Monetary Stabilization Policy by Means of Taylor Rule ina Dynamic Keynesian Model with Capital Accumulation
Toichiro Asada
Faculty of Economics, Chuo University742-1 Higashinakano, Hachioji, Tokyo 192-0393, Japan
E-mail : asada@tamacc.chuo-u.ac.jp

Abstract
In this paper, we investigate the macroeconomic effects of monetary stabilization
policy by means of Taylor interest rate rule by using an analytical framework of the
dynamic Keynesian model, which considers the capital accumulation effect ofinvestment

expenditure

explicitly.

We

prove

analytically

that

the

dynamic

stability/instability of the system depends on the sensitivities of the response of central
bank’s monetary policy and the parameters which characterize the way of expectation
formation, and the cyclical fluctuations occur at the intermediate parameter values. We
also present some numerical simulations which support the above mentioned analyticalresults.
Keywords : Monetary stabilization policy, Taylor rule, Dynamic Keynesian model,
capital accumulation, inflation targeting, employment targeting, numerical simulations

1

1. Introduction
This paper is a sequel to a series of papers by the same author (Asada 2006a, 2006b,
2008, 2009) which dealt with the theoretical analyses of the monetary policy especiallyin reference to the Japanese economy in the period from the late 1990s to the 2000s,that was accompanied by the serious ‘deflationary depression’.1
Asada(2006a, 2006b, 2008) applied the analytical method of the ‘high-dimensional
Keynesian macrodynamics’ which was developed by Asada, Chen, Chiarella and
Flaschel(2006), Asada, Chiarella, Flaschel and Franke(2003) etc. to the problems of themonetary policy in the period of the deflationary depression.2 The spirit of these models
is ‘Keynesian’ in that employment and production are determined by the ‘effective
demand’ in the sense of Keynes(1936), and the possibility of the involuntary
unemployment of labor as well as the under-utilization of capital stock is allowed for.
Moreover, the capital accumulation effect of investment expenditure was explicitly
considered in these models. In other words, these models can deal with the ‘long run’effects as well as the ‘short run’ effects of the monetary policy.
By the way, in these models, the growth rate of nominal money supply was treated asa policy variable of the central bank, and the effectiveness of the inflation targeting aswell as the importance of the inflation expectation formation was considered. It is wellknown, however, that the policy makers of BOJ (Bank of Japan) assert that their policy
variable is not the (growth rate of) nominal money supply but the nominal rate ofinterest like the so called ‘Taylor rule’ that was proposed by Taylor(1993). Asada(2009)
applied the ‘Taylor rule’ to the study of the monetary policy in the period of thedeflationary depression, and showed that we can obtain essentially the same qualitative
policy conclusion as that of the models with an alternative monetary policy rule.
It is worth noting, however, that Asada(2009) is a simplified small scale (in fact,
two-dimensional) model that neglects the capital accumulation effect of the investment
expenditure. In this paper, we study the effect of the monetary stabilization policy bymeans of the ‘Taylor rule’ in a dynamic Keynesian model that explicitly considers thecapital accumulation effect of the investment expenditure.
This paper is organized as follows. In section 2, we present the analytical frameworkof the model. In section 3, we derive a system of fundamental dynamical equations,
1 As for the empirical analyses of the Japanese economy in this period, see

Hamada(2004), Harada and Iwata (eds.)(2002) and Ito and Ban(2006) as well as
Asada(2009). As for the theoretical analyses of the related topics, see Eggertsson andWoodford(2006), Gong(2005), Krugman(1998), McCallum(2000), and Reifschneider andWilliams(2000).
2 ‘High-dimensional’ macrodynamic model means the macrodynamic model with many
variables.
2

which consists of three-dimensional system of nonlinear differential equations, andstudy the properties of its equilibrium solution. In section 4, we provide some analyticalresults on the local stability/instability of the equilibrium point and the existence of thecyclical fluctuations around the equilibrium point. In section 5, we provide somenumerical simulations which support the analytical results in section 4. Section 6 isdevoted to some concluding remarks, which refer to some popular approaches which we
did not adopt in this paper deliberately. Somewhat complicated and lengthy
mathematical proof of a proposition is relegated to the appendix.

2. The model
The model in this paper consists of the following system of equations.

y = c ( y , π e ,τ ) + g ( r − π e ) + h(1)m = φ (r , π ) y
w& / w = ε (e − e ) + n2 + π e ; ε＞0p = z ( wN / Y ) = zw / a ; z＞1

(2)

e

(3)
(4)

e& / e = y& / y + g (r − π ) − (n1 + n 2 ) = y& / y + g (r − π ) − n(e)if r＞0α (π − π ) + β (e − e )⎧r& = ⎨; α ≧ 0, β ≧ 0⎩max[0, α (π − π ) + β (e − e )] if r = 0ee

π& e = γ [θ (π − π e ) + (1 − θ )(π − π e )] ; γ＞0, 0 ≦ θ ≦1

(5)
(6)
(7)

where the meanings of the symbols are as follows. Y = real national income(realoutput). K = real capital stock. y = Y / K = output-capital ratio, which reflects the rateof capacity utilization of capital stock. C = real private consumption expenditure.

c = C / K = consumption-capital ratio. K& = I = real investment expenditure(realcapital accumulation). g = K& / K = I / K = rate of investment(rate of capital
G = real government expenditure. h = G / K = government
accumulation).
expenditure-capital ratio(fixed). τ = marginal tax rate(fixed, 0＜τ＜1). r = nominal
rate of interest ≧ 0. p = price level ＞ 0. π = p& / p = rate of price inflation.

