The non-negative constraint on the nominal interest rate
and the eﬀects of monetary policy∗
Kohei Hasui†
September 25, 2013

Abstract
This paper analyzes the eﬀects of monetary policy shock when there is a nonnegative constraint on the nominal interest rate. I employ two algorithms: the piecewise linear solution and Holden and Paetz’s (2012) algolithm (the HP algorithm).
I apply these methods to a dynamic stochastic general equilibrium (DSGE) model
which has sticky prices, sticky wages, and adjustment costs of investment. The mainﬁndings are as follows. First, the impulse responses obtained with the HP algorithm
do not diﬀer much from those obtained with the piecewise linear solution. Second,
the non-negative constraint inﬂuences the eﬀects of monetary policy shocks under
the Taylor rule under some parameters. In contrast, the constraint has little eﬀects
on the response to money growth shocks. Third, wage stickiness contributes to theeﬀects of the non-negative constraint through the marginal cost of the product.
The result of money growth shock suggests that it is important to analyze theeﬀects of the zero lower bound (ZLB) in a model which generates a signiﬁcant liquidity eﬀect.
Keywords: Zero lower bound; Monetary policy shock; Wage stickiness; Liquidity
eﬀect
JEL Classiﬁcation: E47; E49; E52∗

Graduate School of Economics Kobe University Working Paper Series No. 306
Ph.D student, Graduate School of Economics, Kobe University. E-mail: 100e107e@stu.kobe-u.ac.jp.
I would like to thank Toshiki Jinushi, Shigeto Kitano, Teruyoshi Kobayashi, Junya Masuda, Keiya
Minamimura, Atsushi Miyake, Masahiko Shibamoto, Yuki Teranishi, Shingo Umino and other seminar participants at Kobe University, Kobe Gakuin University and Nagoya University for their helpfulcomments and suggestions.
†1Introduction

After the colossal ﬁnancial crisis in 2008, the short term nominal interest rates stayedat zero. This prompts the question of how the zero lower bound (ZLB) inﬂuences theeﬀects of monetary policy shocks. Many papers have derived the impulse responses andanalyzed the economic behaviors in the dynamic stochastic general equilibrium (DSGE)
literature, but most of that research did not include the non-negative constraint on
nominal interest rates. This paper analyzes the eﬀects of monetary policy shocks when
there is a non-negative constraint on nominal interest rates in a typical DSGE framework.
Several authors have described models including a non-negative constraint on nominal interest rates in the optimal monetary policy literature. Their analyses focused on
how to avoid going into the liquidity trap and on the eﬀectiveness of monetary policy
through expected inﬂation1 .(Eggertsson and Woodford 2003, Jung, et al. 2005, Kato andNishiyama 2005, Adam and Billi 2006, 2007, Nakov 2008). More recent analyses useda strand of occasionally binding constraint to tackle non-linear problem.(Christiano, etal. 2010 , Fernández-Villaverde, et al. 2012, Nakata 2012) The studies in both the DSGE
literature and the optimal monetary policy literature did not analyze the eﬀects of monetary policy shocks when there is a non-negative constraint on the nominal rate of interest.
Holden and Paetz (2012) created an algorithm dealing with the ZLB. Holden andPaetz’s algorithm (henceforth, the HP algorithm) employs news shocks (Holden andPaetz called these the “shadow shock”) to deal with the ZLB. They added this algorithm
which generates impulse responses to news shocks to their Dynare code to derive theextended versions of impulse responses.
The intuition for algorithm is as follows. If there is a ZLB constraint, nominal interest
rates maybe zero for some periods. The boundaries of nominal interest rate aﬀect theeconomic behavior. For example, if the nominal interest rate binds, output and inﬂation
decrease more. These eﬀects are expressed by anticipated components that are createdby news shocks. In other words, if the nominal interest rate binds, the eﬀects which are
created by anticipated shocks are allocated to other macro-variables. Since news shockis the shcok that agents know when the shock materialize, agents can behave rationallyconsidering the occurence of shocks. This feature is used for the behavior of an agent
who knows when the nominal rate of interest will reach the lower bound.
The HP algorithm allocates the anticipated component to macro variables by solv1

Nakajima (2008), Fujiwara, Sudo and Teranishi (2010), and Ida (2013) analyze the optimal monetary
policy with ZLB in an open economy.

1ing the complementary condition with slackness. Impulse responses accommodating
ZLB consist of an unanticipated component and an anticipated component with weightparameters. Solving the complementary problem, we obtain optimal weight-parameters.
In the present analysis, I used both the HP algorithm and the piecewise linear solution, which is an algorithm that interpolates impulse responses with other impulse
responses. The picewise linear solution replace the periods during which the nominal interest rate might hit the lower bound with another impulse response that accommodates
the model structure which nominal interest rate binds. The period which is replaced byanother impulse response is determined by guess and verify method.
Here I use these two algorithms and analyze the eﬀects of monetary policy shocks
when there is a non-negative constraint on the nominal interest rate. I employ a DSGE
model which has sticky prices, sticky wages and adjustment costs of investment. First,
I show that strong reductions in the nominal interest rate play a signiﬁcant role in the
dynamics of the economy after policy shock when there is the ZLB constraint in thismodel. The nominal interest rate decreases signiﬁcantly when the monetary policy ruleis the Taylor type. On the other hand, the nominal interest rate decreases tvery small
when the monetary policy rule is the money growth rule.
Second, I remove the wage stickiness to test how it contributes to the eﬀects of the
ZLB on monetary policy shocks under the Taylor rule. The response of the inﬂation inan economy under ﬂexible wages becomes larger than in an economy under sticky wages.
Then, inﬂation can absorb the relatively large eﬀects of the ZLB on the case without thewage stickiness. The impulse responses results indicate that eﬀects of the ZLB in theeconomy under ﬂexible wages is smaller than in the economy under sticky wages.
Third, I manipulate the persistence of the monetary policy shock under the Taylor
rule. An increase in the persisitece of a monetary policy shock signiﬁcantly reduces theeﬀects of the ZLB by reducing the response of the nominal interest rate. The increase in
the persistence of a monetary policy shock gives a long term feature to nominal interest
rates. This decreases the reduction in the nominal rate of interest in response to thepolicy shock, and then the absence of signiﬁcant easings are mitigated more so than in
the case in which the persistence of shocks is low.
In the remainder of the paper, I explain the model in Section 2, and I derive theimpulse responses by the HP algorithm and the piecewise linear solution in Section 3.
My conclusions and directions for future research are presented in Section 4.

