money demand because it is somewhat simpler. 25With 0>τ > 1,nominalbalancesincreasewhentheinflationrateisabovetarget. Forfurther − discussion of this case, see footnote 27. 11

with the form plotted in figure 4.26 Given the definition of ΩULB in equation (16), it follows that the term in square brackets (Π ) approaches ΩULB/R as m . t t 1 Therefore, the growth rate of real balances (m /m ) approaches RΠ ∗ 1+−τ > → 1 ∞ for t t 1 ΩULB τ > 1.27 − − ¡ ¢ The unique steady-state solution for equation (31) is γ γ m = = (32) ∗ U 0 (C¯ ) − U 0 I ( ∗ C¯) − V 0 I ∗ I− ∗ 1U 0 (C ¯ ) − V 0 If the steady-state level of real balances is to be positive, it must be that ¯ U (C) I > 0 ΩULB (33) ∗ U (C¯ ) V ≡ 0 0 − The money-growth rule may be associated not only with the TE, but also with a range of BTE in which inflation declines forever and approaches the limit ΩULB/R. The equilibria may be indexed by m : 0 1. m = m : Steady state TE with Π = Π = Π . 0 ∗ 0 t ∗ 2. m > m : BTE with positive growth in m , Π 0,Π and Π ΩULB/R t ∗ t 0 ∗ t ∈ h i → from above, t > 0. Themoney-growthrulehasarepresentationas aninterest-raterulethatisrelated to but somewhat different from the one discussed in section 3.2.28 4.2 Indeterminacy with ALB money demand With ALB money demand money-growth rules are also consistent with the existence of both a TE and BTE. In the BTE, real money balances are forever increasing money demand, as with ULB money demand, but the interest rate reaches its lower 26For figure 4, τ = 2, Π = 1.025, γ = 1, U (C¯) = 1, V = 0.001, and 1/R = 0.975, so m = ∗ 0 0 γ/(U (C¯) V )=1.001. 0 0 − 27m isanunstablesteadystatesinceequation(31)andtheexpressionforΠ givenby(29)imply ∗ ∗ that for all τ > 1 − dm dm t − t 1 ¯ ¯ mt=m ∗ =(Π ∗ )1+τ h¡ R 1 ¢ U 0 (C¯) U − 0( m C γ ¯ ∗ ) − V0) i − (1+τ) · 1+ U (1 0 + (C¯ τ ) ) − m∗ m γ ( ∗ m − γ ∗ V )2 0 ¸ = 1+ R m (1 ∗ + U τ 0 ) ( γ C¯ Π ) ∗ >1 ¯ 28Equation (16) can be used to obtain an expression for m in terms of I . Using this expression t t and assuming V =0, it follows from equations (30) and (8) that 0 I 1 I 1 Π 1+τ I 1 I 1+τ I 1 I 1+τ t − = t − 1 − t = t − 1 − t − 1 = t − 1 − t − 1 I I Π I RΠ I I t t − 1 µ ∗¶ t − 1 µ ∗¶ t − 1 µ ∗ ¶ 12

