A Dynamic Analysis of an Economy with a Zero Interest Rate Bound (Mathematical Economics)

147

A

Dynamic Analysis

of

an

Economy with

a

Zero

Interest Rate

Bound*

HideyukiAdachi

and TamotsuNakamura

GraduateSchool ofEconomics

Kobe University

2-1 Rokkodai, Nada-ku

Kobe657-8501, Japan

$\mathrm{E}$-mail:nakmura\copyright econ.Kobe-u.ac.jp

TamotsuNakamura

GraduateSchool ofEconomics

Kobe University

2-1 Rokkodai, Nada-ku

Kobe657-8501, Japan

$\mathrm{E}$-mail:nakmura\copyright econ.kobe-u.ac.jp

Abstract

For

more

than

a

decade the Japanese economy has been in the serious slump with very low

interest rates and low price inflation

or

even deflation. The traditional IS-LM models analyze

thiskind ofsituation

as a

special

case

ofmacroeconomicphenomenaknownas a “liquidity trap”

assuming constantpricelevels. This papertriestoexplainthe dynamicsofinflation andinterest rates incorporatingthe Phillips

curve

into

an

IS-LM model. If the economy is around thesteady

state with not-low nominal interest rates, it may converge to its steady state or exhibit the

cyclicalbehavior of inflationanclinterestrates.Incontrast, ifitsnominalinterest rates

are

close

to zero, the economy’s behavior becomes very unstable. The economy may fall into a sO-called

deflationaryspiral. The effects of fiscal andmonetary policiesarealsoexamined in the economy

withdeflationandlowinterestrates.

1. Introduction

Mostrecentmacroeconomicmodels

assume

economic agents’ intertemporal maximizingbehavior:

utility-maximization by households, and profit-

or

value-maximization by producers. Building models

based upon rigorous and explicit microfoundations has of

course

various merits. For example,

one can

examine the economic effectsoftax policies

more

precisely becauseachangeinthe tax systemaffectsthe

behavior of each agent at themicro level. As Akerlof(2002) points $\mathrm{o}\mathrm{u}l$ however, those models cannot

explainsuchimportantmacroeconomicphenomena

as

(1)the existenceofinvoluntaryunemployment,(2)

the impactof monetary policyonoutput andemployment, (3)the failureofdeflationto accelerate when

.

_{We wouldliketothankProfessorsTakashi Kamihigashi, Torn Maruyama, KojiShimomura,andseminar}

participantsat_{Kobe and Kyoto Universities. The first author grateffilly acknowledges the Kobe University}

21stCentury COEProgram,theresearch grantffom the JapaneseMinistryof EducationandScience. The authors

are

solely responsible forremaining

errors.

148

unemploymentis high.

As iswell-known, the Japaneseeconomyhas been intheserious slump with the above phenomena

for

more

than

a

decade. According to Akerlof’s insight,

one

maynot employthemacroeconomic model based upon maximizing behavior to explain the recent Japanese economy. Instead of intertemporal

maximizing behavior and perfect-foresight assumptions, this paper employs behavioral assumptions

characteringtraditional IS-LMmodelsand naive(oradaptive)expectations toanalyzethe recentJapanese

economy.

The recent Japanese economy is also characterized by very-low nominal

interest

rates and low

inflation

or even

deflation,

as

Figure-l shows. This kind ofsituation is treated

as

a

special

case

as a

“liquiditytrap” intheIS-LM models. But,

one

cannotanalyzethedynamicsofinterestrates and inflation

in

a

traditional IS-LM model since it does not allow price levels

or

inflation rates to change. The assumption of constant prices (or inflation rates) entails the other analytical flaw that

one

cannot

distinguishrealinterestratesffom nominalinterestrates.

