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Working Paper Series
Real Implications of the Zero Bound on
Nominal Interest Rates
WP 03-15
This paper can be downloaded without charge from:
http://www.richmondfed.org/publications/
Alexander L. Wolman
Federal Reserve Bank of Richmond
forthcoming in
Journal of Money, Credit and Banking
Real Implications of the
Zero Bound on Nominal Interest Rates
Alexander L. Wolman
∗
Federal Reserve Bank of Richmond Working Paper No. 03-15
December 2003
JEL Nos. E31, E43, E52
Keywords: zero bound, monetary policy, deflation, interest rates, price-level stationarity
Abstract
If monetary policy succeeds in keeping average inflation very low, nominal interest
rates may occasionally be constrained by the zero lower bound. The degree to which
this constraint has real implications depends on the monetary policy feedback rule
and the structure of price-setting. Policy rules that make the price level stationary
lead to small real distortions from the zero bound. If policy imparts persistence into
the inflation rate, the real implications of the zero bound are large in the presence of
backward looking price-setting, and small if prices are set to maximize profits.
∗
Alexander.Wolman@rich.frb.org. For helpful discussions and comments, I thank two anonymous referees,
Alan Blinder, Mike Dotsey, Marvin Goodfriend, Bob King, Don Kohn, Jeff Lacker, Pierre Sarte, Lars
Svensson, Carl Walsh, Mike Woodford, seminar participants at Princeton, Rutgers, Virginia, and the Banco
de Portugal, and conference participants at the NBER Conference on Monetary Policy, December 1998, the
SCE Conference in Boston, June 1999, and the AEA Meetings in Boston, January 2000. Elise Couper and
John Hejkal provided outstanding research assistance. The views expressed here are those of the author;
they do not represent the views of the Federal Reserve Bank of Richmond or the Federal Reserve System.
1 Introduction
At least since the writings of Irving Fisher, economists have understood that nominal interest
rates are bounded below by some number close to zero. No one would willingly hold a security,
payable in money, which yielded a return below the storage cost of money. As long as storing
money is nearly costless, a lower bound near zero applies. At least since the writings of John
Maynard Keynes, some economists have also been concerned that very low nominal interest
rates present a danger for a modern economy. Keynesian concerns — about a liquidity trap
— involved the behavior of money demand at low nominal interest rates. While the behavior
of money demand is one factor in understanding macroeconomic equilibrium, the simple fact
recognized by Fisher potentially makes it undesirable for monetary policy to keep inflation
very low. Whether or not this is so is equivalent to the question of whether or not the
bound on nominal interest rates has real implications. We study this question in a dynamic
equilibrium model with staggered price-setting.
A confluence of factors has led recently to much attention being focused on the effects of
the nominal interest rate bound. First, since the inflation of the 1970s and early 1980s, the
industrialized countries have begun to approach price stability. Assuming that the average
level of real interest rates is roughly invariant to the average level of inflation, the Fisher
equation implies that low inflation should correspond to low nominal interest rates. The
second factor has been more acute: Japan has had short term nominal interest rates effectively equal to zero since 1998, and in the United States, the Fed Funds target currently
stands at 1.00%. Finally, the evolution of macroeconomics has also stimulated work on the
zero bound. A consensus has been reached that policy analysis must be conducted explicitly
within the context of dynamic models, and the bound on nominal interest rates is a concrete
object that can be analyzed in the context of modern models.1
Summers (1991) takes the position that the lower bound on nominal rates does make very
1
low inflation undesirable, as it would occasionally prevent needed declines in real interest
rates. We refer to this view as emphasizing the direct implications of the zero bound,
and discuss it further in the next section. Fuhrer and Madigan (1997), Orphanides and
Wieland (1998) and Reifschneider and Williams (1999) analyze the implications of the zero
bound in dynamic rational expectations models that use variants of the Fuhrer-Moore (1995)
specification of price-setting behavior. The first two of these papers find that the zero bound
generates a significant real distortion at very low inflation rates, while Reifschneider and
Williams find that certain specifications of the monetary policy reaction function can go a
long way towards eliminating the real distortion.
Rotemberg and Woodford (1997,1998) study the zero bound’s implications indirectly in
a dynamic model that involves explicit optimizing behavior by consumers and firms. While
they find that the zero bound has strong implications for the type of policy rules that
are feasible, the effect on the optimal average inflation rate is small. Benhabib, SchmittGrohe and Uribe (2001) also analyze the zero bound’s implications in an optimizing model;
they argue that in the presence of the zero bound, certain types of policy rules lead to
equilibria with “deflationary spirals.” Eggertsson and Woodford (2003) study optimal policy
in the presence of the zero bound, in an otherwise-linear dynamic New-Keynesian model. A
linearized version of our model would look quite similar to the model used by Eggertsson and
Woodford. Instead of studying optimal policy, however, we compare two simple policy rules.
One is an inflation-targeting Taylor rule that is often described as a close approximation to
actual central bank behavior. The other rule differs only by targeting the price level instead
of the inflation rate.
Krugman (1998) blames Japan’s recent economic problems partly on inappropriate monetary policy in the presence of the zero bound. Buiter and Panigirtzoglou (1999) and Goodfriend (2000) pursue the idea that by imposing a carry tax on money, the lower bound on
nominal interest rates can be driven below zero. Goodfriend also analyzes quantitative mon2
etary policy actions when the nominal interest rate is at its lower bound. McCallum (2000)
studies two factors which mitigate the zero bound’s real effects; first, the steady state real
interest rate may be decreasing in the target inflation rate, and second, there is an exchange
rate channel for monetary policy.