2

2

The model

I use the medium scale DSGE model, presented by Christiano et al. (2005), Smets andWouters (2003), and others, to analyze the eﬀects of the ZLB on monetary policy shocks.
The model economy has the sectors of households, ﬁnal goods ﬁrms, intermediate ﬁrms,
and the government . The ﬁrms in the intermediate goods sector produce diﬀerentiatedgoods and set the price following the Calvo (1987) pricing rule. Workers supply a diﬀerentiated labor force to the intermediate-goods sector. The ﬁrms maximize their proﬁt as
evaluated by marginal utility following the Calvo pricing rule.
Throughout the optimization in each sector, the aggregate demand equation, theaggregate supply with inﬂation dynamics, and the labor supply equation with inﬂation
and wage dynamics are derived.
The ﬁscal policy is Ricardian and the central bank employs the Taylor (1993) ruleand money growth rule. The disturbance term is the only monetary policy shock in thiseconomy.

2.1

Households

I assume that the household is a continuum and indexed by h in (0, 1). The households
get the utility from the consumption Ct (h) and real money balances Mt (h)/Pt and gets
disutility from the labor supply Nt (h).
E0

∞
∑
t=0

(
βt

)
Ct (h)1−σ(Mt (h)/Pt )1−ς+
− ηNt (h)2 .
1−σ
1−ς

(1)

Where, Mt and Pt are nominal money and aggregate price, respectively. The household’sbudget constraint is
Pt Ct (h) + Pt It (h) + Bt (h) + Mt (h)≤ Wt (h)Nt (h) + rtk Pt Kt−1 (h) + Rt−1 Bt−1 (h) + Mt−1 (h) + Dt (h),(2)

where, It , Kt , Bt , and Dt denote the investment, capital, government bond, and dividend
from the proﬁt, respectively. The investment assumed to follow the accumulative processwith adjustment cost.
(
Kt (h) = (1 − δ)Kt−1 (h) + It (h) − S

3It (h)
It−1 (h))
It (h),(3)

where, S(·) denotes the adjustment function of the investment and satisﬁes the property
S(1) = S 0 (1) = 0. The household’s ﬁrst order conditions are as follows.
Ct (h)−σ = λt (h),]λt+1 (h),Πt+1[]
λt+1 (h)λt (h) = βRt Et
,
Πt+1
[
(
)
(
)
]
It (h)It (h)It (h)0
λt (h) = ψt (h) 1 − S−S
It−1 (h)It−1 (h) It−1 (h)
[
(
)(
) ]
It+1 (h) 20 It+1 (h)+ βEt ψt+1 (h)S,It (h)It (h)[]kψt (h) = βEt λt+1 (h)rt+1+ ψt+1 (h)(1 − δ) .

mt (h)−ς = λt (h) − βEt[

where, λt and ψt are lagrange multipliers associated with the household’s budget constraint and capital accumulation equation, respectively. These conditions are reduced
into[
1 = βEt

Rt
Πt+1

(

Ct+1 (h)Ct (h))−σ ],mt (h)−ς
Rt − 1=,Rt
Ct (h)−σ{]}λt+1 [ krt+1 + (1 − δ)qt+1 (h) ,qt (h) = Et
λt
[
(
)
(
)
]
It (h)It (h)It (h)0qt (h) 1 − S−S
It−1 (h)It−1 (h) It−1 (h)
[
(
)(
) ]
λt+1 (h) 0 It+1 (h)It+1 (h) 2= 1 − Et qt+1 (h)S,λt (h)It (h)It (h)

(4)

(5)
(6)

(7)

Eq.(4) is a Euler equation which describes the household’s intertemporal decision rule ofsavings. Eq.(5) is money demand equation showing that the opportunity cost of holding
money equals the nominal interest rate. Eq.(6) shows the asset price determination, andEq.(7) is the process of the investment associated adjustment costs. The term qt denotes
Tobin’s marginal q, and it is deﬁned as ψt /λt .
Following Erceg et, al. (2000), I focus on the symmetric equilibrium, i.e. Ct (h) =Ct , Mt (h) = Mt , It (h) = It , Kt (h) = Kt , qt (h) = qt . Then, I log-linearize the ﬁrst order4

conditions around the steady state. The resulting expressions are as follows.
1(rt − Et πt+1 ),σ(8)σ(1 − β)ς(1 − β)
ct −
mt .,
β
β

(9)

ct = Et ct+1 −rt =qt = Et πt+1 − rt +
it =

rk
1−δ
rk +
Et qt+1
1 + rk − δ t
1 + rk − δ

(10)

β
1
κβ
Et it+1 +it−1 +
qt .
1+β
1+β
1+β

(11)

where, I deﬁne κ ≡ 1/S 00 (1), and all variables are log-deviated from the steady state.
Eq.(11) is the log-linearized version of Eq.(7). It is reduced into the simple form signiﬁcantly since I give the property, S(1) = S 0 (1) = 0, to the adjustment function, S(·).