In order to considerinflation

or

deflation in

an

IS-LMffamework,

we

need

an

equationthat relates

the rate ofinflationtothe level ofunemployment

or

GDP. As“thesingle mostimportant

macroeconomic

relationship isthe Phillips curve” (Akerlof2002,p. 418), nestingthe Phillips

curve

into

a

simple IS-LM

model,

we

willexaminethe dynamicsoftheexpectedinflationrates andnominalinterestrates.

$|\ulcorner^{-}$ – - - , $|$ $|$ $|$ $|$ $|$ $\mathrm{i}!$

$\lfloor---\overline{---...-\underline{\lfloor_{---\cdot\cdot-\cdot\cdot---}-\mathrm{N}\mathrm{o}\min}\underline{\mathrm{a}1\mathrm{G}\mathrm{D}\mathrm{P}\mathrm{G}\mathrm{r}\mathrm{o}\mathrm{w}\mathrm{t}\mathrm{h}\mathrm{R}\mathrm{a}\mathrm{t}\mathrm{e}-\mathrm{l}\mathrm{n}\mathrm{f}1\mathrm{a}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{R}\mathrm{a}\mathrm{t}\mathrm{e}-\overline{\mathrm{N}}}\underline{\underline{\mathrm{o}}}}\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{a}\mathrm{I}---\cdot$Interest Rate

143

There is the growing literature that focuses on the issues

we

address in this paper based upon rigorous _{and explicit microfoundations. For example,} Benhabib, Schmitt-Grohe and Uribe (2002), and Buiter and Panigirtzoglou (2003), capturing the interactions between forward-looking prices and the

agents’ intertemporal maximizing behavior, discuss the possibility of liquidity trap and the economic

policiestoavoidortoescapeffomthe trap.Althoughtheir approaches

are

totallydifferent ffom ours,

we

do notdenytheimportance oftheanalyses, which givemany veryimportanteconomicinsights.Themain

aim of this paper is to examine whether

a

simple IS-LM model

can

explain importantmacroeconomic

phenomena,andtoshow thatitisstill useful for policy debates.

Thispaper isstructuredas follows. Section 2 setsupthe basicmodel, derives the stationarystates,

and analyzes the dynamic properties. Section 3 extends the model to examine the economic policy implications. Section 4 concludesthepaper.

and analyzes the dynamic properties. Section 3extends the model to examine the economic policy implications. Section 4concludesthepaper.

2.The Basic Model

2-1.The Setup

The model to be presented hereis nothing

more

than

a

collectionofeconomicbehavioral equations

provided in such

an

undergraduate textbook

as

Mankiw (2002). Let

us

begin with

an

IS-LM model

as

follows:1

$\mathrm{Y}=C(\mathrm{Y}-T)+I(r)+G=C(\mathrm{Y}-T)$\dagger I(i-w$e$

)$+G$,

$M/P=L(i,\mathrm{Y})$, ₍₂₎

where$\mathrm{Y}$is income,$\mathrm{T}$istaxes,$\mathrm{r}$isreal interestrate,$\mathrm{G}$ isgovernment purchases, $i$isnominal interestrate, $\mathrm{M}$ismoney

supply, and$\mathrm{P}$isprice level. Eq.

(1)is

an

ISequation while(2)isanLMequation. According

to thetextbook, if $P$and $\pi^{e}$, in additionto the policyvaribles, aregiven,

one

canfind the equilibrium

incomeandnominal

or

realinterestrate.

Solving equation(1)for$\mathrm{Y}$givesus

$\mathrm{Y}=F(i-\pi^{e},A)$,

whereAis thevectorof

exogenous

variables including such policy variables

as

$T$,$G$and$M$

.

To simplythe

analysis,

we

$\log$-linearizethe above equation

as

below:

$y=e_{a}a-e_{t}r=e_{a}a-e_{r}(i-\pi^{e})$ ₍₃₎

$\mathrm{Y}=F(i-\pi^{e},A)$,

whereAis thevector

exogenous

variables including such policy variables

as

$T$,$G$and$M$

.