π e = expected rate of price inflation. π = target rate of price inflation.
r − π e = expected real rate of interest. M = nominal money supply.
m = M /( pK ) = real money supply-capital ratio. w = nominal wage rate＞0. N = labor
employment. N s = labor supply. e = N / N s = rate of employment = 1 – rate of

e = ’natural’ rate of employment (0＜e ＜1).
a = Y / N = average labor productivity. n = N& s / N s = growth rate of labor supply.

unemployment(0 ≦ e ≦ 1).

1

n2 = a& / a = growth rate of average labor productivity(rate of technical progress).

3

n = n1 + n2 = n(e) = ’natural’ rate of growth＞0 (ne = dn / de ≧ 0). 3
Eq. (1) is the ‘IS equation’, which is nothing but the equilibrium condition of the goods
market. In this equation, c( y, π ,τ ) is a standard Keynesian consumption functione(0＜c y = ∂c / ∂y＜1, cπ e = ∂c / ∂π e ≧ 0, cτ = ∂c / ∂τ＜0) and g (r − π e ) is a standard
Keynesian investment function ( g r −π e = dg / d (r − π )＜0). Solving this equation witherespect to y, we obtain the following reduced form of the IS equation.

y = y (r − π e , π e , h,τ ) ; y r −π e = ∂y / ∂(r − π e )＜0, yπ e ≧ 0,
y h = ∂y / ∂h＞0, yτ = ∂y / ∂τ＜0(8)This equation shows that the rate of capacity utilization of capital stock is determinedby the effective demand in our model.4
Eq. (2) is the ‘LM equation’, which is the equilibrium condition of the money market.5
The right hand side of this equation is a standard Keynesian money demand function,
where φ r = ∂φ / ∂r＜0 and φπ e = ∂φ / ∂π ≦ 0. Differentiating Eq. (2) with respect toetime, we obtain the following dynamic version of the LM equation.

µ = π + g (r − π e ) + ( y& / y ) − η r (r& / r ) − η π (π& e / π e ) ; µ = M& / M
e

where

ηr = (∂φ φ∂φ φ) /( ) = −( ) /( )
∂r r
∂r r

and

ηπ = (
e

(9)

∂φ
φ
∂φ
φ
) /( e ) = −( e ) /( e )
e
∂π
π
∂π
π

are

elasticities of the real money demand per capital stock with respect to changes of thenominal rate of interest and the expected rate of price inflation respectively.
Eq. (3) is a standard expectations-augmented wage Phillips curve. Eq. (4) represents
the markup pricing principle of the imperfect competitive firms. The parameter z is
the average markup, which is assumed to be a constant that reflects the ‘degree of3 This means that in this model the ‘natural rate of growth’ can be an endogenous

variable that is influenced by the rate of employment. As for the rationale of thisformulation, see Asada(2008) pp. 87 – 88.
4 In this model, the fixed technological coefficient is assumed. Nevertheless, theoutput-capital ratio ( y ) becomes a variable that reflects the rate of capacity utilizationof the capital stock (δ ) through the equation y = δy f , where y f is theoutput-capital ratio in case of full utilization of the capital stock, which is assumed to bea constant(cf. Asada 2008).
5 In case of the so called ‘liquidity trap’ with r = 0, any value of m which is not
smaller than the right hand side of Eq. (2) can become the equilibrium value of m.
4monopoly’ in the sense of Kalecki(1971). Differentiating Eq. (4) with respect to time, wehave

π = ( w& / w) − (a& / a) = ( w& / w) − n2 .

(10)

Substituting this equation into Eq. (3), we have the following expectations-augmented
price Phillips curve.

π = ε (e − e ) + π e ; ε＞0(11)Eq. (5) describes the dynamic of the rate of employment. We can derive this equationby the following way. By definition, we haveN=

(Y / K ) K= yK / a.
Y/N

(12)

Therefore, we have the following expression.

e = N / N s = ( yK ) /(aN s ).

(13)

Differentiating this equation with respect to time, we obtain Eq. (5).
Eq. (6) is a version of the monetary policy by means of the ‘Taylor interest rate policy
rule’ ( so called the Taylor rule) that was introduced by Taylor(1993). In this formulation,
it is assumed that the central bank raises or reduces the nominal rate of interest toward
the realization of the target rate of inflation as well as the ‘natural’ rate of employment.
This is a kind of ‘inflation targeting’ as well as ‘employment targeting’ monetary policy
rule.6 In this formulation, the nonnegative constraint on the nominal rate of interest isalso considered.
Eq. (7) describes a hypothesis concerning the inflation expectation formation, which isa mixture of the ‘forward looking’ and the ‘backward looking (or ‘adaptive’) inflation
expectations. The parameterθ is the weight of the ‘forward looking’ inflation

expectation formation, which can be considered to reflect the ‘credibility’ of theannouncement by the central bank concerning the inflation targeting.7
A dynamic system which consists of six equations (5), (6), (7), (8), (9) and (11) isenough to determine the dynamics of six endogenous variables e, r ,π e , y, µ andπ.