2.2

Wage decision

I give the sticky wage into the labor supply. There are inﬁnite continuum labor N (h, j), h, j ∈(0, 1) and the aggregate labor supply is deﬁned by(∫ 1
Nt (j) =Nt (h, j)

θw −1
θw

) θ θw−1
w

dh

,

0

(12)

∫ 1
Nt =

Nt (j)dj.
0

where, N (h, j) denotes the h type of labor supply to the ﬁrm j. I deﬁne the intratemporal
proﬁt maximization problem as follows.
max Wt Nt (j) −N (h,j)(∫ 1∫ 1
Wt (h)Nt (h, j)dh s.t. Nt (j) =Nt (h, j)

θw −1
θw

) θ θw−1
w

dh

0

0

The ﬁrst order condition is(
N (h, j) =Wt (h)Wt)−θw
Nt (j).

(13)

Eq.(13) is the demand function for h type of labor by ﬁrm j. Substituting Eq.(13) into
the zero-proﬁt condition yields
(∫ 1
Wt =

Wt (h)1−θ

) 1−θ1
dh

w

.

(14)

0

Next, I deﬁne the optimal wage setting for workers. The workers set their wages tomaximize the diﬀerence between disutilty from the labor supply and their real wages

5evaluated by marginal utility. Each worker has an opportunity to change his wage withprobability ω. Then, the worker’s optimization problem is deﬁned by[]
Wt (h)Xt,s2max Et(ωβ) λt+s (h)Nt+s (h) − ηNt+s (h) ,Pt+sWt (h)s=0()−θw
W̃t Xt,ss.t. Nt+s (h) =Nt+s .
Wt+s
∞
∑

s

(15)

where,{
Π1 × Π2 × · · · × Πt
Xt,s =
1t≥1
t=0

I assume the indexation of the unchanged wage. Then, unchanged wages are shifted bypast inﬂation πt−1 . The ﬁrst order condition is
[
]
∞
∑
W̃
X
N
(h)
t
t,s
t+s
Et
(ωβ)s λt+s (h)− Mw
Nt+s (h) = 0,
Pt+s
λt+s (h)

(16)

s=0

where Mw ≡ θw /(θw −1). From Eq.(14), the aggregate wage is given by the Dixit-Stigltz
form.
[
Wt =

(1 − ω)W̃t1−θw + ω(πt−1 Wt−1 )1−θ

] 1

1−θ

(17)

There is πt−1 in the second term of the bracket since I assume the indexation of unchanged
wages. Log-linearinzing Eq.(16) and Eq.(17) around the steady state and combining both
equations yield the wage Philips curve (hereafter, the WPC).
κw wt = βEt wt+1 + wt−1 + βEt (πt+1 − πt ) − (πt − πt−1 )1 − Mw(1 − Mw )σ
+
ct −
nt .
bw ω
bw ω

(18)

where
κw ≡

bw (1 + βγ 2 ) − Mw
2Mw − 1, bw ≡bw ω(1 − ω)(1 − βω)

Since I assume the indexation in the wage setting, there is lagged variable in both wage
and inﬂation in the WPC. The WPC denotes the relationship between wages and thelabor supply.

6

2.3

Final goods sector

There is an inﬁnite continuum of intermediate goods Yt (j), j ∈ (0, 1). The ﬁnal goods
sector produces its output by combining the intermediate goods.
(∫ 1
Yt =

Yt (j)

θ−1
θ

θ
) θ−1

dj

(19)

,

0

The optimization of the ﬁnal goods ﬁrm is deﬁned as the intratemporal proﬁt maximization.
max Pt Yt −

(∫ 1

∫ 1

Yt (j)Pt (j)Yt (j)dj s.t. Yt =0

Yt (j)

θ−1
θ

θ
) θ−1

dj

0

The ﬁrst order condition is(
Yt (j) =Pt (j)
Pt

)−θ
(20)

Yt .

Eq.(20) is the demand function for type j intermediate goods for any j ∈ (0, 1). Substituting Eq.(20) into Eq.(19) yields the aggregate price index.
(∫ 1
1−θ

Pt =

P (j)

1
) 1−θ

dj

.

(21)

0

2.4

Intermediate goods sector

In this subsection, I derive the dynamics of inﬂation log-linearized around the steadystate. The intermediate ﬁrm j has a production technology given by
Yt (j) = Kt−1 (j)α Nt (j)1−α .