To simplythe

analysis,

we

$\log$-linearizethe above equation

as

below:

$y=e_{a}a-e_{t}r=e_{a}a-e_{r}(i-\pi^{e})$ ₍₃₎

1 _{ThemodelisbasedonAdachi}

150

where $y=\ln \mathrm{Y}$ and $a=\ln$A, $e_{a}$ and $e_{r}$

are

positiveconstants.

For the LMequation, itisassumedtotake the following form:

$M/P$$=k(i)\mathrm{Y}$

.

$M/P$$=k(i)\mathrm{Y}$

.

where $k(i)$ is theMarshallian$k$with $k’(i)<0$

.

Takingthe $\log$ofboth sides of the aboveequation and

differentiatingit with respect totime,

we

obtain

$\mu-\pi=(k’(i)\mathit{1}k(i))i+\dot{y}$ (4)

where $\mu$

$\equiv\dot{M}\mathit{1}M$

.

$\pi$$\equiv PIP$

.

We

assume a

liquiditytrap, i.e., there is

some

non-nagativerate such

that, if the nominal intrest rate approches it, then the liquidity preference becomes infinite. Since the

nominalinterestratecannot benagative,let

us assume

the following:

$. \lim(-k’ (i)/k(i))$$=\infty$, and _{$\frac{d}{di}(-k’(i)/k(i))>0$} around $i=0$

.

(5)

We employ two important behaviorial eguations in macroeconomics in determining the rate of inflation $\pi^{e}$

:

Phillips

curve

and Okun’s law. Theprice-price Phillips

curve

relates thegap between $\pi^{e}$

and $\pi$ to theunemploymentrate $u(t)$:

where $\mu$

$\equiv\dot{M}\mathit{1}M$

.

$\pi$$\equiv\dot{P}\mathit{1}P$

.

We

assume a

liquiditytrap, i.e., there is

some

non-nagativerate such

that

if the nominal intrest rate approches it, then the liquidity preference becomes infinite. Since the nominalinterestratecannotbenagative,let

us assume

the following:

$\lim_{iarrow 0}(-k’(i)/k(i))=\infty$, and $\frac{d}{di}(-k’(i)/k(i))>0$ around $i=0$

.

(5)

We employ two important behaviorial eguations in macroeconomics in determining the rate of inflation $\pi^{e}$

:

Phillips

curve

and Okun’s law. Theprice-price Phillips

curve

relates thegap between $\pi^{e}$

and $\pi$ to theunemploymentrate $u(t)$:

$\pi(t)=\pi^{e}(t)-f(u(t))$

or

$\pi(t)-\pi^{e}(t)=-7$$(u(t))$

.

$f’(u(t))>0$

.

Linearizingthe above around the NAIRU

or

the natural rateofunemployment $u_{n}$,

we

have

$\pi(t)-\pi^{e}(t)=$一と $(u(t)-u_{n})$$\backslash \overline{a}=f’(u_{n})>0$

.

(6)

On the otherhand, the Okun’s lawshowsthe important empirial relationshipbetween

a

changein

unemploymentrateand

a

changeinincome,which is expressed in descrete time:

$u_{t+1}-u_{t}=- \overline{\beta}(\frac{\mathrm{Y}_{t+1}-\mathrm{Y}_{t}}{\mathrm{Y}_{t}}-\frac{\mathrm{Y}_{p.t+1}-\mathrm{Y}_{p.t}}{\mathrm{Y}_{p.t}})$,

and incontinuous time

$\mathrm{k}(\mathrm{i})=-\overline{\beta}(\dot{\mathrm{Y}}(t)/\mathrm{Y}(t)-\dot{\mathrm{Y}}_{P}(t)/\mathrm{Y}_{P}(t))$ ,

where $\overline{\beta}>0$,and $\mathrm{Y}_{P}$ isthepotentialrateofoutput.Integarationofthe abovegives

$u(t)=-\beta(y(t)-yp(t))$, (7)

and incontinuous time

$\dot{u}(t)=-\overline{\beta}(\dot{\mathrm{Y}}(t)/\mathrm{Y}(t)-\dot{\mathrm{Y}}_{P}(t)/\mathrm{Y}_{P}(t))$,

where $\beta>0$,and $\mathrm{Y}_{P}$ isthepotentialrateofoutput.Integarationofthe abovegives

151

where $y_{p}(t)\equiv\ln \mathrm{Y}_{P}(t)$

.

