Monetary policy and the adjustment cost between money and assets*
Kohei Hasui†
January, 24, 2014

ABSTRACTThis paper considers monetary policy when an adjustment cost between money and an asset
exists. My ﬁndings are as follows: First, a positive money growth shock generates a signiﬁcantdecrease in the nominal interest rate through households' portfolio rebalances (the liquidity
eﬀect). Second, the determinacy of the model depends on the portfolio adjustment cost of themoney and asset. According to the value of the portfolio adjustment cost, a monetary policy
must raise the nominal interest rate aggressively in response to a rise in inﬂation in order tobring the economy to the locally unique equilibrium. Third, a money velocity shock aﬀects theentire economy even though the monetary policy is based on the Taylor rule. Finally, themonetary authority might be able to mitigate the eﬀects of a negative money velocity shock if
the monetary authority controls the money growth so as to keep the interest rate ﬁxed. If themoney growth rule can be regarded as a strand of a quantitative easing policy, I conclude thatthe quantitative easing rule might be eﬀective to prevent the economy from the eﬀect of
decreasing the money velocity.

Keywords: monetary policy, adjustment cost, liquidity eﬀect, interest rate
JEL Classification: E47; E49; E5*

Rokko Forum Working Paper Series No.1302. The previous title of this paper is ''Monetary policy and imperfect
substitution between money and asset''. The title was changed because the author considers the comments in the
Rokko forum at Kobe University.
†
Ph.D Student, Graduate School of Economics, Kobe University, 2-1 Rokkodai-cho Nada-ku, Kobe 657-8501, Japan.
Email: 100e107e＠stu.kobe-u.ac.jp

11. Introduction
In theory, a central bank can stimulate the economy by expanding the supply of base money
even though the short-term nominal interest rate is close to zero. This is often called
quantitative easing. The rebalancing of investors' portfolio rebalance is one of the importantchannels in quantitative easing. According to Bernanke and Reinhart (2004), when the money isan imperfect substitute for other assets, increases in the monetary base can stimulate theeconomy by encouraging investors' portfolio rebalances and then decreasing the yields of assets.
This channel has been analyzed in several papers in the dynamic stochastic general equilibrium(DSGE) literature: Andrés et al. (2004) described a model with an imperfect substitution
between short-term assets and long-term assets, and they determined the monetary policy
channel through the relative price of alternative assets. Chen et al. (2012) and Harrison (2012)
analyzed the eﬀect of an asset purchase policy on the economy through the portfolio rebalancechannel at the zero lower bound. These papers focused mainly on the economic stimulus
provided by quantitative easing through an imperfect substitution.
Here, however, I considers the implications of the monetary policy as well as that ofstimulating the economy when a strong imperfect substitution between money and assets
exists.1 I introduce the adjustment cost in money and an asset (i.e., a bond) into a household’s
preference, which is similar to the speciﬁcation by Andrés et al. (2004). The reasons for thisassumption of an adjustment cost in money and assets are (i) it provides a short cut, and (ii) it
enables modeling the liquidity of money. The intuition for the second reason is the household'scompensation for liquidity. Households perceive the loss of liquidity as they hold more assets.
They then hold a liquid asset (i.e. money) to compensate for the loss of liquidity. This set-up
leads to the extension of the Euler equation and the money demand equation, and then theseequations include the term of marginal disutility of the money-asset ratio. Harrison (2012)
employed a speciﬁcation similar to that described by Andrés et al. (2004); he introduced theportfolio adjustment cost of short-term bonds and long-term bonds into the household’s
preference. On the other hand, the ﬁrm sector is the same as in the standard sticky price modelin which the monopolistic competitive ﬁrms set their price following the Calvo (1983) pricing
rule.
I analyze the behavior of the economy by giving the money growth shock and the money
velocity shock in both the model with the adjustment cost in money and asset and the model
without. First, I analyze the behavior of the economy in response to a positive money growthshock. The adjustment cost in money and asset ampliﬁes the reduction of the real interest rate in
the Euler equation and this reduction dominates the upward pressure by the expected inﬂation.

