The liquidity eﬀect and tightening eﬀect of the zero
lower bound∗
Kohei Hasui†
May 2014

Abstract
The tightening eﬀect of the zero lower bound on the nominal interest rate is a non-trivial
topic in monetary policy: at the zero lower bound, responding to a rise in money growthby reducing the nominal interest rate — what is called the liquidity eﬀect — is not possible
because the nominal interest rate cannot be further decreased. However, the absence of theliquidity eﬀect caused by the zero lower bound might amplify the tightening eﬀect of thezero lower bound. I call this tightening eﬀect of the zero lower bound through its liquidity
eﬀect on the economy the rebound of liquidity eﬀect, and demonstrate it quantitatively witha simple dynamic stochastic general equilibrium framework.
Keywords: liquidity eﬀect, zero lower bound, monetary policy
JEL Classification: E47; E49; E52∗
Forthcoming in Japanese Journal of Monetary and Financial Economics, Japan Society of Monetary Economics.
†
Doctoral student, Graduate School of Economics, Kobe University, 2-1 Rokkodai-cho Nada-ku, Kobe 6578501, Japan. Email: 100e107e@stu.kobe-u.ac.jp.

1

Introduction

In the short run, a rise in the growth of nominal money lowers the short-run nominal interest
rate. This is called the liquidity eﬀect. The liquidity eﬀect does not appear when the nominalinterest rate hits the zero lower bound, which is the situation that developed countries faced
after the colossal financial crisis of 2008. However, the absence of the liquidity eﬀect causedby the zero lower bound might aﬀect the entire economy through its tightening eﬀect. In thispaper I describe how I examined the tightening eﬀect of the zero lower bound on the economy
quantitatively in a dynamic stochastic general equilibrium (DSGE) framework that generates astrong liquidity eﬀect.
Researchers have studied the liquidity eﬀect both theoretically and empirically. Since standard DSGE models do not generate the liquidity eﬀect,1 researchers have retrofitted the structures of various models to generate the eﬀect (Christiano, 1991; Christiano and Eichenbaum,
1995; Christiano and Gust, 1999; Christiano et al., 1997, 2005; Keen, 2004; Edge, 2007). Christiano (1991), Christiano and Eichenbaum (1995), and Christiano and Gust (1999) modeled theliquidity eﬀect by introducing a structure in which the timing of loan market participation
between households and firms diﬀered (the limited participation model). In the empirical literature, analyses were performed mainly with vector autoregression (VAR) (Christiano et al.,
1999; Leeper and Gordon, 1992; Leeper et al., 1996). Since the liquidity eﬀect does not appear
in the straightforward specification of VAR, Christiano et al. (1999), for example, used thenon-borrowed reserve corresponding to the variable of monetary policy to generate the liquidity
eﬀect. However, these previous studies examined the liquidity eﬀect without considering thezero lower bound on nominal interest rates. Since the liquidity eﬀect does not appear when thenominal interest rate hits the zero lower bound, it has not been studied in situations involving
the zero lower bound.
It is possible, however, that the zero lower bound on nominal interest rates could havea tightening eﬀect through the liquidity eﬀect. If the model economy has a structure which
generates a strong reduction of the nominal interest rate in response to increases of money
growth, a large part of the nominal interest rate which could decrease if the zero lower bounddoes not exist might spill over into another part of the economy, such as the output, inflation,and real money, etc., because the non-negativity constraint does not allow the nominal interest
rate to decrease further. Thus the tightening eﬀect of the zero lower bound could depend on the1strength of reductions in the nominal interest rate if the zero lower bound did not exist. Thisline of reasoning must be explored by a theoretical rather than empirical analysis. I borrow thespecification of households portfolio adjustment cost function developed by Christiano and Gust
(1999) to generate a persistent and strong liquidity eﬀect. This specification enables the model
to generate and adjust the strength of the liquidity eﬀect easily, without a limited participation
formation that includes complicated structures. I show that a positive money growth shock canstimulate the economy more with a strong liquidity eﬀect than without the liquidity eﬀect when
the nominal interest rate does not hit the zero lower bound. I also show that a positive money
growth shock cannot stimulate the economy more with a strong liquidity eﬀect than without
this eﬀect when the nominal interest rate hits the zero lower bound. These results demonstratethat the tightening eﬀect of the zero lower bound might be amplified by a structure which
generates the strong liquidity eﬀect. This possibility has not received much attention.
The remainder of this paper is structured as follows: Section 2 presents a description of themodel, and Section 3 shows the results of the simulation. Section 4 contains my conclusions.

