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Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 795, April 2004 - Screen Reader Version*

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

Ragna Alstadheim, Howard University
Dale W. Henderson^*, Federal Reserve Board

NOTE: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgment that the writer has had access to unpublished material) should be cleared with the author or authors.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 http://www.federalreserve.gov/pubs/ifdp/. This paper can be downloaded without charge from the Social Science Research Network electronic library at http://www.ssrn.com/.

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, monetary policy, monetary rule, fiscal policy, money demand.

JEL Classification: E31, E50, E62, E41

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 expected inflation would be under price flexibility in order to set their contract prices. As regards practice, the current situation in Japan, with deflation and zero short-term 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 steady-state target equilibrium ($TE$) for the inflation rate may have additional equilibria. To be more precise, there may be multiple below-target equilibria ($BTE$), paths along which the inflation rate is always below target and is constant or either asymptotically 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.

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 present two kinds of monetary-policy rules that may preclude $BTE$. 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 $BTE$.

In the next section we lay our model and discuss two specific money-demand functions. Section 3 is a presentation of 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.

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

$$\underset{C_{t},B_{t},M_{t},Y_{t}}{Max}\sum_{t=0}^{\infty}\beta^{... ...U(C_{t})+V(m_{t})-\left[ 1/\left( 2\rho_{t}\right) \right] Y_{t}{} ^{2}\right\}$$ (1)

$$U^{\prime}(C_{t}) >0,\quad U^{\prime\prime}(C_{t})<0,\quad m_{t} =M_{t}/P_{t},\quad V^{\prime}(m_{t})\geq0,\quad V^{\prime\prime}(m_{t} )\leq0$$

subject to the following period budget and positivity constraints:

$$P_{t}Y_{t}+M_{t-1}+I_{t-1}B_{t-1}=T_{t}+P_{t}C_{t}+M_{t}+B_{t}$$ (2) 

$$C_{t}>0,M_{t}>0,P_{t}>0,\rho_{t}>0\quad\ \forall\ t$$

Period utility is increasing in consumption, weakly increasing in real (money) balances $m_{t}$, and decreasing in output.⁷ $\rho_{t}$ represents the level of productivity; as $\rho_{t}$ increases the production of a given level of output causes less disutility.⁸ The agent takes as given the money price of goods ($P_{t}$) and the gross nominal interest rate ($I_{t}$) earned on a bond held from period $t$ to period $t+1$, and chooses holdings of two nominal financial assets, money ($M_{t}$) and bonds ($B_{t}$); consumption ($C_{t}$); and output ($Y_{t}$). 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 period must equal tax payments $T_{t}$ plus the sum of consumption and money and bond holdings for this period. In addition, each agent and, therefore, agents as a group are subject to a no-Ponzi-game condition:

$$\lim_{t\rightarrow\infty}(M_{t}+B_{t})\Pi_{k=0}^{k=t-1}I_{k}^{-1}\geq 0,\quad\Pi_{k=0}^{k=-1}I_{k}^{-1}\equiv1$$ (3)

where $M_{t}+B_{t}$ is their net (nominal) financial assets in period $t$.

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

$$U^{\prime}(C_{t})=\beta U^{\prime}(C_{t+1})\frac{I_{t}}{\Pi_{t+1}}$$ (4)

$$U^{\prime}(C_{t})=V^{\prime}(m_{t})+\beta U^{\prime}(C_{t+1})\frac{1} {\Pi_{t+1}}\;$$ (5)

$$C_{t}=\rho_{t}U^{\prime}(C_{t})\qquad\qquad\qquad\qquad\quad\;$$ (6)

where $\Pi_{t+1}=\frac{P_{t+1}}{P_{t}}$ is the (backward looking) gross inflation rate. A fourth necessary condition (the transversality condition) is that (3) hold with equality

$$\lim_{t\rightarrow\infty}(M_{t}+B_{t})\Pi_{k=0}^{k=t-1}I_{k}^{-1}=0,$$ (7)

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: $Y_{t}$ has been set equal to $C_{t}$ as it must be in equilibrium since there is no government spending, and desired asset stocks have been set equal to actual asset stocks.

