Discussion Paper No.261
Numerical Analysis of Three Time Delays in Monetary
Policy: The Case of a Sticky-Price Model

Eiji Tsuzuki
Chiba Keizai University
Futoshi Kurokawa
Chiba Keizai University
Shunsuke Shinagawa Waseda University

February 2016

INSTITUTE OF ECONOMIC RESEARCH
Chuo University
Tokyo, Japan

Numerical Analysis of Three Time Delays in Monetary
Policy: The Case of a Sticky-Price Model
Eiji Tsuzuki,∗ Futoshi Kurokawa,† and Shunsuke Shinagawa‡

Abstract
In this study, we develop a New Keynesian model that includes the policy rulewith which the nominal interest rate’s responses are induced according to fluctuations in three economic variables, namely output, the inflation rate, and asset prices.
In this model, we also assume that there is a time lag in the interest rate’s responseto each variable. The model economy is represented as a “diﬀerential equation system with three delays.” For the determinacy analysis, we use the numerical method
developed by Gu and Naghnaeian [K. Gu and M. Naghnaeian, Stability crossing set
for systems with three delays, IEEE Transactions on Automatic Control 56 (2011),
pp. 11–26] to find the parameter regions that achieve local determinacy in order toexamine the eﬀects of the three policy lags on local equilibrium determinacy. This is
the first such application of this method to New Keynesian economics. We demonstrate that implementations of monetary policy should be “purposefully” delayed toachieve local equilibrium determinacy. Hence, the central bank should determine its
target variables by considering not only the responsiveness of the nominal interest
rate to output, the inflation rate, and asset prices but also the lag lengths associatedwith policy implementations.

JEL Classification Codes: E32; E52
Keywords: New Keynesian model, policy lag, interest rate rule, equilibrium determinacy, delay-diﬀerential equation∗

Corresponding author. Faculty of Economics, Chiba Keizai University, Chiba 263-0021, Japan; Tel.:+81-43-253-9745; Fax: +81-43-254-6600; E-mail: tsuzukie5@gmail.com†
Faculty of Economics, Chiba Keizai University, Chiba 263-0021, Japan‡
Faculty of Political Science and Economics, Waseda University, Tokyo 169-8050, Japan1

1

Introduction

Optimizing models that consider price stickiness are referred to as New Keynesian (NK)models. The standard NK model comprises three equations: an Euler equation, Phillips
curve, and monetary policy rule.1 Benhabib et al. (2003) assume the backward-looking
monetary policy rule wherein the nominal interest rate responds to the weighted mean ofpast inflation rates (i.e., lagged inflation rates).
Generally, time lags can be classified into two types, namely distributed lag models
and fixed lag models, of which Benhabib et al.’s (2003) model can be considered to be
the former. In this study, we develop an NK model that includes fixed time lags. Thisis, we assume that the nominal interest rate responds to target variables evaluated atfixed points in time. Conceptually, fixed lag models can be regarded as limiting cases of
distributed lag models wherein the weighting factor for each target variable is unity at a
certain point in time. Algebraically, however, it is impossible to treat fixed lags as special
cases of distributed lags. Accordingly, spatial methods must be used.
In the simplest case where the system includes only one fixed lag, its characteristic
function can generally be expressed as follows: ∆(λ) = p0 (λ)+p1 (λ)e−λτ , where λ is a root,τ > 0 is the lag length, and p0 (λ) and p1 (λ) are polynomials of λ. Owing to the existenceof the exponential function e−λτ , the equation ∆(λ) = 0 includes an infinite number ofroots. Bellman and Cooke (1963), among others, have characterized the solutions of thistype of function.
Recently, more complex configurations of characteristic functions have been examined:• Beretta and Kuang (2002): lag-dependent parameters; ∆(λ) = p0 (λ, τ )+p1 (λ, τ )e−λτ .
• Gu et al. (2005): two lags; ∆(λ) = p0 (λ) + p1 (λ)e−λτ1 + p2 (λ)e−λτ2 .
• Deng et al. (2006): time-dependent lag; ∆(λ) = p0 (λ) + p1 (λ)e−λτ (t) .
• Gu and Naghnaeian (2011): three lags; ∆(λ) = p0 (λ) + p1 (λ)e−λτ1 + p2 (λ)e−λτ2 +p3 (λ)e−λτ3 .
• Lin and Wang (2012): two lags (where one of the exponential functions includes thesum of lags); ∆(λ) = p0 (λ) + p1 (λ)e−λτ1 + p2 (λ)e−λτ2 + p3 (λ)e−λ(τ1 +τ2 ) .
1

Introductory textbooks for NK economics are presented by Woodford (2003), Walsh (2010), and Galı́
(2015).

2

In economics (especially in the context of NK economics), however, few works have examined the eﬀects of fixed time lags,2 especially compared with the many applications ofthese mathematical studies in biology.
Tsuzuki (2014, 2015) develops NK models that include only one fixed lag in the monetary policy rule. Here, the nominal interest rate is assumed to change in response tofluctuations in the inflation rate with a time lag, which represents an inflation targeting
policy with delay. Tsuzuki’s models can thus be considered to be fixed lag versions ofBenhabib et al.’s (2003) model. According to Tsuzuki (2014, 2015), an increase in thelag time necessarily increases the number of roots with positive real parts. Accordingly,a policy lag may resolve the problem of equilibrium indeterminacy. However, in a systemwith two fixed lags, as in the model of Tsuzuki et al. (2015) that includes monetary andfiscal policy lags, an increase in the lag may raise the number of roots with negative realparts. This finding implies that a policy lag may resolve the problem of instability rather
than that of indeterminacy.
In the present study, we consider the coexistence of three fixed lags in monetary policy
responses. The model economic system is developed based on the model of Carlstrom andFuerst (2007). In their model, the central bank was assumed to manipulate the nominalinterest rate according to variations in both the inflation rate and asset prices. We add
output as the third variable of the policy rule in accordance with the formulation in
Bullard and Mitra (2002).
In the simplest NK model that does not include a time lag and in which the nominalinterest rate responds only to the inflation rate, the necessary and suﬃcient condition
for achieving local equilibrium determinacy is that the nominal interest rate increases bymore than one unit when a one-unit increase in the inflation rate occurs; in other words,a monetary policy must be “active.” This is the well-known policy norm referred to as
the “Taylor principle.” However, some notable studies have demonstrated that even if
the Taylor principle is not satisfied, equilibrium determinacy can be achieved. Meng andYip (2004), Bilbiie (2008), and Gliksberg (2009) show that equilibrium determinacy canbe achieved when endogenous investment, limited asset market participation, and the2

In economics, Guerrini and Sodini (2013) and Tsuzuki and Shinagawa (2015) apply Beretta andKuang’s (2002) method, while Matsumoto and Szidarovszky (2015b) apply Lin and Wang’s (2012)
method. In addition, Matsumoto and Szidarovszky (2015a) present an application of Gu et al.’s (2005)
method.

