Board of Governors of the Federal Reserve System
International Finance Discussion Papers
Number 795, April 2004 - Screen Reader Version*
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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
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 () for the inflation rate may have additional equilibria. To be more precise, there may be multiple below-target equilibria (), 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 ; in particular, a balanced-budget fiscal policy precludes 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 .
We also consider monetary-policy rules that preclude . 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 . 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 . 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 debt6 There is always a growth rate of debt high enough to preclude no matter what the monetary policy because 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 is the unique equilibrium because the deficit precludes .
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 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.
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
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
We assume that productivity is constant 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:
We assume that fiscal policy determines the total amount of nominal government bonds outstanding, , 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, , or directly by the public, , through control of open-market operations. The consolidated government balance sheet implies that . 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 () with a target value for inflation ( ) given by because of considerations not included in the model. Since the target nominal interest rate () must satisfy , it is given by . Absent such considerations, the optimal values for and would be and , respectively. Our main focus is on the possible existence of below-target equilibria We define as weakly increasing or decreasing paths for inflation and the interest rate along which they are always below and respectively, and are either constant at, asymptotically approach, or eventually reach values represented by and , respectively, where .
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
The lower bound may be unattainable or attainable. To model an attainable lower bound () for , we assume that the utility of real balances is given by
To model an unattainable lower bound (), we assume that the utility of real balances is given by
Money-demand functions with an and an are represented in figure 1.10 In both cases, if and only if . With an 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 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 (from (4)) we know that there exists a minimum level for real money balances denoted by : with an model, and with an . In order for in the model, we need .
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
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, , associated with an money-demand function:12
The Fisher equation (8), the interest-rate rule (18), and imply a log-linear difference equation in when :
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 .
However, equation (20) applies only when it calls for an interest rate at or above . The inflation rate cannot decline forever. If the inflation rate given by (20) calls for an interest rate below the inflation rate is determined by the Fisher equation (8) together with instead of by (20). That is, the inflation rate will stop declining when it is equal to its lower bound:
The list of equilibria as indexed by is
2. decreases to in finite time
3. Steady-state equilibrium.
4. Equilibria with , .15
Along any path, the inflation rate eventually reaches so that reaches its liquidity-trap value 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 at price level , given a particular initial There is an infinity of equilibria associated with each initial Once the liquidity trap is reached, the set of possible paths for includes all paths for which since agents are indifferent between money and bonds.
There may be more equilibria than those listed above. Beginning on any initial inflation rate above 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 and the difference equation in figure 2 is a horizontal line. Suppose it is announced that will equal in period and all future periods. The Fisher equation (8), implies that would be associated with from period 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 . 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
Now consider interest-rate rules designed to keep the gross nominal interest rate from falling below a policy-determined lower bound, , that may be above . The policy-determined lower bound may be attainable, , or unattainable, . For example, with the interest-rate rule
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 :
With the rule (24), there must be two steady-state equilibrium inflation rates: one is and the other is which is below and above 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
There is a continuum of equilibria, indexed by . Each is associated with one of the two possible steady-state inflation rates.
2. Non-steady-state ; from above
3. Steady-state .
4. Non-steady-state ; from below.23
When the money-demand function has an , money-growth rules are consistent with the existence of both a and in which real money balances are forever increasing and the interest rate is approaching its lower bound.24
Consider the money-growth rule:
The Fisher equation (8) and the money market equilibrium condition (9) with the functional form for in equation (15) imply an expression for the inflation rate in terms of real balances:
The unique steady-state solution for equation (31) is
The money-growth rule may be associated not only with the , but also with a range of in which inflation declines forever and approaches the limit . The equilibria may be indexed by :
2. with positive growth in , and from above .
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
With money demand money-growth rules are also consistent with the existence of both a and . In the real money balances are forever increasing money demand, as with money demand, but the interest rate reaches its lower bound. The Fisher equation (8) and the -part of the money-demand function (14) imply an expression for inflation in terms of real balances:
There exists a steady-state equilibrium with positive real balances equal to
Figure 5 is a diagram for the case where because .31
The list of equilibria indexed by is now
2. with , reaches from above, . reaches from below,
3. with and . The growth rate is constant at .
There is a second steady-state equilibrium with zero real balances when so that .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 model also has a representation as an interest-rate rule as long as .33 However, for all . Hence, the interest-rate rule representation requires an additional specification of policy when in order to uniquely pin down the path of real money balances for each initial . For example, Eggertson and Woodford (2003) add a rule for money growth that applies whenever the interest rate reaches its lower bound.
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
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 were given by it would be less than or equal to the lower bound, . In such a situation, jumps up to . In the next period, the inflation rate must be higher than the target inflation rate given the Fisher equation and the fact that . 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.
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,
The rule (38) implies asymmetric responses under the following assumptions:
If , , and , so that and .