2

Then the nominal interest rate decreases in response to the positive money growth (the liquidity
eﬀect). I show next that the liquidity eﬀect is a strand of the portfolio rebalance channel in mymodel.
Second, I show that monetary policy must raise the nominal interest rate aggressively under
the Taylor rule in response to the increase in inﬂation in order to bring the economy to the local
equilibrium. Because of the strong imperfect substitution, the real interest rate ﬂuctuates greatly.
This leads the monetary authority to give a stronger response of the nominal interest rate so as
not to lower the real interest rate too much. Third, I analyze the economic behavior in response
to a money velocity shock under the Taylor rule. I show that the money velocity shock aﬀects
the entire economy in a model with the strong imperfect substitution even though the monetary
policy is based on the Taylor rule. Since the model without the adjustment cost is the same as
the standard sticky price model, money responds to the nominal interest rate passively under the
Taylor rule. On the other hand, if a strong imperfect substitution exists, the money velocity
aﬀects the entire economy through the marginal disutility of the money-bond ratio in the Euler
equation.
Lastly, I consider the monetary policy toward a negative velocity shock under both the money
growth rule and the Taylor rule. I show that the monetary policy can mitigate the eﬀects of thevelocity by ﬁxing the nominal interest rate under the money growth rule. In contrast, if theinterest rate is ﬁxed under the Taylor rule, the money velocity shock ampliﬁes the eﬀect to thereal economy. In addition, the eﬀects of the velocity shock are the large under the money
growth rule if the nominal interest rate is not ﬁxed.
The remainder of this paper is as follows: I describe the model in Section 2, and show theresults of a simulation in Section 3. Section 4 presents my conclusions.

2. The model
Here I describe the model economy. The model is the standard sticky price model except for the
adjustment cost in money and asset. Following Andrés et al. (2004), I assume an imperfect
substitution by introducing the cost of a money-bond ratio into a household’s preference.

2.1. Households
The representative household gets utility from consumption Ct and real money mt and gets
disutility from the labor supply Nt and the ratio between real money mt and real asset bt .
E02
 C1  σ( v m )1  ς
N1  ηυ δ βt  t t t
 t  t  1  ,
1 ς
1 η
2δ
 
 1  σ
t 0

Σ

3

(1)

where β ∊ (0,1), σ, ς , η, δ and υ are greater than zero, vt denotes the money demand shock, δtdenotes the ratio between the money and the bond, mt/bt , and δ denotes the steady-state value ofδt . The last term in the bracket shows the imperfect substitution between the money and theasset. The household pays the costs of changing the ratio between the money and the asset in theform of disutilities. At the steady-state, the costs of the imperfect substitution become zero. The
reasons for this assumption are (i) to provide a short cut in the model, and (ii) modelling theliquidity of money is achieved. The intuition for the second reason is the household'scompensation for liquidity. As the household hold more bonds, it loses the liquidity. Then thehousehold tries to make up the liquidity by increasing mt, but the household must pay the costsin the form of disutility for raising the real money. Thus, the marginal disutility of the moneybond ratio describes the value of money liquidity.
Ct is the aggregate consumption expressed by the Dixit-Stiglitz bundle. There is an inﬁnite
continuum fo diﬀerentiated goods Ct (i) on the i ∊ (0,1).
θ

θ 1
 1
 θ 1,
Ct    Ct (i ) θ di 
 0


(2)

where θ > 1 and θ denotes the elasticity of substitution between diﬀerentiated goods. The
household determines the allocation of diﬀerentiated goods given the price Pt (i) for eachdiﬀerentiated goods so that the allocation achieves the minimum cost. This cost minimization
leads to the relation as follows: P (i ) C t (i )   t 
 Pt 

θ

Ct ,

(3)

where aggregate price Pt is
1

 1
 1θ
,
Pt    Pt (i )1  θ di 
 0


(4)

Eq. (3) denotes the household’s demand of Ct (i) toward aggregate consumption Ct . Eq. (3) andthe deﬁnition of the aggregate price yields the following relation:
1


Pt Ct   Pt (i ) Ct (i ) di .
0

The household’s budget constraint is
Pt Ct + Mt + Bt = Wt Nt + Rt − 1 Bt − 1 + Mt − 1 + Dt + Tt ,4(5)

where Pt , Rt , Bt , Mt , Wt , Dt and Tt denote the aggregate price, gross nominal interest rate,
nominal bond, nominal money, nominal wage, dividend from the ﬁrm sector, and lump sum tax,
respectively. Eq. (5) shows that the household allocates the income from wages, bond yields,
lump sum tax and a ﬁrm’s proﬁt to the consumption, money and bond. The household’s ﬁrstorder conditions are as follows: R  Ct  1   σ 
Ωδ
t


  tσ t ,1  βE t  Π t  1  Ct  Ct bt(6)

vt1  ς mt1  ς Rt  1
Ω t δtΩ σt ,σσ
RCt
Rt Ct bt Ct btt(7)