6 It is assumed that the central bank announces the target rate of inflation(π ) to the

public, so that the public can use this information to form the inflation expectation (cf.
Eq. (7)). As for the empirical and theoretical arguments on inflation targeting, see, forexample, Asada(2006 a,b), Asada(2008), Asada(2009), Bernanke, Laubach, Mishkin and
Posen(1999), Galí(2008), Ito and Ban(2006), Krugman(1988), and Woodford(2003).
7 This type of ‘mixed’ expectation formation hypothesis was introduced by Asada,
Chiarella, Flaschel and Franke(2003), Asada(2006 a,b), Asada(2008), and Asada(2009).
53. A system of fundamental dynamical equations and the properties of its equilibrium
solutionIn this section, we shall reduce a system of equations that was presented in theprevious section to the more compact form.
Differentiating Eq. (8) with respect to time, we have the following expression.

y& = y r −π e (r , π e )r& + {− y r −π e (r , π e ) + yπ e (r , π e )}π& e(14)Substituting equations (8), (11) and (14) into equations (5), (6) and (7), we have thefollowing complete three dimensional system of nonlinear differential equations.

⎧
α {ε (e − e ) + π e − π } + β (e − e )ifr& = f1 (π e , e) = ⎨e⎩max[0, α {ε (e − e ) + π − π } + β (e − e )] if

(i)

r＞0
r=0

( ii ) π& = f 2 (π , e) = γ [θ (π − π ) + (1 − θ )ε (e − e )]eee

( iii ) e& = f 3 (r , π , e) = e[{1 / y (r − π , π )}{ y r −π e (r , π ) f1 (π , e)
e

e

e

e

e

+ {− y r −π e (r , π e ) + yπ e } f 2 (π e , e)} + g (r − π e ) − n(e)](15)

We can consider that this is a system of ‘fundamental dynamical equations’ in ourmodel. This system is supplemented by the dynamic LM equation (Eq. (9)), which
determines the dynamic of the growth rate of the nominal money supply ( µ )endogenously. However, this equation does not feed back to the dynamics of the systemof equations (15), so that we can safely neglect Eq. (9) when we consider thefundamental dynamical system.
Next, let us consider the ‘normal’ equilibrium solution of the system (15) such thatr& = π& e = e& = 0 and e = e . If we neglect the nonnegative constraint on the nominalerate of interest, we have the following ‘normal’ equilibrium solution (r*, π *, e*).

e* = e(16)π * = π* = πg ( r * −π ) = n(e )

(17)

e

(18)

In this case we also have y& = 0 from Eq. (14). Substituting the relationships

y& = r& = π& e = 0 and equations (17) and (18) into Eq. (9), we obtain the following
equilibrium growth rate of nominal money supply.

µ * = π + n (e )(19)Equations (16) and (18) mean that the rate of employment and the rate of capital
accumulation are settled down to their ‘natural’ values at the normal equilibrium point.
Eq. (17) means that the ‘actual’ and the ‘expected’ rates of inflation are equal to the6‘target’ rate of inflation at the normal equilibrium point. Eq. (19) implies that theequilibrium growth rate of nominal money supply is determined by the ‘target’ rate of
inflation and the equilibrium value of the ‘natural’ rate of growth, and not the other way
round in this model of endogenous money supply.
We can interpret the determination of the equilibrium nominal rate of interest (r *)as follows. The equilibrium real rate of interest ( ρ *) is determined byg ( ρ *) = n(e ).

(20)Then, r * is determined byr* = ρ * +π .

(21)

Needless to say, we have r *＞0 if and only if the inequalityπ＞ − ρ *(22)

is satisfied. This inequality can always be satisfied if the target rate of inflation (π ) is
set to be sufficiently large, even if ρ *＜0. 8 On the other hand, the deflationary-biased
central bank such as BOJ in the late 1990s and the 2000s may fail to satisfy theinequality (22). In case of π ＜ − ρ *, the economically meaningful ‘normal’ equilibrium
does not exist.

4. Stability/instability analysis and the existence of cyclical fluctuations
Next, we shall investigate the local stability/instability of the ‘normal’ equilibrium
point of the system (15) by assuming that the inequality (22) is in fact satisfied. We canwrite the Jacobian matrix of this system that is evaluated at the equilibrium point asfollows.

⎡0
J = ⎢⎢ 0⎢⎣ f 31ααε + β ⎤− γθ γε (1 − θ )⎥⎥
f 32

f 33

where f 31 = e g * r −π e ＜0,
(−)

(23)

⎥⎦

f 32 = e [{ y * r −π e α + ( y * r −π e − y *π e )γθ } / y * − g * r −π e ], and
(−)

(−)

( + or 0 )(−)

f 33 = e [ y *r −π e (αε + β ) / y * +{− y * r −π e + y *π e }γε (1 − θ ) / y * − ne ].
(−)

(−)

( + or 0 )( + or 0 )Then, the characteristic equation of this system at the equilibrium point becomes asfollows(cf. mathematical appendix of Asada, Chiarella, Flaschel and Franke 2003).

8 The equilibrium real rate of interestρ * can be negative if the equilibrium ‘natural’rate of growth n(e ) is too large.
7∆(λ ) = λI − J = λ3 + a1λ2 + a 2 λ + a3 = 0

(24)

where

a1 = −traceJ = γθ − f 33 ,

(25)

(?)

a 2 = sum of all principal second-order minors of J=0 αε + β− θ ε (1 − θ )
α
+
+γ
f 31
f 33
f 32
f 33
0 − γθ
0= −(αε + β ) f 31 − γ {θ f 33 + ε (1 − θ ) f 32 },(−)

a3 = − det J = − f 31γ

(?)

(26)

(?)

α αε + β− θ ε (1 − θ )= − f 31 γ {αε (1 − θ ) + (αε + β )θ }＞0.