Our interest here is in understanding the conditions under which the zero bound has
significant real implications, and thus could constitute an argument against very low inflation.
We study a closed economy in which the steady state real interest rate is invariant to inflation,
and we assume that the monetary authority follows an interest rate feedback rule. The paper
is closest in spirit to the work of Reifschneider and Williams (2000). However, the framework
here is a dynamic equilibrium model in which prices are set in the staggered fashion of Taylor
(1980), and there is a demand for money motivated by a shopping time requirement, as in
McCallum and Goodfriend (1987). The model is solved with a nonlinear method, so that
the zero bound’s implications can be analyzed directly. Wolman (1998), showed that for
one parameterization of this model the zero bound did not constitute a significant real
distortion. Here we show that the crucial determinants of the zero bound’s real implications
are the manner in which firms set their prices and the policy rule followed by the monetary
authority. If prices are set to maximize expected present discounted profits, then a policy rule
which keeps the price level stationary can keep the zero bound from significantly interfering
with the behavior of the real economy. This result does not appear to be sensitive to the
source of shocks hitting the economy.
The paper proceeds as follows. Section 2 heuristically discusses both the zero bound’s
direct implications and the reason that price level stationarity might be important for the
equilibrium implications. Section 3 describes the model to be used for the formal analysis.
Section 4 presents the main results, and section 5 presents additional results on uncertainty.
Section 6 concludes.
3
2 A Heuristic Discussion
Partly because of the nonlinearity associated with the bound on nominal interest rates, the
model laid out below is too complicated to solve analytically. As such, the results we will
present are based on computation. To give the reader some background for interpreting
those results, it will be useful to consider in isolation two components of the model — indeed
components of all the models used to study this issue. Fisher’s equation, relating the nominal
interest rate to the real interest rate and expected inflation, is the source of the zero bound’s
direct implications. A general feature of the monetary policy rule — whether it keeps the
price level trend-stationary — helps us to move from the direct implications to thinking about
equilibrium outcomes.
2.1 Direct Implications of the Zero Bound
The Fisher equation states that the gross nominal interest rate (1 + R ) is approximately
t
equal to the product of the gross real interest rate (1 + r ) and expected inflation (E P +1/P ):2
t
1 + Rt
≈ (1 +
t
t
t
+1 /Pt ) .
rt ) (Et Pt
Suppose we treat inflation and expected inflation as fixed. Then the lower bound on the
nominal interest rate translates directly into a lower bound on the real interest rate. Taking
as given that there is some underlying “natural” real rate behavior which would be optimal,
the zero bound could inhibit that behavior. At very low levels of inflation the zero bound
would almost surely inhibit that behavior. This direct implication is the basis for Summers’
(1991) argument that monetary policy should not aim for a very low (zero) inflation rate.
Implicit in the argument is the idea that inflation would be stabilized in a narrow band. In
practice, however, it is difficult to imagine that policy would stabilize inflation in a narrow
band; this has not occurred in countries with explicit inflation targeting policies. One is
then led to ask whether certain policy rules might generate temporary variation in expected
4
inflation that would allow the desired variation in real interest rates to occur even in the
presence of low average inflation.
2.2 The Potential Importance of Price-Level Trend-Stationarity3
Anticipating results to be presented below, we consider here one general feature of policy
rules: their implications for whether the price level is trend stationary. Among rules that
promise to achieve zero inflation on average, one class keeps the price level stationary around
a constant level, and the other class allows base drift in the price level but always brings
inflation back toward zero. To simplify the discussion, consider feedback rules that set the
nominal interest rate as an increasing linear function of either the inflation rate or the log
price level:
R = max {r∗ + fp · πt, 0} ,
t
where
(1)
πt = Pt /Pt−1 , or
Rt
= max {r∗ + fp · (ln (Pt) − ln (P ∗)) , 0} .
(2)
Now consider a situation where inflation is negative in the current period, the nominal
interest rate is at zero, and the natural real rate is negative. This is a situation where the
direct implications of the zero bound would suggest that the natural real rate cannot be
achieved. Figure 1.A plots the long run path of inflation for rules that allow base drift in
the price level, and the point marked with an asterisk is the current period being described.
Because there is deflation in the current period, if the rule successfully stabilizes inflation it
will cause next period’s expected deflation rate to be lower. But with the current nominal
rate at zero, there needs to be positive expected inflation in order to achieve a negative real
rate.
Figure 1.B plots the long run path of the price level under rule (2), and the point marked
with an asterisk indicates a current period, where the nominal rate is zero, the price level
5
is below its average, and the natural real rate is negative. Because the price level is below
average in the current period, if the rule successfully stabilizes the price level it will cause an
expected increase in the price level next period, meaning positive expected inflation. Positive
expected inflation is precisely what is needed to accomplish a negative real rate when the
nominal rate is zero.
This simple analysis suggests rules that keep the price level stationary may make it
possible for low average inflation to coexist with unimpeded behavior of the real interest
rate. Alternatively, more complicated policies which allowed base drift in the price level could
be consistent with natural real rate behavior if they generated “overshooting” of expected
inflation under certain circumstances.
Before moving to the formal model, note that the discussion above was not framed
in terms of particular types of shocks; what matters is the shocks’ implications for the
appropriate real rate of interest. Note also that for a policy rule to be successful in allowing
appropriate real rate variability, there must be some flexibility in expected inflation. If the
structure of the economy is such that expected inflation is insufficiently flexible, it will be
impossible to prevent real implications of the zero bound.