We assume that productivity is constant $\left( \rho_{t}=\rho\right)$ except in section 6.3. Under 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:

$$I_{t}=R\Pi_{t+1}$$ (8)

$$I_{t}=\frac{U^{\prime}(\bar{C})}{U^{\prime}(\bar{C})-V^{\prime}(m_{t})}$$ (9)

$$\bar{C}=\rho U^{\prime}(\bar{C})$$ (10)

$$\lim_{t\rightarrow\infty}(M_{t}+B_{t})\Pi_{k=0}^{k=t-1}I_{k}^{-1}=0,$$ (11)

where $R\equiv\frac{1}{\beta}>1$ is the constant gross real interest rate and $\bar{C}$ is the constant flexible-price value of consumption. $I_{t}\leq 0$ is ruled out by (8), given $R,\Pi_{t+1}>0$. In turn, (9) implies that $U^{\prime}(\bar{C})>V^{\prime}(m_{t})$, so depending on the functional form of $V^{\prime}(m_{t})$ there may be a lower bound on $m_{t}$ that is greater than zero. Since $U^{\prime}(\bar{C})>0$ and $V^{\prime} (m_{t})\geq0$, (9) implies $I_{t}\geq\Omega\geq1,$ where $\Omega$ is the lower bound on $I_{t}$ implied by $V^{\prime}(m_{t})$. Since $R>1$ and $I_{t} \geq\Omega$, (8) also implies $\Pi_{t}\geq\frac{\Omega}{R}$.

2.2  Policy

We assume that fiscal policy determines the total amount of nominal government bonds outstanding, $D_{t}$, through control of the budget deficit inclusive of interest payments. Monetary policy determines whether these bonds are held by the monetary authority as a match for the money supply, $M_{t}$, or directly by the public, $B_{t}$, through control of open-market operations. The consolidated government balance sheet implies that $D_{t}=M_{t}+B_{t}$. Most of this paper is devoted to the analysis of determinacy 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.⁹Fiscal policy and equation (11), therefore, may be 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 ( $\Pi^{\ast}$) given by $\Pi^{\ast}>\frac{\Omega}{R}$ because of considerations not included in the model. Since the target nominal interest rate ($I^{\ast}$) must satisfy $I^{\ast}=R\Pi^{\ast}$, it is given by $I^{\ast}>\Omega$. Absent such considerations, the optimal values for $\Pi _{t}$ and $I_{t}$ would be $\frac{\Omega}{R}$ and $\Omega$, respectively. Our 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 $\Pi^{\ast}$ and $I^{\ast}$ respectively, and are either constant at, asymptotically approach, or eventually reach values represented by $\Pi^{BTE}$ and $I^{BTE}$, respectively, where $I^{\ast}>I^{BTE}=R\Pi^{BTE}\geq\Omega$.

2.3  Money demand and lower bounds on the interest rate

We consider two particular specifications of money demand. Under both specifications the gross nominal interest rate has a lower bound (possibly one). Equation (9) implies

$$I_{t}=\frac{U^{\prime}(\bar{C})}{U^{\prime}(\bar{C})-V^{\prime}(m... ...\bar{C})-\underset{m_{t} \rightarrow\infty}{\lim}V^{\prime}(m_{t})}=\Omega\geq1$$ (12)

Of course, a lower bound of one for $I_{t}$ implies a lower bound of zero for the net nominal interest rate.

The lower bound $\Omega$ may be unattainable or attainable. To model an attainable lower bound ($ALB$) for $I_{t}$, we assume that the utility of real balances is given by

$$V(m_{t})=\left\{ \begin{array}[c]{cc} \underline{V}^{\prime}m_{t}... ...m_{t} & \text{for }m_{t}>\overline{m} \end{array} \right. ,\qquad\overline{m}>0$$ (13)

As before, $\underline{V}^{\prime}\geq0$ represents the minimum marginal utility of real balances. Equation (12) implies

$$I_{t}=\left\{ \begin{array}[c]{cc} \frac{U^{\prime}(\bar{C})}{U^{... ...U^{\prime}(\bar{C})}{U^{\prime}(\bar{C})-\underline {V}^{\prime}}=\Omega^{ALB},$$ (14)

To model an unattainable lower bound ($ULB$), we assume that the utility of real balances is given by

$$V(m_{t})=\underline{V}^{\prime}m_{t}+\gamma\ln m_{t}\qquad\gamma ... ...me}\geq0,\quad m_{t}>\frac{\gamma}{U^{\prime} (\bar{C})-\underline{V}^{\prime}}$$ (15)

where $\underline{V}^{\prime}$ represents the lower limit of the marginal utility of real balances. With this functional form, equation (9) implies

$$I_{t}=\frac{U^{\prime}(\bar{C})}{U^{\prime}(\bar{C})-\frac{\gamma... ...{U^{\prime}(\bar{C})}{U^{\prime}(\bar{C})-\underline {V}^{\prime}}=\Omega^{ULB}$$ (16)