N tη Wt
,

Pt
Ctσ

(8)

where Ωt denotes the marginal utility of δt and Πt denotes the gross inﬂation rate:PδυΩ t   t  1 , Π t  t .
Pt  1δδ

Eq. (6) denotes the Euler equation which describes a household’s intertemporal decision forsavings. Since I assume the adjustment cost in money and asset, the marginal cost of the moneybond ratio appears in the last term of the equation. Eq. (7) is the money demand equation. Eq.
(7) diﬀers from the standard money demand equation in that the marginal cost of the moneybond ratio tacks on the nominal interest rate. Terms derived from adjustment cost in Eqs. (6)and (7) are key factors contributing to the ampliﬁcation of an economic disturbance and policy
shock.
Finally, Eq. (8) is the labor supply equation; it shows that the opportunity cost of supplying
labor equal to the real wage.

2.2. Firms
The ﬁnal goods sector inputs the diﬀerentiated goods Yt (i) from the intermediate goods sector
and produces the ﬁnal goods Yt . The ﬁnal goods sector’s proﬁt maximization yields the demand
for Yt (i), i ∊ (0, 1) toward the aggregate output Yt :
 P (i ) 
Yt (i )   t 
 Pt 

θ

Yt ,

(9)

The intermediate goods sector produces the diﬀerentiated goods Yt (i), i ∊ (0,1) by the productYt (i) = Nt (i).

5where Nt = ʃ 01 Nt (i) di 2. By the cost minimization, the real marginal cost equals to the real
wage:
st 

Wt
.
Pt

The monopolistically competitive intermediate goods sector sets the price facing the staggered
price. Here I apply the speciﬁc model introduced by Calvo (1983). In each period, ﬁrms can set
the price with probability 1 − α and cannot change it with probability α. Following Calvo (1983),
intermediate goods ﬁrms maximize their proﬁts dynamically evaluated by households' marginalutility and set their optimal price considering that they cannot change their price forever withprobability α. I deﬁne Pt* as the price which can be set as optimal in the period t. The ﬁrst-ordercondition yieldsk θ  Et Σ k 0 α Λ t ,t k st k Yt k (i ),
Pt*  k
 θ  1  Et Σ k 0 α Λ t ,t k Yt k (i ) / Pt k(10)

where Λt,t+k denotes the stochastic discount factor, βk (Ct+k /Ct)−σ.
The deﬁnition of the aggregate price index Eq. (4) satisﬁes the following recursiveform.
Pt1  θ  (1  α) ( Pt* )1  θ  α Pt11θ .

(11)

2.3. Fiscal policy
The government’s budget constraint is:
Rt −1 Bt −1 + Mt −1 + Tt = Bt + Mt .
Because Rt ≥ 0, the government budget constraint might diverge. To prevent the model from thisdivergence, I assume the taxation policy as follows:τt = −φbt −1,(12)where τt = Tt /Pt and φ > 0.
2.4. Equilibrium
The goods market clearing condition is:
Yt = Ct ,and I deﬁne the money growth as μt = Mt / Mt − 1.

6βTable 1. Baseline parameter values
subjective discount factor

0.99

σ

intertemporal substitution for consumption2ςintertemporal substitution for real money

2

η

Frisch elasticity of labor supply

1αprice stickiness

0.8

δ
υ~

Steady-state ratio between m and b

0.1

adjustment cost

0.045

φ

taxation policy

0.2

3. Simulation
This section derives impulse responses to the money growth shock and money velocity shock.
All impulse responses are from a log-linearized model. First, I compare the results from themodel with the adjustment cost with the result from the model without the adjustment cost. The
money growth shock generates the reduction of the nominal interest rate (the liquidity eﬀect),
and the money velocity shock eﬀects the entire economy even though the monetary policy isunder the Taylor rule. Second, I derive the impulse responses with the ﬁxed interest rate to thenegative money velocity shock and see the eﬀectiveness of the ﬁxed interest rate under both the
Taylor rule and the money growth rule on the negative velocity shock when a strong imperfect
substitution exists.