3 The Model
As in other recent work on the zero bound, the model used here is one in which there is
assumed to be sluggish price adjustment. The model is in the tradition of Taylor (1980),
in that price-setting is staggered: each firm sets its price for 2 periods, with 1/2 of the
firms adjusting each period. There are a continuum of firms, and they produce differentiated
consumption goods using, as the sole input, labor provided by consumers at a competitive
wage. It has been more common in recent work with sticky-price models to use the Calvo
framework, in which firms face an exogenous, constant probability of being able to adjust
6
their price. We use the two-period Taylor framework for two reasons. First, it seems to
be a closer approximation to reality at a microeconomic level: in the Calvo environment,
a positive fraction of firms charge prices that were set arbitrarily far in the past. Second,
although the Calvo model is known for its analytical tractability, the Taylor model with
two-period pricing has fewer state variables than the Calvo model.
Consumers are infinitely lived, and purchase consumption goods using income from their
labor, which is supplied elastically. Consumers hold money in order to economize on transactions time, as in McCallum and Goodfriend (1987). Monetary policy is given by an exogenous
rule which is known to all agents. Aside from the internal dynamics associated with price
stickiness, the model is driven by exogenous shocks to labor supply or consumption demand.
The analysis will concentrate on the internal dynamics.
3.1 Consumers
Consumers have preferences over a consumption aggregate (c ) and leisure (l ) given by
∞
Et
�
t
t
β t · (ln (ct + ϑt ) + χt · lt ) ,
t=0
(3)
where ϑ and χ are random variables which follow persistent two-state Markov processes.
t
t
The discount factor β is set to 0.995, and the means of ϑ and χ are set to
respectively.4 The consumer’s budget constraint is
t
t
0
and 3,
ct + τ t + Mt /Pt + (1 + Rt )−1 (Bt/Pt ) = Mt−1 /Pt + Bt−1/Pt + wt nt + dt ,
and the time constraint is
n
t
+
l
t
+
h(M /(P c )) = E,
t
(4)
t t
where τ is a real lump-sum tax (or transfer if negative), P is the price level, M is nominal
money balances chosen in period t, to carry over to t + 1, B is holdings of one-period nominal
t
t
t
t
zero-coupon bonds, issued by the government, maturing at t + 1, R is the interest rate on
t
7
nominal bonds, w is the real wage, n is time spent working, d is real dividend payments
t
t
t
from firms, h(M /(P c )) is time spent transacting, and E is the time endowment. Defining
real balances to be m ≡ M /P , the function h(·) is parameterized as in Wolman (1997):
t
t t
t
t
t
h(m /c ) = φ · (m /c ) − 1+ν ν A−1/ν (mt/ct) 1+ν ν + Ω,
t
t
t
t
for
m /c < A φν ,
t
t
·
(5)
h(mt/ct) = 1+1 ν Aφ1+ν + Ω,
for
mt/ct ≥ A · φν ,
ν = −0.7695 5 Transactions time is thus decreasing
φ = 1.4 × 10−3 A = 1.7 × 10−2
in real balances and increasing in consumption, up to a satiation level of the ratio of real
balances to consumption.
Goods market structure. Following Blanchard and Kiyotaki (1987), we use the Dixit
and Stiglitz (1977) monopolistic competition framework. The consumption aggregate is an
�
integral of differentiated products, c = [ c(ω) ε−ε 1 dω] ε−ε 1 . We set ε = 10, which implies that
every producer faces a downward sloping demand curve with constant elasticity 10.
Since all producers that adjust their prices in a given period choose the same price, it is
easier to write the consumption aggregate as
with
,
, and
.
t
ε−1
ct = c(c0,t , c1t ) = (0.5 · c0,tε
+05
.
−
ε 1
ε
· c1,t
ε
) ε−1 ,
(6)
where c is the quantity consumed in period t of a good whose price was set in period t − j .
The constant elasticity demands for each of the goods take the form:
j,t
cj,t
=
�
∗
Pt− j /Pt
�−ε
(7)
· ct ,
where Pt∗−j is the nominal price at time t of any good whose price was set j periods ago, and
Pt is the price index at time t, given by
Pt
=
�
∗ 1−ε
0.5 · (Pt )
�
+ 0 5 P ∗−1
.
·
t
�1−ε � 1−1 ε
.
(8)
Optimization. If we attach a Lagrange multiplier λ to the budget constraint, so that λ is
t
t
the marginal value of real wealth, the first order conditions for the individual’s maximization
8
problem, with respect to c , l , B and M , are
t
t
t
(ct +
t
ϑ )−1 = λ − χ h ( )( mc2 ) ,
t
�
t ·
t
t
·
(9)
t
χt = wt · λt ,
λt
Pt
(10)
= β (1 + R ) E (λ +1/P +1) ,
t
·
t
t
(11)
t
and
λt /Pt) · (1 + wt · h (·)(1/ct )) = βEt (λt+1 /Pt+1 ) .
�
(
(12)
In choosing consumption optimally (9), the individual weighs the benefit of consuming a
marginal unit, which is the left hand side of (9), against the cost, which consists of both
forfeited real wealth (the first term on the right hand side) and time spent transacting (the
second term on the right hand side). In choosing leisure optimally (10), the individual
balances the marginal utility of leisure against the wage earnings that the time would yield.
The choice of bond holdings (11) balances the marginal value of nominal wealth today against
(1 + R ) times the marginal value of nominal wealth tomorrow. And finally, optimal money
holdings (12) imply that the individual balances the transactions-facilitating benefit against
t
the foregone interest cost of holding money.6
3.2 Firms
Each firm produces with an identical technology:
cj,t
=
nj,t ,
j
=
(13)
0, 1,
where n is the labor input employed in period t by a firm whose price was set in period
t − j . Given the price that a firm is charging, it hires enough labor to meet the demand for
j,t
its product at that price. Firms that do not adjust their price in a given period can thus be
thought of as passive. Given that it has set a relative price P ∗− /P , real profits for a firm of
t
9
j
t
type j are
�
Pt∗−j /Pt
�
· cjt
− ·
wt
(14)
njt ,
that is, revenue minus cost.