3

existence of capital adjustment costs, respectively, are assumed.3 Furthermore, Bullard
and Mitra (2002) and Carlstrom and Fuerst (2007) demonstrate that in discrete-timesticky-price models, an increase in the responsiveness of the nominal interest rate to boththe inflation rate and output increases the possibility of the Taylor principle holding,whereas an increase in the responsiveness of the nominal interest rate to asset prices
decreases this possibility.
In this study, we investigate the eﬀects of monetary policy lags (i.e., the interaction
between multiple policy lags) on the condition for local equilibrium determinacy. First,
we consider the case where no policy lags are present. We demonstrate that the responsesof the nominal interest rate to both the inflation rate and output increase the possibility ofequilibrium determinacy, whereas responses to asset prices decrease that possibility, in linewith the results shown by Carlstrom and Fuerst (2007) and Bullard and Mitra (2002).
Next, we introduce three time lags to the monetary policy rule. For the determinacy
analysis, we use the numerical method developed by Gu and Naghnaeian (2011) to findthe parameter regions that achieve local determinacy. This is the first such application ofthis method to NK economics.
An extensional part of the discussion about the rights and wrongs of the central bank’s
response to fluctuations in asset prices pertains to the diﬀerence between the views of theFederal Reserve and the Bank for International Settlements. The Fed’s view, which derives
from a monetary policy perspective, is that the central bank should be dedicated to thestabilization of commodity prices and that it does not need to respond to fluctuations inasset prices. Bernanke and Gertler (2001) provide a theoretical ground with this view.
Conversely, the Bank for International Settlements’ view advocates a need for vigorous
austerity measures when confronted with excessive increases in asset prices. Carlstrom
and Fuerst (2007) establish the validity of the Fed’s view in the presence of sticky prices.
This is also confirmed in our model without a lag. However, if positive policy lags exist,
how would this result change, if at all? The present study provides an answer to thisquestion.
The remainder of this paper is organized as follows. Section 2 presents a diﬀerential
equation system that describes the dynamics of the model economy. Section 3 examines
the case with no policy lags. Section 4 examines the case with three policy lags. Section
3

Buﬃe (2013) demonstrates that Bilbiie’s (2008) result strongly depends on the assumption of real
wage flexibility. If the real wage rate is suﬃciently sticky, the Taylor principle reasserts itself as thecondition for determinacy.

45 concludes.

2

The model

In this section, we present the standard NK model. The model economy is constructed byfirms, households, and the public sector. Each firm produces diﬀerentiated goods by using
workers under monopolistic competition. Likewise, each household supplies diﬀerentiatedworkers to firms under monopolistic competition and consumes goods.

2.1

Intratemporal optimization of firms

Various types of diﬀerentiated workers indexed by j (j ∈ [0, 1]) exist. Firms first aggregate
the diﬀerentiated workers lj via the Dixit–Stiglitz function as follows:η[∫ 1 η−1 ] η−1
η
lj dj
,
(1)
l=
0

where l is the composite labor and η > 1 is the elasticity of substitution among workers.
The first-order condition for cost minimization yields the demand function for worker
lj as follows:4
( )−η
Wj
lj =
l,
(2)
W
where Wj is the nominal wage rate of worker lj and W is the nominal wage rate of the∫1
1
whole economy, defined as W = [ 0 Wj1−η dj] 1−η .

2.2

Intratemporal optimization of households

Various types of diﬀerentiated consumption goods indexed by i (i ∈ [0, 1]) exist. Households first aggregate their diﬀerentiated goods and then consume them as a composite
good. As in the previous section, we express the aggregation of goods as the Dixit–Stiglitz∫ 1 ϕ−1ϕfunction: y = [ 0 yi ϕ di] ϕ−1 , where y is the amount of the composite good and ϕ > 1 isthe elasticity of substitution among goods. The first-order condition for cost minimization
yields the demand function for good i as follows:
( )−ϕ
pi
y,
(3)
yi =
p
4

See Blanchard and Kiyotaki (1987).

5

∫1
1
where pi is the price of good i and p is the price level, represented as p = [ 0 p1−ϕdi] 1−ϕ .
i

2.3Intertemporal optimization of firms

We assume a linear technology and specify the production function of good i as follows:yi = li ,(4)where yi is the output of good i and li is the amount of composite labor used to produce
good i.
Considering the constraints expressed in Equations (3) and (4), firm i solves the profit
maximization problem as follows:
] ∫
∫ ∞[
t
pi yi − W li γ
∗ 2
M aximizeπi− (πi − π ) y e− 0 r(s)ds dt,
p20
subject to ṗi = πi pi ,where πi = ṗi /pi is the price change rate of good i, r is the real interest rate, and π ∗ isthe steady-state value of the inflation rate. In addition, γ2 (πi − π ∗ )2 y represents a pricerevision cost. Owing to the existence of this cost, the price becomes sticky. Hence, γ > 0
can be interpreted as a parameter that reflects the price stickiness measure; the larger
the value of γ, the greater the stickiness. Here, we formulate the price revision cost ina quadratic function consistent with that outlined by Rotemberg (1982). Moreover, forsimplicity, we assume that the price revision cost is a firm’s payment to households that
do not spend on goods. For example, it is considered to be a lump-sum payment toworkers who handle price replacement tasks.
In the following discussion, we examine a “symmetric equilibrium” in which all firms’
behavior is based on the same equations. In this case, we can drop subscript i fromall the variables. Furthermore, as the number of goods is normalized to unity (see the
Dixit–Stiglitz function), the following expressions hold: pi = p, πi = π, and yi = y. By
using these expressions along with the solutions to the optimization problem above, we
can obtainẏ
1−ϕ ϕWπ̇ + (π − π ∗ ) = r(π − π ∗ ) −
−
.
(5)
y
γ
γ p
This equation is the NK Phillips curve. Furthermore, economically significant solutions∫twould also require satisfying the transversality condition expressed as limt→∞ p(t)e− 0 r(s)ds dt =
0.
6