3.1. Liquidity effectFirst, I derive the impulse response to the positive money growth shock. The money growth ruleis given by AR(1) process:μˆ t  ρ μ μˆ t  1  εtμwhere, μ̂t = ln µt − ln µ and εtμ , denotes an independently identically distributed (i.i.d.)
disturbance term. I set ρµ = 0.5 following to Christiano et al. (1998) 3. Table 1 shows thebaseline parameters. The value of δ is 0.1 following Woodford (1996). Woodford (1996) used
the ratio of monetary base and debt. I set υ so that υ~ ≡ υ ( 1 − β ) / (m − ςb) becomes 0.045
following Andrés et al. (2004). Setting a value of υ~ , the household's cost of the money-bondratio, is diﬃcult. Andrés et al. (2004) estimated the value υ by the maximum likelihood methodwith U.S. data, and they indicated 0.045. Harrison (2012) set parameter similar to υ as 0.09. I set
φ, the taxation parameter, as 0.2.

7

% from ss

0.5
0.4
0.3
0.2
0.1
0.0
-0.1

(b) Inflation

0

5

10

15

20

25

30

0

5

10

15

20

25

Quarters

Quarters(c) Nominal interest rate(d) Money growth

0.00
-0.04
-0.08
-0.12

0.25
0.20
0.15
0.10
0.05
0.00
-0.05

% from ss% from ss% from ss(a) Output

0

5

10

15

20

25

30

without adjustment cost

0.12
0.10
0.08
0.06
0.04
0.02
0.00

0

Quarters

30

5

10

15

20

25

30

Quarters

with adjustment cost

Fig 1. Impulse responses to a positive money growth shock.
Figure 1a-c illustrates the impulse response to the positive money growth shock that is
illustrated by Figure 1d.4 Figure 1a and Figure 1b illustrate the responses of output and inflation,
respectively. Since the price responds incompletely, the positive money growth shock raises
both output and inﬂation. However, the strengths are different for the cases with and without
adjustment cost. Both variables respond more strongly in the case with adjustment cost.
Figure 1c illustrates the responses of the nominal interest rate. The nominal interest rate
shows the negative response in the model with the adjustment cost and it’s response is muchstronger than in the case without adjustment cost.
The intuition for these results is as follows: Because of the adjustment cost, the imperfect
substitution in the money and asset is stronger. The household then pays the costs of changing
real money and asset in the form of disutilities. These costs appear as the real interest in the
Fisher equation.
ˆ t  bˆt ) ]  Et πˆ t  1 .
r̂t  [ σ ( Et yˆ t  1  yˆ t )  υ~ ( m(13)

Eq. (13) is log-linearized form of Eq. (6). Since the term of the real interest rate includes
marginal disutilities, the positive response in money and the negative response in real bondaﬀect the nominal interest rate more negatively than the model without the adjustment cost.5

8

Thus, the positive money growth shock lowers the real interest rate signiﬁcantly through themarginal dis-utilities from the adjustment cost in money and asset. The reduction in the realinterest rate dominates the positive pressure of expected inﬂation and the nominal interest rate
decrease (the liquidity eﬀect).
This result is consistent with the quantitative easing eﬀects through the imperfect substitutionof the money presented by Bernenke and Reinhart (2004). The adjustment cost denotes thestrong imperfect substitution of the money for non-money assets, and then the large increase inmoney leads to the household’s portfolio rebalance. Consequently, the yield of the asset, i.e., thenominal interest rate, decreases. Thus, the liquidity eﬀect can be a strand of the portfolio
rebalance eﬀect by the quantitative easing in my model. This implication is somewhat diﬀerent
from the limited participation model presented by Christiano (1991).6 The limited participation
model generates the liquidity eﬀect by the increase in loans from the ﬁnancial intermediary to
the ﬁrm. Since the household cannot participate in the asset market after the central bank’s cash
injection, all of the eﬀect of the cash injection is absorbed by the supply side. This leads toincreases in labor, output and loans. The ﬁnancial intermediary supplies the loan as long as thenominal interest rate is positive, and consequently the nominal interest rate falls.
This mechanism is similar to the portfolio rebalance but it is indirect, and the eﬀect of the
cash injection goes through the supply side. However, my model specifies the portfolio
rebalance by introducing the cost function into the household’s utility directly, and then theeﬀects of the cash injection are absorbed largely by the demand side.

