This study examines the expectational stability of the rational expectation equilibria(REE) under Taylor rules when trend inflation is non-zero. We find that whether or not a higher (lower) trend inflation makes the REE more (less) unstable depends largely on the data (such as contemporaneous data, forecasts and lagged data) used in the conduct of monetary policy.adaptive learning, E-stability, Taylor rule, trend inflation

Kobayashi, Teruyoshi

Muto, Ichiro

English

Research Papers in Economics

Expectational stability under non-zero trend inﬂation∗
Teruyoshi Kobayashi† Ichiro Muto‡
September 3, 2009
Abstract
This study examines the expectational stability of the rational expectation equilib-
ria (REE) under Taylor rules when trend inﬂation is non-zero. We ﬁnd that whether
or not a higher (lower) trend inﬂation makes the REE more (less) unstable depends
largely on the data (such as contemporaneous data, forecasts and lagged data) used in
the conduct of monetary policy.
JEL Classiﬁcation: D84, E31, E52
Keywords: Adaptive learning, E-stability, Taylor rule, trend inﬂation
∗The views expressed here are those of the authors and do not necessarily reﬂect the oﬃcial view of the
Bank of Japan.
†Corresponding author. Department of Economics, Chukyo University, 101-2 Yagoto-honmachi, Showa-
ku, Nagoya 466-8666, Japan. E-mail: kteru@mecl.chukyo-u.ac.jp. Tel, fax: +81-52-835-7943.
‡Bank of Japan, 2-1-1 Hongokucho Nihonbashi Tokyo, 103-8660, Japan.
E-mail: ichirou.mutou@boj.or.jp. Tel: 81-3-3277-1045, Fax: +81-3-5255-7455.1I n t r o d u c t i o n
Many of the New Keynesian analyses have virtually neglected the existence of non-zero
trend inﬂation. However, some point out that non-zero trend inﬂation greatly alters the
implications for monetary policy making. Among them, Kiley (2007) and Ascsari and
Ropele (2009) show that, under Taylor rules, the parameter region that satisﬁes the de-
terminacy of the rational expectation equilibria (REE) substantially depends on the level
of trend inﬂation. Since the determinacy is viewed as a normative requirement for mone-
tary policy rules, their results suggest that the central bank should choose the policy rule
parameters by correctly recognizing the relationship between the determinacy and level of
trend inﬂation.
However, REE determinacy is not the sole requirement for the monetary rules. In the
literature of adaptive learning, which is based on the framework of Evans and Honkapohja
(2001), Bullard and Mitra (2002) propose the expectational stability (E-stability) of REE
as another requirement for monetary policy rules. They compute the parameter regions
that satisfy the E-stability as well as the determinacy of REE under alternative versions
of Taylor rules (such as contemporaneous, forecast-based and lagged-based rules). Their
analysis provides a benchmark in the literature, but their study focuses on a relatively
speciﬁc environment in which the trend inﬂation is exactly equal to zero. An important
extension is to generalize their work introducing non-zero trend inﬂation.
In this study, we examine the E-stability (as well as the determinacy) of REE in a
New Keynesian model under non-zero trend inﬂation. In doing so, we pay attention not
only to positive, but also to negative trend inﬂation. Although the existing studies have
not examined the E-stability and determinacy of REE under negative trend inﬂation, we
consider this issue an important one for recent monetary policy making because deﬂation
has become a more pressing concern in the major developed countries. We show how the
parameter combination of the Taylor rule coeﬃcients that guarantee both E-stability and
REE determinacy varies with the level of trend inﬂation.
2 The model
2.1 A New Keynesian model under non-zero trend inﬂation
Some previous studies, such as Kiley (2007), Sbordone (2007), Cogley and Sbordone (2008)
and Ascari and Ropele (2007, 2009), provide alternative expressions of New Keynesian
models under non-zero trend inﬂation. Among them, we employ the model of Sbordone
(2007) and Cogley and Sbordone (2008), which is given as follows:
yt = Etyt+1 − σ(it − Etπt+1 − rn
t ), (1)
1πt = κyt + b1Etπt+1 + b2
∞ X
j=2
φ
j−1
1 Etπt+j, (2)
rn
t = ρrrn
t−1 + εt. (3)
πt is the percentage deviation of inﬂation from the (possibly non-zero) rate of trend in-
ﬂation, which is assumed to be constant. yt, it and rn
t are the output gap, the nominal
interest rate and the natural rate of real interest, respectively.
Since ﬁrms take into account the inﬂuence of trend inﬂation on their future relative
prices, they are more forward-looking under non-zero trend inﬂation than under zero trend
inﬂation. As a result, the third term of (2), which is absent under zero trend inﬂation,
arises and thus the parameters such as κ, b1, b2 and φ1 are aﬀected by the level of trend