From(6)and(7),

we

have

$\pi(t)-\pi^{e}(t)=a[y(t)-y_{n}(t)]$, (8)

where $\alpha$ $=\overline{a}\overline{\beta}>0$ ,and _{$y_{n}=y_{p}(t)-(1/\overline{\beta})u_{n}$} isthenaturalrate ofoutput. From

now

on,

we assume

that $y_{n}$ isconstant because

our

analysisislimitedto theshort-runormedium-run.

Toclose the model,

we

needthe equation that determines the expectedrate ofinflation. Here,

we

assume

anaiveoradaptiveexpectation:

$\dot{\pi}^{e}=\gamma(\pi-\pi^{e})$ (9)

where $\gamma$ isapositiveconstant.

where $\alpha$ $=\overline{a}\overline{\beta}>0$ ,and _{$y_{n}=y_{p}(t)-(1/\overline{\beta})u_{n}$} isthenaturalrate ofoutput. From

now

on,

we assume

that $y_{n}$ isconstant because

our

analysisis$\mathrm{l}\mathrm{i}\cdot\dot{\mathrm{u}}\mathrm{t}\mathrm{e}\mathrm{d}$ to theshort-mnormedium-mn.

Toclose the model,

we

needthe equation that detemines the expectedrate ofinflation. Here,

we

assume

anaiveoradaptiveexpectation:

$\dot{\pi}^{e}=\gamma(\pi-\pi^{e})$ (9)

where $\gamma$ isapositiveconstant.

The system consists offour equations-(3), (4), (8) and $(9)-$ and four unknowns-},$i,$_” and $\pi^{e}$

Fromaneasymanupulation,

one

canobtainthe simpliedsystemoftwodifferential equations:

$i=\phi(i)\{\alpha(\mu_{r}+1)[e_{a}a-e_{r}(i-\pi^{e})-y_{n}]-(\mu-\pi^{e})\}$ ₍₁₀₎

$\dot{\pi}^{e}=a\gamma[e_{a}a-e,(i-\pi^{e})-y_{n}]$ (11)

where $\phi(i)=[e_{r}-(k’(i)/k(i))]^{-1}>0$ _with $\lim_{iarrow 0}\emptyset(i)=0$ and

0

$\mathrm{W}$$\phi(i)<1/e_{r}$

.

Equation(11) islinear

whileequation(10)_{is non-linear.}But,

we

shouldnotethat $\phi(i)$ is

an

onlynon-linearterm,which

comes

ffom themoney demand ffinction. Inotherwords,the LMffiction playsakey rolein

our

$\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}.2$

2-2. Stationary StatesandStability

As Figure-2shows,thesystemhas two stationarystates:onewithinflation $E$

,

andthe other with

deflation $E_{D}$

.

At $E_{I}$ thenominalinterestrate $i$ andthe expectedrate ofinflation $\pi^{e}$,whichisequal

to the actual rateofinflation $\pi$,is determined by

$\mu=\pi^{e*}$ and _{$e_{a}a$}-_{$e_{r}(i-\pi^{e})$ $=y_{n}$},

At $E_{D}$,

on

the otherhand,thefollowingrelationshipsholds:

$i=0$ _and $\pi^{p}=-r$ $<$ $\mathrm{u}$,

$\dot{\pi}^{e}=a\gamma[e_{a}a-e,(i-\pi^{e})-y_{n}]$ (11)

where $\phi(i)=[e_{r}-(k’(i)/k(i))]^{-1}>0$ _with $\lim_{iarrow 0}\emptyset(i)=0$ and $0\leq\phi(i)<1/e_{r}$

.