2.4

Intertemporal optimization of households

In every period, household j obtains utility from consumption cj and real money holding
mj , and disutility from labor supply lj . We specify the utility function as follows:u(cj , mj , lj ) ≡ ln cj + ln mj −

lj1+ψ
,
1+ψ

(6)

where ψ > 0 is the disutility elasticity of labor supply.
Households possess assets that comprise money Mj ≡ pmj , bond Bj , and a constantvolume of stock. We standardize the measure of stock to unity. The stock price relative
to the commodity price is denoted as Q and the stock yields dividend D in every period.
Nominal asset level Aj can be expressed as follows: Aj = Mj + Bj + pQ. Assets can beincreased based on income Wj lj , bond interest RBj (where R ≡ r + π is the nominalinterest rate), dividend D, capital gain Q̇, lump-sum income from firms T ≡ γ2 (πi −π ∗ )2 y, and benefits from the government X, whereas assets can be decreased based on
consumption cj . Thus, we obtainȦj = Wj lj + RBj + pD + pQ̇ + pT + pX − pcj .

(7)

By using the non-arbitrage condition, the nominal yields on stock represented as Q̇+D+
Q
π must equal the nominal interest rate for bonds R, i.e.,Q̇ = RQ − D − πQ.

(8)

By using this expression, Equation (7) can be rewritten as follows:ȧj = wj lj + raj + T + X − cj − Rmj − πQ,(9)where aj ≡ Aj /p is the real asset level of household j and wj ≡ Wj /p is the real wage
rate of worker lj .
Considering Equations (2) and (9), household j solves the utility maximization problem as follows:
]
∫ ∞[
lj1+ψ
M aximizecj ,mj ,Wj
ln cj + ln mj −e−ρt dt,1+
ψ0
subject to ȧj = wj lj + raj + T + X − cj − Rmj − πQ,
where ρ > 0 is the subjective discount rate of households.
7

Under the assumption of the symmetry condition, the following equations hold: cj =c, Wj = W , and lj = l. By using these expressions along with the solutions to the
optimization problem above,5 we obtainċ= r − ρ,
c
η
w=
clψ .
η−1

(10)
(11)

Equation (10) is an Euler equation, which is one of the optimal conditions for a Ramseytype utility maximization problem. Equation (11) is a labor supply function. If the labormarket is perfectly competitive, the real wage rate w should equal the marginal disutilityof labor measured in terms of goods clψ .6 However, in the present model, the labormarket is monopolistically competitive. Hence, the real wage rate becomes equal to themarginal disutility of labor multiplied by the markup η/(η − 1) > 1, as in Equation (11).
Furthermore, economically significant solutions would require satisfying the transversality
condition expressed as limt→∞ a(t)e−ρt dt = 0.
Finally, the profits of firms are paid to households in the form of a dividend; therefore,
D=y−

2.5

W
γ
l − (π − π ∗ )2 y.
p
2

(12)

Central bank

The central bank manipulates nominal interest rate R according to fluctuations in output
y, inflation rate π, and asset price Q, which implies that it targets output, inflation, andasset prices simultaneously. In this case, the monetary policy rule is expressed as follows:
R = R(y, π, Q); αy ≡

∂R
∂R
∂R
> 0; απ ≡> 0; αq ≡> 0; R̄ = R(y ∗ , π ∗ , Q∗ ), (13)
∂y
∂π
∂Q

where αy , απ , and αq are the responsiveness of the nominal interest rate to output, theinflation rate, and asset prices, respectively. In addition, R̄ > 0 is the target level of the
One of the conditions for optimality is given by m1j − µR = 0. The dynamics of the model economic
system can be examined without using this condition. Therefore, we ignore it from the simultaneous
equation system of the model economy.
6
The inverse of the marginal utility of consumption is expressed as ∂c/∂u = c. In addition, themarginal disutility of labor is expressed as ∂u/∂l = −lψ . Therefore, the marginal disutility of labor∂u/∂lψmeasured in terms of goods is given by dc
dl = − ∂u/∂c = cl .
58

nominal interest rate that corresponds to the target levels of these three variables (weregard their steady-state values here).
When delays are present in the central bank’s responses to economic fluctuations,
Equation (13) can be rewritten as follows:
R(t) = R(y(t − τ1 ), π(t − τ2 ), Q(t − τ3 )),(14)

where τ1 , τ2 , and τ3 are delays in the output, inflation rate, and asset-price targeting
policies, respectively.

3

Case with no policy lags

To emphasize the eﬀects of policy lags on equilibrium determinacy, we first consider thecase with no policy lags (i.e., τ1 = τ2 = τ3 = 0).
In this case, the model economic system comprises Equations (4), (5), (8), and (10)–(13). By using the goods market equilibrium condition y = c, the system can be summarized in the following three equations:ẏ = [R(y, π, Q) − π − ρ]y,1−ϕϕηπ̇ = ρ(π − π ∗ ) −−y 1+ψ ,γγ(η − 1)
(
)
η
γ
1+ψ
∗ 2
Q̇ = R(y, π, Q)Q − 1 −y− (π − π ) y − πQ.
η−1
2

(15)

The non-trivial solutions to this system are expressed as follows:[(ϕ − 1)(η − 1)
y =
ϕη
∗

1
] 1+ψ

,

π ∗ = R̄ − ρ,
(
) ∗
η
y
∗
∗ 1+ψ
Q = 1−y.
η−1ρThe Jacobian matrix of System (15) evaluated at the steady state can be given by
αy y ∗(απ − 1)y ∗αq y ∗J ≡  −P1
ρ
0
,
∗
∗
∗
αy Q + P2 (απ − 1)Q αq Q + ρ
9(16)whereϕη(1 + ψ) ∗ ψ
y > 0,γ(η − 1)()ηη(1 + ψ) ∗ 1+ψ
∗ 1+ψ
P2 ≡ − 1 −
y
+
y
η−1
η−1
(
)
η(2 + ψ) ∗ 1+ψ
=− 1−
y
η−1
)
(
(2 + ψ)(ϕ − 1)
=− 1−
.
ϕ
P1 ≡