bound. The Fisher equation (8) and the m m-part of the money-demand function t ≤ (14) imply an expression for inflation in terms of real balances: 1 U (C¯ ) 0 Π = (34) t R U (C ¯ ) m+m V µ ¶· 0 − t − 1 − 0 ¸ Combining the money-growth rule (30) with (34) we obtain a difference equation in real balances: m = m 1 U 0 (C¯ ) − (1+τ) Π (1+τ), m m (35) t t − 1 ·µ R ¶ U 0 (C¯ ) − m+m t − 1 − V 0 ¸ ∗ t − 1 ≤ Given the definition of ΩALB in equation (14) it follows that as m increases and t 1 reachesm, theterminsquarebrackets(Π )increasesandreachesΩAL−B/R. Therefore, t the growth rate of real balances (m /m ) increases and reaches RΠ ∗ 1+τ > 1 for t t 1 ΩALB − τ > 1. − ¡ ¢ There exists a steady-state equilibrium with positive real balances equal to U (C¯ ) I 1 m = 0 +m+V U (C¯ ) = m+V ∗ − U (C¯ ) (36) ∗ 0 0 0 0 RΠ − − I ∗ µ ∗ ¶ under weak conditions that we assume are met.29 This equilibrium is a TE in which Π = Π and I = I .30 Equation (35) applies only when m m. In cases in which ∗ ∗ t ≤ m > m, the difference equation in real balances is given by (30) with the inflation t rate fixed at ΩALB/R. Figure 5 is a diagram for the case where m > 0 because m+V U (C¯ ) > 0.31 0 0 − The list of equilibria indexed by m is now 0 1. m = m = m : TE steady state with Π = Π 0 t ∗ 0 ∗ 2. m > m > m : BTE with Π ΩALB,Π , Π reaches ΩALB from above, 0 ∗ 0 ∈ R ∗ t R t > 0. m t reaches m from below, D E Th 2 a 9 t Th is e , p t o h s e it t iv a e rg s e t t ea i d n y te s r t e a s t t e r e a x t i e st m so u n s l t y b if e m sm + a V ll 0 e − r I t ∗ I h − ∗ a 1 n U t 0 ( h C e ¯) m > ax 0 im or u e m qu p iv o a s l s e i n b t l l e y i U nt 0 ( e C¯ r U e ) 0 −s ( t C m ¯) r−a V t 0 e. > F I o ∗ r . parameter values for which lim I = , m+V U (C¯) = m > 0, so the condition is definitely t 0 0 mt→ m ∞ − met. In order to exist, the strictly positive steady state equilibrium must also satisfy m m m, ∗ ≤ ≤ which means that m+V 0 − U 0 (C¯) ≤ m ∗ =m+V 0 − I∗ I− ∗ 1 U 0 (C¯) ≤ m. This condition is always satisfied as long as V 0 is not too large. ³ ´ 30m is an unstable steady state since (36) and the definition of I imply ∗ ∗ dm I t =1+(1+τ)m Π ∗ >1 ∗ ∗ dm t − 1¯m¯=m ∗ U 0 (C) ¯ 31For figure 5, V 0 =0.001, τ =2 ¯ ¯, m=1.5,U 0 (C¯)=1, 1/R=0.975, Π ∗ =1.025, so m=0.5. 13

3. m m : BTE with Π 0, ΩALB and Π = ΩALB, t > 0. The m growth 0 ≥ 0 ∈ R t R t rate is constant at I ∗ 1+τD. i ΩALB ¡ ¢ There is a second steady-state equilibrium with zero real balances when m+V 0 U (C¯ ) 0 so that m = 0.32 This case is not of particular interest to us becaus − e 0 ≤ we want to focus on equilibria in which inflation is below target (real balances are above target). However, it is quite important for those considering the existence of hyperinflation equilibria. Themoney-growthruleintheALB modelalsohasarepresentationasaninterestrate rule as long as m m.33 However, I = ΩALB for all m m. Hence, the t t ≤ ≥ interest-rate rule representation requires an additional specification of policy when I = ΩALB in order to uniquely pin down the path of real money balances for each initialm . Forexample, EggertsonandWoodford(2003)addaruleformoneygrowth 0 that applies whenever the interest rate reaches its lower bound. 5 Monetary-policy rules that imply uniqueness 5.1 A nonmonotonic interest-rate rule First, we show that making the interest-rate rule a nonmonotonic function of the inflation rate may insure a unique equilibrium, following up on an observation by Benhabib, Schmitt-Grohe, and Uribe (2001b). Consider the piecewise log-linear interestrate rule I Πt λ if I Πt λ > Ω t∗ Π ∗ Π ∗ ∗ I = (37) t  ¡ ¢ ¡ ¢   I > I ∗ if I ∗ Π Π ∗ t λ ≤ Ω This rule is similar to the on  e described in section¡3.2¢, but the interest rate is pegged  e at I > I instead of at the policy-determined lower bound value ΛALB. It is feasible ∗ with both ALB and ULB money demands, since it never calls for I = Ω. t e The difference equation in the inflation rate that follows from (37) and the Fisher equation (8) is illustrated in figure 6.34 Consider a situation where the inflation rate is so low that if I were given by I Π λ it would be less than or equal to the lower ∗ Π ∗ 32See section 2.3. In this case (not sh¡own)¢, the line representing the difference equation would start at the origin and have a slope that is below one at the origin, increases until it exceeds one, and is constant for m >m. t 33Using (30),(8) and (14), we get I 1 I 1 I 1+τ t − U 0 (C¯)= t − 1 − U 0 (C¯) (V 0 +m) ∗ +(V 0 +m) I I − I t · t − 1 ¸· t − 1¸ 34For figure 6, Π ∗ =1.025,1/R=0.975, Ω=1.001, λ=2, and I˜=1.05/0.975≈1.077. 14