If , and , so that and .
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 no matter how low the value of . is a unique steady-state equilibrium, and is the only equilibrium.37 If the initial inflation rate is in the interval , the economy embarks on a path with ever-increasing inflation. The part of the difference equation that applies when implies , and the inflation rate continues to increase because the part that applies when is now relevant.
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
The top relationship in equation (41) is a rewrite of the money-growth rule (28) and is the steady-state level of real money balances when following that relationship:
There is always a fiscal policy that precludes no matter what the monetary policy rule. Fiscal policy determines the path of total nominal government debt measured as total nominal government bonds, . The consolidated government balance sheet implies that must equal the money supply (which equals the government bonds held by the monetary authority) plus government bonds held by the public:
Fiscal policy and the transversality condition (11) taken together have implications for the possibility of . First, consider a candidate steady-state in which the interest rate is constant at . The path can be a steady-state equilibrium only if
One important implication of the previous paragraph, is that if , the Friedman rule ( ) cannot be implemented exactly under any type of monetary policy. This result for the case of 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 , there may be with small deficits ( ).
Next, consider a candidate path on which the interest-rate path is some weakly decreasing sequence with . Let where is a weakly decreasing sequence with . Let the value of at some time given by be arbitrarily close to one. This sequence of interest rates can be an equilibrium only if
The class of fiscal policies for which is especially interesting. As just shown, this class precludes 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 is the unique equilibrium value for the inflation rate. These observations suggest that 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 . This possibility is yet another reminder of the importance of analyzing monetary and fiscal policy jointly.40
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 can take on any value implied by the open-market purchases used to increase the money supply. In particular, 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 designated by where ; that is, government claims on the public cannot exceed the finite amount 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 are to be precluded, the transversality condition (11) implies that (the target rate of inflation and money growth) must be chosen so that
In section 6.1, we show that a balanced-budget policy precludes when the interest rate is always at a lower bound of one. Here we show that a balanced-budget policy may not preclude 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 in which the interest rate fluctuates between one and a value greater than one.
Consider the following period utility function with time-varying productivity, :
Equations (50), (4) and (6) imply that the nominal and real interest rates are given by
We consider a simple example in which (a) the attainable lower bound on the nominal interest rate is unity, , (b) alternates between a high value ( ) in even periods and a low value ( ) in odd periods,
If is at the lower bound of one, equations (51) and (54) imply that . From the interest-rate rule (18) with a time varying and equation (52), it follows that is above the lower bound if
is at the lower bound again. Given and , is given by the difference equation (19) that applies when :
By induction, interest rates, real interest rates, and inflation rates are
It follows that 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, and Uribe (2001a), a balanced-budget policy precludes only equilibria in which the central bank commits to keep the interest rate equal to one no matter what happens to inflation.
Under many specifications of monetary policy, standard models may exhibit price-level indeterminacy. That is, they may have multiple equilibria that include both a locally-unique steady-state target (inflation-rate) equilibrium () and multiple below-target equilibria (), equilibria in which the inflation rate is always below target and is constant or eventually reaches or asymptotically approaches a below-target 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 policy-determined lower bound. Second, with either an interest-rate rule or a money-growth rule, money demand may imply a lower bound on the interest rate so that the interest 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 when an interest-rate rule does not.
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 interest-rate 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 no matter what the monetary policy because with those policies paths violate the transversality condition. Balanced budget fiscal policy can preclude 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.
As stated in the introduction, it is not yet clear whether the existence of is an important problem or a curiosum. Some have argued that 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, he shows that the locally-unique meets this criterion while the multiplicity of equilibria associated with the below-target steady state, often referred to as `sun-spot' equilibria, do not. Both McCallum (2001, 2003) and Evans and Honkapohja (2002) use stability under a particular type of learning as an alternative criterion. In terms of our model, they show that the is stable under learning while the 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 The combination of any of a range of small deficits with a standard interest-rate or money-supply rule guarantees that the is the unique equilibrium because the deficit precludes . This observation suggests that 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 . This possibility provides another illustration of the importance of analyzing monetary and fiscal policy jointly.
It might be argued that Japan has been in a 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 government debt. That is, according to our analysis, fiscal policy was too expansionary to be consistent with a .
Many analysts have concluded that the Japanese situation is better described as a standard `liquidity-trap equilibrium' (). A 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 or , but, for others, it does not. For example, it is often suggested that Japan should announce a higher target inflation rate. Raising the target inflation rate (or increasing the money supply even though the interest rate is at the lower bound) can help an economy escape from an by raising expected future inflation thereby lowering the real interest rate.47 However, we have shown that this policy cannot help escape from a 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 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 .