Money-demand functions with an $ALB$ and an $ULB$ are represented in figure 1.¹⁰ In both cases, $\Omega=1$ if and only if $\underline{V}^{\prime}=0$. With an $ALB$ there is a liquidity trap, 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 $\underline{m}$: $m\geq\underline {m}=\max\{\overline{m}+\underline{V}^{\prime}-U^{\prime}(\bar{C}),0\}$ with an $ALB$ model, and $m>\underline{m}=\frac{\gamma}{U^{\prime}\left( \bar {C}\right) -\underline{V}^{\prime}}$ with an $ULB$. In order for $\lim_{m\rightarrow\underline{m}}I=\infty$ in the $ALB$ model, we need $\overline{m}+\underline{V}-U^{\prime}(\bar{C})\geq0$.

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

$$I_{t}=g(I^{\ast},\frac{Y_{t}}{\bar{Y}},\frac{\Pi_{t}}{\Pi^{\ast}})$$ (17)

where $\Pi^{\ast}$ is the target-equilibrium ($TE$) inflation rate and $\bar{Y}=\bar{C}$ is the flexible-price output level. With flexible prices, output is always at its flexible-price level , $Y_{t}=\bar{Y}$, so from now on we omit $\frac{Y_{t}}{\bar{Y}}$. We assume that the elasticity of the interest rate with respect to the inflation rate evaluated at the target inflation rate is greater than one. If $\Pi_{t}=\Pi^{\ast}$, then $I_{t}=I^{\ast}.$ In this and the following section, we assume that the interest-rate rule is at least weakly increasing in the inflation rate and that the interest rate has a lower bound, either a preference-determined lower bound or a policy-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 $\Pi_{t}=\Pi^{\ast}$ and a $BTE$, where $\Omega/R\leq\Pi^{BTE}<\Pi^{\ast}\$and $\Omega\leq I^{BTE}<I^{\ast}$.

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 way to the preference-determined lower bound, $\Omega^{ALB}$, associated with an $ALB$ money-demand function:¹²

$$I_{t}=\max\left[ \Omega^{ALB},I^{\ast}\left( \frac{\Pi_{t}}{\Pi^{\ast} }\right) ^{\lambda}\right] ,\qquad\lambda>1$$ (18)

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 $\lambda>1$.¹³ We refer to this case as the liquidity-trap case.

The Fisher equation (8), the interest-rate rule (18), and $I^{\ast}=R\Pi^{\ast}$ imply a log-linear difference equation in $\Pi _{t}$ when $I_{t}>\Omega^{ALB}$:

$$\begin{tabular}[c]{l} $\Pi_{t+1}=\frac{I^{\ast}}{R}\left( \frac{\... ...\Pi}_{t+1}=(1-\lambda)\hat{\Pi}^{\ast}+\lambda\hat{\Pi}_{t}$ \end{tabular} \ \$$ (19)

where variables with hats over them represent logarithms. The solution is

$$\hat{\Pi}_{t+k}=\hat{\Pi}^{\ast}+(\lambda)^{k}[\hat{\Pi}_{t}-\hat{\Pi}^{\ast}]$$ (20)

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. Deviations from the inflation target would result in explosive or implosive paths of inflation since $\lambda>1$.

However, equation (20) applies only when it calls for an interest rate at or above $\Omega^{ALB}$. The inflation rate cannot decline forever. If the inflation rate given by (20) calls for an interest rate below $\Omega^{ALB},$ the inflation rate is determined by the Fisher equation (8) together with $I_{t}=\Omega^{ALB}$ instead of by (20). That is, the inflation rate will stop declining when it is equal to its lower bound:

$$\Pi_{t+1}=\frac{\Omega^{ALB}}{R}$$ (21)

The difference equation reflecting the lower bound has the form shown in figure 2:¹⁴

$$\Pi_{t+1}=\max\left[ \frac{\Omega^{ALB}}{R},(\Pi^{\ast})^{1-\lambda}(\Pi _{t})^{\lambda}\right]$$ (22)

The list of equilibria as indexed by $\Pi_{0}$ is

1. 1. $\Pi_{0}=\Pi^{\ast}.$ Steady-state $TE$.