bound, Ω. In such a situation, I jumps up to I. In the next period, the inflation t rate must be higher than the target inflation rate given the Fisher equation and the fact that I > I . But such a path is not a possibele solution, because it implies that t∗ the inflation rate increases without limit. Hence, the economy cannot start out on a path of deeclining inflation. 5.2 A monotonic but asymmetric interest-rate rule Next, wedemonstratethat there is an asymmetricinterest-rate rule thatisassociated with a unique steady-state inflation rate and a determinate price level. Consider the following rule35, Π Π γ I = (I )( t )λ t+1 (38) t t∗ Π Π ∗ µ ∗ ¶ Combining equation (38) with the Fisher equation (8) and using I = RΠ , give a t∗ ∗ first-order difference equation in the inflation rate, 1 λ γ λ Π t+1 = Π ∗ −1 − −γ Π t 1 − γ (39) Taking logs, we have Π = α Π +α Π (40) t+1 0 ∗ 1 t where α = 1 λ γ and α = λ . With λ and γ chosen so that α > 1 , the equation is unstab 0 le an − 1 d− − γ the uniqu 1 e so 1 l−u γ ti b on for Π b is Π . b | 1 | t ∗ The rule (38) implies asymmetric responses under the following assumptions: If Π < Π , 2 > γ > 1 and λ > 1, so that α > 1 and α < 1. t ∗ 0 1 • − If Π > Π , γ = 0 and λ > 1, so that α < 0 and α > 1. t ∗ 0 1 • Under these assumptions, the rule calls for a stronger response to expected future inflation when current inflation is below target. With the asymmetric rule the difference equation for inflation (39) has the form shown in figure 7.36 In contrast with the symmetric rule (19), the asymmetric rule (39) implies a difference equation for inflation for which Π Π no matter how low 1 ∗ ≥ the value of Π . Π is a unique steady-state equilibrium, and Π = Π is the only 0 ∗ 0 ∗ equilibrium.37 If the initial inflation rate is in the interval Π 0,Π , the economy 0 ∗ ∈ h i embarks on a path with ever-increasing inflation. The part of the difference equation that applies when Π 0,Π implies Π > Π , and the inflation rate continues to t ∗ 1 ∗ ∈ h i increase because the part that applies when Π [Π , is now relevant. t ∗ ∈ ∞i 35Benhabib, Schmitt-Grohe, andUribe(2001c) alsoconsiderinterest-raterules wheretheinterest rate responds to both current and future inflation, but in a different context. 36For figure 7, λ=2, Π =1.025, and 1/R=0.975. γ =1.5 when Π <Π , and γ =0 otherwise. ∗ t ∗ 37All it takes for equation (39) to be associated with a unique steady-state TE is that α < 1 1 − when Π <Π and α >1 when Π >Π . One could, for example, let γ =0 so that there was zero t ∗ 1 t ∗ response to future inflation, and vary λ to meet these requirements. 15