2

The modelThis section describes the economic model. The model includes households, the firm sector
and the government. The firm sector consists of intermediate goods producers and final goods
producers; firms in the intermediate goods sector produce diﬀerentiated goods following the
Calvo (1987) sticky pricing rule.
Optimization problems for households and firms derive the aggregate demand equation, themoney demand equation and the aggregate supply equation. I describe details for the derivationof these model equations below.

2.1

Households

Households obtain utility from consumption Ct and real money balances mt , and disutility fromthe labor supply Nt .
In addition, I introduce the portfolio adjustment cost function of real money balances into
households utility function in order to generate the strong liquidity eﬀect.
[
]
∞
1+η
1−σ
1−ς
∑
C
m
N
t
E0
βt
+ t − t− Ψ(mt , mt−1 ) .
1−σ
1−ς
1+η
t=0

2

(1)

where β denotes the subjective discount factor, σ denotes the relative risk aversion of consumption, ς denotes the relative risk aversion of real money, and η denotes Frisch elasticity of labor
supply. The last term in the bracket is the adjustment cost function represented by Christiano
and Gust (1999).
{
[ (
)]
[
(
)]
}
mt
mt
Ψ(mt , mt−1 ) = δ exp α− 1 + exp −α−1 −2 ,
mt−1
mt−1
Ψ(m, m) = 0
where α and δ are non-negative parameters and m is the steady-state value of mt . It is unusual
to introduce the cost function into the utility. The reason for this assumption is (i) it provides
a short-cut to generating a strong and persistent liquidity eﬀect, and (ii) it is tractable toanalyzing the impacts of the liquidity eﬀect on the economy by adjusting parameters δ and α.2
The households budget constraint is
Ct + bt + mt ≤ wt Nt + Rt−1

bt−1 mt−1++ τt + d t .
Πt
Πt

(2)

where bt , wt , Rt , Πt , τt , and dt are the real bond, real wage, gross nominal interest rate, grossinflation rate, lump-sum tax, and profit from the firm sector, respectively. Equation (2) showsthat households allocate income from wages, bond yields, lump-sum tax and firms profits to
consumption, money and bonds. Households maximize lifetime utility (1) subject to Eq. (2)dynamically. The first-order conditions are as follows:

where

C −σ
Ct−σ
= βEt t+1 ,
Rt
Πt+1

(3)

[
] −σ
−1
m−ς
Ct ,
t − Ψ1 (mt , mt−1 ) − βEt Ψ2 (mt+1 , mt ) = 1 − (Rt )(4)

Ntη Ctσ = wt ,

(5)

{
[ (
)]
[
(
)]}
mt
mt
exp α
− 1 − exp −α
−1
mt−1
mt−1
{
[ (
)]
[
(
)]}
(δα)mtmt
mt
Ψ2 (mt , mt−1 ) = −exp α− 1 − exp −α−1(mt−1 )2
mt−1
mt−1
δα
Ψ1 (mt , mt−1 ) =mt−1

Equation (3) is the Euler equation, which describes households optimal intertemporal saving
decisions. Equation (4) is the money demand equation, which describes the opportunity costsof holding money equal to the nominal interest rate. Equation (4) is diﬀerent from standard3

money demand equations in that it depends on the expected real money balance and lagged
real money balance. Since I assume the portfolio adjustment cost of the real money balance,
the opportunity costs of holding money means that marginal adjustment costs are tacked onto
the nominal interest rate. Finally, Eq. (5) is the labor supply equation, which shows that themarginal disutility of labor must equal the real wage.

2.2

Final goods sector

The final goods sector inputs diﬀerentiated goods, Yt (i), i ∈ (0, 1), from the intermediate goods
sector. The final goods sector then aggregates Yt (i) and sells it to households. The production
function of the final goods sector is thus:
[∫ 1
Yt =

Yt (i)

θ−1
θ

θ
] θ−1

di

,

θ>1

0

where θ denotes the elasticity of substitution between diﬀerent goods.
The intratemporal profit maximization of the final goods sector yields the goods for firm isdemand equation and aggregate price definition:
Yt (i)
=
Yt

(

)−θ

Pt (i)
Pt

,

[∫ 1
1−θ

Pt =

Pt (i)

1
] 1−θ
di
.