   2. $\Pi_{0}\in\langle\frac{\Omega^{ALB}}{R},\Pi^{\ast}\rangle.$ $\Pi _{t}$ decreases to $\frac{\Omega^{ALB}}{R}$ in finite time$.$

   3. $\Pi_{0}=\frac{\Omega^{ALB}}{R}.$ Steady-state $BTE$ equilibrium.

   4. $\Pi_{0}\in\langle0,\frac{\Omega^{ALB}}{R}\rangle.$ Equilibria with $\Pi _{0}<\Pi_{t}=\frac{\Omega^{ALB}}{R}$, $t>0$.¹⁵

Along any $BTE$ path, the inflation rate eventually reaches $\frac{\Omega^{ALB}}{R}$ so that $I_{t}$ reaches its liquidity-trap value $\Omega^{ALB}$ 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 $P_{n}$, given a particular initial $P_{0}.$ There is an infinity of equilibria associated with each initial $P_{0}\in\langle0,\Pi^{\ast}P_{-1}\rangle.$ Once the liquidity trap is reached, the set of possible paths for $M_{k},$ $k\geq n$ includes all paths for which $\frac{M_{k}}{P_{k}}\in\lbrack\overline{m},\infty\rangle,$ $k\geq n$ since agents are indifferent between money and bonds.

There may be more equilibria than those listed above. Beginning on any initial inflation rate above $\Pi^{\ast}$ 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 $\lambda=0$ and the difference equation in figure 2 is a horizontal line. Suppose it is announced that $I_{t}$ will equal $I^{\ast}$ in period $t$ and all future periods. The Fisher equation (8), implies that $I^{\ast}$ would be associated with $\Pi^{\ast}$ 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 $\Pi_{t}\in\langle0,\Pi^{\ast}\rangle$. 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 $I_{t}=I^{\ast}.$

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, $\Lambda$, that may be above $\Omega$. The policy-determined lower bound may be attainable, $\Lambda^{ALB}$, or unattainable, $\Lambda^{ULB}$. For example, with the interest-rate rule

$$I_{t}=\max\left[ \Lambda^{ALB},I^{\ast}\left( \frac{\Pi_{t}}{\Pi^{\ast} }\right) ^{\lambda}\right] ,\qquad\Lambda^{ALB}\geq\Omega,\qquad\lambda>1$$ (23)

the policy-determined lower bound is attainable. This rule and the difference equation 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 ( $\Lambda^{ALB}\geq\Omega^{ALB}$) and with $ULB$ money demand ( $\Lambda^{ALB}>\Omega^{ULB}$). The entire list of possible equilibria is given by items 1 through 4 in the last subsection except that $\Lambda^{ALB}$ replaces $\Omega^{ALB}$ everywhere. In contrast to the liquidity trap case, when $\Pi _{t}$ reaches $\frac{\Lambda^{ALB}}{R}$ along a $BTE$ path, real balances and nominal balances are uniquely 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 $BTE$:

$$I_{t}=\left( I^{\ast}-\Lambda^{ULB}\right) \left( \frac{\Pi_{t}}{... ...\frac{\lambda I^{\ast}}{I^{\ast}-\Lambda^{ULB}}}+\Lambda ^{ULB},\qquad\Pi_{t}>0$$ (24)

$$I^{\ast}=R\Pi^{\ast}>\Lambda^{ULB}\geq\Omega\geq1,\qquad\lambda>1$$

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 $\Pi^{\ast}$ is $\lambda>1$.

With the rule (24), there must be two steady-state equilibrium inflation rates: one is $\Pi^{\ast}$ and the other is $\Pi^{BTE}$ which is below $\Pi^{\ast}$ and above $\Lambda^{ULB}/R$ 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

$$\Pi_{t+1}=\frac{1}{R}\left[ \left( I^{\ast}-\Lambda^{ULB}\right) ... ...\ast}}\right) ^{\frac{\lambda R}{I^{\ast}-\Lambda^{ULB}} }+\Lambda^{ULB}\right]$$ (25)

This equation is a convex function that has a lower bound of $\frac {\Lambda^{ULB}}{R}\geq\frac{\Omega}{R}$ and that crosses the $45^{\circ}$ degree line from below at $\Pi_{t+1}=\Pi_{t}=\Pi^{\ast}$ where its slope is greater than one:

$$\left. \frac{d\Pi_{t+1}}{d\Pi_{t}}\right\vert _{\Pi_{t}=\Pi^{\ast}} =\lambda>1,$$ (26)

that is, $\Pi^{\ast}$ is an unstable steady state equilibrium. In addition its slope approaches zero as $\Pi _{t}$ approaches 0 from above and rises continuously with $\Pi _{t}$:

$$\frac{d\Pi_{t+1}}{d\Pi_{t}}=\lambda\left( \frac{\Pi_{t}}{\Pi^{\ast}}\right) ^{\frac{\lambda R}{I^{\ast}-\Lambda}-1}>0$$ (27)