5.3 A nonmonotonic money-growth rule Finally, we demonstrate that a nonmonotonic money-growth rule can work analogously to the nonmonotonic interest-rate rule in section 5.1. We assume a moneygrowth rule of the form Π Π ∗ t − τ Π ∗ M t − 1 τ > − 1 if m t − 1 < m M = (41) t  ¡ ¢ αΩALBP m α < 1 if m m  R t − 1 ∗ t − 1 ≥ and an ALB moneydemand. The nonmonotonic money-growth rule implies a discontinuous difference equation in real money balances of the formshown in figure 8.38 The top relationship in equation (41) is a rewrite of the money-growth rule (28) and m is the steady-state level of real money balances when following that relation- ∗ ship: I 1 m m+V ∗ − U (C¯ ) (42) ∗ 0 0 ≡ − I µ ∗ ¶ The second part of (41) tells us that the level of real balances in period t will be lower than the steady-state m level if m m. To see this, note that the lower bound on ∗ t 1 the inflation rate is ΩALB/R. Hence − , P ≥ ΩALBP . With α < 1, real balances must t ≥ R t − 1 then be lower than m in period t. BTE are precluded because real balances jump ∗ below m as soon as they would reach or exceed the level that applies in a liquidity ∗ trap. And when the real balances have reached such a low level, they continue to decline according to the difference equation in real balances, equation (35). Hence, the initial price level at t will be uniquely pinned down at P = Π P . t ∗ t 1 − 6 Fiscal policy and BTE 6.1 Fiscal policy can always preclude BTE There is always a fiscal policy that precludes BTE no matter what the monetary policy rule. Fiscal policy determines the path of total nominal government debt measured as total nominal government bonds, D . The consolidated government balance t sheet implies that D must equal the money supply (which equals the government t bonds held by the monetary authority) plus government bonds held by the public: D M +B , D = 0 t 0 (43) t t t 0 ≡ 6 ≥ For simplicity, we devote most of our attention to fiscal policies under which there is a constant gross growth rate (Γ) for total government bonds: D = ΓD (44) t+1 t 38For figure 8, α = 0.9, V =0.001, τ =2, m= 1.5, U (C)= 1, 1/R = 0.975, and Π =1.025 so 0 0 ∗ that m=0.5. 16

where Γ = 1 is the case of a balanced-budget policy. Fiscalpolicyandthetransversalitycondition(11)takentogetherhaveimplications for the possibility of BTE. First, consider a candidate steady-state BTE in which the interest rate is constant at IBTE. The path I = IBTE t can be a steady-state t ∀ equilibrium only if Γ t 1 − lim ΓD = 0 (45) t IBTE 0 →∞µ ¶ ThereforeifΓ IBTE,fiscalpolicyprecludessuchapath. Withourparameterization, ≥ Γ < IBTE is a necessary condition for fiscal policy to be consistent with steady-state BTE.39 One important implication of the previous paragraph, is that if Γ 1, the Fried- ≥ man rule (I = 1 t) cannot be implemented exactly under any type of monetary t ∀ policy. This result for the case of Γ = 1 is obtained by Schmitt-Grohe and Uribe (2000) in cases in which the monetary-policy rule is either an interest-rate rule or a money-growth rule. Another important implication is that if IBTE > 1, there may be BTE with small deficits (IBTE > Γ > 1). Next, consider a candidate BTE path on which the interest-rate path is some weaklydecreasingsequence I withlim I = IBTE. Let I Θ IBTE ,where k k k k k { } →∞ { }≡ { } Θ is a weakly decreasing sequence with lim Θ = 1. Let the value of Θ at k k k k s { ome } time tˆgiven by Θ > 1 be arbitrarily clos → e ∞ to one. This sequence of interest tˆ rates can be an equilibrium only if lim D ΓtΠk=t 1 Θ 1(IBTE) 1 = 0 (46) t 0 k=tˆ− −k − →∞ £ ¤ The condition (46) cannot hold if Γ > IBTE and Θ is arbitrarily close to one. Theretˆ fore, Γ > IBTE precludes BTE in which the interest rate asymptotically approaches or eventually reaches IBTE. In addition, Γ = IBTE precludes equilibria in which IBTE is approached asymptotically if lim Πk=t 1Θ 1 > 0. This condition is satisfied t k=tˆ− −k →∞ if the nominal interest rate approaches IBTE (and hence Θ approaches one) fast k enough, for example if the nominal interest rate reaches IBTE (Θ reaches one) in k finite time. The class of fiscal policies for which I > Γ > IBTE is especially interesting. ∗ As just shown, this class precludes BTE for any monetary policy. It follows that a combination of a member of this class with a standard interest-rate rule like equation (18)ormoney-supplyrulelikeequation(28)issufficienttoinsurethatΠ istheunique ∗ equilibrium value for the inflation rate. These observations suggest that BTE may not be a matter for concern in OECD countries. Most, if not all, of these countries run positive but relatively small government deficits on average so their fiscal policies 39Of course, there are many other parameterizations of fiscal policy which are consistent with the existence of BTE. 17