(27)

(−)

It is well known that the equilibrium point of the system (15) is locally asymptoticallystable if and only if the following set of inequalities is satisfied(cf. Gandolfo 1996
p.221).9a1＞0, a3＞0, a1 a 2 − a3＞0(28)

In our model, the inequality a 3＞0 is always satisfied, so that we can reduce the local
stability conditions to the following set of two inequalities.

a1＞0, a1 a 2 − a3＞0(29)

Now, we can easily prove the following proposition that describes a set of sufficientconditions for local instability.
Proposition 1.
Suppose that the monetary policy parameter values α ≧ 0 and β ≧ 0 are fixed
arbitrarily. Furthermore, suppose that the value of the credibility parameter θ ∈ [0,1]is sufficiently close to zero (including the case of θ = 0) and the adjustment speed of
expectation adaptation γ＞0 is sufficiently large. Then, the equilibrium point of thesystem (15) is locally unstable.
(Proof.)
9 In this case, the inequality

a 2＞0 is automatically satisfied.
8

First, let us consider the case of θ = 0. In this case, it is easy to see that we havea1＜0 for all sufficiently large values of γ＞0, which violates one of the local stability
conditions (29). By continuity, this conclusion also applies to the case of θ＞0 as long as

θ is close to zero. □On the other hand, the following two propositions present some economicallymeaningful sufficient conditions for local stability.
Proposition 2.
(1) Suppose that the parameter values of β ≧ 0,

θ ∈ [0,1], and γ＞0 are fixed

arbitrarily. Then, the equilibrium point of the system (15) is locally asymptoticallystable for all sufficiently large values of α＞0.
(2) Suppose that the parameter values of α ≧ 0, θ ∈ [0,1], and γ＞0 are fixed
arbitrarily. Then, the equilibrium point of the system (15) is locally asymptoticallystable for all sufficiently large values of β ＞0.
(Proof.)(1) We can easily see that a1 is a linear increasing function of α , and a1 a 2 − a3 is aquadratic function of α in which the coefficient of the term α 2 is positive. Thismeans that both of two inequalities in (29) are satisfied for all sufficiently large
values of α＞0.
(2) The method of the proof of Proposition 2 (1) is essentially the same as that of
Proposition 2 (1). □Proposition 3.
Suppose that the parameter values α＞0, β ≧ 0, and γ＞0 are fixed arbitrarily.
Then, the equilibrium point of the system (15) is locally asymptotically stable for allθ ∈ (0,1] which are sufficiently close to 1 (including the case of θ = 1).
(Proof.)
Suppose that θ = 1. In this, case, we have f 33＜0 and the following relationships
are satisfied.

a1 = γ − f 33 ＞0

(30)

( −)

9

a 2 = −(αε + β ) f 31 − γ f 33 ＞0(31)

a3 = − f 31 γ (αε + β )＞0

(32)

(−)

( −)

( −)

a1 a 2 − a3 = f 33 f 31 (αε + β ) + ( f 33 − γ )γ f 33 ＞0(−) ( −)

( −)

(33)

(−)

Inequalities (30) and (33) mean that all of the local stability conditions (29) are satisfiedin case of θ = 1. By continuity, these inequalities are satisfied even if 0＜θ＜1, as longas θ is sufficiently close to 1. □Next, we shall consider the bifurcation analysis by choosing the monetary policy
parameter α ≧ 0 as a bifurcation parameter.10
Proposition 4.
Suppose that the parameter values β ≧ 0 and θ ∈ [0,1) are fixed at the levels suchthat a1＜0 when α = 0. Then, we have the parameter value α 0＞0 such that thefollowing properties (1) and (2) are satisfied.
(1) The equilibrium point of the system (15) is locally unstable for all α ∈ [0, α 0 ), and
it is locally asymptotically stable for all α ∈ (α 0 ,+∞).
(2) There exists a family of non-constant closed orbits around the equilibrium point of
the system (15) for some range of the parameter value α which is sufficiently closeto α 0 .
(Proof. ) See Appendix.
Proposition 1 means that the high adjustment speed of price expectations that is
combined with the public’s highly backward-looking (adaptive) expectation formation isa destabilizing factor of the macroeconomic system. We can represent this destabilizing
positive feedback mechanism schematically as follows.

e ↓⇒ π ↓⇒ π e ↓⇒ (r − π e ) ↑⇒ g ↓⇒ y ↓⇒ e ↓(M 1 )

Proposition 2 says that the quick response of the monetary authority (central bank) to
10 We can obtain essentially the same conclusion as Proposition 4 even if we choose theparameter β ≧ 0 or θ ∈ [0,1] as a bifurcation parameter.

10the macroeconomic conditions is a stabilizing factor. This stabilizing negative feedback
mechanism can be expressed as follows.

e ↓⇒ r ↓⇒ (r − π e ) ↓⇒ g ↑⇒ y ↑⇒ e ↑(M 2 )

Proposition 3 implies that the public’s highly forward-looking expectation formation(high credibility of the monetary authority’s announcement on inflation targeting) is astabilizing factor. This stabilizing negative feedback mechanism may be expressed asfollows.11

e ↓⇒ π ↓⇒ π e ↓⇒ π ＞π e ⇒ π e ↑⇒ (r − π e ) ↓⇒ g ↑⇒ y ↑⇒ e ↑(M 3 )

Proposition 4 means that the cyclical fluctuations occur at the intermediate values ofthe monetary policy parameters.