We will analyze two specifications for the determination of Pt∗. Firms either choose Pt∗ to
maximize the present discounted value of profits over the two periods that the price will
be charged, or they choose Pt∗ according to the partially backward-looking specification of
Fuhrer and Moore (1995). The former specification is the natural one suggested by the rest
of the model. The latter has been found useful in generating inflation persistence similar to
that seen in the data. Note that for both specifications, we assume that the firm’s nominal
price is fixed for two periods. An alternative assumption — made for example by Yun (1996)
— is that in between adjustments, a firm’s nominal price automatically adjusts at the steady
state inflation rate. Our results for the case of a zero inflation target would be unchanged
by this assumption. For the case of a positive inflation target analyzed below, the overall
message would not change. We will return to this point in section 4.
Profit maximizing pricing : Maximization of present value implies that a firm chooses its
current relative price taking into account the effect on current and expected future profits.
Substituting into (14) the demand curve (7) and the technology (13), the present discounted
value of expected profits is given by
ct
·
�
Pt∗/Pt )1−ε − wt · (Pt∗ /Pt)−ε
(
�
+
(15)
βEt λλt+1
t
·
ct+1
·
�
1−ε
( t∗/Pt+1 )
P
− w +1
t
·
−ε � ,
(Pt∗ /Pt+1 )
for the two periods over which a price will be in effect. Differentiating (15) with respect to
P ∗ and setting the resulting expression equal to zero, the optimal relative price satisfies
t
Pt∗
Pt
ε
= ε−1
·
�λ c w
t t
t
+ βEt λt+1 ct+1 (Pt+1/Pt )ε wt+1
λt ct + βEt λt+1 ct+1 (Pt+1/Pt )ε−1
10
�
.
(16)
It is easiest to understand this expression under perfect foresight, in which case the optimal
relative price is a constant markup (ε/ (ε − 1)) over a weighted average of current and future
real marginal cost (here real marginal cost is equal to the real wage). The weights represent
marginal revenue shares associated with the current and future periods. In a non-inflationary
steady state, the firm would choose a markup over current marginal cost of (ε/ (ε − 1)). In an
inflationary or deflationary steady state, an adjusting firm’s markup over current marginal
cost differs from (ε/ (ε − 1)), as adjusting firms take into account the future erosion (or
increase) of their relative price. Out of steady state, the markup over current marginal cost
becomes time varying: it depends on the current and future marginal utility of wealth, price
level, and aggregate demand. With uncertainty, the weighted average representation of (16)
would need to be augmented by a covariance term.
Backward-looking pricing : The second specification we will consider has firms choosing
their relative price as a function of the relative price chosen by firms in the previous period,
and that expected to be chosen in the next period, as well as “output gaps” in the current
and immediate future periods:
∗
ln (Pt
/Pt ) = 0.5 · (qt + γ ln (ct /c)) + 0.5 · Et (qt+1 + γ ln (ct+1 /c)) + zt,
(17)
where
qt
=05
.
·
∗
ln (Pt
/Pt )
+ 0.5 · ln
�
Pt∗−1/Pt−1
�
,
(18)
and c is the steady state level of consumption. This specification is close to that in Fuhrer
and Moore (1995), although the term z — an increasing function of the real wage — is absent
from that paper and others using the same general specification. This term is necessary in the
t
context of the optimizing model used here, as without it the model would not be completely
specified.7 The important feature of (17) is that the relative price chosen by adjusting firms
in the current period depends directly on the relative price chosen by adjusting firms in the
previous period. This feature has been useful in generating aggregate price dynamics similar
11
to those in U.S. data. However, there has thus far been little success in deriving an equation
like (17) from optimal firm behavior.
3.3
Monetary and Fiscal Policy
The government issues non-interest bearing currency and interest bearing discount bonds,
imposes lump sum taxes or distributes lump sum transfers, and purchases some amount of the
same basket of differentiated products that individuals purchase. It is also natural to consider
the nominal interest rate as one of the policy variables. As Woodford (1995) explains, it is
not feasible for the government to choose each of these variables independently: individuals’
optimal choice of money and bond holdings together with the government’s budget constraint
limits to three the number of policy variables the government can independently determine.
We assume that the government fixes the real quantity of interest-bearing debt and the real
quantity of government purchases at constant levels. The quantity of money is endogenously
determined by money demand, given the nominal interest rate implied by the government’s
feedback rule. Lump sum taxes or transfers are determined as a residual, to satisfy the
government’s flow budget constraint:
M + B / (1 + R ) = M −
t
t
t
t
1 + Bt−1 + pt gt
−p τ .
t
(19)
t
The nominal interest rate feedback rules are similar to the ones already discussed, but
allow for a response to real activity as well as prices. They are in the spirit of Taylor (1993).
We consider the following rule that makes the price level trend-stationary,
�
�
R = max R∗ + fp ln P
t
·
t
− ln Pt
�
�
+ fc · (ln ct − ln c) , 0
,
(20)
and a rule that allows for base drift in the price level:
Rt
= max {R∗ + fp · ((Pt/Pt−1) − π∗) + fc · (ln ct − ln c) , 0} .
12
(21)
In these rules, R∗ is a “target” nominal interest rate, which is equal to the steady state
real interest rate plus the targeted inflation rate. In the targeted steady state, the net
nominal interest rate will equal R∗, and the gross inflation rate will equal π∗.8 Rules similar
to the inflation-based rule (21) have been widely argued to approximate actual central bank
behavior. The price-level based rule is a simple alternative, and the reasoning given earlier
suggests that the two rules may lead to quite different implications of the zero bound.