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* 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, email@example.com and firstname.lastname@example.org . Return to text
1. Of course, inflation-rate determinacy and price-level determinacy are linked. If the inflation rate (defined as the percentage change in the price level between today and yesterday) is determined and yesterday's price level is known, then today's price level is determined. Return to text
2. A 1999 version of Woodford (2003) cited by Benhabib, Schmitt-Grohe, and Uribe (2001b) contains an explanation of the possible existence of BTE with both interest-rate rules and money-supply rules. In Woodford (2003) and Benhabib, Schmitt-Grohe, and Uribe (2001a) the inflation rate may asymptotically approach a below-target steady state; in Schmitt-Grohe and Uribe (2000), Benhabib, Schmitt-Grohe, and Uribe (2001b) and Eggertsson and Woodford (2003) it may eventually reach a below-target steady state. Return to text
4. McCallum (2001) refers to a `liquidity trap situation' as a situation in which `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 zero [and] the quantity of money becomes irrelevant because money and bonds are essentially perfect substitutes'. Our definition is the same as Krugman's except that, like Sargent (1987) among others, we explicitly allow for a liquidity trap at a positive interest rate. Return to text
5. Since 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 (2001a) show that a lower bound can give rise not only to nominal indeterminacy but also to real indeterminacy. Return to text
8. Increases in can be interpreted either as increases in productivity or decreases in the disutility 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 be the disutility associated with labor () and let be the production function. Inverting the production function yields
9. This 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. Return to text
13. We 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). Return to text
15. The initial inflation rate is not constrained by the lower bound . The reason is that the constraint on the inflation rate is implied by the Fisher equation in combination with the interest rate given by the interest-rate rule. The Fisher equation has no implications for the initial price level or the inflation rate , but it has implications for the inflation rate . Hence, Return to text
18. Consider the family of interest-rate rules given by
19. The continuous-time policy rule used in Benhabib, Schmitt-Grohe, and Uribe (2001a) and the main text of Benhabib, Schmitt-Grohe, and Uribe (2001b) has a policy-determined unattainable lower bound. The discrete-time analogue of this rule is
We employ the rule (24) because it is directly comparable to the other interest-rate rules and the money-supply rules (28) used in this paper. Rule (24) is also used by Evans and Honkapohja (2003), who assume that . Return to text
20. That is
24. This possibility is pointed out, for example, in Woodford (1994), Woodford (2003), Christiano (2001), and Benhabib, Schmitt-Grohe, and Uribe (2001b)). We begin with the case of money demand because it is somewhat simpler. Return to text
27. is an unstable steady state since equation (31) and the expression for given by (29) imply that for all
28. Equation (16) can be used to obtain an expression for in terms of . Using this expression and assuming , it follows from equations (30) and (8) that
29. The positive steady state exists only if or equivalently That is, the target interest rate must be smaller than the maximum possible interest rate. For parameter values for which , , so the condition is definitely met. In order to exist, the strictly positive steady state equilibrium must also satisfy which means that . This condition is always satisfied as long as is not too large. Return to text
30. is an unstable steady state since (36) and the definition of imply
32. See section 2.3. In this case (not shown), 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 Return to text
33. Using (30),(8) and (14), we get
37. All it takes for equation (39) to be associated with a unique steady-state is that when and when . One could, for example, let so that there was zero response to future inflation, and vary to meet these requirements. Return to text
41. Woodford (2003) and Woodford (1994) restrict attention to the case of (and ) and argue that the restriction on the range of the target inflation rate, necessary to preclude is a limitation of money-supply rules relative to interest-rate rules. However, the condition in equation (48) implies that if , the authorities can follow a money-supply rule that implies uniqueness and still target a gross inflation rate below . In particular, with one can get arbitrarily close to the Friedman rule by choosing large enough. Return to text
42. In 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 is associated with uniqueness as long as gross money growth is equal to or larger than one ( in our notation). When the public holds no government bonds, money must be increased by money-transfers. A money-growth rule with then corresponds to the class of fiscal policies discussed in is subsection with , since the nominal growth rate of money is equal to the nominal growth rate of government bonds. Benhabib, Schmitt-Grohe, and Uribe (2001b) show that multiple equilibria are precluded under money-growth rules when As we have shown the lower bound on can be negative. What is needed is that money be supplied with lump-sum transfers instead of open-market operations after some point Return to text
43. Woodford (1994) (2003) restricts attention to the case of (and ) and argues that the restriction on the range of the target inflation rate, necessary to preclude is a limitation of money-supply rules relative to interest-rate rules. Return to text
45. A cannot be attained because would be below a lower bound of one. As noted by Krugman (1998) being in a liquidity trap in which is at its lower bound has only nominal consequences in a flexible-price model but has real consequences in a sticky-price model because output has to adjust. For example, Svensson (2000) considers what happens when is too low and how it can credibly be raised. Return to text
46. For example, the combination , , , and satisfy both (54) and (56), since . and (
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