(6)

(7)

0

2.3

Intermediate goods sector

Firms in the intermediate goods sector are an infinite continuum for the interval (0, 1), and firmi produces intermediate goods Yt (i) with the product function:
Yt (i) = At Nt (i),where At is productivity and is independent from the indexi, and A = 1 at the steady state.
By the cost minimization for firm i, the real marginal costs equal the real wage per unit ofproductivity:
st =

wt
At

st denotes the real marginal cost, and firm is real marginal product is independent from theindex i.
4

The intermediate goods sector is monopolistically competitive. Moreover, each firm sets theprices for their goods following Calvos (1983) pricing rule: the firm has a 1 − ω probability ofchanging the price, and a ω probability of not changing the price. Given these probabilities andthe demand equation for goods i, intermediate goods firms set their optimal prices to maximize
yields,
Pt∗ =

(

θ
θ−1

)

∑
Et ∞
ωk Λ
s Y (i)∑∞k=0 k t,t+k t+k t+k
Et k=0 ω Λt,t+k Yt+k (i)/Pt+k(8)

where Λt,t+k denotes the stochastic discount factor, β k (Ct+k /Ct )−σ . The intuition of Equation(8) is that the optimal price equals the expected marginal cost weighted by markup θ/(θ − 1).
Applying Calvos (1983) pricing rule to Equation (7), the aggregate price takes a recursive
form, which consists of the changed price component and the unchanged price component.
1−θ
Pt1−θ = (1 − ω)(Pt∗ )1−θ + ωPt−1(9)

All prices that cannot be changed in the current period are reduced into the second term on
the right-hand side of Equation (9).

2.4Government

The government has the following budget constraint:
Rt−1 bt−1 mt−1+
+ τt = bt + mt
ΠtΠt
The monetary authority injects money into the system following the money growth rule.
µt = m̂t − m̂t−1 + πt ,µt = ρµt−1 + ϵt .
m̂t denotes the percent deviation form of real money, πt denotes the percent deviation form of
inflation, µt is the growth rate of nominal money, and ϵt is an i.i.d. disturbance term.

5

Table 1: Baseline parameter values
β
σ
ς
η
ω
α
δ
ρ

3

subjective discount factor
intertemporal substitution for consumption
intertemporal substitution for real money
Frisch elasticity of labor
price stickinessparameter in cost function
parameter in cost function
persistence of money growth

0.99
2
2
1
0.8
2
2
0.5

Simulation

Now I present the complete log-linearized model.3 All variables are the percent deviation fromthe steady state.
xt = Et xt+1 −1(rt − Et πt+1 ) + ut
σπt = βEt πt+1 + κxt ,

(10)
(11)

m̂t + ψ∆m̂t − ψβEt ∆m̂t+1 =σβ(xt + ytf ) −rt ,ςς(1 − β)(12)µt = m̂t − m̂t−1 + πt ,(13)µt = ρµt−1 + ϵt .

(14)

where ψ = 2δα2 /(ςm1−ς ), m = (1 − β)1/ς , ytf = (1 + η)/(η + σ) ln At is the flexible price output,f
xt = yt − ytf is the output gap, and ut = Et yt+1
− ytf . If ψ = 0, the model becomes thestandard New Keynesian model. Assume the baseline parameters are set as shown in Table 1;
the parameters of the adjustment cost function, α = δ = 2, were set following Christiano andGust (1999), and the parameters of the persistence of money growth, ρ = 0.5, were set following
Christiano et al. (1998).4,5 The simulation model was run and the results of impulse responses
are analyzed below.
In Subsection 3.1, impulse responses of the model are derived with and without adjustment
costs, and the occurrence of the strong liquidity eﬀect is checked with the adjustment cost model.
Subsection 3.2 derives impulse responses considering the zero lower bound on the nominalinterest rate, and analyzes the tightening eﬀect of the non-negativity constraint on the nominalinterest rate in the model in which the strong liquidity eﬀect occurred.
6

(b) Inflation% ∆ from SS

1
0

0

5

10
15
Quarters(c) Nominal interest rate20

% ∆ from SS

0.5
0
−0.5

% ∆ from SS% ∆ from SS% ∆ from SS(a) Output gap
2

−1

0

5

10
15
Quarters(e) Money growth

20

0.4
0.2
0

0

5

10
15
Quarters(d) Real money balance

20

1.5
1
0.5
0

0

5

10
Quarters

15

20

1
without adjustments costwith adjustment costs

0.5
0

0

5

10
Quarters

15

20

Figure 1: Impulse responses to a positive money growth shock. Solid lines: without the adjustment cost; dashed lines: with the adjustment cost