For plausible parameter values, P2 > 0 holds.7 Hence, in the following discussion, we
assume that this inequality holds.
Thus, the characteristic equation of the above system can be represented as∆1 (x) ≡ |xI − J| = x3 + b1 x2 + b2 x + b3 = 0,(17)where x is a characteristic root, I is an identity matrix,
b1 ≡ −traceJ= −(αy y ∗ + αq Q∗ + 2ρ) < 0,b2 ≡ sum of the second-order principal minors of J
= −(P2 y ∗ − ρQ∗ )αq + ρ2 + 2αy y ∗ ρ + P1 (απ − 1)y ∗ ,b3 ≡ −detJ= αq y ∗ ρP2 − αy y ∗ ρ2 − P1 (απ − 1)y ∗ ρ.
If αq = 0, Jacobian matrix J becomes decomposable. In other words, the dynamics ofy and π are unaﬀected by those of Q (whereas the dynamics of Q are aﬀected by y and π).
Therefore, in this case, the dynamic structure of the system is consistent with that of asimple NK model. Hence, the Taylor principle would hold as the necessary and suﬃcientcondition for determinacy. However, as demonstrated by Bullard and Mitra (2002), when
the nominal interest rate responds not only to the inflation rate but also to output,
the condition for determinacy becomes more complicated; in this case, the condition isexpressed as αy ρ + P1 (απ − 1) > 0. If αy = 0, then the well-known determinacy condition,απ > 1, can be obtained. When αq > 0, the following lemma holds:
7

If ψ > ϕ/(ϕ − 1) − 2, P2 > 0; therefore, if specifically ϕ > 2, P2 > 0 holds for all ψ > 0.

10

Lemma 1 The equilibrium is locally determinate if and only if
αq < αq3 ≡αy ρ + P1 (απ − 1)
.
P2

(18)

Proof. Matrix J includes three roots. For equilibrium determinacy, all these roots musthave positive real parts. The necessary and suﬃcient conditions for all roots to havepositive real parts (Inverse Routh–Hurwitz theorem) are given by b1 < 0, b2 > 0, andb3 < 0.8 Irrespective of the value of αq , b1 < 0 holds. Furthermore, b3 < 0 if and only
if the condition in Equation (18) is satisfied. A necessary condition for Equation (18) tohold is that αy ρ + P1 (απ − 1) > 0. Under this condition, b2 is ensured to become positive
if P2 y ∗ < ρQ∗ ; however, if P2 y ∗ > ρQ∗ , it becomes positive only when
αq < αq2 ≡ρ2 + 2αy y ∗ ρ + P1 (απ − 1)y ∗.
P2 y ∗ − ρQ∗

As αq2 > αq3 , if Equation (18) is satisfied, αq < αq2 is also satisfied. Thus, b2 > 0 andb3 < 0 if and only if αq < αq3 .
□The economic implications of this result can be explained as follows. In our model, an
increase in the inflation rate increases the real wage rate, which is a marginal cost of firms.
This change further results in decreases in profits and asset prices (by P2 > 0). The central
bank responds not only to the variation in the inflation rate but also to the variation in
asset prices. Therefore, the response of the nominal interest rate to the inflation rate ispartly oﬀset by the response to asset prices. This fact implies that the Taylor principle
becomes less likely to be satisfied. Thus, the larger the value of αq , the more likely is theoccurrence of indeterminacy. Carlstrom and Fuerst (2007) use a discrete-time sticky-pricemodel to demonstrate the equivalent result.9

8

See Asada et al. (2007) and Tsuzuki (2013).
See Proposition 1 in Carlstrom and Fuerst (2007). They also consider the cases where (i) nominal
wages are sticky and (ii) both prices and nominal wages are sticky. In this study, we examine the standard
case where only prices are sticky.
9

11

4

Case with positive policy lags

The model economic system when Equation (14) is used as the policy rule is expressed asfollows:ẏ(t) = [R(y(t − τ1 ), π(t − τ2 ), Q(t − τ3 )) − π(t) − ρ]y(t),1−ϕϕηπ̇(t) = ρ(π(t) − π ∗ ) −−y(t)1+ψ ,γγ(η − 1)Q̇(t) = R(y(t − τ1 ), π(t − τ2 ), Q(t − τ3 ))Q(t)
(
)
η
γ
1+ψ
∗ 2
− 1−
y(t)
− (π(t) − π ) y(t) − π(t)Q(t).
η−1
2

(19)

This is a diﬀerential equation system with three delays.
The steady-state values of System (19) are given by Equation (16). We linearize thissystem at the steady state to obtain
˙ = [αy ŷ(t − τ1 ) + απ π̂(t − τ2 ) + αq Q̂(t − τ3 ) − π̂(t)]y ∗ ,
ŷ(t)
˙
π̂(t)
= ρπ̂(t) − P1 ŷ(t),˙Q̂(t) = αy Q∗ ŷ(t − τ1 ) + απ Q∗ π̂(t − τ2 ) + αq Q∗ Q̂(t − τ3 ) + P2 ŷ(t) − Q∗ π̂(t) + ρQ̂(t),(20)
where ŷ(t) ≡ y(t) − y ∗ , π̂(t) ≡ π(t) − π ∗ , and Q̂(t) ≡ Q(t) − Q∗ . Assuming that theexponential functions ŷ(t) = C1 ext , π̂(t) = C2 ext , and Q̂(t) = C3 ext are the solutions tothis system, where C1 , C2 , and C3 are arbitrary constants, and plugging these functions
into System (20), we get the following:


  
x − αy y ∗ e−τ1 xy ∗ − απ y ∗ e−τ2 x−αq y ∗ e−τ3 x
ŷ(t)
0


  
P1
x−ρ
0

  π̂(t)  = 0 .
Q̂(t)0−P2 − αy Q∗ e−τ1 x Q∗ − απ Q∗ e−τ2 x x − ρ − αq Q∗ e−τ3 x
The determinant of the left-hand side matrix must equal zero for non-trivial solutions toexist:∆2 (x) ≡ s0 (x) + s1 (x)e−τ1 x + s2 (x)e−τ2 x + s3 (x)e−τ3 x = 0,
(21)