(22)

By the cost minimization problem, the total cost of ﬁrm’s product is)1−α ( )α(11(rtk )α (wt )1−α Yt (j),1−ααand the marginal cost is
(
st =

1
1−α

)1−α ( )α1(rtk )α (wt )1−α .
α(23)The real marginal cost st is independent of the index j. The intermediate ﬁrm’s proﬁts
at t are replaced into[]Pt (j)− st Pt Yt (j).
Pt

7The intermediate ﬁrms dynamically maximize their proﬁts evaluated by the household’s
marginal utility by setting their optimal price considering that they cannot change theirprice forever with probability γ. I deﬁne P̃t as the price which can be set optimal in the
period t. Then, the intermediate ﬁrm’s optimal price setting is[]∞∑P̃t Xt,ss(γβ) λt+s
max Et− st+s Pt+s Yt+s (j),
Pt+s
P̃t
s=0
(
)−θ
P̃t Xt,ss.t. Yt+s (j) =Yt+s .
Pt+s(24)

The ﬁrst order condition is
Et

∞
∑

[
(γβ)s vt+s

s=0

(
]
)−θ
P̃t Xt,sP̃t Xt,s− Mst+s (j) Pt+s
Yt+s = 0,
Pt+s
Pt+s

(25)

where, M ≡ θ/(θ − 1). From Eq.(21), the aggregate price is given by the Dixit-Stiglitz
type CES aggregator and it is divided into the changed price component and the unchanged price component.
[
] 1
1−θ
Pt = (1 − γ)P̃t1−θ + γ(πt−1 Pt−1 )1−θ

(26)

Since I assume price indexation on the unchanged prices, there is the past inﬂation in
the second term of the bracket. Log-linearizing Eq.(25) and Eq.(26) and combining thetwo equation yield the dynamic equation of inﬂation.
πt =

β
1
(1 − γ)(1 − γβ)Et πt+1 +πt−1 +st ,1+β1+βγ(1 + β)(27)

Equation (27) is the New Keynesian Philips curve (hereafter, the NKPC), which describes
the supply side of the economy, the terms of inﬂation appear because of the sticky pricein the intermediate ﬁrm sector. There is lagged inﬂation in Eq. (27) since I assumed
indexiation of the unchanged prices. The eﬀects of stickiness are on st , which is themarginal cost of intermediate ﬁrms. As γ becomes large, the coeﬃcient of st becomes
small. Moreover, the log-linear version of real maginal cost is given by
st = αrtk + (1 − α)wt .

(28)

The inﬂation dynamics may become small since the sticky wage is present in this economy.
The sticky wage lowers the dynamics of st and then the inﬂation dynamics becomes
smaller.

8

2.5

Monetary policy

I derive the impulse responses to both the money growth rule and the Taylor (1993) rule.
First, when the monetary policy rule is the Taylor type,
rt = ψπ πt + ψy yt + vtr ,r
vtr = ρr vt−1
+ ²rt , ²rt ∼ i.i.d (0, σr2 ).

where, ψπ ≥ 1 is called ‘the Taylor principle’.
Second, the monetary policy is the money growth ruleµt = ρµ µt−1 + ²µt , ²µt ∼ i.i.d (0, σµ2 ),(29)

where, µt ≡ Mt − Mt−1 denotes the money growth rate; Eq.(29) is the log-linearized
form. The relationship between money growth and the real money rate can be describedas follows by using the deﬁnition of real balances, mt = Mt − pt .
µt = mt − mt−1 + πt .
Finally, I give the ZLB constraint explicitly2 .
rt ≥ − ln(1/β).

2.6

(30)

Aggregation

I aggregate the household’s budget constraint and the intermediate ﬁrms’ budget constraint in h and j.
Pt Ct + Pt It + Bt + Mt = Wt Nt + Pt rtk Kt−1 + Rt−1 Bt−1 + Mt−1 + Dt .
Dt = Pt Yt − Wt Nt − Pr rtk Kt−1 .
From the above equations and the government’s budget constraint, the goods market
clearing condition can be obtained.
Yt = Ct + It .

2

(31)

Since impulse responses indicate percent deviates from the steady state, the lower bound of thenominal interest rate becomes − ln(1/β) in the Figures.

9

2.7

Steady-state conditions

Next, I derive the steady-state conditions. First, from the equler equation,R=

1β

The capital rental rate is given by
rk = R + δ − 1.
The optimal price equation at the steady state gives the steady state value of real wage,(w=(1 − α)1−α ααM(rk )α

1
) 1−α

.

From the intermediate ﬁrm’s cost minimization problem, the output-capital ratio at thesteady state is derived as follows.
K
=
Y

(

(1 − α)rk
w

)α−1
.

Then, the output-investment ratio for the steady state is given by capital accumulation.
I
K
=δ
Y
Y
Finally, the steady state value for C/Y is
C
I
=1− ,
Y
Y

3

Simulation

In this section, I derive the impulse responses to the monetary policy shock dealing with
the non-negative constraint on the nominal interest rate. First, I show the intuitionof the HP algorithm and the piecewise linear solution. Second, I show that strong
reductions in the nominal interest rate play a signiﬁcant role in the dynamics of theeconomy in response to the policy shock when there is a ZLB constraint in this model.
The nominal interest rate decreases signiﬁcantly when the monetary policy rule is the
Taylor type under the benchmark parameters. On the other hand, the nominal interest
rate decreases to a very low rate in response to the money growth shock.
Third, I remove the wage stickiness to investigate how it contributes to the eﬀects
of the ZLB in response to monetary policy shocks under the Taylor rule. The response10

of the inﬂation in the economy under ﬂexible wages becomes larger than that in theeconomy under sticky wages. Then, inﬂation can absorb the relatively large eﬀects of
the ZLB in the case without wage stickiness. The impulse responses results indicate thatthe eﬀects of the ZLB in an economy under ﬂexible wages is smaller than that in an
economy under sticky wages.
Fourth, I change the persistence of the monetary policy shock under the Taylor rule.
An increase in the persisitece of monetary policy shock signiﬁcantly reduces the eﬀects
of the ZLB by reducing the response of the nominal interest rate. The increase in thepersistence of the monetary policy shock gives a long term feature to nominal interest
rates. This decreases the reduction in the nominal rate of interest in response to thepolicy shock, and then the absence of signiﬁcant easings is mitigated more than in thecase which the persistence of shocks is low.