3.2. Determinacy under the Taylor rule and the money velocity shock
In this subsection, I consider a model economy under the Taylor rule when an adjustment costin money and asset exists. First, I consider the achievement of local uniqueness by a monetary
policy. Second, I derive the impulse responses to the negative shock of money velocity under
the Taylor rule.
I choose the standard type of the Taylor rule:r̂t  ψπ π̂ t + ψy ŷ t .

where, r̂t , π̂ t and ŷ t denote the percent deviation from the steady state for Rt , Πt and Yt .In
general, the Taylor rule accommodates the one percent increase in inﬂation by raising thenominal interest rate more than one percent. If the monetary policy raises the nominal interest
rate insuﬃciently, the economy cannot be stabilized since the increase in expected inﬂation
leads to a large decrease in the real interest rate and the output could increase more. In standard

9

5
4.5
4

Determinacy



3.5
3
Indeterminacy

2.5
2
1.5
1

0

0.05

0.1

0.15

tilder()Fig. 2 Determinacy and indeterminacy regions for ψπ and υ~ .
sticky price models, monetary policy can prevent this stabilization by setting ψπ > 1. In theory,ψπ > 1 satisﬁes the Blanchard and Kahn condition.
Analogous to standard sticky price models, setting ψπ > 1 satisﬁes the Blanchard and Kahn
condition under baseline parameters. However, the determinacy of the model depends on thevalue of υ~ in my model. From Eq. (13), the change in the real interest depends on υ~ . Figure 2
illustrates the areas of the determinacy and indeterminacy on the combination of ψπ and υ~ .7
Figure 2 shows that the high υ~ requires a high response to the inﬂation, which necessitates anaggressive policy in order to not decrease the real interest too much in response to the inﬂation.
Thus the strong imperfect substitutiability in money and asset might lead to indeterminacy
according to the substitution cost in my model.
Now I derive the impulse responses to the money velocity shock. First, I describe thedeﬁnition of velocity in my model. If υ = 0, Eq. (7) becomes:
M tς Vt  Ptς Ytσ ,(14)

where Vt  vtς  1 (1  Rt1 ) . If we regard Eq. (14) as the quantity equation of money, Vt denotes
the velocity of money. Then, the household’s preference shock on money, vt , is the part of themoney velocity as long as ς > 1. Because the reduction in the velocity of money might lead to arecession of the economy, it is valuable to examine how the negative velocity shock aﬀects the
economy with the imperfect substitution. The money velocity shock follows the AR(1) process:vˆt  ρ μ vˆt  1  εtv ,10

x10-3(a) output

0
-2
-4
-6
-8
-10

-2

% from ss% from ss

x10-3

0

2

4

6

8

10 12 14 16(b) Inflation

0
-1
-2
-3

-2

0

2

4

Quarters

0

2

4

6

8

10 12 14 16

5
0
-5

-2

0

2

4

% from ss% from ss

-4
-8

2

4

6

8

10 12 14 16

10 12 14 16

0
-5
-10
-15
-20

-2

0

2

4

6

8

10 12 14 16

Quarters

Quarters
without adjustment cost

8(f) Money velocity

0

0

6

Quartersx10-3 (e) Nominal interest rate-2

10 12 14 16

10Quarters-128(d) Money growth% from ss% from ss(c) Real money
10
8
6
4
2
0

-2

6

Quarters

with adjustment cost

Fig. 3 Responses to a negative velocity shock under the Taylor rule.

where, v̂ t = ln vt − ln v and ε tv follows the i.i.d. process, ρv = 0.5, ψπ = 2 and ψy = 0.5.
Figure 3 illustrates the impulse responses to the negative velocity shock. Only the real money(Fig. 3c) and the money growth (Fig. 3d) respond to the velocity shock in the model without theadjustment cost. Because the model without the adjustment cost is the standard sticky pricemodel, the money responds passively to the nominal interest rate when the monetary policy is
the Taylor rule: the shock originating from the money demand equation does not aﬀect theentire economy because output and inﬂation are determined by only the system (except for themoney demand equation in the standard sticky price model). Thus the real money and themoney growth absorb the velocity shock completely and then all of the other variables do notrespond to the velocity shock.
However, in the model with the adjustment cost, all variables respond to the velocity shock.
The strong imperfect substitution in the money and asset allocate the eﬀects of the velocity
shock to output, inﬂation and the nominal interest rate. An intuition for this is as follows:
Because the Euler equation includes money, the velocity shock aﬀects the output negatively.
The negative response of the output also lowers the inﬂation through the aggregate supply. The
response of the nominal interest rate depends on whether the real interest rate dominates the