3.1

The liquidity eﬀect

First, the assumption that the adjustment costs would generate the liquidity eﬀect under thebaseline parameters was confirmed. Figure 1 illustrates impulse responses to a positive money
growth shock. The solid lines in the figure indicate impulse responses without adjustment costs,
and the dashed lines are impulse responses with adjustment costs; the solid line in Figure 1e
represents the money growth shock common to both models, with and without the adjustment
cost. Figure 1 shows that the positive money growth shock stimulated the economy: the outputgap responded positively, and since the nominal rigidity of price levels existed, the inflation andreal money balance increased, and the nominal interest rate decreased. There was a remarkablediﬀerence between the case with adjustment costs and the case without adjustment costs interms of the output-gap and the nominal interest rate. The nominal interest rate decreased
more when adjustment costs were present. This was the liquidity eﬀect. Through the Euler

7equation for households, the strong liquidity eﬀect raised the output gap more with adjustment
costs than without. Thus, the monetary stimulus raised the economy to a greater extent throughthe strong liquidity eﬀect.

3.2

The zero lower bound on nominal interest rate

Next a non-negativity constraint was imposed on the nominal interest rate:rt ≥ − ln(1/β)(15)

The zero lower bound in the log-linearized model became − ln(1/β), considering that Rt was
1/β at the steady state and that all the variables became zero at the steady state in the
log-linearized model. The derivation of impulse response considering the zero lower bound iscomplicated because of its nonlinear nature. I used the MATLAB Toolbox provided by Guerrieri
and Iacoviello (2014), which derives impulse responses while occasionally considering binding
constraints by piecewise linear solutions.6
Figure 2 illustrates the impulse responses of the model with adjustment costs. Solid lines
indicate responses with the constraint, and dashed lines represent the responses without theconstraint. Figure 2 shows that the positive money growth shock stimulated the economy, andthat the non-negativity constraint reduced its eﬀect; the nominal interest rate hit the zero lower
bound (Fig. 2c) and this contracted upward responses of output-gap and inflation (Fig. 2a,b).
Since adjustment costs existed, the nominal interest rate decreased strongly toward the positive
money growth shock. This allowed the nominal interest rate trend to hit the zero lower boundmore intensely, with the potential to cause a more severe tightening eﬀect on the economy. Thiseﬀect is analyzed in Figure 3, which illustrates impulse responses to the positive money growthshock with the non-negativity constraint on the nominal interest rate. The solid lines in Figure
3a-d indicate the case with adjustment costs, and the dashed lines indicate the case without
adjustment costs. Figure 3e illustrates real money growth ∆mt (dashed line) and expected
real money growth Et ∆mt+1 (solid line) in cases in which adjustment costs and the zero lower
bound exist. Figure 3f illustrates the diﬀerence between the discounted expectation of realmoney growth and the current real money growth in Figure 3e, βEt ∆mt+1 − ∆mt . The resultspresented in Figure 3ad are the opposite of those shown in Figure 1. A money growth shock
stimulates the economy less with adjustment costs than without adjustment costs—–the solidlines of Figure 3a-d are below the dashed lines in output gap and inflation—–but the diﬀerence8(b) Inflation% ∆ from SS

1
0

0

5

10
15
Quarters(c) Nominal interest rate20

% ∆ from SS

0.5
0
−0.5

% ∆ from SS% ∆ from SS% ∆ from SS(a) Output gap
2

−1

0

5

10
15
Quarters(e) Money growth

20

0.4
0.2
0

0

5

10
15
Quarters(d) Real money balance

20

1.5
1
0.5
0

0

5

10
Quarters

15

20

1
with ZLB
without ZLB

0.5
0

0

5

10
Quarters

15

20

Figure 2: Impulse responses to a positive money growth shock. Solid lines: with the nonnegativity constraint on the nominal interest rate; dashed lines: without the non-negativity
constraint on the nominal interest rate.