3.1
3.1.1

Algorithms dealing with the ZLBThe HP algorithm

I explain the HP algorithm intuitively in this subsection.3
First, I need to solve the rational expectation model. In this paper I use the ‘Sims(2002) form’. Then, I derive impulse responses to unanticipated policy shocks. I deﬁne
impulse responses as Ul — where, Ul is the T × 1 matrix, T is the period of simulationand l corresponds to each variables in the model.
Second, I derive the impulse responses to anticipated shocks. I introduce the news
shock to the equation in which I want to set an inequality. If the ZLB constraint ispresent, the nominal rates maybe zero for some periods. The HP algorithm introduces
news shocks to accommodate this. Agents know when a neews shock will materialize,
and thus the agents can behave rationally given the information about the time thatshock will occur. This structure is applied to explorations of how an economy behaves
if it knows when the nominal rate of interest binds. In other words, the HP algorithm
replaces ‘future ZLB’ with ‘anticipated shock’. Holden and Paetz (2012) call this ‘shadow
shocks’. Shadow shocks are added into equations which include inequality-constrained
variables because we want to know how the economy behaves in response to the dynamics
of the nominal interest rate.
In this paper, I add the shadow shock term to the Taylor rule and to the money3

See Holden and Paetz (2012) for more details.

11

growth rule.
rt = ψπ πt + ψy yt + vtr +

∗ −1
T∑

n
vt,s

(32)

s=0

or
rt = σct + ςmt +

∗ −1
T∑

n
vt,s

(33)

s=0
n denotes news shocks. v n is a shock which is known at t − s and materializes
where, vt,st,sn is expressed as follows in an AR(1) system when s = 3.
at t. For example, vt,snxt = ρx xt−1 + vt + vt,3,n
vt,3= news3t









(34)

news3tnews2t−1
 news2t  =  news1t−1 
news1t²
Thus∑T ∗ −1s=0n means there are T ∗ systems like (34). In other words, we must derivevt,s

all impulse responses to shocks ² since we need all behavior of an agent when nominal
rates are binding, where, T ∗ ≤ T . Using a news shock algorithm, I derive the extended
version of impulse responses. I deﬁne impulse responses to news shock as Al , where ldenotes each variable, and Al are T × T ∗ matrices.
Next, I derive the impulse responses dealing with ZLB. The HP algorithm uses theidea of complementary conditions employing slack variables. First, it is necessary todeﬁne the impulse responses. The result of the impulse response of the nominal rate
must satisfy
rtz = max{− ln(1/β), rt }.

(35)

Holden and Paetz (2012) converted the form of Eq.(35) into the following parameterweighted form.
rtz = Ur,t + ln(1/β) + Ar,t α.

(36)where, Ur is the impulse response of the nominal interest rate to an unanticipated policy
shock and Ar denotes the impulse responses to T ∗ news shocks. α is a T ∗ × 1 vector.
Thus, the impulse responses accommodating the ZLB constraint consist of unanticipated12and anticipated components. The anticipated component is ampliﬁed by α to deal withrt + ln(1/β) ≥ 0.
Eventually, the problem is replaced to ﬁnd the optimal value of parameter vector α.
Holden and Paetz (2012) uses the idea of a complmentary slackness type condition,
α> (Ur + ln(1/β) + Ar α) = 0(37)

and then deﬁne the problem;α∗ =arg min α> (Ur + ln(1/β) + Ar α)
s.t. α ≥ 0, Ur + ln(1/β) + Ar α ≥ 0.

(38)

If the objective function is close to zero, it regards α∗ as satisfying the complementary
condition. MATLAB has a quadratic optimization function quadprog.m in its optimization toolbox. Since quadprog.m requires the initial value of α and α∗ is obtained
only when the objective function converges to zero, it is necessary to change the initialvalue of α or the number of news shocks impulse responses if the objective function does
not converge zero.
Finally, the responses dealing with ZLB for each variables are obtained as follows.
Ul + Al α∗ ,
3.1.2

(39)

The piecewise linear solution

Here, I explain piecewise linear solution. Guerierri and Iacoviello (2013) created the
MATLAB codes for the piecewise linear solution. They provide the codes on the web.4
The piecewise linear solution is an algorithm that replaces periods in which the nominal rate might bind with another impulse responses at which the nominal interest ratebinds. The algorithm needs two regimes as follows:
AEt Xt+1 + BXt + CXt−1 + F Et = 0,(40)

A∗ Et Xt+1 + B ∗ Xt + C ∗ Xt−1 + D∗ = 0,(41)

where, Xt is a vertical vector of variables. D∗ is a vertical vector which includes thedeviated threshold value from the steady state at which the nominal interest rate binds.
A, B, C, F, A∗ , B ∗ and C ∗ are structual matrices that include coeﬃcients. I suppose that4

See Guerrieri and Iacoviello (2013) for more details. They provide Dynare codes, occbin 20130531.zipon https://www2.bc.edu/˜iacoviel/

13

α
0.3

β
0.99

γ
0.7

ω
0.8

θ
6

θw
6

ψπ
1.5

ψy
0.5

ρµ
0.5

ρr
0.5

Table 1: Calibration

Eq.(40) satisﬁes the Blanchard-Kahn condition and the inequality constraint does notbind. The rational expected solution for this regime isXt = ΦXt−1 + ΨEt .