12

wheres0 (x) ≡ x3 − 2ρx2 + (ρ2 − P1 y ∗ )x + P1 y ∗ ρ,
s1 (x) ≡ −αy y ∗ x2 + 2αy y ∗ ρx − αy y ∗ ρ2 ,s2 (x) ≡ P1 απ y ∗ x − P1 απ y ∗ ρ,
s3 (x) ≡ −αq Q∗ x2 + (ραq Q∗ − P2 αq y ∗ )x + αq y ∗ ρP2 .
Equation (21) is the characteristic equation of System (19).
Owing to the existence of the terms that contain the exponential functions e−τ1 x ,e−τ2 x , and e−τ3 x , Equation (21) has an infinite number of roots.10 Furthermore, unlike inthe case of ordinary diﬀerential equations, delay diﬀerential equations require the initialvalues of y(t), π(t), and Q(t) evaluated not only at time zero (t0 ; present time) but also
at t0 − τ1 ≤ t < t0 , t0 − τ2 ≤ t < t0 , and t0 − τ3 ≤ t < t0 . However, the only values that
economic agents can determine at time t0 are (y(t0 ), π(t0 ), Q(t0 )) because “past values”
must be considered as given. Therefore, equilibrium determinacy can be achieved only
when there are exactly three roots with positive real parts among the infinite number ofroots. The equilibrium is indeterminate if fewer than three roots have positive real parts
and it is unstable if more than three roots have positive real parts (i.e., an equilibrium
path does not exist).

4.1

Method of analysis

The following analysis is performed based on the numerical method developed by Gu andNaghnaeian (2011). The procedure is given as follows:
1. The set of the imaginary part of the roots that generates pure imaginary roots(crossing frequency set: Ω) is characterized. The crossing frequency set is broadlygrouped into two types: Grashof and Non-Grashof sets.
2. The set of lags corresponding to the crossing frequency set (stability crossing set:T ∈ (τ1 , τ2 , τ3 )) is characterized.
3. By assuming plausible parameter values, we depict the stability crossing set as asurface in a 3D space.
10

See Chapter 3 in Bellman and Cooke (1963).

13

The notations used in the following sections are defined as follows: R = set of all
real numbers; C = set of all complex numbers; Z+
3 = {1, 2, 3}; i = imaginary unit.
+′
Furthermore, let u = (u mod 3) + 1 for u ∈ Z3 ; that is, 1′ = 2, 2′ = 3, and 3′ = 1. In
addition, let u′′ = (u′ )′ = [(u + 1) mod 3] + 1. Then, {u, u′ , u′′ } = Z+
3.

4.2

Preliminaries

First, to apply Gu and Naghnaeian’s (2011) method, some preconditions should be examined. The polynomials su (x), u = 0, 1, 2, 3, must satisfy the following conditions (nontriviality assumptions):
Assumption 1 deg (s0 (x)) ≥ max {deg (su (x))|u ∈ Z+
3 },Assumption 2 s0 (0) + s1 (0) + s2 (0) + s3 (0) ̸= 0,|s1 (x)| + |s2 (x)| + |s3 (x)|< 1.
x→∞|s0 (x)|Assumption 3 lim

Assumption 1 ensures the existence of a set (τ1 , τ2 , τ3 ) ∈ R3+ that establishes perfectstability, wherein a root with a positive real part does not exist. However, this study
examines equilibrium determinacy, which is diﬀerent from the concept of stability used inmathematics. Hence, the existence of a perfectly stable state is not necessarily required;nevertheless, we adopt this assumption only to apply Gu and Naghnaeian’s (2011) method.
Because deg (s0 (x)) = 3 and max {deg (su (x))|u ∈ Z+3 } = 2, this assumption is satisfied.
Assumption 2 implies that ∆2 (0) ̸= 0; i.e., zero is not a root. For the same reason as forAssumption 1, we adopt this assumption, while it also plays the only role of a necessary
condition to ensure the existence of a perfectly stable state. We should assume detJ ̸= 0
because ∆2 (0) = −detJ. Finally, Assumption 3 is required to ensure the continuity of
2 (x)|+|s3 (x)|roots over (τ1 , τ2 , τ3 ). Because limx→∞ |s1 (x)|+|s= 0, Assumption 3 is satisfied.
|s0 (x)|Under these assumptions, we can state the following: as the lags (τ1 , τ2 , τ3 ) continuously vary within R3+ , the number of roots of ∆2 (x) = 0 lying on C+ can change if a root
appears on or crosses the imaginary axis (Lemma 1 in Gu and Naghnaeian (2011)).

14

4.3

Crossing frequency set

Denote a pure imaginary root as x = iω, where ω > 0 is the imaginary part.11 In thissection, we characterize the values of ω > 0 that satisfy ∆2 (iω) = 0 (crossing frequency).
Letsu (x)
au (x) =, u = 0, 1, 2, 3.
(22)
s0 (x)
Obviously, a0 (x) = 1. Moreover, we also define thatf0 (x) = |a1 (x)| + |a2 (x)| + |a3 (x)|,fu (x) = |au′ (x)| + |au′′ (x)| − |au (x)|,(23)
u = 1, 2, 3.

(24)

By using Equation (22), ∆2 (iω) = 0 can be rewritten as follows:∆3 (iω) = 1 + a1 (iω)e−iωτ1 + a2 (iω)e−iωτ2 + a3 (iω)e−iωτ3 = 0.

(25)

Furthermore, by considering each term of Equation (25) as a vector depicted on thecomplex plane, the four terms form a rectangle, as shown in Figure 1.12

a2 (iω)e−iωτ2
a1 (iω)e−iωτ1a3 (iω)e−iωτ3

a0 (iω) = 1
Figure 1: Rectangle formed by the four vectors
From the geometric fact that a rectangle can be formed only when the length of one
side does not exceed the sum of the other lengths, we can postulate that a crossing
frequency set comprises ω > 0 that satisfy the following four inequalities:
f0 (iω) ≥ 1
fu (iω) ≥ −1,
11

u = 1, 2, 3.