5. A numerical illustration
In this section, we shall present some numerical simulations which support theanalytical results in the previous section. Let us assume the following parameter values
and the functional form of the investment function.12

e = 0.97, π = 0.03 = 3% per year, n = 0.05 = 5% per year(34)

g (r − π ) = −0.2(r − π − 0.03) + 0.05 = 0.2(π − r ) + 0.056(35)eThese data are enough to determine the equilibrium values (e*, π *, r*, µ *) which areeeegiven by equations (16) – (19). In fact, we obtain the following results.

e* = e = 0.97, π e * = π * = π = 0.03 = 3% per year, r* = 0.06 = 6% per year,(36)µ * = π + n = 0.03 + 0.05 = 0.08 = 8% per year.
In this case, the equilibrium real rate of interest ρ * becomesρ * = r * −π = 0.06 − 0.03 = 0.03 = 3% per year.

(37)

Next, let us derive the explicit functional form of the function y (r − π , π , h,τ ) ineeEq. (8) under the simplified specification of the consumption function. For simplicity, we
11 In this schematic representation, it is assumed that at the initial stage of expectation
formation, the ‘backward-looking’ factor is strong, but at the latter stage, the‘forward-looking’ factor becomes strong because of the increase of the credibility of thecentral bank’s announcement of the target rate of inflation.
12 In this example, it is assumed that the ‘natural’ rate of growth n does not depend on
the rate of employment for simplicity.

11

assume the linear consumption function c = σ (1 − τ ) y + c neglecting the effect of π e
on c, where σ ∈ (0,1) is the marginal propensity to consume from the disposable
income. In this case, we obtain the following relationship solving Eq. (1) with respect toy.

y (r − π e , π e , h,τ ) =1{g (r − π e ) + c + h}
1 − σ (1 − τ )(38)Furthermore, let us assume the following parameter values.

σ = 0.7, τ = 0.3, c = 0.03, h = 0.03

(39)

Substituting equations (35) and (39) into Eq. (38), we obtain the following approximate
solution.

y (r − π e , π e , h,τ ) ≅ 1.96{−0.2(r − π e − 0.03) + 0.11}= 0.392(π e − r ) + 0.22736(40)

In this case, we havey r −π e = −0.392, yπ e = 0.

(41)

Substituting r = r* = 0.06 and π e = π e * = 0.03 into Eq. (40), we have the following
approximate solution of the equilibrium value y * .

y* ≅ 1.96 × 0.11 = 0.2156(42)

Suppose, furthermore, thatβ = 0, ε = 0.2, γ = 0.3.

(43)Then, the system of equations (15) becomes as follows.13(i)⎧
ϕ1 (π e , e; α )ifr& = f1 (π e , e; α ) = ⎨e⎩max[0, ϕ1 (π , e; α )] if

r＞0
r=0

( ii ) π& e = f 2 (π e , e;θ ) = 0.3[θ (0.03 − π e ) + 0.2(1 − θ )(e − 0.97 )]⎧ ϕ 2 (r , π e , e; α ,θ )if
( iii ) e& = f 3 (r , π , e; α , θ ) = ⎨e⎩min[0, ϕ 2 (r , π , e; α ,θ )] if
e

0＜e＜1
e =1

(44)

where

ϕ1 (π e , e; α ) = α {0.2(e − 0.97) + π e − 0.03},
ϕ 2 (r , π e , e; α ,θ ) = e[

− f 1 (π , e; α ) + f 2 (π , e;θ )+ 0.2(π e − r ) + 0.006].
e
π − r + 0.58
e

(45)

e

(46)

Figures 1 – 5 are the results of our numerical simulation of the ‘out of equilibrium’
13 In this formulation, an additional natural nonlinearity, which means that the rate of

employment cannot exceed 1, is introduced.
12

dynamics corresponding to the following initial conditions.

r (0) = r * +0.01 = 0.07, π e (0) = π e * −0.01 = 0.02, e(0) = e * −0.02 = 0.95 (47)
We consider the following three alternative scenarios, where t denotes the time
period, and the unit time period is interpreted as a year.14
Case A : α = 0 and θ = 0 for 0 ≦ t＜5,

α = 0.1 and θ = 0 for t ≧ 5.
Case B : α = 0 and θ = 0 for 0 ≦ t＜5, α = 0.13 and θ = 0 for t ≧ 5.
Case C : α = 0 and θ = 0 for 0 ≦ t＜5, α = 0.13 and θ = 0 for 5 ≦ t＜10,
α = 0.13 and θ = 0.1 for t ≧10.
Case D : α = 0 and θ = 0 for 0 ≦ t＜5, α = 0.13 and θ = 0 for 5 ≦ t＜10,
α = 0.13 and θ = 0.3 for t ≧10.
Comparison of the cases A and B suggests that the increase of the speed of the
response of central bank’s monetary policy (α ) has a stabilizing effect, but slightincrease of the value of the parameter α may not sufficient to restore the ‘normal’equilibrium because of the existence of the nonnegative constraint on the nominal rateof interest, if the central bank does not announce the target rate of inflation at all or theannouncement of the inflation targeting by the central bank is highly incredible for the
public.15
Comparison of the cases B, C, and D suggests that even the slight increase of the‘credibility’ of the inflation targeting by the central bank has surprisingly strong powerof macroeconomic stabilization. In fact, in our numerical example the economy moves,although relatively slowly, toward the ‘normal’ equilibrium point, if only 10％ of the
population think that the central bank’s announcement of the 3％ target rate of
inflation per year is credible.16 As our numerical example shows, this conclusion is true
even when the economy is stuck to the so called ‘liquidity trap’ with the lower bound of
the nominal rate of interest like the Japanese economy in the late 1990s and the 2000s.
In our numerical example, the economy moves to the state of full employment fairly
rapidly if 30％ of the population believe the central bank’s target rate of inflation.
14 For the numerical simulation, we adopted Euler’s algorithm and the time interval∆t = 0.1 (years).