Therefore, it must intersect the $45^{\circ}$ line a second time at a point below $\Pi^{\ast}$ represented by $\Pi^{BTE}$ where its slope is less than one.²²That is, $\Pi^{BTE}$ is a stable equilibrium with deflation ( $\Lambda ^{ULB}/R<\Pi^{BTE}<1)$, stable prices ( $\Lambda^{ULB}/R<\Pi^{BTE}=1$), or inflation $(1<\Pi^{BTE}$).

There is a continuum of equilibria, indexed by $\Pi_{0}$. Each $\Pi_{0}$ is associated with one of the two possible steady-state inflation rates.

1. 1. $\Pi_{0}=\Pi^{\ast}:$ Steady-state $TE$.

   2. $\Pi_{0}\in\langle\Pi^{BTE},\Pi^{\ast}\rangle:$ Non-steady-state $BTE$; $\Pi _{t}$ $\rightarrow$ $\Pi^{BTE}$ from above

   3. $\Pi_{0}=\Pi^{BTE}:$ Steady-state $BTE$.

   4. $\Pi_{0}\in\left\langle 0,\Pi^{BTE}\right\rangle :$ Non-steady-state $BTE$; $\Pi_{t}\rightarrow\Pi^{BTE}$ from below.²³

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:

$$\frac{M_{t}/M_{t-1}}{\Pi^{\ast}}=\left( \frac{\Pi_{t}}{\Pi^{\ast}}\right) ^{-\tau},\qquad\tau>-1$$ (28)

If $\tau=0$, money grows at the constant target gross growth rate of money which is equal to $\Pi^{\ast}$. In that case, since $M_{-1}$ and $P_{-1}$ are given, $M_{0}$ is determined by $\Pi^{\ast}$. If $\tau\neq0$, there is one $M_{0\text{ }}$ associated with each $P_{0}$.²⁵ In either case, there will be a range 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 $V(m_{t})$ in equation (15) imply an expression for the inflation rate in terms of real balances:

$$\Pi_{t}=\left( \frac{1}{R}\right) \frac{U^{\prime}(\bar{C})}{U^{\prime} (\bar{C})-(\frac{\gamma}{m_{t-1}}+\underline{V}^{\prime})}$$ (29)

Furthermore, the money-growth rule can be rewritten as

$$m_{t}=m_{t-1}\left( \Pi_{t}\right) ^{-(1+\tau)}\Pi^{\ast\;1+\tau}$$ (30)

Combining (30) and (29), yields a difference equation in real balances,

$$m_{t}=m_{t-1}(\Pi^{\ast})^{1+\tau}\left[ \left( \frac{1}{R}\right... ...e}(\bar{C})-\frac{\gamma}{m_{t-1} }-\underline{V}^{\prime}}\right] ^{-(1+\tau)}$$ (31)

with the form plotted in figure 4.²⁶ Given the definition of $\Omega^{ULB}$ in equation (16), it follows that the term in square brackets ($\Pi _{t}$) approaches $\Omega^{ULB}/R$ as $m_{t-1}\rightarrow\infty$. Therefore, the growth rate of real balances ( $m_{t}/m_{t-1}$) approaches $\left( \frac{R\Pi^{\ast}}{\Omega^{ULB}}\right) ^{1+\tau}>1$ for $\tau >-1$.²⁷

The unique steady-state solution for equation (31) is

$$m^{\ast}=\frac{\gamma}{U^{\prime}(\bar{C})-\frac{U^{\prime}(\bar{... ...\gamma}{\frac{I^{\ast}-1}{I^{\ast}}U^{\prime }(\bar{C})-\underline{V}^{\prime}}$$ (32)

If the steady-state level of real balances is to be positive, it must be that

$$I^{\ast}>\frac{U^{\prime}(\bar{C})}{U^{\prime}(\bar{C})-\underline{V}^{\prime }}\equiv\Omega^{ULB}$$ (33)

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 $\Omega^{ULB}/R$. The equilibria may be indexed by $m_{0}$:

1. 1. $m_{0}=m^{\ast}:$ Steady state $TE$ with $\Pi_{0}=\Pi_{t}=\Pi^{\ast}$.