We can assume that ω > 0 without loss of generality because the complex roots will always be
conjugated.
12
Note that the length of each vector is independent of (τ1 , τ2 , τ3 ). ((τ1 , τ2 , τ3 ) determine the directionof the vector.)

15

Thus, when we depict fu (iω), u = 0, 1, 2, 3, against ω, the crossing frequency set Ω
comprises all ω > 0 that the curve f0 (iω) is above 1 and the curves f1 (iω), f2 (iω), andf3 (iω) are above −1.
Moreover, if the sum of the longest side and the shortest side of a rectangle is smaller
than that of the other two sides, the rectangle is recognized as a Grashof rectangle. By
using this definition, we can classify ω ∈ Ω into the following four types:
Type 0 Grashof set: Ω0G
Set of ω ∈ Ω that satisfies
fu (iω) > 1,

u = 1, 2, 3.

(26)

fu (iω) > 1,

u = 1, 2, 3,(27)Type u Grashof set: ΩuG
Set of ω ∈ Ω that satisfies

fu′ (iω) < 1,

(28)

fu′′ (iω) < 1.

(29)

Type 0 Non-Grashof set: Ω0N
Set of ω ∈ Ω that satisfies
fu (iω) ≤ 1,

u = 1, 2, 3.

(30)

u = 1, 2, 3,(31)Type u Non-Grashof set: ΩuN
Set of ω ∈ Ω that satisfies
fu (iω) ≤ 1,
fu′ (iω) ≥ 1,

(32)

fu′′ (iω) ≥ 1.

(33)

ΩuG , u = 0, 1, 2, 3, are open intervals and ΩuN , u = 0, 1, 2, 3, are closed intervals.
(However, ΩuN , u = 0, 1, 2, 3, become semi-open intervals when ω = 0 is a solution of
Equations 31–33 or Equation 30. See Proposition 3 in Gu and Naghnaeian (2011) for
details.)
Following Fujiwara (2008), we set the structural parameter values as shown in Table
1. Further, we set the target inflation rate at a realistic value of 2%, which implies that16R̄ = 0.03. Finally, the monetary policy reaction parameters are set at their typical values,
as shown in Table 2, where the value of τq is set so that Equation 18 is satisfied (i.e., if a
policy lag is not present, the equilibrium is determinate).
ηϕγ

21 6.0ψρ27.454 1.0 0.01

Table 1: Structural parameters

αy

απ

αq

0.5 1.5

0.01

Table 2: Policy reaction parameters
Under these assumptions, we investigate the eﬀects of policy lags on equilibrium determinacy. The functions fu (iω), u = 0, 1, 2, 3, are depicted as shown in Figure 2. This
figure indicates that
Ω = [ω1 , ω4 ] = [0.2203, 0.8176].

2f0 (i ω)1.5
f (i ω)1

1

f (i ω)

0.5

3

0

-0.5
f (i ω)
2

-1

-1.5
0

ω ω
1
2

ω3

ω4

1
ω

2

Figure 2: fu (iω), u = 0, 1, 2, 3
Moreover, the crossing frequency set Ω can be resolved into the following three com-17

ponents:
Ω=

3
∪

Ωh

h=1

Ω = [ω1 , ω2 ] = [0.2203, 0.3054] ⊂ Ω2N ,1Ω2 = (ω2 , ω3 ) = (0.3054, 0.5922) ⊂ Ω3G ,Ω3 = [ω3 , ω4 ] = [0.5922, 0.8176] ⊂ Ω0N .
Gu and Naghnaeian (2011) show the geometric configurations of the stability crossing
sets (T h , h = 1, 2, 3) that correspond to all possible types of crossing frequency sets (Ωh ,h = 1, 2, 3) (see Theorem 1 in Gu and Naghnaeian (2011)). According to their study, T 1
and T 3 shape caps and T 2 shapes wavy sheets in the (τ1 , τ2 , τ3 ) space.

4.4

Concretization of the stability crossing sets

To illustrate the stability crossing sets T h , h = 1, 2, 3, in the (τ1 , τ2 , τ3 ) space, we present
their specific expressions.
Let {u, v, w} = Z+
3 ; then, Equation (25) can be rewritten as follows:∆3 (iω) = av (iω)e−iωτv + aw (iω)e−iωτw + ad (iω, τu ) = 0,(34)

ad (iω, τu ) = 1 + au (iω)e−iωτu .

(35)

where

Considering the three terms in Equation (34) as vectors on the complex plane and depicting them, a triangle can be formed, as shown in Figure 3.13
This figure indicates that θv and θw can be expressed as follows:∓θv = arg (av (iω)) − ωτv − arg (ad (iω, τu )) + 2rv π,±θw = arg (aw (iω)) − ωτw − arg (ad (iω, τu )) + 2rw π,

13

For details of the discussion here, see Section 3 in Gu et al. (2005).

18

av (iω)e−iωτv
θv
aw (iω)e−iωτw
θw

ad (iω, τu )au (iω)e−iωτu1Figure 3: Triangle formed by |ad (iω, τu )|, |av (iω)|, and |aw (iω)|
where rv , rw = 0, 1, 2, 3, · · · . Thus, we obtain
τv = τv± (ω, τu , rv )arg (av (iω)) − arg (ad (iω, τu )) ± θv + 2rv π,ωτw = τw∓ (ω, τu , rw )==arg (aw (iω)) − arg (ad (iω, τu )) ∓ θw + 2rw π
.
ω

(36)

(37)

By using the cosine theorem, θv and θw can also be expressed as()|ad (iω, τu )|2 + |av (iω)|2 − |aw (iω)|2−1
θv = cos,2|ad (iω, τu )| · |av (iω)|()|ad (iω, τu )|2 + |aw (iω)|2 − |av (iω)|2−1
θw = cos,2|ad (iω, τu )| · |aw (iω)|
which are substituted into Equations (36) and (37), respectively.
A similar procedure derives the following expression from Equation (35):τu = τu (ω, ru )=arg (au (iω)) + (2ru − 1)π,ω

ru = 0, 1, 2, 3, · · · .