11

0

5

10

15

20

25

0.05
0.00
-0.05
-0.10
-0.15
-0.20

30

0

5

10

15

20

25

Quarters

Quarters(c) Nominal interest rate(d) Money velocity

0.20% from ss% from ss(b) Inflation% from ss% from ss(a) Output

0.1
0.0
-0.1
-0.2
-0.3
-0.4
-0.5

0.15
0.10
0.05

30

0
-5
-10
-15
-20

0.00
0

5

10

15

20

25

30

0

5

Quarters

fixed

10

15

20

25

30

Quarters

no fixed

Fig. 4 Responses to a negative velocity shock under the money growth rule with a fixedinterest rate.
expected inﬂation or not. Under the baseline parameter, the nominal interest rate responds
negatively to the negative velocity shock.
This is a strand of the portfolio rebalance eﬀect as well as the liquidity eﬀect. Since thenegative decrease in the money velocity lowers the liquidity per unit of money, the householdmust hold more money to keep the liquidity. However, holding more money is costly because of
the adjustment cost of the money and asset. This leads to the decrease in output and inﬂation.
Thus, the money velocity shock aﬀects all of the economy when there is a strong imperfect
substitution in money and assets even though the monetary policy employs the Taylor rule.
3.3. A negative money velocity shock under the ﬁxed interest rate
In this subsection I derive the impulse responses to a negative money velocity shock andconsider the eﬀectiveness of a ﬁxed interest rate under both the Taylor rule and the money
growth rule.8
Figure 4 illustrates the impulse responses to a negative money velocity shock under themoney growth rule.9 In the model without the ﬁxed interest rate, output and inflation respondnegtively to the money velocity shock. This is similar to the result of the Taylor rule, illustrated

12x10-3x10-3(a) Output% from ss% from ss

-5
-10
-15
-20
-25
-2 0

(b) Inflation

0

2

4

6

-2
-4
-6
-8
-10
-2 0

8 10 12 14 16

2

4

6

8 10 12 14 16

Quarters

Quartersx10-3 (c) Nominal interest rate(d) Money velocity

0
-2
-4
-6
-8
-10
-12
-2 0

% from ss% from ss

0

0
-5
-10
-15
-20

2

4

6

8 10 12 14 16-2 0

Quarters

fixed

2

4

6

8 10 12 14 16

Quarters

no fixed

Fig. 5 Responses to a negative money demand shock under the Taylor rule with a fixedinterest rate.
by Figure 3. However, the nominal interest responds positively to the negative money velocity
shock under the money growth rule. Since the output responds large negatively under the money
growth rule, the positive response of the real interest rate dominate the downward pressures ofthe expected inﬂation. This leads to a strong positive response of the nominal interest rate.
The result obtained with the ﬁxed interest rate changes dramatically. The ﬁxed interest ratecompletely mitigates the tightening eﬀects from the negative money velocity shock on the
output and inﬂation under the money growth rule. In other words, the nominal interest rate is thedoor to the propagation from the money velocity shock to the output and the inﬂation under themoney growth rule. Because the upward pressure works on the nominal interest rate in response
to a negative money velocity shock, the ﬁxed interest rate gives easing eﬀects to the economy.
This leads to the reduction of the downward pressures on output and inﬂation. Further, since thenominal interest rate responds strongly under the money growth rule, the ﬁxed interest rate lets
the output and the inﬂation respond positively.
Now I apply the same experiment to the model under the Taylor rule. Figure 5 illustrates theimpulse responses to a negative velocity shock under the Taylor rule. Since the model has the13Table 2. Variances for output and inflation
Money growth rule

Fixed

Not Fixed

Fixed

No Fixed

t 0 yt2

2.3582e-07

35.3554

7.1056

1.4998t 0 π t26.5949e-08

2.03851.316

0.2842

1
T 1
1
T 1

Taylor ruleT

T

Variances of output and inflation from the steady state for a simulated period. ‘Fixed’ denotes the modelwith a fixed interest rate policy. ‘No Fixed’ denotes the model without the fixed interest rate policy. I
multiplied by 104 all of the results and set T = 15.