may preclude BTE. This possibility is yet another reminder of the importance of analyzing monetary and fiscal policy jointly.40 6.2 Uniqueness with money growth through transfers In subsection 4.2 we show that with an ALB money-demand function, the moneygrowth rule (28) is associated with multiple equilibria. There, as well as in all of the paper before section 6, we assume that B can take on any value implied by the t open-market purchases used to increase the money supply. In particular, B can be t as negative as necessary, or, in other words, government interest-bearing claims on the public can increase without limit. In this subsection, we assume that there is a finite lower bound on B designated t by B where < B; that is, government claims on the public cannot exceed the −∞ finite amount B. Once this limit is reached, the money supply can be increased − only by money-financed transfers (‘money rain’). The required policy is best viewed as a combination of fiscal policy and monetary policy: the fiscal authority makes a bond-financed transfer to the private sector, and the monetary authority buys the bonds. If BTE are to be precluded, the transversality condition (11) implies that Π (the ∗ target rate of inflation and money growth) must be chosen so that t 1 t − 1 ΩALB − τ 1 t − 1 lim (Π )1+τ M + lim B = 0 (47) t ΩALB " ∗ R # 0 t ΩALB 6 →∞µ ¶ µ ¶ →∞µ ¶ where (Π )1+τ(ΩALB) τ is the constant rate of growth of nominal balances implied by ∗ R − the money-growth rule when the interest rate is at ΩALB so that the inflation rate is ΩALB. Thecondition(47)canalwaysbemetunderthenotveryrestrictiveassumption R τ that (Π )1+τ ΩALB − M +B = 0. Since M > 0, the first term is strictly positive ∗ R 0 6 0 τ and either un ³ bound ´ ed or bounded at (Π )1+τ ΩALB − M if Π is chosen so that41 ∗ R 0 ∗ ³ ´ ΩALB − τ (Π )1+τ ΩALB (48) ∗ R ≥ µ ¶ or, equivalently, so that ΩALB 1 Π ∗ R1+τ (49) ≥ R µ ¶ 40An early joint analysis of monetary and fiscal policy is Leeper (1991); more recent analyses include Canzoneri, Cumby, and Diba (2001) and Evans and Honkapohja (2002). 41Woodford (2003) and Woodford (1994) restrict attention to the case of τ =0 (and ΩALB =1) andarguethattherestrictionontherangeofthetargetinflationrate,Π >1,necessarytopreclude ∗ BTE is a limitation of money-supply rules relative to interest-rate rules. However, the condition in equation (48) implies that if τ > 0, the authorities can follow a money-supply rule that implies uniquenessandstilltarget agross inflationratebelowΩALB. In particular, with ΩALB =1onecan get arbitrarily close to the Friedman rule by choosing τ large enough. 18