Equation(11) islinear

whileequation(10)_{is non-linear.}But,

we

shouldnotethat $\phi(i)$ is

an

onlynon-linear$\mathrm{t}\mathrm{e}\mathrm{m}$,which

comes

ffom themoney demand ffinction. Inotherwords,the LMffiction playsakey rolein

our

$\mathrm{m}\mathrm{o}\mathrm{d}\mathrm{e}\mathrm{l}.2$

2-2. Stationary StatesandStability

As Figure-2shows,thesystemhas two stationarystates:onewithinflation $E$

,

andthe other with

deflation $E_{D}$

.

At $E_{I}$ thenominalinterestrate $i$ andthe expectedrate ofinflation $\pi^{e}$,whichisequal

to the actual rateofinflation $\pi$,is detemined by

$\mu=\pi^{e*}$ and _{$e_{a}a-e_{r}(i-\pi^{e})=y_{n}$,}

At $E_{D}$,

on

the otherhand,thefollowingrelationshipsholds:

$i=0$ _and $\pi^{\Phi}=-r_{n}<\mu$,

2 _{Incontrasttoourmodel,Romer}

152

where $r_{n}=$$(eaa-y_{n})\mathit{1}e_{r}$ istherealinterestrateconsistentwith the naturalrateof output $y_{n}$

Figure-2: Two Stationary States

Linearizingthe systeminthe neighborhood of $E_{l}$,

we

have thefollowingJacobianmatrix:

$J,$ $\equiv|^{-\phi(i,)\mathit{0}oe_{r}(\gamma e_{r}+1)}-a_{f}e$

,

$\phi(i,)[\mathit{0}oe_{r}(\mu_{r}+1)+1]a\mu,\rfloor$,

and hence its$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$anddeterminant

are

$trJl=\mathit{0}oe_{r}[\gamma-\phi(i_{I})(\gamma \mathrm{e}_{r}+1)]$ and $\det J_{I}=\phi(i,)\alpha\mu_{r}>0$

.

Since the determinant isalways positive, in orderforthe system to be asymptotically stable around $E$

,

the$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$mustbenegative,whichisequivalent to

$\gamma$$<k(i_{J})/k’(i,)$,

where $i$

,

is the interestrate at $E_{l}$

.

Therefore,the stationarystate with the highnominal interest rate is

asymptotically stable(a) if the adjustmentspeed $\gamma$ of expectedinflationrate$(\pi^{e})$ withrespect toactual

inflation rate$(\pi)$is sufficientlylow,

or

(b) if theelasticityofmoneydemandwith respect tothe nominal

interest rate $(i_{l}’(i_{l})/k(i,)$ $)$ is sufficiently small,

or

both. One

can

also establish the following

where $i_{l}$ is the interestrate at $E_{l}$

.

Therefore,the stationarystate with the highnominal interest rate is

asymptotically stable(a) if the adjustmentspeed $\gamma$ of expectedinflationrate$(\pi^{e})$ withrespect toactual

inflation rate$(\pi)$is sufficientlylow,

or

(b) if theelasticityofmoney demandwith respect tothe nominal

153

proposition.

Proposition 1.

Hopf-Bifurcation

occurs

at $\gamma$$=\gamma_{0}$, and hence thereexist

some

non-constantperiodic solutions

of

thesystem at

some

parameter values, $\gamma\in(0,\infty)$ aresufficiently closeto $\gamma_{0}$

.

Proof$\cdot$

.

ThecharacteristicequationoftheJacobian, $\lambda^{2}$

(traceJ )$\lambda+(\det J_{l})=0$,has

a

pairof

pure

imaginary

roots when $traceJ_{l}$ $=0$ and $\det J,$ $>0$

.