9

0.5
0

0

5

10
15
Quarters(c) Nominal interest rate

% ∆ from SS

0.01
0
−0.01
−0.02

0

5

10
15
Quarters(e) ∆m(t) and E∆m(t+1)% ∆ from SS

0.50% ∆ from SS

−0.5

0

5

10
15
Quarters(g) Money growth

0

5

10
15
Quarters(d) Real money balance

20

5

1.5
1
0.5
0

10
15
Quarters(f) βE∆m(t+1)−∆m(t)

20

0

5

20

0.01
0
−0.01

20

1

10
Quarters

15

with adjustment costs
without adjustment costs

0.5
0

0.2

0

20

1

0.4

0

20

% ∆ from SS% ∆ from SS(b) Inflation% ∆ from SS% ∆ from SS(a) Output gap
1∆m(t)E∆m(t+1)
0

5

10Quarters

15

20

Figure 3: Impulse responses to a positive money growth shock with the non-negativity constraint
on the nominal interest rate. a-d: Solid lines indicated cases with the adjustment cost; dashedlines indicate those without the adjustment cost. f: Expected real money growth minus realmoney growth with the adjustment cost (zoomed for interval [0.01, 0.01])

10

(b) Inflation% ∆ from SS

0.5

0

5

10
15
Quarters(c) Nominal interest rate20

% ∆ from SS

0

0.02
0
−0.02

% ∆ from SS% ∆ from SS% ∆ from SS(a) Output gap
1

0

5

10
15
Quarters(e) Money growth

20

0.4
0.2
0

0

5

10
15
Quarters(d) Real money balance

20

1.5
1
0.5
0

0

5

10
Quarters

15

20

1
with adjustment costs
without adjustment costs

0.5
0

0

5

10
Quarters

15

20

Figure 4: Impulse responses to a positive money growth shock with the non-negativity constraint
on the nominal interest rate when α = δ = 3. Solid lines: with the adjustment cost; dashedlines: without the adjustment cost.
is not as large as in the previous simulation (Fig. 3a,b). This implies that the zero lower boundmight tighten the economy more through the strong liquidity eﬀect. I call this the rebound of
liquidity eﬀect. The nominal interest rate hits the zero lower bound for six quarters and thenrises higher than without adjustment costs. In Figure 3f, βEt ∆mt+1 − ∆mt , which is the termin the money demand equation, shows dynamics similar to the behavior of the nominal interest
rate. Since the term ψ(βEt ∆mt+1 − ∆mt ) is derived from the adjustment cost specification,
the behavior of the nominal interest rate depends on the adjustment cost. Thus the liquidity
eﬀect occurs through adjustment costs, and the rebound of liquidity eﬀect is expected to growas adjustment costs increase. Let us now set the parameter α = δ = 3; Figure 4 illustrates theresults with these parameters. As can be seen in the figure, the diﬀerence between the solidlines and dashed lines becomes large. The dynamics of the nominal interest rate are similar to
the results shown in Figure 3 but are greater. Since the parameter α = δ = 3, the value of ψ

11

increases. This increase amplifies the eﬀect of βEt ∆mt+1 − ∆mt to the nominal interest rate.
According to the results shown in Figure 4, if the model has a structure which could generate
the stronger liquidity eﬀect when the zero lower bound does not exist, the great contractioneﬀect occurs if the nominal interest rate hits the zero lower bound.
These results suggest some considerations for monetary policy. First, monetary policy canstimulate the economy by producing a strong liquidity eﬀect when the nominal interest rate
does not hit the zero lower bound. However, once the nominal interest rate hits the zero lower
bound, the existence of the strong liquidity eﬀect might have the opposite outcome and amplify
the tightening eﬀect of the zero lower bound. Second, monetary policy should pay attention to
the contraction eﬀect of the zero lower bound when an economy that possesses the structures togenerate a strong liquidity eﬀect falls into the zero lower bound. It could then be important todevelop policies to mitigate the structures of the strong liquidity eﬀect. For example, a policy
mitigating wage stickiness might be eﬀective, because it would contribute to the dynamics of
expected inflation, which puts upward pressure on the nominal interest rate through the Fisher
relationship (Hasui, 2013). However, too much mitigation of inflation dynamics is risky, because
once inflation diverges, the central bank cannot stop it.