The second term is either zero or bounded at B as ΩALB 1. Therefore, (47) is ≥ satisfied no matter whether the first term is bounded or not.42 Note that equation (49) implies uniqueness can be consistent with Π < ΩALB if τ > 0 and, in particular, ∗ with Π arbitrarily close to its Friedman-rule value of 1 if ΩALB = 1 and τ is large ∗ R enough.43 6.3 An illustration of the insufficiency of balanced-budget fiscal policy for precluding BTE In section 6.1, we show that a balanced-budget policy precludes BTE when the interest rate is always at a lower bound of one. Here we show that a balanced-budget policy may not preclude BTE even if the interest rate is at a lower bound of one infinitely often. We assume that there is regular and perfectly foreseen variation in productivity, e.g. seasonal variation. In our example, there is a BTE in which the interest rate fluctuates between one and a value greater than one. Consider the following period utility function with time-varying productivity, ρ : t (1/2)(ψ C )2 [1/(2ρ )]Y2 +V(m ) (50) − − t − t t t Solutions for the model with this utility function can be obtained using the procedure for the constant-productivity case. We report only the results needed to make our point. Equations (50), (4) and (6) imply that the nominal and real interest rates are given by 1 1+ρ I = R Π , R = t+1 (51) t t t+1 t β 1+ρ µ ¶ t Today’s real interest rate, R , is high if productivity increases between today and t tomorrow and vice versa. Therefore, the ‘target’ interest rate, I , must vary if the t∗ inflation rate is always to be at its fixed target, Π :44 ∗ I = R Π (52) t∗ t ∗ 42In a model in which the lower bound on the gross interest rate is one, Woodford (1994) shows that a money-growth rule in combination with the condition that B = 0 t is associated with t ∀ uniqueness as long as gross money growth is equal to or larger than one (Π 1 in our notation). ∗ ≥ When the public holds no government bonds, money must be increased by money-transfers. A money-growth rule with Π > ΩALB then corresponds to the class of fiscal policies discussed in ∗ is subsection with Γ > ΩALB , since the nominal growth rate of money is equal to the nominal growth rate of government bonds. Benhabib, Schmitt-Grohe, and Uribe (2001a) show that multiple equilibriaareprecludedundermoney-growthruleswhenB 0.Aswehaveshownthelowerbound t ≥ on B can be negative. What is needed is that money be supplied with lump-sum transfers instead t of open-market operations after some point 43Woodford (1994, 2003) restricts attention to the case of τ =0 (and ΩALB =1) and argues that the restriction on the range of the target inflation rate, Π > 1, necessary to preclude BTE is a ∗ limitation of money-supply rules relative to interest-rate rules. 44A similar relation is discussed in Woodford (2001b). 19

We consider a simple example in which (a) the attainable lower bound on the nominal interest rate is unity, ΩALB = 1, (b) ρ alternates between a high value t (ρ ) in even periods and a low value (ρ ) in odd periods, even odd 1 ∆ 1+ρ R = < R = = R ∆2, ∆ = even > 1 (53) even odd even β∆ β 1+ρ odd and (c) Π is set high enough that I ΩALB = 1 for all R which according to ∗ t∗ ≥ t equations (52) and (53) implies that Π must be chosen so that45 ∗ 1 1 Π > (54) ∗ ≥ R R even odd IfI isatthelowerboundofone, equations(51)and(54)implythat Π = 1/R = 0 1 0 1/R Π . From the interest-rate rule (18) with a time varying I and equation even ∗ t∗ ≤ (52), it follows that I is above the lower bound if 1 Π λ 1 R I = I 1 = R Π ( )λ = Π 1 λ odd > 1 (55) 1 1∗ Π 1 ∗ R Π ∗ − Rλ µ ∗¶ 0 ∗ even From equation (53) it follows that the condition (55) is satisfied if R odd = R1 λ∆2 > Π λ 1 (56) Rλ ev−en ∗ − even Given Π , condition (56) is met if ∆ is high enough. Both (54) and (56) are met for ∗ some admissible parameter values, and we assume that they are met in the rest of this subsection.46 I is at the lower bound again. Given I > 1 and Π = 1/R , Π is given by 2 1 1 even 2 the difference equation (19) that applies when I > 1: t Π = (Π )1 λ(Π )λ = (Π )1 λ(1/R )λ, (57) 2 ∗ − 1 ∗ − even Given the interest-rate rule (18) and inequality (54), I is 2 Π (1 λ)(1/R )λ λ I = max 1,(R Π ) ∗ − even = max 1,(R Π )1 λ2 = 1 (58) 2 even ∗ even ∗ − Π " µ ∗ ¶ # h i 45A Π < β/∆ cannot be attained because I would be below a lower bound of one. As noted ∗ ∗ by Krugman (1998) being in a liquidity trap in which I is at its lower bound has only nominal t consequences in a flexible-price model but has real consequences in a sticky-price model because outputhastoadjust. Forexample,Svensson(2000)considerswhathappenswhenΠ istoolowand ∗ how it can credibly be raised. 46For example, the combination β = .95, ∆ = 1.1, Π = 1.06, and λ = 1.5 satisfy both (54) and ∗ (56), since Π =1.06>0.95 1.1=1.045=1/R and (1/R )1 λ∆2 =(1/.95 1.1)0.5 1.12 ∗ even even − ∗ ∗ ∗ =1.2369>(Π )λ 1 =1.0296. ∗ − 20