The latter $\det J,$ $>0$ always holds. Suppose that $J’ \mathit{0}$ isthe

critical value of parameter

₇

such that $\gamma_{0}=-k(i,)\mathit{7}k’(iI)$

.

Then, traceJ$=0$ when $\gamma$$=\gamma_{0}$,

so

that

the equation has

a

pair of pure imaginary roots. In addition, if the characteristic roots $\mathrm{X}(\mathrm{y})$

are

imaginary,thentherealpart: ${\rm Re}\lambda(\gamma)$ $=$ traceJ/2.Hence, $\frac{d({\rm Re}\lambda(\gamma)}{d\gamma}|_{\gamma=\gamma_{0}}=\frac{ooe_{r}\cdot(1-\phi(i_{l})e_{r})}{2}$

.

Since $trJ,$ $=ae_{r}[\gamma-\phi(i,)(\gamma e_{r}+1)]=0$ when $\gamma=\gamma_{0}$, $d({\rm Re}\lambda(\gamma))/d\gamma|_{r\overline{-}\gamma_{0}}=ae_{r}\phi(i_{l})/2>0$ , which

means

thatallconditions of the Hopf-Bifurcation theorem

are

satisfied. (Q.E.D.)

Proof$\cdot$

.

ThecharacteristicequationoftheJacobian, $\lambda^{2}$

(traceJ )$\lambda$\dagger_{$(\det J_{l})=0$},has

a

pair

pure

imaginary

roots when $traceJ_{l}=0$ and $\det J,$ $>0$

.

The latter $\det J,$ $>0$ always holds. Suppose that $\gamma_{0}$ isthe

critical value of parameter $\gamma$ such that $\gamma_{0}=-k$(i, )1$k’(i_{I})$

.

Then, traceJ$=0$ when

$\gamma$$=\gamma_{0}$,

so

that

the equation has

a

pair of pure imaginary roots. $\ln$ addition, if the characteristic roots $\lambda(\gamma)$

are

imaginary,thentherealpart: ${\rm Re}\lambda(\gamma)=$ traceJ/2.Hence, $\frac{d({\rm Re}\lambda(\gamma)}{d\gamma}|_{\gamma=\gamma_{0}}=\frac{ooe_{r}\cdot(1-\phi(i_{l})e_{r})}{2}$

Since $trJ,$ $=ae_{r}[\gamma-\phi(i,)(\gamma e_{r}+1)]=0$ when $\gamma=\gamma_{0}$, $d({\rm Re}\lambda(\gamma))/d\gamma|_{r\overline{-}\gamma_{0}}=ae_{r}\phi(i_{l})/2>0$ , which

means

thatallconditions of the Hopf-Bifurcation theorem

are

satisfied. (Q.E.D.)

—-.!

8

$!|$

.

Figure-3(a)TheLimitCycle Figure-3 (b) InterestRates and Expected

154

Let specifythe function $k(i)$

so

thatthe elasticity ofmoney demand with respect to the nominal interest rate $c=ik’(i)/k(i)$ is constant or $k(i)=ai^{-c}$, where $a$ and $c$

are

positive constants. At

a

certainvalue of 7,thereexistthelimit cycle

as

Figure-3(a)shOws.3 Our observations

are

consistentwith

the movements ofnominal interestrates and expected rates ofinflation in the period oflate 1970’s to 1980’sin Japan,

as

Figure-3(b)shows.