(42)

I also suppose that regime (41) does not always satisfy the Blanchard-Kahn condition
and the inequality constraint always binds. Now I suppose that an agent guesses that
regime (41) starts from Ts and ﬁnishes at Tf . The guessed solution for [Ts , Tf ] is obtained
as follows: Since the agent guessed that regime (41) ﬁnishes at Tf , the solution (42) isapplied after Tf . Then, Et XTf +1 = ΦXTf . Substituting this into Eq.(41) yields
XTf = ΦTf XTf −1 + ΓTf .

(43)

ΦTf ≡ −(A∗ Φ + B ∗ )−1 C ∗ , ΓTf ≡ −(A∗ Φ + B ∗ )−1 D∗(44)where,

Iterating this process, we obtain Φt and Γt for all t ∈ [Ts , Tf ]. The path can then be
simulated and veriﬁed. If the guessed solution is not veriﬁed, another guess can be triedby changing the value of Ts , Tf or both.

3.2

Benchmark impulse response

I set the deep parameters as the listed in Table 1. First, I simulate the model with the
Taylor rule. I construct the vector of variables as follows.
[
Xt ≡ rt , ct , it , yt , πt , nt , wt , kt−1 , rtk , st , qt ,k
Et ct+1 , Et πt+1 , Et wt+1 , Et rt+1
, Et it+1 , Et qt+1 , vtr

]>

(45)

Figure 1 illustrates the impulse response to the monetary easing shock when the policy
rule is the Taylor type. I set the value of the policy shock so that the nominal interest
rate responds to −1 at minimum. The solid blue line in the ﬁgure indicates the impulse
responses without the ZLB constraint. Since the easing policy stimulate the economy,
the decrease in the nominal interest rate raises the labor supply, output, inﬂation andinvestment. The responses of these variables, especially inﬂation and investment, are14

Nominal rate (r)Output (y)0.51percent deviation

0.8
0

0.6
0.4

−0.5

0.2
0

−1

0

5

10
quarters

15

20

−0.2

0

5

Inflation (π)

10

15

20

15

20

Labor supply (n)

0.1

1.5

0.08
1
0.06
0.04

0.5

0.02
0
0
−0.02

0

5

10

15

20

−0.5

0

5

r

Investment (i)Policy shock (v )

0.6

0

0.4

−0.5

0.2

−1

0

−1.5

−0.2

0

5

10

10

15

20

−2

without ZLB
piecewise linear
HP

0

5

10

15

20

Figure 1: Impulse responses to monetary policy shocks under the Taylor rule. Solid blue
line: the impulse response without the ZLB; Solid purple line: the impulse response with
the ZLB by the piecewise linear solution; Dashed green line: the impulse response with
the ZLB by the HP algorithm.

15

relatively small compared to the response of the nominal interest rate. Even though the
responses of the labor supply and the output are larger than that of the nominal interest
rate, they are not as large as twice the response of the nominal interest rate. Thus,
the response of the nominal interest rate is not greatly diﬀerent from those of the othervariables.
In Figure 1, the dashed green and solid purple line show the impulse responses obtained by the HP algorithm and the piecewise linear solutions. The result from these two
methods are extremely close. Figure 1 indicates that the nominal interest rate stays at
the lower bound until the 4th quarter and then becomes small positive. This describes
the zero lower bound on the nominal interest rate. The dynamics of the other variables
change dramatically toward the result without the ZLB constraint. The responses of allof the variables decrease markedly. The interpretation maybe as follows. The central
bank gives the monetary policy shock to stimulate the economy and then the nominalinterest rate decreases. If there is the ZLB constraint, however, the nominal interest rate
can no longer decrease under zero. This means that the nominal interest rate is not able
to ease signiﬁcantly because of the ZLB. The absence of signiﬁcant easing aﬀects theother macro variables delaying the positive response of the nominal interest rate further
and further. The nominal interest rate’s return to a positive status is delayed one quarterin this model toward the case without the ZLB constraint.
Next, I simulate the model with the money growth rule.
[
Xt ≡ rt , mt , ct , it , yt , πt , nt , wt , kt−1 , rtk , st , qt ,k
Et ct+1 , Et πt+1 , Et wt+1 , Et rt+1
, Et it+1 , Et qt+1 , µt

]>

(46)

Figure 2 shows the impulse responses to the positive shock under the money growth
rule. Similar to the result obtained with the Taylor rule, the easing policy in the money
growth rule stimulates the economy and increase all of the variables indicated in Figure
2. The positive money growth shock lowers the nominal interest rate. This is called the‘liquidity eﬀect’, which is deﬁned as the negative relationship between money growth andthe nominal interest rate. In the theoretical literature, the occurrence of the liquidity
eﬀect depends negatively on the persistence of money growth rate (Christiano et al. 1997).
I set the money growth persistence, ρµ , at 0.5 to generate as strong a liquidity eﬀect
as possible. The response of the nominal interest rate is very small relative to thoseof the other macro variables. Both the dashed green line and the solid purple line
in Figure 1 indicate the impulse responses dealing with the non-negative constraint.
The impulse response of the nominal rate binds the zero lower bound for ﬁrst 1 to16

Nominal rate (r)Output (y)percent deviation

0.5

150
100

0
50
−0.5
0
−1

0

5

10
quarters

15

20

−50

0

5

Inflation (π)
200

30

150

20

100

10

50

0

0
0

5

10

15

20

15

20

Labor supply (n)

40

−10

10

15

20

−50

0

5

10
Money growth (µ)

Investment (i)
300

140
120

200

100

without ZLB
piecewise linear
HP

80

100

60
40

0

20
−100

0

5

10

15

20

0

0

5

10

15

20

Figure 2: Impulse responses to monetary policy shocks under the money growth rule.
Solid blue line: the impulse response without the ZLB; Solid purple line: the impulse
response with the ZLB by the piecewise linear solution; Dashed green line: the impulse
response with the ZLB by the HP algorithm.