15 In fact, we can easily see that the normal equilibrium point becomes stable in both ofthe cases A and B if the negative nominal rate of interest is allowed for, but Figures 1 –3 show that it becomes unstable if the nonnegative constraint on the nominal rate ofinterest is explicitly considered.
16 In this explanation, we interpret the parameter value θ as the ratio of the people
who believe the central bank’s announcement of the target rate of inflation to the total
amount of population.

13

14

15

16

It seems that the numerical example in this section suggests the importance of the‘regime switching’ of monetary policy, especially the importance of the change of the‘credibility’ of the central bank’s announcement, for the drastic change of themacroeconomic performance.17

6. Concluding remarks
In this final section, we shall briefly comment on two popular mainstream approaches
which we did not adopt deliberately in this paper, namely, the ‘New Keynesian’ dynamic
model and the ‘optimal’ monetary policy model.18 19
The so called ‘New Keynesian’ dynamic model, that is explained extensively in
Woodford(2003), Galí(2008) and others, also adopts a kind of IS curve (New Keynesian
IS curve) and a kind of Phillips curve (New Keynesian Phillips curve) together withsome version of the Taylor interest rate monetary policy rule by the central bank. In thissense, the building blocks of our model are apparently similar to those of the ‘New
Keynesian’ dynamic model. However, our approach is more traditional (so to speak, ‘Old
Keynesian’) than ‘New Keynesian’ approach, so that we can avoid the notorious
anomalies of the dynamic behavior in ‘New Keynesian’ dynamic model, which waspointed out by Mankiw(2001), Flaschel and Schlicht(2006), Asada, Chen, Chiarella and
Flaschel(2006) etc. It is well known that ‘New Keynesian’ dynamic model is aforward-looking ‘rational expectation’ model with the so called microfounded
optimization behavior of the single ‘representative agent’ other than the central bankand the government. However, as Mankiw(2001) pointed out correctly, this ‘New
Keynesian’ formulation causes the counter-factual ‘sign reversal’ problem, which means

17 Miyao(2006) chap. 3 provided an argument against the inflation targeting policy

which was proposed by Krugman(1998). Miyao(2006) admits that the increase of the
expected rate of inflation is effective to increase the production and employment
through the increase of current aggregate demand. He points out, however, that theincrease of the expected future income also induces the increase of current aggregate
demand in Krugman(1998)’s model, and asserts that the so called ‘structural reform’can increase the expected future income and it is better than the inflation targeting.
However, we can point out following two important facts. First, Miyao(2006)’s argument
by no means imply the ineffectiveness of the inflation targeting. Second, the so called‘structural reform’ by Koizumi’s administration in the early 2000s in Japan, the mostnotable one is the privatization of the postal service, had nothing to do with the increaseof the expected future income of the Japanese public. See also chap. 2 by A. Noguchi in
Harada and Iwata(eds.)(2002).
18 Obviously, these two approaches are sometimes closely related to each other.
19 The more detailed discussions on the topics in this section are contained in
Asada(2009).
17

that the rate of change of the inflation rate becomes the decreasing function of thecurrent output level and the actual rate of employment. In other words, in this model
the rate of inflation accelerates whenever the actual output level is below the ‘natural’
output level. 20 Another problem is that the ‘New Keynesian’ rational expectation
approach treats the expected and the actual rates of inflation as well as the nominal
rate of interest as the ‘jump variables’ or ‘not pre-determined variables’ which can makethe unstable equilibrium point (in the traditional sense) ‘stable’ by allowing for the
discontinuous ‘jump’ to the convergent path. However, as Mankiw(2001) pointed out,
this postulate also contradicts the empirical fact.21
On the other hand, in our model the more traditional ‘Old Keynesian’ postulate isadopted, which means that the expected and actual rates of inflation as well as thenominal rate of interest are treated as the ‘pre-determined variables’, although in ourformulation some ‘forward looking’ element as well as the ‘backward looking’ element of
the inflation expectation formation is allowed for. In our rather traditional formulation,
the anomalies of the behavior of the main macroeconomic variables, which are peculiar
to the ‘New Keynesian’ formulation, do not occur. Next, let us quote our generalevaluation of the ‘New Keynesian’ dynamic model from Asada, Chiarella, Flaschel andFranke(2009) chap. 5.22
“ While the microfoundation of economic behavior is per se an important desideratum
to be reflected also by behaviorally oriented macrodynamics, the use of ‘representative’consumers and firms for the explanation of macroeconomic phenomena is too simplistic
and also too narrow to allow for a proper treatment of what is really interesting on theeconomic behavior of economic agents – the interaction of heterogeneous agents –, and
it is also not detailed enough to discuss the various feedback channels present in thereal world. Market Clearing, the next ingredient of such approaches, may however be aquestionable device for studying the macroeconomy in particular on its real side.” “Yet,neither microfoundations per se nor market clearing assumptions are the true dividing
20 Mankiw(2001) wrote as follows. “Although the new Keynesian Phillips curve has

many virtues, it also has one strikingly vice : It is completely at odd with the facts.”(Mankiw 2001, p. C52)
21 Mankiw(2001) wrote as follows. “In these models of staggered price adjustment, theprice level adjusts slowly, but the inflation rate can jump quickly. Unfortunately for themodel, that is not what we see in the data.” (Mankiw 2001, p. C54)
22 This chapter is entitled as “New Keynesian equilibrium vs. Keynesian disequilibrium
dynamics : Two competing approaches”. Incidentally, the inflation expectation formation
hypothesis that is expressed by Eq. (7) in our model can be considered to be one of thesimplest formulations of the ‘heterogeneous’ expectations hypothesis that is described inthis quotation.
18

line between the approaches we are advocating and the ones considered in this section.
It is the ad hoc, that is not behaviorally microfounded assumption of Rational