Because money is assumed costless to hold, there is a potential indeterminacy at a zero
nominal interest rate. Individuals demand real balances at least equal to cAφυ , but they
are indifferent among any levels of real balances at least that large. We resolve this indeterminacy by assuming that in states of the world where the nominal interest rate is zero,
the government accommodates nominal money demand at exactly P cAφυ . An alternative
assumption that would lead to virtually identical results would be for the monetary rule to
impose a bound on the nominal interest rate of > 0, where is arbitrarily small.
3.4
Solving the Model
We solve the model using the finite element method (see McGrattan (1996)). This involves
picking a grid of points for the model’s endogenous state variables, and then finding values of
the “control” variables (nonpredetermined variables) numerically for each grid point and each
value of the discrete random forcing variable such that the model’s equations are satisfied.
The method uses interpolation to calculate expectations of future variables, which in general
will not fall on the original grid. The solution consists of functions mapping from the states
to the controls. Those functions can be used in conjunction with the Markov process for the
forcing variable to simulate the model.
One feature of this solution method is that it restricts consideration to equilibria in which
policies are functions of only the economy’s natural state variables. That is, the equilibria
13
we find are Markovian. This description makes it clear that the method cannot be used
to investigate certain types of multiplicity. For example, Benhabib, Schmitt-Grohe and
Uribe (2001) describe equilibria where initial conditions are not uniquely pinned down; our
solutions always involve unique initial conditions. In Section 6 we will return to the work of
Benhabib, Schmitt-Grohe and Uribe.
Depending on the combination of policy regime and pricing specification, and depending
on whether we are considering variation in the preference parameters, the size of the system
of equations needed to solve the model varies between 3 × N and 6 × N1 × N2, where N is the
number of grid points for a case with one endogenous state variable, and N1 and N2 are the
natural analogues for cases with two endogenous state variables. The other element of the
expression (3 or 6) comes from the fact that we can reduce the model to three fundamentally
nonlinear equations in three nonpredetermined variables, plus a law of motion for the state
variable(s). In the case of shocks, the number of equations is doubled to account for the
two-state Markov process.
Bound in a Perfect Foresight Equilibrium
(Transitional Dynamics)
4 The Zero
Concern about the zero bound typically involves unusual economic conditions that one would
not think of modelling using perfect foresight. However, it turns out that our main findings
can in fact be conveyed in an environment without uncertainty, and without variation in
exogenous state variables. Studying the model’s transitional dynamics — its behavior starting
from arbitrary values of the endogenous state variable(s) — reveals the same flavor of results
as studying the response to shocks to exogenous state variables. This is the framework
we will use to illustrate the importance of the policy rule and price-setting in determining
14
whether the zero bound significantly interferes with the real economy. Later we will provide
an example illustrating how similar results are obtained in a stochastic environment. In
each of the cases that follow, the parameters of the policy rule are fp = 1.5 and fc = 0.125.
These correspond to the standard Taylor-rule values, because the real activity argument of
the policy rule is a quarterly measure. We will compare the model’s transitional dynamics
at 5% average annual inflation and 0% average annual inflation.
4.1 A Case Where the Direct Implications Dominate
The direct implications of the zero bound are that if inflation and expected inflation are
taken as given, then the zero bound prevents negative real rates from occurring. Our earlier
discussion suggests that this scenario is most likely to occur in equilibrium if the monetary
authority allows drift in the price level and firms set their prices in a backward-looking
manner. Both of these features impart inertia into the inflation rate.
For the policy rule that makes the price level nonstationary after linear detrending (21),
and the backward-looking price-setting rule (17), Figure 2.A displays equilibrium behavior —
or “policy functions” — for key variables, when the average inflation rate is 5% annually.9 The
x-axis
in each panel is the relative price chosen by firms in the previous period (P ∗−1/P −1).
This is the single state variable needed to solve the model. While consumption, real and
nominal interest rates and expected inflation are the variables of primary interest, the policy
t
t
function for the current relative price (P ∗/P ) and the 45o line allow one to trace out dynamics
for an arbitrary initial condition. The fact that current relative price is increasing with
respect to the lagged relative price, but with a slope less than unity, means that the relative
t
t
price of adjusting firms converges monotonically to its steady state level. The policy function
for expected inflation (panel iii) then implies similar persistence: expected inflation converges
monotonically to its steady state level.
15
Monotonic convergence of inflation suggests that if steady state inflation is zero, then the
zero bound may affect the behavior of real rates (this follows from our discussion of trendstationarity). Our suspicion is confirmed by Figure 2.B, which displays the same policy
functions as Figure 2.A, for an average inflation rate of zero. In states where the nominal
interest rate is zero, the real rate “should be” negative (that is, it would be in the absence
of the zero bound), but in fact the more extreme the initial condition is in demanding a
low real interest rate, the higher the equilibrium real interest rate is. The combination of
backward-looking price-setting and a policy rule that imparts some inertia into inflation
creates an environment in which the direct implications of the zero bound are a reasonable
approximation to the equilibrium implications.
The results illustrated in Figure 2 are consistent with previous work by Fuhrer and Madigan (1997) and Orphanides and Wieland (1998): Fuhrer-Moore type price-setting combined
with a policy rule that stabilizes inflation (not the price level) leads to significant real distortions associated with the zero bound if the average inflation rate is kept near zero. Unlike
the models used by those authors however, our model has a nontrivial demand for money.
That feature does not significantly exacerbate or mitigate the real effects of the zero bound.