17

Table 2:
Variables
yt
it
nt
πt

max(PW)
0.075585
0.124390.107420.020795

Taylor rulemax(HP) max(IRF)
0.081866
0.856860.13501
0.579
0.11695
1.2241
0.021413
0.082309

max(PW): maximum responses with the ZLB by the piecewise linear solution; max(HP): maximum responses with the ZLB by the HP algorithm; max(IRF): maximum responses without the
ZLB to monetary policy shocks for each variables.

Table 3:
Variables
yt
it
nt
πt

Money growth rulemax(PW) max(HP) max(IRF)
115.3241115.8718
117.2244
228.1931
233.1472
230.1629
163.5437
164.3163
166.2422
37.4943
36.7371
37.7284

max(PW): maximum responses with the ZLB by the piecewise linear solution; max(HP): maximum responses with the ZLB by the HP algorithm; max(IRF): maximum responses without the
ZLB to monetary policy shocks for each variables.

7 quarters and reaches the lower bound again for 17 quarters. The nominal interest
rate could not ease signiﬁcantly because of the ZLB. The result of this insigniﬁcanteasing in the nominal interest rate spills over and then lowers the response of the othervariables. However, the reductions in responses are extremely small. This is reﬂected by
the extreme closeness of the solid blue line, the solid purple line and the dashed green
line.

Next I compare the eﬀect of zero lower bound between the Taylor rule and themoney growth rule. First, I compare the maximum responses of four variables. Tables
2 and 3 indicate the maximum responses of yt , it , nt and πt to each monetary policy
shock. The columns labeled max(PW) and max(HP) indicate the maximum value of
responses to the monetary policy shock in the HP algorithm and the piecewise linear
solution. The colunmns of max(IRF) indicate the maximum value of responses to themonetary policy shock without constraint. The result from max(PW) and max(HP) areclose to each other. In Table 2, the results from max(PW) and max(HP) are less thanthose of max(IRF). On the other hand, the results from max(PW) and max(HP) are18

Table 4: The Eﬀects of ZLB indicate
variables
Eﬀects of ZLB
yt
it
nt
πt

∑
∑
|IRF−ZIRF|/ |IRF| for each variables.

Taylor rulePW
HP
80.4885% 78.2123%
74.3258% 71.8679%
82.1438% 80.4424%
70.6225% 69.5775%

Money growth rule
PW
HP
1.6348% 4.2977%
1.0829% 4.5272%
1.5695% 3.8598%
1.0385% 3.1222%

close to max(IRF) in Table 3. Thus, the maximum responses to monetary policy shocks
are more greatly aﬀected by the non-negative constraint on the nominal interest rate
under the in Taylor rule than under the money growth rule.
Second, I compare the eﬀect of the zero lower bound in impulse responses for 20
periods.
20∑|IRFt − PWt |/

t=0

20
∑

|IRFt | and

t=0

20
∑
t=0

|IRFt − HPt |/20∑|IRFt |

t=0

Table 4 indicates that the eﬀects of constraint are much larger under the Taylor rulethan the money growth rule. The responses with constraint were about 70%-80% of
the size of the responses without constraint under the Taylor rule. The responses withconstraint were about 1%-4% of the size of the responses without constraint under themoney growth rule.

3.3

Flexible wages

Next, I derive the impulse responses without the wage stickiness. Since there is no
stickiness in wage, the version of the Philips curve equation is substituted for the labor
supply equation from which the household’s ﬁrst order conditions are derived. The loglinearized version isnt + σct = wt .

(47)

Figure 3 indicates the impulse responses to the negative monetary policy shock under
the Taylor rule. Some standard responses without sticky wages become larger than the
responses with sticky wages and others do not. The response of the nominal interest
rate with a ﬂexible wages rate is smaller than that of the nominal interest rate withsticky wages. Table 5 indicates the maximum responses to the monetary policy shock19

percent deviation

Nominal rate (r)Output (y)

0.5

1

0

0.5

−0.5

0

−1

0

5

10
quarters

15

20

−0.5

0

5

Inflation (π)

10

15

20

15

20

Labor supply (n)

0.8

1.5

0.6

1

0.4
0.5
0.2
0

0
−0.2

0

5

10

15

20

−0.5

0

5

r

Investment (i)Policy shock (v )

0.6

0
−0.5

0.4

without ZLB
piecewise linear
HP

−1
0.2
−1.5
0
−0.2

10

−2
0

5

10

15

20

−2.5

0

5

10

15

20

Figure 3: Impulse responses to monetary policy shocks under the Taylor rule and ﬂexible
wages. Solid blue line: the impulse response without the ZLB; Solid purple line: theimpulse response with the ZLB by the piecewise linear solution; Dashed green line: theimpulse response with the ZLB by the HP algorithm.