(38)Equations (36)–(38) are used to express the stability crossing sets.
Incidentally, for the triangle shown in Figure 3 to exist, the following condition musthold:||av (iω)| − |aw (iω)|| ≤ |ad (iω, τu )| ≤ |av (iω)| + |aw (iω)|.
(39)
19

These inequalities define the motion range of τu for the given values of ω. The motion
range depends on the type of Ωh . Hence, the representation of T h also diﬀers accordingly.
∪
When Ωh is a Grashof set (ΩG = 3u=0 ΩuG ), the motion range of τu can be defined asfollows (see Appendix A.1):
Tu (ω, ru ) = [τu (ω, ru ), τu (ω, ru + 1)].
On the contrary, when Ωh is a Non-Grashof set (ΩN =can be defined as follows (see Appendix A.2):∪3uu=0 ΩN ), the motion range of τuTu (ω, ru ) = [τum (ω, ru ), τuM (ω, ru )],
(

where
−1

θum = cos

1 + |au (iω)|2 − (|av (iω)| + |aw (iω)|)2
2|au (iω)|

)
,

(40)

arg (au (iω)) + 2ru π − θum,ωarg (au (iω)) + 2ru π + θum
τuM (ω, ru ) =.
ωτum (ω, ru ) =Thus, by defining the set T h± asτu ∈ Tu (ω, ru ), 


h±
±
T
= (τ1 , τ2 , τ3 ) τv = τv (ω, τu , rv ),
,




∓
τw = τw (ω, τu , rw )(41)

the stability crossing sets T h , h = 1, 2, 3, can be represented for the given (ru , rv , rw ) and{(ω, τu )|ω ∈ Ωh , τu ∈ Tu (ω, ru )}, as follows:∪∩
T h = (T h+ T h− ) R3+ .
(42)

4.5

Drawing the stability crossing sets

Under the assumptions in Tables 1 and 2, T 1 , T 2 , and T 3 are drawn, as shown in Figures
4–6, respectively.14
In the case of Ω1 ⊂ Ω2N , u = 2, v = 1, and w = 3. Figure 4 shows the case of (ru , rv , rw ) = (0, 0, 0).
In the case of Ω2 ⊂ Ω3G , u = 3, v = 2, and w = 1. Figure 5 shows the case of ru = 0, 1, 2; rv = 1;
and rw = 1. Finally, in the case of Ω3 ⊂ Ω0N , u = 1, v = 3, and w = 2. Figure 6 shows the case of(ru , rv , rw ) = (1, 1, 1).
14

20

15

60
50

10
40
5
τ3

3

τ

30
20
0
10
-5
0
-10

-10
25
1
20

-10

30
25

0

15

-15

20
15
10

-20
-1

10

τ2

τ2

τ1

-25

Figure 4: T 1 : Cap

12

τ3

10
8
6
4
2
12
11
10
10

9
8
9
7
8
6
7

5

5

τ1

Figure 5: T 2 : Wavy sheets

14

τ2

5

τ1

Figure 6: T 3 : Cap

21

60

τ3

40

20

0

60
40

60
40
20
20
0
0
-20
τ2

-20

τ1

-40

Figure 7: Stability crossing sets
The complete figure of T h , h = 1, 2, 3, corresponding to all combinations of ru = 0, 1, 2;
rv = 0, 1, 2; and rw = 0, 1, 2 (27 patterns) can be drawn as shown in Figure 7. At least
for positive values of (τ1 , τ2 , τ3 ) lying inside the surface of T h , h = 1, 2, 3, that enclose theorigin, the equilibrium is determinate. To capture the size and shape of the determinacy
region more clearly, we show some cross-section diagrams when τ1 , τ2 , or τ3 is fixed atcertain values (0, 10, and 20), as in Figures 8–16. Whenever τ1 , τ2 , or τ3 crosses thesecurves (which we call the stability crossing curves), the sign of the real part of the complex
roots changes.
The direction of the change (i.e., whether it runs from positive to negative or vice versa)
can be examined as follows. In the case of a change in τ1 , the direction is confirmed bydetermining the sign of the following expression:
[
]
∂x
Re
,
(43)
∂τ1 (τ1 ,τ2 ,τ3 )=(τ ∗ ,τ ∗ ,τ ∗ ),x=iω∗123where (τ1∗ , τ2∗ , τ3∗ ) is a point lying on the surface of a stability crossing set and x = iω ∗ isa pure imaginary root that corresponds to the point. In addition, ∂x/∂τ1 can be derived

22

from the implicit function ∆2 (x, τ1 , τ2 , τ3 ) = 0 as follows:s1 (x)xe−τ1 x
∂x
.
= ′
∑3
∂τ1
s0 (x) + j=1 {s′j (x) − sj (x)τj }e−τj x

(44)

[ ]
∂x
If Re ∂τ
> 0, a pair of complex roots moves from left to right on the complex plane1
when τ1 crosses τ1∗ ; therefore, the real part of the complex roots changes from negative
to positive. Conversely, if the inequality runs in the opposite direction, a pair of complex
roots moves from right to left on the complex plane; in this case, the real part of thecomplex roots changes from positive to negative.
For example, at the point (τ1∗ , τ2∗ , τ3∗ ) = (6.9689, 12.0174, 0) (see Figure 8), ω ∗ = 0.5395
and Re[∂x/∂τ1 ] = 0.0158 > 0. Accordingly, when τ1 crosses this point from left to right,
the number of roots with positive real parts increases by two. We have already shownthat there are exactly three roots with positive real parts in the left-hand side of thispoint (the equilibrium is determinate when policy lags are not present). Therefore, in theright-hand side region of that point, there are five roots with positive real parts, implyingthat the equilibrium is unstable. As for the other areas, we can reveal the number of roots
with positive real parts by using the same procedure. The values written in each area of
Figures 8–16 indicate the number of roots with positive real parts.