By induction, interest rates, real interest rates, and inflation rates are I = 1 I = (R Π )1 λ∆2 > 1 even odd even ∗ − R R = R ∆2 > R (59) even odd even even Π = 1/R Π = (R Π )1 λ(1/R ) < 1/R odd even even even ∗ − even even since we are assuming that R Π 1. The interest rate in odd periods, I = even ∗ odd I (Π /Π )λ, is high because both t ≥ he target interest rate and actual inflation rate o∗dd odd ∗ are high: I = R Π is high because R is high, and Π = 1/R is high even o∗dd odd ∗ odd odd even though I is at the lower bound because R is low. even even It follows that BTE are possible in our example with variable productivity and a balanced budget. The transversality condition is met because the interest rate alternates between one and a value greater than one. As pointed out by Benhabib, Schmitt-Grohe, andUribe(2001a), abalanced-budgetpolicyprecludesonlyequilibria in which the central bank commits to keep the interest rate equal to one no matter what happens to inflation. 7 Concluding remarks Under many specifications of monetary policy, standard models may exhibit pricelevel indeterminacy. That is, they may have multiple equilibria that include both a locally-unique steady-state target (inflation-rate) equilibrium (TE) and multiple below-target equilibria (BTE), equilibria in which the inflation rate is always below target and is constant or eventually reaches or asymptotically approaches a belowtarget value. Rules with either the (nominal) interest rate or money growth as instruments are consistent with price-level indeterminacy in the presence of a conventionally-defined liquidity trap, a situation in which bonds and money are perfect substitutes at either a positive or zero nominal interest rate. Interest-rate and money-growth rules may also be consistent with indeterminacy even if there is no liquidity trap for one of two reasons. First, an interest-rate rule may keep the interest rate above a policydetermined lower bound. Second, with either an interest-rate rule or a money-growth rule, moneydemandmayimplyalowerboundontheinterestratesothattheinterest rate cannot be lowered ‘enough’ even though it can always be lowered somewhat. We have found that implementing monetary policy by setting money growth instead of the interest rate makes a difference only if the economy is in a liquidity trap and money is injected without open-market operations, e.g. by lump sum transfers. Under these conditions, a money-growth rule, more accurately viewed as a combination of a money-growth and fiscal-policy rules, may preclude BTE when an interest-rate rule does not. 21

The above results all apply when the interest rate and money growth respond monotonically to eliminate deviations in the current inflation rate from its target value. We have expanded on the observation made in Benhabib, Schmitt-Grohe, and Uribe (2001b) that a unique steady-state inflation rate may be implied by interestrate rules that are non-monotonic in the current inflation rate and have shown that an analogous result holds for non-monotonic money-growth rules. We also show that a unique steady state inflation rate is implied by an asymmetric interest-rate rule that calls for a response not only to current inflation but also to expected future inflation, with the response to expected future inflation being stronger when the current inflation rate is below target. Conclusions about determinacy under alternative monetary rules depend on fiscal policy as measured by the growth rate of total nominal government debt. There is always a class of fiscal policies that can preclude BTE no matter what the monetary policy because with those policies BTE paths violate the transversality condition. Balanced budget fiscal policy can preclude BTE if the interest rate is always at a zero lower bound. However, it cannot do so if the lower bound on the interest rate is positive or if productivity is variable even though the interest rate is at a lower bound of zero infinitely often. Asstatedintheintroduction,itisnotyetclearwhethertheexistenceofBTE isan important problem or a curiosum. Some have argued that BTE are of little interest for theoretical reasons. McCallum (2001, 2003) uses his ‘minimum state variable’ criterion as one way of determining which equilibria are of interest. In terms of our model, heshowsthatthelocally-uniqueTE meetsthiscriterionwhilethemultiplicity of BTE equilibria associated with the below-target steady state, often referred to as ‘sun-spot’equilibria,donot. BothMcCallum(2001,2003)andEvansandHonkapohja (2002) use stability under a particular type of learning as an alternative criterion. In terms of our model, they show that the TE is stable under learning while the BTE are not. Although these theoretical arguments are attractive, not all analysts are completely convinced by them. Our analysis suggests another more practical reason for focusing on the TE. The combinationofanyofarangeofsmalldeficitswithastandardinterest-rateormoneysupply rule guarantees that the TE is the unique equilibrium because the deficit precludes BTE. This observation suggests that BTE may not be a matter for concern in most OECD countries. Most of these countries run positive but relatively small government deficits on average so their fiscal policies may preclude BTE. This possibilityprovidesanotherillustrationoftheimportanceof analyzingmonetaryandfiscal policy jointly. It might be argued that Japan has been in a BTE for the last several years. Although this argument cannot be rejected out of hand, there is a good reason to question it. Japan has had near-zero interest rates and deflation despite the fact that the growth rate of nominal government debt has exceeded nominal interest rates on 22