Figure-4: Saddle-Path around $E_{D}$

Linearizing thesysteminthe neighborhood of $E_{D}$,

we

havethefollowingJacobianmatrix:

$J_{D}\equiv\lfloor_{-a_{?}\mathrm{e}_{r}}^{-\phi’(0)(\mu-}\mathrm{r}^{e})$ $a\mu_{r}0\rfloor$,

andhence its$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$and determinant

are:

$trJ_{D}=\phi’(0)(\mu-\pi^{e})+a\mu_{r}$ ancl $\det J_{D}=- arr\emptyset’(0\mathrm{X}\mathrm{u}-\pi e)$

.

andhence its$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{e}$and deteminant

are:

$trJ_{D}=\phi’(0)(\mu-\pi^{e})+a\mu_{r}$ and $\det J_{D}=-al\mathrm{e}_{r}\phi’(0)(\mu-\pi^{e})$

.

3 _{Inthefollowing numericalsimulations, all theparametersother than}

$\gamma$

are

adjusted

so

thatthe steady

155

It is evidentfrom (5)that $\det J_{D}<0$

.

Therefore the associated characteristic equationhas

a

pair ofreal

roots which have different signs. As is shown in Figure-4, the stationary state with deflation $E_{D}$ is

a

saddle-point. If the

economy

is in theregion below$\mathrm{t}$he saddle-point path, it will fall into

a

deflationary

spiral, which is characterized by deflationand

a

continued decline inincome.

Proposition 2

Once the economy enters theregion below the saddle-point path, it will

_fall

into thedeflationary

spirals.

Proof. See Figure-4.

A Figure-5(a) shows the The results derived in the above may help understanding of the behavior of

nominal

interest

ratesandexpectedratesof inflationin 1990’sinJapan

as

Figure-5(b)shows. Proof. See Figure-4.

AFigure-5(a) shows the The results derived in the above may help understanding of the behavior of

nominal

interest

ratesandexpectedratesof inflationin 1990’sinJapan

as

Figure-5(b)shows.

$99\mathfrak{l}$.1to20mO. $|\mathrm{i}$ $6$ $|_{1}($

.

$|$

Figure-5(a)SimulationResult Figure-3 (b) InterestRates and Expected

when $\gamma$$=0.15$ Inflation Rates between 1991 and2000

2-3. The Effects of Economic Policies

Inthis section,

we

willexamine the effectsoffiscalandmonetarypolicies. If the

economy

isaround thestationarystate withinflation,i.e. $E$

,

,andthisstationarystateisstable, thenthepolicyeffects

are

the

156

same

as found in the textbook. Incontrast, the economy is around thestationary state with deflation, i.e.

$E_{D}$, it will not converge to neither $E_{D}$

nor

$E_{l}$

.

Hence,

one

cannot appeal to the usual comparative

statics method for evaluating the policy effects in this

case.

Instead,

we

will examine thepossibility that

the policies

are

able to pull the

economy

out of the deflationary spiral if the

economy

fell into the spiral withoutthem.

$\ovalbox{\tt\small REJECT}_{-}’\pi-$ $\ovalbox{\tt\small REJECT}_{\prime\prime\prime}\prime\prime F^{J}’\prime\prime\prime\prime\prime\prime\prime\prime\prime*$

(a)Expansionary Monetary Policy (b)Expansionary Fiscal Policy

Figwe-6Effects of Economic Policies

If the economy is at point A in Figure-6, it will probably go into the deflationary spirals. An

expansionary monetarypolicy shifts up $i(t)=0$ locus,but does not change the law ofmotions around

point $\mathrm{A}$,

as

Figure-6(a) shows. Hence, the policy is not effective inthe

sense

that it cannot salvage the

economy

outof the deflation.

Incontrast,a fiscalpolicy shifts down $\dot{\pi}^{e}(t)=0$ locus

as

wellas $i(t)=0$ locus. AsFigure-6(b),

the law ofmotionscould be changed duetothis policy. Hence,the policyiseffective inthatit

can

pull the

economy

out of the deflation. However, it should be noted that the economy is very unstable

157

3.TwoExtensions

So far

we

have assumed

a

very simple naive expectation and simple economic policies. In next

subsection,wewillmodifytheexpectation formula according toMalinvaud(2000).Insubsection 3-2, we

willbriefly discuss the effectiveness of inflationtargetpolicy using thebasic model.