inﬂation.1 To see how non-zero trend inﬂation inﬂuences ﬁrms’ forward-lookingness, it
would be useful to check the values of b1 and b2 for alternative levels of trend inﬂation.
Under our parameter values, (b1,b 2) takes (.968,−.009), (.99,0) and (1.033,.017) for the
rate of (annualized) trend inﬂation -1%, 0%, 2%, respectively. Thus, the sum of coeﬃcients
on inﬂation expectations increases with the level of trend inﬂation. We also ﬁnd that the
sign of b2 is positive for ¯ Π > 1 and negative for ¯ Π < 1. This implies that the “additional
forward-lookingness” stemming from the presence of non-zero trend inﬂation works in the
opposite direction, depending on whether the trend inﬂation is positive or negative.
As for monetary policy rules, we introduce some versions of Taylor rules in which the
central bank responds to (i) the contemporaneous data (yt, πt), (ii) the forecast (Etyt+1,
Etπt+1), and (iii) the lagged data (yt−1, πt−1). The policy rule is generically given as
it = FlYt−1 + FcYt + FfEtYt+1 (4)
where Yt =[ yt πt]0, Fi =[ Fiy Fiπ]f o ri = c,f,l. c, f and l represent the contemporaneous
rule, the forecast-based rule, and the lagged-based rule, respectively.
2.2 Adaptive learning
We assume that agents estimate the structural parameters by recursive least squares with
decreasing gain, which is the most standard algorithm of adaptive learning (Evans and
Honkapohja, 2001). The perceived law of motion (PLM) is generally given as
Yt = At + CtYt−1 + Dtrn
t , (5)
1Cogley and Sbordone (2008) derive the parameters as follows: φ1 = αβΠ
(θ−1)
, φ2 = αβΠ
θ(1+ω)
,
χ =
1−αΠ
(θ−1)
α(1+θω)Π
(θ−1),b 1 =( 1+( 1+ω)θχ)φ2 − (θ − 1)χφ1,b 2 =( θ − 1)χ(φ2 − φ1), κ = χ(1 − φ2), where α
is the probability of not changing prices, β is the discount factor, θ is the elasticity of substitution among
diﬀerent goods, ω is the responsiveness of real marginal cost to output, and Π is the trend inﬂa t i o ni ng r o s s
term. The parameter values used in our numerical exercises follow those of Cogley and Sbordone (2008):
α = .588, θ =9 .8a n dω = .429. β and σ are set at .99 and 6.25, respectively.
2where ai,t and di,t are the i-th element of At and Dt, respectively, and cij,t denotes the
ij-th element of Ct. It follows that
EtYt+1 = At + CtYt + Dtρrrn
t
EtYt+2 =( I + Ct)At + C2
t Yt +( CtDtρr + Dtρ2
r)rn
t
EtYt+3 =( I + Ct + C2
t )At + C3
t Yt +( C2
t Dtρr + CtDtρ2
r + Dtρ3
r)rn
t ,
. . .
To eliminate the expectation terms from (2), we ﬁrst express the inﬁnite summation term
P∞
j=2 φ
j−1
1 EtYt+j as a function of Yt and rn
t :
∞ X
j=2
φ
j−1
1 EtYt+j =( 1 − φ1)−1φ1At +( 1− φ1)−1(I − φ1Ct)φ1CtAt +( I − φ1Ct)−1φ1C2
t Yt
+(I − φ1Ct)−1[φ1ρrCtDt +( 1− φ1ρr)−1φ1ρ2
rDt)]rn
t .
The second element of this vector can be written as
∞ X
j=2
φ
j−1
1 Etπt+j = a3,t + c31,tyt + c32,tπt + d3,trn
t (6)
By inserting (6) into (2), we obtain the following actual law of motion (ALM):
QtYt = SEtYt+1 + Nit + Utrn
t + Pt, (7)
where
Qt =
"
10
−κ − b2c31,t 1 − b2c32,t
#
,S=
"
1 σ
0 b1
#
,N =
"
−σ
0
#
,
Ut =
"
σ
b2d32,t +1
#
,P t =
"
0
b2a3,t
#
.
It follows that
Yt = Γ−1
t [(S + NFf)At + Pt + NFlYt−1 +( ( S + NFf)ρrDt + Ut)rn
t ], (8)
where Γt ≡ Qt − NFc − (S + NFf)Ct. T h eT - m a p sf r o mt h eP L Mt ot h eA L Ma r et h e n
given as
T(At)=Γ−1
t [(S + NFf)At + Pt], (9)
T(Ct)=Γ−1
t NFl, (10)
T(Dt)=Γ−1
t [(S + NFf)ρrDt + Ut]. (11)
Here, let DT(A,C,D) be the Jacobian matrix of the T-maps evaluated at the corresponding
RE values:
DT(A,C,D)=
∂vec(T(A),T(C),T(D))
∂(vec(A,C,D))0 .
Then the E-stability of REE can be attained if and only if all of the eigenvalues of DT
have real parts less than one.
33 The E-stability under non-zero trend inﬂation
The combinations of the Taylor rule coeﬃcients, Fiπ and Fiy, i = c,f,l, that ensure the
E-stability and the determinacy of REE are presented in Figures 1, 2 and 3. In all ﬁgures,
the upper-right panel corresponds to the case of zero trend inﬂation, which is equivalent to
the situation analyzed by Bullard and Mitra (2002). The other panels show the E-stable
and the determinate regions under non-zero trend inﬂation. Although the determinate
regions (except for the case of negative trend inﬂation) are essentially the same as those
presented by Ascari and Ropele (2009), the E-stable regions are novel in our study.
Our main ﬁnding is that the relationship between the E-stability of REE and the
level of trend inﬂation largely depends on the speciﬁcation of Taylor rule. Under the
contemporaneous rule, the E-stable region, which is exactly the same as the determinate
region, shrinks as the rate of trend inﬂation increases. Under the forecast-based rule, the
E-stable region is broader than the determinate region. Nevertheless, the E-stable region
is just the same as it is in the case of the contemporaneous rule, so that the implications