   2. $m_{t}>m^{\ast}:$ $BTE$ with positive growth in $m_{t}$, $\Pi_{0} \in\left\langle 0,\Pi^{\ast}\right\rangle$ and $\Pi_{t}\rightarrow \Omega^{ULB}/R$ from above$,$ $t>0$.

The money-growth rule has a representation as an interest-rate rule that is related 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 bound. The Fisher equation (8) and the $m_{t} \leq\overline{m}$-part of the money-demand function (14) imply an expression for inflation in terms of real balances:

$$\Pi_{t}=\left( \frac{1}{R}\right) \left[ \frac{U^{\prime}(\bar{C} )}{U^{\prime}(\bar{C})-\overline{m}+m_{t-1}-\underline{V}^{\prime}}\right]$$ (34)

Combining the money-growth rule (30) with (34) we obtain a difference equation in real balances:

$$m_{t}=m_{t-1}\left[ \left( \frac{1}{R}\right) \frac{U^{\prime}(\b... ...^{\prime}}\right] ^{-(1+\tau)}\Pi^{\ast(1+\tau)},\qquad m_{t-1}\leq\overline{m}$$ (35)

Given the definition of $\Omega^{ALB}$ in equation (14) it follows that as $m_{t-1}$ increases and reaches $\overline{m}$, the term in square brackets ($\Pi _{t}$) increases and reaches $\Omega^{ALB}/R$. Therefore, the growth rate of real balances ( $m_{t}/m_{t-1}$) increases and reaches $\left( \frac {R\Pi^{\ast}}{\Omega^{ALB}}\right) ^{1+\tau}>1$ for $\tau >-1$.

There exists a steady-state equilibrium with positive real balances equal to

$$m^{\ast}=\frac{U^{\prime}(\bar{C})}{R\Pi^{\ast}}+\overline{m}+\un... ...rline{V}^{\prime}-\left( \frac{I^{\ast}-1}{I^{\ast}}\right) U^{\prime}(\bar{C})$$ (36)

under weak conditions that we assume are met.²⁹ This equilibrium is a $TE$ in which $\Pi=\Pi^{\ast}$ and $I=I^{\ast}.$ ³⁰ Equation (35) applies only when $m_{t} \leq\overline{m}$. In cases in which $m_{t}>\overline{m}$, the difference equation in real balances is given by (30) with the inflation rate fixed at $\Omega^{ALB}/R.$

Figure 5 is a diagram for the case where $\underline{m}>0$ because $\overline{m}+\underline{V}^{\prime}-U^{\prime}(\bar{C})>0$.³¹

The list of equilibria indexed by $m_{0}$ is now$\$

1. 1. $m_{0}=m_{t}=m^{\ast}:$ $TE$ steady state with $\Pi_{0}=\Pi^{\ast}$

   2. $\overline{m}>m_{0}>m^{\ast}:$ $BTE$ with $\Pi_{0}\in\left\langle \frac{\Omega^{ALB}}{R},\Pi^{\ast}\right\rangle$ , $\Pi _{t}$ reaches $\frac{\Omega^{ALB}}{R}$ from above, $t>0$. $m_{t}$ reaches $\overline{m}$ from below,

   3. $m_{0}\geq\overline{m}$ $:$ $BTE$ with $\Pi_{0}\in\left\langle 0,\frac{\Omega^{ALB}}{R}\right]$ and $\Pi_{t}=\frac{\Omega^{ALB}}{R} ,$ $t>0$. The $m_{t}$ growth rate is constant at $\left( \frac{I^{\ast} }{\Omega^{ALB}}\right) ^{1+\tau}$.

There is a second steady-state equilibrium with zero real balances when $\overline{m}+\underline{V}^{\prime}-U^{\prime}(\bar{C})\leq0$ so that $\underline{m}=0$.³² This case is not of particular interest to us because 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.

The money-growth rule in the $ALB$ model also has a representation as an interest-rate rule as long as $m_{t} \leq\overline{m}$.³³ However, $I=\Omega^{ALB}$ for all $m_{t}\geq\overline{m}$. Hence, the interest-rate rule representation requires an additional specification of policy when $I=\Omega^{ALB}$ in order to uniquely pin down the path of real money balances for each initial $m_{0}$. For example, Eggertson and Woodford (2003) add a rule for money growth 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 interest-rate rule

$$I_{t}=\left\{ \begin{array}[c]{cc} \begin{array}[c]{c} I_{t}^{\as... ...i_{t}}{\Pi^{\ast}}\right) ^{\lambda} \leq\Omega \end{array} \end{array} \right.$$ (37)

This rule is similar to the one described in section 3.2, but the interest rate is pegged at $\widetilde{I}>I^{\ast}$ instead of at the policy-determined lower bound value $\Lambda^{ALB}.$ It is feasible with both $ALB$ and $ULB$ money demands, since it never calls for $I_{t}=\Omega$.