4.2
The Role of Policy
Maintaining backward-looking price setting, we now ask whether the zero bound’s direct implications are counteracted significantly by a policy rule that keeps the price level stationary.
Setting the stage, Figure 3.A displays the equilibrium for 5% average inflation, under the
stationary price level rule (20) and backward-looking price setting (17). Both the previous
period’s detrended price level (P −1/ (π∗) −1) and the previous period’s relative price set by
adjusting firms are state variables in this case. Rather than plotting three-dimensional policy functions, the figure plots variables as functions of P ∗−1/P −1, for a value of P −1/ (π∗) −1
t
t
t
16
t
t
t
near the steady state. Plotting the three-dimensional policy functions would not change the
basic message. From panels iii and iv, note that inflation does not converge monotonically
to steady state. Correspondingly, expected inflation is higher when the nominal interest rate
is lower. These features suggest that under zero average inflation, policy rule (20) may be
more effective than (21) in allowing necessary variation in real interest rates. And indeed
Figure 3.B shows this to be the case. There is a slight distortion apparent in the behavior
of consumption and real interest rates under zero average inflation: real rates cannot fall as
low as they would at a higher average inflation rate. However, the distortion is much less
extreme than under the rule allowing base drift, where real rates are prevented from falling
below zero at all.
The intuition for how the stationary price level rule helps to mitigate real effects of the
zero bound is essentially that given in section 2 above, and by Duguay (1994) and Coulombe
(1998): by promising to return the price level to a fixed path, policy automatically generates
expected inflation in situations where the nominal rate hits the zero bound and the real
rate needs to be negative. One could question the relevance of this case, given that central
banks in the industrialized countries generally focus on inflation rather than the price level.
However, this analysis is aimed at generating a better understanding of what factors affect
the zero bound’s real consequences. Our finding about the importance of the policy feedback
rule suggests one rationale for why central banks should focus on the price level rather than
the inflation rate.10
4.3
The Role of Price Setting
Figures 2 and 3 are both based on an ad-hoc, partially backward-looking pricing specification.
The advantage of this specification is that it has been successful in explaining aggregate U.S.
data (Fuhrer and Moore (1995)). However, the natural assumption about price setting in
17
our model is that adjusting firms set a price which maximizes the expected value of their
present discounted profits. Profit maximizing price-setting in this class of model was first
described by Yun (1996). Unlike the alternative, profit maximizing pricing is derived from
optimizing behavior, and as such we can more confidently conduct policy analysis with it.11
Figure 4.A shows the usual set of policy functions for 5% average annual inflation, when
policy keeps the price level stationary (rule (20)) and firms set their prices to maximize
expected present discounted profits. For this specification, the single state variable is the
detrended nominal price set by adjusting firms in the previous period ((P −1/ (π∗) −1). The
policy rule makes expected inflation high when the nominal interest rate is low. Not surprisingly then, we see in Figure 4.B that at zero average inflation, the bound on nominal interest
t
t
rates causes only a small distortion in real variables. In contrast, a rule that allows base
drift in the price level prevents the real rate from going negative under profit maximizing
pricing. Figure 5 shows that in this case, states where the nominal interest rate is zero all
involve the real interest rate being equal to some barely negative number.
Note that it would be incorrect to describe any of the examples here as involving nominal
and real rates being “stuck.” Tracing out the transitional dynamics in Figure 5, for example,
we find that if the nominal interest rate starts at zero it only stays there for one period. The
closest thing to a trap that we observe is in Figure 2, Fuhrer-Moore pricing with base drift
in the price level. There, if the initial condition is significantly below the steady state, then
the economy can experience several consecutive periods of zero nominal interest rates and
high but declining real interest rates.
Comparing the two pricing specifications, conditional on the policy rule, the real effects
of the zero bound are greater under the backward looking pricing scheme. This accords
with the intuition developed in section 2: any factor that contributes to the inflation rate
being persistent will exacerbate the zero bound’s distortionary effect. Recall from above
that all of the results for a zero inflation target would carry over to a world where the price
18
charged by “nonadjusting” firms automatically increased at the steady state inflation rate,
as in Yun (1996). Results for the five percent inflation target would not be identical, and
thus the quantitative real implications of the zero bound would differ under this alternative
assumption. However, to a great extent the real implications are visible from looking at the
zero inflation results in isolation; for example, comparing figures 2.B and 3.B reveals the
importance of a stationary price level in mitigating the zero bound’s real implications. This
sort of comparison would be unchanged with automatic price changes for nonadjusting firms.
5 Incorporating Uncertainty
We claimed earlier that the same flavor of results would be conveyed by analyzing an environment with uncertainty as by studying the model’s pure transitional dynamics. To support
that claim, we present examples in which there is random variation in first ϑ and then χ ;
ϑ is a consumption demand shifter, and χ is a labor supply shifter. The examples involve
profit maximizing pricing with a stationary price level. This case yielded small real effects
t
t
t
t
of the zero bound under certainty, so it is an interesting one to pursue in more detail.
Figure 6 shows policy functions for the two realizations of ϑ , under the assumption that
ϑ follows a symmetric, persistent process with probability 0.2 of exiting the current state.
t
t
The distortion in real rate and consumption behavior that occurs in the region where the
nominal rate is zero is not qualitatively different than for the transitional dynamics alone.
Figure 7 repeats this information for random variation in χ , with similar results. In short,
t
the nature of the shocks does not seem to affect conclusions about the zero bound’s real
implications.
We have not considered a “price shock,” something that affects firms’ pricing decisions
in a way that the monetary authority cannot respond to within the period. One might
suspect that the zero bound would be a more intractable problem in the presence of price
19
shocks. An unusual string of such shocks might keep the economy stuck at the zero bound.