3.3

Considering empirical evidence

It must be noted that the implications described in the previous subsection are based on theoretical and quantitative results. It is also important to consider empirical evidence. Nelson(2002) chose the parameter value α = δ = 0.43 considering the evidence analyzed by Taylor(1993) and Janssen (1998). Nelson (2002)s choice of parameters implies α = δ = 0.7937 in
my model.7 Figure 5 illustrates impulse responses to a positive money growth shock with the
non-negativity constraint on the nominal interest rate under α = δ = 0.7937. The result implies
that the rebound of the liquidity eﬀect is extremely small. By reducing the cost parameters,
α and δ, decreases of the nominal interest rate become small in the money demand equation.
The zero lower bound tightens the economy by constraining the nominal interest rate, but its
eﬀect is quietly small because the decrease of the nominal interest is small to begin with. Thisempirically assessed result implies that it is not necessary for monetary authorities to pay attention to the contraction eﬀect of the zero lower bound. As described in the previous subsection,
since the tightening eﬀect of the zero lower bound through the liquidity eﬀect is not large underα = δ = 2, it might not necessarily be important to develop policies to mitigate the structures12

(b) Inflation% ∆ from SS

0.5

0

5

10
15
Quarters(c) Nominal interest rate20

% ∆ from SS

0

0
−0.01
−0.02

% ∆ from SS% ∆ from SS% ∆ from SS(a) Output gap
1

0

5

10
15
Quarters(e) Money growth

20

0.4
0.2
0

0

5

10
15
Quarters(d) Real money balance

20

1.5
1
0.5
0

0

5

10
Quarters

15

20

1
with adjustment costs
without adjustment costs

0.5
0

0

5

10
Quarters

15

20

Figure 5: Impulse responses to a positive money growth shock with the non-negativity constraint
on the nominal interest rate when α = δ = 0.7937. Solid lines: with the adjustment cost; dashedlines: without the adjustment cost.

13

of the liquidity eﬀect generated through the money demand equation.

4

Conclusion

In this paper, I quantitatively analyzed the tightening eﬀect of the zero lower bound on thenominal interest rate. The main findings are as follows. (1) When the nominal interest rate
hits the zero lower bound, the existence of a structure which generates the strong liquidity
eﬀect mitigates the eﬀects of a money growth shock. (2) These results show that the tightening
eﬀect of the zero lower bound on a monetary policy shock might depend on the strength of the
reduction in the nominal interest rate, i.e., the potential liquidity eﬀect. (3) In my assessment of
the empirical literature, however, the model implied that the tightening eﬀect of the zero lower
bound through the rebound of the liquidity eﬀect is quite small. Thus, while the tightening
eﬀect of the zero lower bound is non-trivial in monetary policy, it may not be necessary to pay
a great deal of attention to it in monetary policy.

Notes
1. To say that the model does not generate the liquidity eﬀect means that the model does not
generate a suﬃcient reduction of the nominal interest rate toward a rise in money growth.
2. Christiano and Gust (1999) introduced the adjustment cost function to provide a persistent
response of the liquidity eﬀect. Andrés et al. (2004) and Nelson (2002) also used this costfunction for a similar reason.
3. See Walsh (2010), Chapter 8 for details of the derivation.
4. Nelson (2002) chose α = α = 0.43 considering an empirical assessment; however, since I am
interested in the eﬀects of the zero lower bound in the situation where a strong liquidity
eﬀect exists, I choose the parameter value α = α = 2 following Christiano and Gust (1999).
To consider the empirical assessment, model behavior with the zero lower bound underα = δ = 0.7937 is analyzed in subsection 3.3.
5. The higher the persistence in money growth becomes, the more positively the nominalinterest rate tends to respond to the positive money growth.
6. See Guerrieri and Iacoviello (2014) for details about piecewise linear solutions.
14

7. Taylor (1993) and Janssen (1998) estimated the coeﬃcient of the real money balance inthe money demand equation for the U.S. and U.K., respectively. Nelson (2002) set ψ = 10
following these results. This setting is consistent with my model since the money demand
equation is mostly identical (see Nelson 2002, p.700). Then, ψ = 10 implies α = δ = 0.7937
in my model.

Acknowlegdments
The author would like to thank two anonymous referees for helpful comments and suggestions.
The author also would like to thank Eiji Ogawa for his quick and proper response. Part of thispaper was written when I was a master course student at Kobe University. The author wishes
to thank professor Toshiki Jinushi for his helpful comments and suggestions. The author also
would like to thank Agata Wierzbowska for helpful comments and suggestions. All remaining
errors are my own.

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