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 $I^{\ast}\left( \frac{\Pi} {\Pi^{\ast}}\right) ^{\lambda}$ it would be less than or equal to the lower bound, $\Omega$. In such a situation, $I_{t}$ jumps up to $\widetilde{I}$. In the next period, the inflation rate must be higher than the target inflation rate given the Fisher equation and the fact that $\widetilde{I}>I_{t}^{\ast}$. But such a path is not a possible solution, because it implies that the inflation rate increases without limit. Hence, the economy cannot start out on a path of declining inflation.

5.2  A monotonic but asymmetric interest-rate rule

Next, we demonstrate that there is an asymmetric interest-rate rule that is associated with a unique steady-state inflation rate and a determinate price level. Consider the following rule³⁵,

$$I_{t}=(I_{t}^{\ast})(\frac{\Pi_{t}}{\Pi^{\ast}})^{\lambda}\left( \frac {\Pi_{t+1}}{\Pi^{\ast}}\right) ^{\gamma}$$ (38)

Combining equation (38) with the Fisher equation (8) and using $I_{t}^{\ast }=R\Pi^{\ast}$, give a first-order difference equation in the inflation rate,

$$\Pi_{t+1}=\Pi^{\ast\frac{1-\lambda-\gamma}{1-\gamma}}\Pi_{t}^{\frac{\lambda }{1-\gamma}}$$ (39)

Taking logs, we have

$$\widehat{\Pi}_{t+1}=\alpha_{0}\widehat{\Pi}^{\ast}+\alpha_{1}\widehat{\Pi}_{t}$$ (40)

where $\alpha_{0}=\frac{1-\lambda-\gamma}{1-\gamma}$ and $\alpha_{1} =\frac{\lambda}{1-\gamma}.$ With $\lambda$ and $\gamma$ chosen so that $\vert\alpha_{1}\vert>1$ , the equation is unstable and the unique solution for $\Pi _{t}$ is $\Pi^{\ast}.$

     The rule (38) implies asymmetric responses under the following assumptions:

     $\cdot$ If $\Pi_{t}<\Pi^{\ast}$, $2>\gamma>1$, and $\lambda>1$, so that $\alpha_{0}>1$ and $\alpha_{1}<-1$ .

     $\cdot$ If $\Pi_{t}>\Pi^{\ast}$, $\gamma=0$ and $\lambda>1$, so that $\alpha_{0}<0$ and $\alpha_{1}>1$ .

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 $\Pi_{1}\geq\Pi^{\ast}$ no matter how low the value of $\Pi_{0}$. $\Pi^{\ast}$ is a unique steady-state equilibrium, and $\Pi_{0}=\Pi^{\ast}$ is the only equilibrium.³⁷ If the initial inflation rate is in the interval $\Pi_{0}\in\langle0,\Pi^{\ast}\rangle$, the economy embarks on a path with ever-increasing inflation. The part of the difference equation that applies when $\Pi_{t}\in\left\langle 0,\Pi^{\ast }\right\rangle$ implies $\Pi_{1}>\Pi^{\ast}$, and the inflation rate continues to increase because the part that applies when $\Pi_{t}\in\left[ \Pi^{\ast},\infty\right\rangle$ is now relevant.

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 money-growth rule of the form

$$M_{t}=\left\{ \begin{array}[c]{ccc} \begin{array}[c]{c} \left( \f... ...rline{m}\\ \\ \text{if }m_{t-1}\geq\overline{m} \end{array} \end{array} \right.$$ (41)

and an $ALB$ money demand. The nonmonotonic money-growth rule implies a discontinuous difference equation in real money balances of the form shown in figure 8.³⁸

The top relationship in equation (41) is a rewrite of the money-growth rule (28) and $m^{\ast}$ is the steady-state level of real money balances when following that relationship:

$$m^{\ast}\equiv\overline{m}+\underline{V}^{\prime}-\left( \frac{I^{\ast} -1}{I^{\ast}}\right) U^{\prime}(\bar{C})$$ (42)

The second part of (41) tells us that the level of real balances in period $t$ will be lower than the steady-state $m^{\ast}$ level if $m_{t-1}\geq \overline{m}$. To see this, note that the lower bound on the inflation rate is $\Omega^{ALB}/R$. Hence, $P_{t}\geq\frac{\Omega^{ALB}}{R}P_{t-1}$. With $\alpha<1$, real balances must then be lower than $m^{\ast}$ in period $t.$ $BTE$ are precluded because real balances jump below $m^{\ast}$ 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 $P_{t} =\Pi^{\ast}P_{t-1}$.