Expectations that by the chosen analytical methods makes the world in generalloglinear (by construction) and the generated dynamics convergent (by assumption) to
its unique steady state which is the root of the discontent that this chapter tries to makeexplicit. Indeed, agents are heterogeneous, form heterogeneous expectations along otherlines than suggested by the rational expectations theory, and have differentiated shortand long-term views about the economy.”
Another popular approach that was not adopted in our model is the ‘optimal’monetary policy approach. Although this approach is sometimes combined with the‘New Keyneian’ dynamic model, this approach can be combined with any othermacrodynamic model. In fact, we can also formulate such an approach in the context ofour model. For example, let us suppose that the central bank tries to control the time
path of r ≧ 0 that minimizes the following ‘loss function’ subject to the constraints ofthe equations (1) – (5) and (7) in this paper.
∞

L = ∫ {ξ (π − π ) 2 + (1 − ξ )(e − e ) 2 }e −νt dt ; 0＜ξ＜1, ν＞00(48)

where ξ and ν as well as π and e are constant parameters.
We can solve such a dynamic optimization problem by means of Pontryagin’s
maximum principle(cf. Gandolfo 1996 chap. 22), and in a typical case the optimal pathconverges to the steady state with ‘natural’ rate of employment, positive nominal rate ofinterest and positive rate of inflation.23 In this case, the ‘deflationary depression’ that isaccompanied by the ‘liquidity trap’ with the lower bound of the nominal rate of interest
cannot occur at the ‘optimal’ path.
However, such an ‘optimal’ monetary policy approach requires that the central bankhas omniscient power with the accurate knowledge of the macroeconomic structure(rational expectations), even if it is not assumed that the public has such an omniscient
power.24 In fact, in Japan in the late 1990s and the 2000s the deflationary depression
with the liquidity trap did occur, which means that not only the Japanese public but
also BOJ had no such omniscient power at that period.
It is not practical to stick to the ‘optimal’ monetary policy to evaluate the performanceof actual monetary policy because of the above mentioned reason. Instead, we would like23 See also Asada(2009).

24 It is worth noting that in the ‘New Keynesian’ dynamic model it is assumed that the

public as well as the central bank has such a power.
19

to propose to use more practical criterion of the policy evaluation, namely, the‘permissibility’ criterion. Asada(2009) proposed the following simple and practical
criterion of the ‘permissibility’ of the monetary policy.
“The monetary policy that can avoid the deflationary depression with the ‘liquidity
trap’ is permissible, while the monetary policy that cannot avoid it is impermissible.”
We can easily see that the ‘optimal’ policy in the above formulation is one of the‘permissible’ policies, but there are infinite numbers of ‘permissible’ but non-optimal
policies. The approach in this paper presented an analytical apparatus to evaluate the‘permissibility’ rather than ‘optimality’ of the monetary policy. It is clear that themonetary policy of BOJ during the period from the late 1990s to the 2000s was not only
‘non-optimal’ but also ‘impermissible’ in view of our criterion.

Acknowledgment
This paper was written while the author was granted to make research under the“Chuo University Leave Program for Special Research Project” in 2009. This research
was financially supported by the Japan Society for the Promotion of Science (Grant-in
Aid (C) 20530161) and Chuo University. Needless to say, however, only the author is
responsible for possible remaining errors.

Appendix : Proof of Proposition 4.
(1) It is easy to see that a1 is a linear increasing function of α , and by assumption
we have a1＜0 when α = 0. Hence, there exists the unique parameter value

α 1＞0 that satisfies the following property.
( P1 ) We have a1＜0 for all α ∈ [0, α 1 ), we have a1 = 0 at α = α 1 , and we havea1＞0 for all α ∈ (α 1 ,+∞).
On the other hand, a1 a 2 − a3 is a quadratic function of α in which the coefficientof α 2 is positive, and we have a1 a 2 − a3 = − a3＜0 when α = α 1 . Therefore, there
exists the unique parameter value α 0＞α 1 that satisfies the following property.

20

( P2 ) We have a1 a 2 − a3＜0 for all α ∈ [α 1 , α 0 ), we have a1 a 2 − a3 = 0 at

α = α 0 , and we have a1 a 2 − a3＞0 for all α ∈ (α 0 ,+∞).
Property ( P1 ) means the equilibrium point of the system (15) is locally unstable forall α ∈ [0, α 1 ), because one of the local stability conditions (29) is violated in thisα . Property ( P2 ) means that the equilibrium point of this system
becomes locally unstable also in the range α ∈ [α 1 , α 0 ) because of the same reason.
In sum, the equilibrium point of this system is locally unstable for all α ∈ [0, α 0 ).
range ofOn the other hand, properties ( P1 ) and ( P2 ) mean that the equilibrium point of
this system is locally asymptotically stable for all α ∈ (α 0 ,+∞), because all of thelocal stability conditions (29) are satisfied in this rage of the parameter value α .
(2) At the point α = α 0 , we have the following property.