government debt. That is, according to our analysis, fiscal policy was too expansionary to be consistent with a BTE. Many analysts have concluded that the Japanese situation is better described as a standard ‘liquidity-trap equilibrium’ (LTE). A LTE could arise, for example, if there were a large negative demand shock in the presence of wage and price inertia. In order for output to be equal to potential it would be necessary for the real interest rate to fall below zero. However, the nominal interest rate could not be driven below zero because of a liquidity trap as conventionally defined. Furthermore, expected inflation might not move in the right direction or by the right amount to reduce the real interest rate as much as is required. For some policies, appropriateness depends crucially on whether one is concerned about LTE or BTE, but, for others, it does not. For example, it is often suggested thatJapanshouldannounceahighertargetinflationrate. Raisingthetargetinflation rate (or increasing the money supply even though the interest rate is at the lower bound)canhelpaneconomyescapefromanLTE byraisingexpectedfutureinflation thereby lowering the real interest rate.47 However, we have shown that this policy cannot help escape from a BTE because the target inflation rate is irrelevant for this type of equilibrium. In contrast, no matter which type of unintended equilibrium one is trying to avoid, there is a strong case for a more aggressive response to inflation when it is below target. It has been stressed that more aggressive easing reduces the chances of falling into LTE in which policymakers must rely on less familiar instruments with more uncertain effects.48 We have shown that responding more aggressively to expected inflation when current inflation is below target makes it possible to avoid BTE. 47See, for example, Krugman (1998). 48See, for example, Orphanides and Wieland (2000). 23

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Obstfeld, M., and K. Rogoff (1996) Foundations of International Macroeconomics, MIT Press, Cambridge. Orphanides, A., andV. Wieland(2000) “EfficientMonetaryPolicyDesign Near Price Stability”, Journal of the Japanese and International Economies, 14, 327—365. Sargent, T. J. (1987) Macroeconomic Theory, Academic Press, New York, Second Edition. Schmitt-Grohe, S., and M. Uribe (2000) “Price Level Determinacy and Monetary Policy under a Balanced-Budget Requirement”, Journal of Monetary Economics, 45, 211—246. Svensson, L. E. O. (2000) “The Zero Bound in an Open Economy: A Foolproof Way ofEscapingfromaLiquidityTrap”, NBERWorkingPaper7957, National Bureau of Econonmic Research. Woodford, M. (1994) “Monetary Policy and Price Level Determinacy in a Cash-in- Advance Economy”, Economic Theory, 4, 345—380. Woodford, M. (2001a) “Fiscal Requirements for Price Stability”, Journal of Money, Credit and Banking, 33, 669—728. Woodford, M. (2001b) “The Taylor Rule and Optimal Monetary Policy”, processed, Princeton University. Woodford, M. (2003) Interest and Prices: Foundations of a Theory of Monetary Policy, Princeton University Press, Princeton. 25

Figure 1: ULB and ALB money demands Figure 2: Interest-rate rule and preference-determined lower bound 26

Figure 3: Interest-rate rule and policy-determined lower bound Figure 4: Money-growth rule with unattainable lower bound 27

Figure 5: Money-growth rule with attainable lower bound Figure 6: Non-monotonic interest-rate rule 28

Figure 7: Asymmetric interest-rate rule Figure 8: Non-monotonic money-growth rule 29