Recall however that whether an economy can “withstand” the zero bound depends crucially
on whether policy can create expected inflation, not whether it can affect the price level
instantaneously. Thus it is not clear that price shocks would affect our results qualitatively.
6 The Zero Bound and Multiple Steady States
Benhabib, Schmitt-Grohe and Uribe (2001) have argued that the zero bound on nominal
interest rates interacts with policy rules similar to (21) to generate both multiple steady
state equilibria and the likelihood that equilibrium paths will converge to a steady state
different from the one targeted by the policy rule. In the models they analyze, there are
no state variables (i.e. predetermined variables), there are only two dynamic variables, and
it is feasible — though nontrivial— to describe the models’ global dynamics. Here there is a
state variable (or two), and there are three dynamic variables. Thus, while we will determine
whether the zero bound leads to the presence of a second steady state, we will not undertake
global analysis of the model’s dynamics.
To determine whether a second steady state equilibrium exists at zero nominal interest
rates, we begin by imposing such a steady state on all of the equations in the model apart
from the monetary policy rule. From (11), this steady state involves an inflation rate of
P /P −1 = β − 1; from (16) and (8) we can calculate the real wage (wzb ), and then (9), (10)
and (12) yield λzb, real balances and, most importantly czb, where a variable with subscript
t
t
zb indicates the value that variable would take on in a steady state with a zero net nominal
interest rate. The final step is to evaluate the policy rule at this candidate steady state; if
the rule returns a nominal interest rate of zero, then there is a second steady state at the
zero bound.
20
For the stationary price level rule, a second steady state at the zero bound exists if
R∗ + fp ·
�
ln
�
P0 · β t
�
− ln
�
P0 · (π ∗)t
��
+ fc · (ln czb
− ln
c) <
0.
(22)
In this expression, P0 is the initial price level, and c is the steady state level of consumption in
the targeted steady state, where the gross inflation rate π = π∗. Note that R∗ = (π∗/β ) − 1.
Condition (22) holds if
ln czb
< ln c − (1/fc ) · ((π∗ /β ) − 1) + t · [(fp /fc ) · (ln π∗ − ln β )] .
If (fp/fc) > 0, then for large enough t this condition will hold, because by assumption β < π∗.
Thus, for the stationary price level rule, there exists a second steady state with zero nominal
interest rates for the parameterization we considered.
For the rule that allows for base drift in the price level, a steady state at the zero bound
exists if
(23)
R∗ + fp · (β − π ∗) + fc · (ln czb − ln c) < 0.
Note that czb and c are independent of the policy rule parameters fp and fc . For fp large
enough then, condition (23) holds because β < π∗. If fc = 0, then a second steady state
exists if fp > 1/β. If fc > 0, then fp needs to be higher in order for there to be a steady
state equilibrium, because czb is generally higher than c. For the values of fp and fc that we
chose, there does exist a steady state equilibrium with zero nominal interest rates.
Knowing that multiple steady states exist is only a first step toward understanding their
implications for the model’s dynamics. It is beyond the scope of the current paper to analyze
global dynamics in the models used here. However, as explained by Eggertsson and Woodford
(2003), a generalization of the monetary policy rules we use would eliminate the zero nominal
interest rate steady state as an equilibrium. The generalization involves a commitment by the
monetary authority not to let the nominal stock of money decline when the nominal interest
rate is zero. This is a modification of our assumption at the end of section 3.3; instead of
21
putting agents exactly at their satiation point for real balances when R = 0, the monetary
authority supplies additional money. With the real quantity of interest-bearing government
debt fixed, in a zero nominal interest rate steady state the ever-increasing quantity of real
balances would violate the agent’s transversality condition for nominal assets.
Our results on nonlinear dynamics presented above show that, even when a nontargeted
steady state exists at the zero bound, it is not inevitable that the dynamics will lead to that
steady state. McCallum (2001) makes a stronger claim, that the deflationary spiral equilibria
of Benhabib, Schmitt-Grohe and Uribe (2001) are implausible, because they are not leastsquares learnable. Thus, even without a commitment not to let the money stock decline,
there is reason to believe that zero nominal interest rate steady states may be unlikely. We
should note however, that the model here differs somewhat from that studied by McCallum,
and the nonlinearity makes it less straightforward to study learnability.
7 Concluding Remarks
The economy’s underlying dynamics (the nature of price-setting) interact with the policy
rule to determine whether the zero bound has significant real effects in equilibrium. With
nominal rigidities, the policy rule is an important factor in determining equilibrium dynamics
(see Dotsey (1999)). In choosing a policy rule, policymakers should be well-informed about
what kinds of rules enable the economy to function normally when inflation is low enough
that the zero bound might be encountered. Intuitively, the crucial feature is that a rule
should generate positive expected inflation in situations where the nominal interest rate is
zero and the real interest rate needs to be negative.12 While we have found that price level
stabilization rules such as (20) have this feature, there are undoubtedly other rules which
do not result in a stationary price level but nonetheless generate the appropriate temporary
expected inflation. Reifschneider and Williams (1999) show that in the Federal Reserve
22
Board’s FRB/US model, rules which are close to (21), but deviate downward in periods
when rates have been zero in the recent past, help to mitigate real implications of the zero
bound.
Any discussion of “good” policy naturally leads one to think about optimal policy. King
and Wolman (1998) show that in the model discussed here, if there is no distortion associated
with money demand, then optimal monetary policy involves keeping the price level constant,
so that the nominal interest rate moves identically with the real interest rate. Their analysis
does not take into account the zero bound, and this omission matters if the real interest
rate ever needs to fall below zero. In the model used here, which includes a money demand
distortion, optimal policy will almost certainly be affected by the zero bound: abstracting
from price stickiness, optimal policy would involve a zero nominal interest rate in every
period. This is the Bailey (1956) and Friedman (1969) welfare cost of inflation argument.