adjustment cost in money and assets, the negative shock of money velocity tightens the entire
economy under the Taylor rule.
In the model with the ﬁxed interest rate, the eﬀects of a money velocity shock are ampliﬁed.
The output and the inﬂation decrease more than in the case without a ﬁxed interest rate. Since
the nominal interest rate moves negatively in response to the negative money velocity shock
under the Taylor rule, the ﬁxed interest rate has tightening eﬀects on the economy. Thus, if themonetary authority ﬁxes the interest rate under the Taylor rule, the money velocity shock
ampliﬁes the eﬀect on the real economy.
Table 2 shows the variance of the output and the inﬂation from the steady state for thesimulated periods.10 The column ‘Fixed’ denotes the model with the ﬁxed interest rate. The
column ‘not Fixed’ denotes the model without the ﬁxed interest rate.
The results show that the ﬁxed interest rate under the money growth rule indicates thesmallest value of the variances for both the output and the inﬂation. The money growth rule also
generates the highest variance of the output if the monetary policy does not take the ﬁxed
interest rate. On the other hand, if the negative money velocity shock hit an economy under the
Taylor rule with a ﬁxed interest rate policy, the recession of the inﬂation and the output
becomes large.
Thus, monetary policy might be able to calm down the ﬂuctuation of the output and theinﬂation toward a negative money velocity shock by employing the money growth rule rather
than the Taylor rule with a ﬁxed nominal interest rate.

4. Conclusion
This paper has analyzed a model with an adjustment cost in money and asset. My ﬁndings are as
follows: First, a positive money growth shock generates a signiﬁcant liquidity eﬀect throughhouseholds' portfolio rebalances. Second, the determinacy of the model depends on the value of
portfolio cost between money and assets. According to the value of imperfect substitution in14

money and assets, monetary policy must raise the nominal interest rate aggressively in response
to a rise in inﬂation in order to bring the economy to the locally unique equilibrium. Third, amoney velocity shock aﬀects the entire economy even though the monetary policy is the Taylor
rule. Finally, the monetary authority might be able to mitigate the eﬀects of a negative money
velocity shock if the monetary policy adopts the money growth rule with a ﬁxed interest rate. If
the money growth rule can be regarded as one strand of a quantitative easing policy, I conclude
that a quantitative easing policy might be eﬀective in stabilizing the economy toward the
reduction of the money velocity even though the economy does not stay near the zero lower
bound.
ACKNOWLEDGMENTS
I would like to thank Professors Toshiki Jinushi, Shigeto Kitano, and Teruyoshi Kobayashi for
their helpful comments and suggestions in the Rokko Forum at Kobe University. I also thankDaisuke Ida, Keiya Minamimura, and Kenya Takaku for their helpful comments andsuggestions.

NOTES
1. I assume the adjustment cost in money and assets. However, money and assets are imperfect
substitute to begin with because of the interests on assets. For this reason, I describe thestrong -imperfect substitution concerning the model with the adjustment costs in money
and assets.
2. This assumption leads to Nt = dtYt ≠ Yt where dt = ʃ 01 (Pt (i)/Pt)−θdi. However, dt does notappear in the loglinearlized model. For details, see Galí (2008).
3. The higher the persistence in money growth becomes, the more positively the nominalinterest rate responds to the positive money growth.
4. I set the positive money growth shock so that the nominal interest rate responds to −0.1
percent at minimum in the imperfect substitution model.
5. The decrease in the real bond is unrealistic. However, since the nominal term of the bond isdefined as Pt bt, the price of the bond increase. Since the rise of price is greater, the realbond is decrease.
6. However, since Christiano (1991) generates the liquidity effect with the flexible price model,
the result cannot be compared with my model directly.
7. I fix ψy at 0.50.
8. I fix the responses of the nominal interest rate at the steady state for all periods of thesimulation.
9. I apply the piecewise linear solution to the impulse responses with a fixed nominal interest
rate. Guerrieri and Iacoviello (2013) provided the MATLAB codes and details of thealgorithm for the piecewise linear solution. See also Bodenstein et al. (2009).
10. The order of results does not depend on the length of T.

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