23

60

60

50

50
+7

+7

40

40
+5
τ2

2

τ

+5
30

30

+5
+5
20

20
+5
+5

+3
10

10
+3
+1

+5

+5
0
5

10

15

20

25

30

35

0

τ1

45
+7

35
+5

2

30

25
+5
20
+5
15

10
+3
5
+5

+5
0
0

5

10

15
τ1

10

15

20

25

Figure 9: τ3 = 10

50

40

5

τ1

Figure 8: τ3 = 0

τ

+5

+5

0
0

20

25

30

Figure 10: τ3 = 20

24

30

35

20

25

18
+5
16

20
+5

+5

+5

+5

14

12

15
τ3

3

τ

+1
10

+3
+3

+3

+5

8

10
+5

6

4

5

2
+5

+5

0
5

10
τ1

15

20

0

20
+5
+7
+7
15
3

+5
+5
+7
+7
+5

5
+3
+3

+7

+5
0
0

5

10

10

15

Figure 12: τ2 = 10

25

10

5

τ1

Figure 11: τ2 = 0

τ

+5

0
0

15

20

25

τ1

Figure 13: τ2 = 20

25

20

25

40

30

35
25
30
+7
20
+7

τ3

3

τ

25

20

15
+5

+3

+5

+3
15

10
10
5
5

0

0
0

5

10

15

20
τ2

25

30

35

40

0

20

+3

3

τ

+5
+3
+7

+5
10
+7

+5

5

0
0

5

10

15

20
τ2

15

Figure 15: τ1 = 10

25

+5

10
τ2

Figure 14: τ1 = 0

15

5

25

30

35

40

Figure 16: τ1 = 20

26

20

25

In each of these figures, an increase in the value of some lag parameter may cause any
one of the following: determinacy, indeterminacy, or instability. For example, in Figure
11, if τ3 = 10, the dynamic property of the system changes with an increase in τ1 asfollows: determinate—indeterminate—determinate—unstable. Specifically, the regionsof τ1 that achieve determinacy are given by τ1 ∈ [0, 5.13] and τ1 ∈ [6.73, 16.4], which
implies that if an implementation lag does not present in inflation targeting and a lagof approximately five years exists in asset-price targeting, then lags in output targetingmust be approximately 0–2.5 or 3.4–8.2 years.
Furthermore, moving on the vertical axis of Figures 11–16 does not change the numberof roots with positive real parts. This finding implies that a lag in asset-price targeting
does not aﬀect equilibrium determinacy, at least when τ1 or τ2 is suﬃciently small.
In addition, as shown in Section 3, in the case where a lag is not present, if thecondition in Equation (18) does not hold, the equilibrium is indeterminate. In this case,the signs of the real parts of the three roots are necessarily + + − (the case of + − −cannot occur). Accordingly, in the above case, there is no possibility that equilibrium
determinacy is achieved by introducing a lag in a policy response because the number ofroots with positive real parts necessarily changes by two in any case.
Moreover, cycles or other complex fluctuations may exist around the steady statebecause on the stability crossing curves and thus all conditions for a Hopf bifurcation aresatisfied.

5

Conclusion

In this study, we used the NK model to analyze the eﬀects of three policy lags on local
equilibrium determinacy. Unlike Tsuzuki (2014, 2015), who only studies the eﬀect ofinflation targeting, the existence of multiple target variables in monetary policy (i.e., notonly the inflation rate but also output and asset prices) provides a new possibility fora lag. In other words, even if the system includes only one lag, the lag can resolve theproblem of instability. For example, when τ1 , which represents a lag in output targeting,moves on the horizontal axis of Figures 8 and 11, the equilibrium changes as follows:
determinate—unstable—determinate. However, increases in τ2 and τ3 , which denote lags
in inflation and asset-price targeting, respectively, do not have such an eﬀect. In thiscase, an increase in τ2 necessarily causes instability, whereas that in τ3 does not aﬀect the
dynamic property.
27

In cases where multiple policy lags coexist, the analysis becomes considerably more
complicated. All lags can have a stabilizing eﬀect on the equilibrium. Specifically, in
Figures 13 and 16, τ1 or τ3 must be positive to achieve determinacy. If the values of theselags are zero, the equilibrium is unstable. This finding suggests that the central bank may
be required to “purposefully” delay its policy implementation.
Moreover, the above results are valid only for the plausible parameter values assumed
in Section 4. They do not have anything like generality. Depending on the parametervalues, the type of Grashof set may change. Accordingly, the configuration of a stability
crossing set may also dramatically alter.
A more theoretical investigation of the eﬀects of policy lags requires an algebraic
approach to diﬀerential equation systems with multiple delays. Unfortunately, such amethod has not thus far been established. Nonetheless, the analysis performed in thisstudy is helpful for policymakers. Indeed, the presented findings suggest that the central
bank should determine its target variables by considering not only the responsiveness ofthe nominal interest rate to these variables but also the lag lengths associated with policy
implementations.
In addition, the present study argues that (i) if a delay exists only in asset-pricetargeting, the Fed’s view would assert its validity at least for a slight change in policy
responsiveness, as in the case with no policy lags; and (ii) if multiple policy lags coexist,
the Bank of International Settlements’ view can become valid policy, depending on thelag parameter set.

A

Appendix

A.1

Motion range of τu : the case of Grashof sets

In the case of Ωh ⊂ ΩuG , u = 0, 1, 2, 3, the expression in Equation (39) holds with strict
inequalities for all τu ∈ R+ (see Equations 26 and 27–29). Therefore, T h can be definedfor (ω, τu ) ∈ Ωh × R+ . Thus, the motion range of τu can be defined as
Tu (ω, ru ) = [τu (ω, ru ), τu (ω, ru + 1)].

28

A.2

Motion range of τu : the case of Non-Grashof sets

In the case of Ωh ⊂ ΩuN , u = 0, 1, 2, 3, the first inequality in Equation (39) holds for allτu ∈ R+ (see Equations 30 and 31–33). However, the second inequality, |ad (iω, τu )| ≤|av (iω)| + |aw (iω)|, is not ensured to hold. This inequality is equivalent to the following
expression: π − θum ≤ arg (au (iω)e−iωτu ) + 2ru π ≤ π + θum , where)(
1 + |au (iω)|2 − (|av (iω)| + |aw (iω)|)2−1
θum = cos.
2|au (iω)|
Therefore, we obtainarg (au (iω)) + 2ru π − θum,ωarg (au (iω)) + 2ru π + θum
τuM (ω, ru ) =.
ωτum (ω, ru ) =By using these expressions, the motion range of τu can be represented as follows:
Tu (ω, ru ) = [τum (ω, ru ), τuM (ω, ru )].

Acknowledgments
This paper is a part of the outcome of research performed under a Waseda University
Grant for Special Research Projects (Project number: 2015B-014).

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