With both price stickiness and money demand, there is a tug of war between zero inflation,
which eliminates the sticky price distortion, and a zero nominal interest rate, which eliminates
the money demand distortion.13 The steady state welfare costs of small inflation or deflation
associated with sticky prices tend to be small, so the optimal policy might well occasionally
involve zero nominal interest rates. Khan, King and Wolman (forthcoming) describe optimal
policy in a model close to the one in this paper — the money demand specification is different
— but their analysis does not address the zero bound issue because they approximate the
dynamics under optimal policy with a linear system.
Wolman (1998) uses the model in the current paper to conduct a limited welfare analysis
that explicitly deals with the zero bound. He compares moderate inflation and moderate
deflation under a rule that keeps the price level trend-stationary, and finds that the latter
generates higher welfare even though the zero bound is encountered regularly. We have
conducted similar analyses for the other cases in this paper (policy rule and price-setting
specification) and found the same qualitative results. That is, even though low inflation
23
targets exacerbate the frictions associated with hitting the zero bound, these costs are outweighed by the benefits of zero nominal interest rates associated with eliminating money
demand frictions. Clearly, the specification of money demand is important for these results.
Eggertsson and Woodford (2003) take into account the zero bound in their analysis of
optimal monetary policy in a Calvo staggered pricing model. Their findings reinforce the
message that price-level targeting has benefits from the standpoint of the zero bound on
nominal interest rates. Optimal policy is not exactly price-level targeting, but they show
that a simple price-level targeting rule approximates optimal policy much more closely than a
simple inflation targeting rule that would achieve identical outcomes absent the zero bound.
It would be worthwhile to study optimal policy in our model, which contains nonlinearities
and money demand distortions absent from Eggertsson and Woodford’s analysis. This would
be feasible using the same nonlinear method we applied. Policy would be derived from an
optimization problem rather than an exogenous rule, but this does not present significant
computational obstacles. We chose to focus on a comparison of two simple rules rather than
to compute optimal policy. These simple rules represent popular prescriptions for monetary
policy, and it is important that policymakers understand how such rules behave in the face
of the zero bound on nominal rates.
24
8 Footnotes
1. The fact that the zero bound involves a nonlinearity makes computation more difficult,
and may have delayed work in this area.
2. In general, the equality is not exact because of covariance between inflation and the
growth rate of the marginal utility of consumption. See for example Sarte [1998].
3. The basic idea in this section is already present in Duguay [1994] and Coulombe [1998].
4. We interpret the model as describing quarterly data. This value of β thus implies a
steady state real interest rate of roughly 2% per annum.
5. These parameters were estimated using U.S. data on M1.
6. The transactions facilitating benefit is given by wPλ · h (·)( c1 ), and the foregone interest
t t
�
t
t
− β E Pλ +1+1 (see (11)). A conventional money demand equation can be derived
by combining (11)-(12): mt/ct = A (φ + (R / (1 + R )) (c /w ))ν .
cost is
λt
Pt
t
t
t
·
t
t
·
t
t
7. To see this, note that in steady state, (17) collapses to 0 = 0 without zt. We specify zt
so that in steady state (17) is equivalent to (16):
zt = kf · (wt − w) ,
where w is the steady state real wage implied by profit maximizing price-setting, and
kf
is a positive constant.
8. Note that in contrast to the earlier discussion, we now allow the average inflation rate
to be non-zero, for both the trend-stationary and nonstationary cases. That is, π∗ may
not be equal to one, and P may be growing at a constant rate π∗.
t
9. The figures show net quarterly interest rates and gross quarterly inflation rates.
25
10. For a different argument in favor of price level targeting, see Svensson [1999].
11. This criticism is especially relevant with respect to the zero bound on nominal interest
rates, as the U.S. economy has not approached the zero bound during the sample in
which Fuhrer-Moore price-setting specifications have been estimated.
12. Mishkin [1996] makes this point in arguing that monetary policy can work through
expanding the money supply when the nominal interest rate is zero.
13. Uhlig [2000] highlights the conflicting nature of conventional views about zero nominal
interest rates. On one hand there is the optimality of the Friedman rule, while on the
other there is the danger of a deflationary trap.
26
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30
Figure 1.A
Stabilizing Inflation
1
0.9
0.8
inflation in the long run
0.7
0.6
*
0.5
expected increase in inflation
0.4
0.3
current period inflation
0.2
0.1
0
0.10
1.10
2.10
3.10
4.10
5.10
time
6.10
7.10
8.10
9.10
Figure 1.B
Stabilizing the price level
1
0.9
0.8
price level in the long run
0.7
0.6
0.5
expected inflation
*
0.4
current period price level
0.3
0.2
0.1
0
1
11
21
31
41
51
time
61
71
81
91
Figure 2.A. 5% inflation, backward pricing, nonstationary price level
Figure 2.B. Zero inflation, backward pricing, nonstationary price level
Figure 3.A. 5% inflation, backward pricing, stationary price level
Figure 3.B. Zero inflation, backward pricing, stationary price level
Figure 4.A. 5% inflation, optimal pricing, stationary price level
Figure 4.B. Zero inflation, optimal pricing, stationary price level
Figure 5. Zero inflation, optimal pricing, nonstationary price level
Figure 6. Random theta, zero inflation, optimal pricing, stationary price level
Figure 7. Random chi, zero inflation, optimal pricing, stationary price level
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