about the E-stability of REE are the same for these two rules: higher trend inﬂation
makes the REE more unstable. However, this is not necessarily true for the lagged-based
rule because the relationship between the E-stability and level of trend inﬂation is more
complex. When the responsiveness to output gap (Fly)i sb e l o ws o m et h r e s h o l dv a l u e
(around 0.3 in our numerical example), then the higher trend inﬂation makes the REE more
unstable. On the other hand, when Fly is larger than the threshold, the E-stable region
becomes broader for higher values of trend inﬂation. This ﬁnding is parallel to Ascari and
Ropele’s (2009) result that a higher trend inﬂation always narrows the determinate region
under the contemporaneous rule and forecast-based rule but either narrows or broadens
the determinate region under the lagged-based rule, depending on the value of Fly.
Next, let us check the case of negative trend inﬂation. As in the cases of positive
trend inﬂation, we ﬁnd that whether or not a higher trend inﬂation makes the REE more
likely to be E-stable and determinate depends on the versions of Taylor rules. Under the
contemporaneous rule, the E-stable region, which is the same as the determinate region,
is quite large under negative trend inﬂation. However, things are diﬀerent in the cases of
the forecast-based rule and the lagged-based rule. When using the forecast-based rule, the
E-stable region is broad, but the determinate region is quite narrow under negative trend
inﬂation. As a result, the REE under negative trend inﬂation is likely to be E-stable but
indeterminate for a wide range of policy parameters. As for the lagged-based rule, the
region in which both the E-stability and determinacy are satisﬁed is quite narrow, and the
REE is likely to be explosive for wide combinations of the Taylor rule coeﬃcients.
Therefore, when the trend inﬂation is negative, the central bank can more easily guar-
antee both the E-stability and the determinacy of REE by adopting the contemporaneous
rule instead of the other two rules. Although the existing studies have not reported this
4point, we consider this property to have an important implication for modern monetary
policy making because, in an environment of low trend inﬂation, the degree of freedom for
the central bank to control the nominal interest rate is inevitably small due to the presence
of the zero lower bound (ZLB) of the nominal interest rate. Our result suggests that as
long as the central bank adopts the contemporaneous rule, the REE is likely to be E-stable
and determinate under negative trend inﬂation even if the coeﬃcients of the Taylor rule
are relatively small. It means that the necessity of cutting interest rates against downward
shocks is removed to some extent when trend inﬂation is negative. This will allay the fear
of ZLB that the central banks have in an era of very low inﬂation.
4C o n c l u d i n g r e m a r k s
We have shown that the relationship between the E-stability of REE and the level of trend
inﬂation depends largely on the data used in the conduct of monetary policy. The REE
is more likely to be both E-stable and determinate under the contemporaneous rule than
under the other two alternative rules when the trend inﬂa t i o ni sv e r yl o w . T h i sr e s u l t
implies that the availability of current economic data for the central bank is especially
important in a low inﬂation environment.
Acknowledgements
We would like to thank Klaus Adam for comments and suggestions. Kobayashi acknowl-
edges ﬁnancial support from KAKENHI 21243027.
References
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[2] Ascari, G. and T. Ropele, 2007, Optimal monetary policy under low trend inﬂation,
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[3] Bullard, J. and K. Mitra, 2002, Learning about monetary policy rules, Journal of
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[4] Cogley, T. and A.M. Sbordone, 2008, Trend inﬂation, indexation, and inﬂation persis-
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6Figure 1: The E-stability and determinacy regions under the contemporaneous rule
Figure 2: The E-stability and determinacy regions under the forecast-based rule
7Figure 3: The E-stability and determinacy regions under the lagged-based rule
8

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Optimal monetary policy under low trend inﬂation,

Trend inﬂation, indexation, and inﬂation persistence in the new Keynesian Phillips curve,

Trend inﬂation, Taylor principle and indeterminacy, Economic Working Papers no. 708, Economic Research Department, Bank of Italy (forthcoming in

Expectational stability under non-zero trend inflation

Expectational stability under non-zero trend inflation

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