( P3 ) a1＞0, a3＞0, a1 a 2 − a3 = 0

We can show that the characteristic equation (24) has a pair of pure imaginary roots
and one negative real root when the property ( P3 ) is satisfied (cf. Theorem A 13 in
the mathematical appendix of Asada, Chiarella, Flaschel and Franke 2003).
Furthermore, we can see that the real part of the complex roots is a decreasing
function of α at the point α = α 0 , since we have∂ (a1 a 2 − a3 )＞0 at α = α 0 .
∂αThis means that the point α = α 0 is in fact the ‘Hopf bifurcation’ point, so that it isensured that there exists a family of non-constant closed orbits for some range of theparameter value α which is sufficiently close to α 0 (cf. Theorem A 10 of themathematical appendix of Asada, Chiarella, Flaschel and Franke 2003). □References[1] Asada, T.(2006a) : “Inflation Targeting Policy in a Dynamic Keynesian Model with
Debt Accumulation : A Japanese Perspective.” In Chiarella, C., R. Franke, P.
Flaschel and W. Semmler (eds.) pp. 517 – 544.
[2] Asada(2006b) : “Stabilization Policy in a Keynes-Goodwin Model with Debt
Accumulation.” Structural Change and Economic Dynamics 17, pp. 466 – 485.
[3] Asada, T.(2008) : “Inflation, Deflation and Employment : A Macrodynamic

21

Approach.” In D. T. Bentley and E. P. Nelson (eds.) Inflation : Roles, Targeting, andDynamics, Nova Science Publishers, New York, pp. 77 – 99.
[4] Asada, T.(2009) : “Central Banking and Deflationary Depression : A Japanese
Perspective.” In Central Banks and Banking in an Era of Turmoil, Nova Science
Publishers, New York, forthcoming.
[5] Asada, T., P. Chen, C. Chiarella and P. Flaschel(2006) : “Keynesian Dynamics andthe

Wage-PriceSpiral:A

Baseline

Disequilibrium

Model.”

JournalofMacroeconomics 28, pp. 90 – 130.
[6] Asada, T., C. Chiarella, P. Flaschel and R. Franke(2003) : Open Economy

Marodynamics : An Integrated Disequilibrium Approach. Springer, Berlin.
[7] Asada, T., C. Chiarella, P. Flaschel and R. Franke(2009) : Monetary Macrodynamics.
Routledge, London, forthcoming.
[8] Bernanke, B., T. Laubach, F. Mishkin and A. Posen(1999) : Inflation Targeting :Lessons from the International Experience. Princeton University Press, Princeton.
[9] Chiarella, C., R. Franke, P. Flaschel and W. Semmler (eds.)(2006) : Quantitative andEmpirical Analysis of Nonlinear Dynamic Macromodels. Elsevier, Amsterdam.
[10] Eggertsson, G. S. and M. Woodford(2006) : “The Zero Bound on Interest Rates andOptimal Monetary Policy.” Brookings Papers on Economic Activity 1, pp. 139 – 233.
[11] Flaschel, P. and E. Schlicht(2006) : “New Keynesian Theory and the New Phillips
Curves : A Competing Approach.” In Chiarella, Franke, Flaschel and Semmler (eds.)
pp. 113 – 145.
[12] Galí, J.(2008) : Monetary Policy, Inflation, and Business Cycle : An Introduction to

the New Keynesian Framework. Princeton University Press, Princeton.
[13] Gandolfo, G.(1996) : Economic Dynamics (Third Edition). Springer, Berlin.
[14] Gong, G.(2005) : “Modeling Stabilization Policies in a Financially Unstable
Economy. Metroeconomica 56, pp. 281 – 304.
[15] Hamada, K.(2004) : “Policy Making in Deflationary Depression.” Japanese

Economic Review 55, pp. 221 – 239.
[16] Harada, Y. and Iwata, K. (eds.)(2002) : Empirical Analysis of the Deflationary

Depression. Toyo Keizai Shinpo-sha, Tokyo. (in Japanese)[17] Ito, T. and T. Ban(2006) : Inflation Targeting and Monetary Policy. Toyo Keizai
Shinpo-sha, Tokyo. (in Japanese)[18] Kalecki, M.(1971) : Selected Essays on the Dynamics of the Capitalist Economy.
Cambridge University Press, Cambridge.
[19] Keynes, J. M.(1936) : The General Theory of Employment, Interest and Money.
Macmillan, London.

22[20] Krugman, P.(1998) : “It’s Baaack : Japan’s Slump and Return of the Liquidity Trap.”

Brookings Papers on Economic Activity 2, pp. 11 – 114.
[21] Mankiw, G.(2001) : “The Inexorable and Mysterious Tradeoff between Inflation andUnemployment.” Economic Journal 111, pp. C45 – C61.
[22] McCallum, B. T.(2000) : “Theoretical Analysis Regarding Zero Lower Bound on
Nominal Interest Rates.” Journal of Money, Credit and Banking 32, pp. 870 – 904.
[23] Miyao, R.(2006) : Time Series Analysis of Macroeconomic Monetary Policy. Toyo
Keizai Shinpo-sha. (in Japanese)[24] Reifschneider, D. and C. Williams(2000) : “Three Lessons for Monetary Policy in a
Low-Inflation Era.” Journal of Money, Credit and Banking 32, pp. 936 – 966.
[25]

Taylor,J.

B.(1993):“Discretion

versus

PolicyRules

in

Practice.”

Carnegie-Rochester Conference Series on Public Policy Vol. 39, pp. 195 – 214.
[26] Woodford, M.(2003) : Interest and Prices : Foundations of a Theory of Monetary

Policy. Princeton University Press, Princeton.

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