20

Table 5:variables
yt
it
nt
πt

Maximum responses
max(PW) max(HP) max(IRF)
0.46998
0.51360.999720.30450.325130.43061
0.671390.73371
1.42820.45146
0.47694
0.62137

Eﬀects of ZLB
PW
HP
42.2759% 38.4409%
25.9485% 24.1256%
41.3739% 37.1486%
23.2359% 27.5527%

max(PW): maximum responses with the ZLB by the piecewise linear solution; max(HP): maximum responses with the ZLB by the HP algorithm; max(IRF): maximum responses without the
ZLB to monetary policy shocks under ﬂexible wages. The columns of Eﬀects of ZLB indicate∑∑|IRF−ZIRF|/ |IRF| for each variable, where, the monetary policy rule is the Taylor type.

in some variables. The standard maximum response of inﬂation becomes much larger
than that with sticky wages. This feature is consistent with the study by Christiano etal. (2005) in that the nominal rigidities, especially the wage rigidity, contributes to theinitial dynamics of inﬂation.
The responses with the ZLB constraint become larger than those with sticky wage.
The intuitive reason for the reduction in the eﬀects of the ZLB is that the inﬂation
dynamics becomes larger because of the absence of sticky wage. The inﬂation can abosorb
the eﬀects of the constraint more than before since the dynamics of inﬂation becomelarger. Table 5 provides the sticky wage versions of Tables 2, 3 and 4. First, themaximum responses of variables with the constraint are more close to max(IRF) thanthe case with sticky wages. Second, the eﬀects of the constraint decrease in all variables.
Thus, the eﬀects of monetary policy shock under the Taylor rule with frexible wages arelarger than those with the sticky wages.

3.4

The persistency in policy shock under the Taylor rule

The persistency in policy shock under the Taylor rule, ρr , contributes to the dynamics
of the nominal interest rate. Figure 4 indicates the impulse responses when ρr is 0.9.
The impulse response of the nominal intrerest rate with the ZLB stays zero for initialperiods, but it departs from zero earlier than the case without the ZLB. The responsesof other variables with the ZLB became close to those without the ZLB. An increasein the persistence of policy shock lowers the negative response of the nominal interest
rate because high persistence of policy shock indicates that monetary easing continues
longer and the nominal interest rate takes on a long term aspect in the model. These21

percent deviation

Nominal rate (r)Output (y)

2

5

1.5

4

1

3

0.5
2

0

1

−0.5
−1

0

5

10
quarters

15

20

0

0

5

Inflation (π)
6

1.5

4

1

2

0.5

0

0

5

10

15

20

−2

0

5

6

−1

4

−2

2

−3

0

−4
5

10

10

15

20

Policy shock (v )
0

0

20

r

Investment (i)
8

−2

15

Labor supply (n)

2

0

10

15

20

−5

without ZLB
piecewise linear
HP
0

5

10

15

20

Figure 4: Impulse responses to monetary policy shocks under the Taylor rule whenρr = .9. Solid blue line: the impulse response without the ZLB; Solid purple line: theimpulse response with the ZLB by the piecewise linear solution; Dashed green line: theimpulse response with the ZLB by the HP algorithm.

22results indicate that an increase in the persistence of policy shock under the Taylor rule
dramatically mitigates the eﬀects of ZLB through the reduction of the response of thenominal interest rate.

4

Conclusion and Future Task

The main ﬁnding of this paper is that the inﬂuence of the ZLB on the eﬀects of monetary
policy shock under the Taylor rule is larger than under the money growth rule in a typical
DSGE model. The reduction in the nominal interest rate is small compared to the money
growth shock. The ZLB constraint on the nominal interest rate has little eﬀect on theother variables because of the insigniﬁcant negative responses of the nominal interest
rate to money growth shocks. In other words, the ZLB does not aﬀect the economy somuch because of the weak liquidity eﬀect. However, this result is unrealistic because
there might be a strong liquidity eﬀect in the actual economy. It is thus important touse models that can generate a strong liquidity eﬀect. This implication is related to thethird analysis described herein, which assessed the eﬀects of ZLB in high persistence ofshocks to Taylor rule. The ZLB might have a signiﬁcant eﬀect on the impact of monetary
policy shocks under the Taylor rule in models that generate strong liquidity eﬀects eventhough shock persistence is high.
Second, ﬂexible wages reduce the eﬀects of the ZLB and increase some variables’
responses. Wage stickiness aﬀects the dynamics of inﬂation through the marginal costsof the intermediate goods sector. A reduction in the stickiness of wages raises inﬂationand then reduces the response of the nominal interest rate. A decrease in the response of
nominal rate of interest mitigates the ampliﬁcation of the eﬀects of anticipated bindings inthe HP algorithm, but this result does not necessally indicate that reduce wage stickinessis good method of decreasing the eﬀects of the ZLB. This issue should be explored further
in the ﬁeld of optimal monetary policy.
Third, the results obtained with the HP algorithm and those obtained with thepiecewise linear solution are close to each other in all analyses in this model.
Finally, these results are not necessarily consistent with the traditional IS-LM literature since monetary policy shocks aﬀect the economy under some cases. This result
may indicate that the model cannot explain the real economy or that monetary policy
is eﬀective even though the nominal interest rate cannot decrease further than zero. It
is, however, diﬃcult to suggest the latter implication. As noted in above, it is important23

to search for a way to greatly reduce response of of the nominal interest rate to themoney growth shocks. The eﬀects of monetary policy employing the Taylor rule might
become close to completely ineﬀective in such models. The liquidity eﬀect might be more
important, because of the ZLB.

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25