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, $D_{t}$. The consolidated government balance sheet implies that $D_{t}$ must equal the money supply (which equals the government bonds held by the monetary authority) plus government bonds held by the public:

$$\begin{tabular}[c]{lllll} $D_{t}\equiv M_{t}+B_{t},$\ & & $D_{0}\neq0$\ & & $t\geq0$ \end{tabular}$$ (43)

For simplicity, we devote most of our attention to fiscal policies under which there is a constant gross growth rate ($\Gamma$) for total government bonds:

$$D_{t+1}=\Gamma D_{t}$$ (44)

where $\Gamma=1$ is the case of a balanced-budget policy.

Fiscal policy and the transversality condition (11) taken together have implications for the possibility of $BTE$. First, consider a candidate steady-state $BTE$ in which the interest rate is constant at $I^{BTE}$. The path $I_{t}=I^{BTE}$ $\forall$ $t$ can be a steady-state equilibrium only if

$$\lim_{t\rightarrow\infty}\left( \frac{\Gamma}{I^{BTE}}\right) ^{t-1}\Gamma D_{0}=0$$ (45)

Therefore if $\Gamma\geq I^{BTE},$ fiscal policy precludes such a path. With our parameterization, $\Gamma<I^{BTE}$ 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 $\Gamma\geq1$, the Friedman rule ($I_{t}=1$ $\forall$ $t$) cannot be implemented exactly under any type of monetary policy. This result for the case of $\Gamma=1$ 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 $I^{BTE}>1$, there may be $BTE$ with small deficits ( $I^{BTE}>\Gamma>1$).

Next, consider a candidate $BTE$ path on which the interest-rate path is some weakly decreasing sequence $\{I_{k}\}$ with $\lim_{k\rightarrow\infty} I_{k}=I^{BTE}$. Let $\{I_{k}\}$ $\equiv\{\Theta_{k}I^{BTE}\},$where $\{\Theta_{k}\}$ is a weakly decreasing sequence with $\lim_{k\rightarrow \infty}\Theta_{k}=1$. Let the value of $\Theta_{k}$ at some time $\hat{t}$ given by $\Theta_{\hat{t}}>1$ be arbitrarily close to one. This sequence of interest rates can be an equilibrium only if

$$\lim_{t\rightarrow\infty}D_{0}\Gamma^{t}\Pi_{k=\hat{t}}^{k=t-1}\left[ \Theta_{k}^{-1}(I^{BTE})^{-1}\right] =0$$ (46)

The condition (46) cannot hold if $\Gamma>I^{BTE}$ and $\Theta_{\hat{t}}$ is arbitrarily close to one. Therefore, $\Gamma>I^{BTE}$ precludes $BTE$ in which the interest rate asymptotically approaches or eventually reaches $I^{BTE}$.$\$In addition, $\Gamma=I^{BTE}$ precludes equilibria in which $I^{BTE}$ is approached asymptotically if $\lim_{t\rightarrow\infty} \Pi_{k=\hat{t}}^{k=t-1}\Theta_{k}^{-1}>0$ . This condition is satisfied if the nominal interest rate approaches $I^{BTE}$ (and hence $\Theta_{k}$ approaches one) fast enough, for example if the nominal interest rate reaches $I^{BTE}$ ( $\Theta_{k}$ reaches one) in finite time.

The class of fiscal policies for which $I^{\ast}>\Gamma>I^{BTE}$ 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) or money-supply rule like equation (28) is sufficient to insure that $\Pi^{\ast}$ is the unique 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 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 money-growth rule (28) is associated with multiple equilibria. There, as well as in all of the paper before section 6, we assume that $B_{t}$ can take on any value implied by the open-market purchases used to increase the money supply. In particular, $B_{t}$ can be 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 $B_{t}$ designated by $\underline{B}$ where $-\infty<\underline{B}$; that is, government claims on the public cannot exceed the finite amount $-\underline {B}.$ 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 $\Pi^{\ast}$ (the target rate of inflation and money growth) must be chosen so that