3-1.NormalLong-Run Expectations

Ourbasic model

assumes

the naive(oradaptive)expectations. Theyareofcoursetoo simple. Infact,

accordingto Nakayama and Ooshima(1999), 50 to 60 /0 _{ofJapanese firms have adaptive expectations}

over

inflation and 30 to 40 /0 _{of households’s inflation expectations}

are

_{adaptive. Here, according to} Malinvaud(2000), $\theta$ portionofeconomic

agentsformadaptive expectations

over

inflationwhile

1-0

have long-run normalexpectations,which

are

consistentwith the long-run equilibrium. Thisis expressed

in discrete-time

as

follows:

$xr_{t+1}^{e}=\psi(\pi_{\iota}-\pi_{t}^{e})+(1-\theta)(\pi^{e}. -\pi_{t}^{e})+\pi_{l}^{e}$

andin continuous-time:

$\dot{\pi}^{e}(t)=\psi(\pi(t)-\pi^{e}(t))+(1-\theta)(\pi^{e}. -\pi^{e}(t))$,

where $\pi^{e}$

.

is the long-run normal

rate ofinflation rate, which is equal to the growth rate of money

where $\pi^{e}$

.

is the long-run nomal

nte ofinflation rate, which is equal to the growth rate of money

supply $4\mathrm{J}$

.

Realizing that $\pi^{e*}=//$,

we

have the followingsystemofequations:

$i=\emptyset(i)\{a(\gamma \mathrm{e}_{r}+1)[e_{a}a-e_{r}(i-\pi^{e})-y_{n}]-(\mu-\pi^{e})\}$

.

(12)

$\dot{\pi}^{e}=$\mbox{\boldmath$\theta$}\mbox{\boldmath$\alpha$}Aeaa_{$-e_{r}(i-\pi^{e})-y_{n}]+(1-\theta)$}(p-rr’), (13)

The dynamicproperties of the systemof(12) and(13)

are

essentiallythe

same

as

those of thesystem of

(10)and(11).Hence

one

can come

tothefollowingproposition.

Proposition3

Hopf-Bifurcation

occurs

at $\gamma=\gamma_{0}^{\mathrm{I}}$, and hence thereexist

some

non-constantperiodicsolutions

of

thesystem at

some

parametervalues, $\gamma\in(0,\infty)$

are

sufficientlycloseto $\gamma_{0}’$

.

158

The monetary policy could be effective under the composition oflong-run normal and adaptive

expectations although it is unable to pull the economy out of the deflation under the

pure

adaptive

expectations. As Figure-7 shows, the expansionary monetary policy shifts not only $\dot{\pi}^{e}(t)=0$ locus up

but also $i(t)=0$ locus down. Hence,the policy

can

be

an

“effective”

one

inthe

sense

thatit is ableto

pull theeconomyoutofthedeflation.

Figure-7: ExpansionaryMonetaryPolicy under Composite Expectations 3-2. Inflation Target Policy

Here

we

willintroduce

a

simple inflationtargetpolicy: the monetary authorityincreasesthegrowth ofmoneysupplyiftheactualinflation is lessthan the target level $\pi^{T}$

,andvice

versa.

Namely,

$\dot{\mu}=\beta( rr-\pi)$

where

7

is

a

positiveconstant. Substituting(9)and(11)intotheabove,

we

have where $\beta$ isapositiveconstant. Substituting(9)and(11)intotheabove,

we

have

$\dot{\mu}=\beta\{(\pi^{T}-\pi^{e})-a[e_{a}a-e_{r}(i-\pi^{e})-y_{n}]\}$

.

(14)

The system

now

consists ofequations (10), (11) and (14). At the stationary state, the following equality holds

$15\theta$

Hence, this target inflation policyworks if the stationary state is stable. But, the questionis the stability.

Linearizingthe system around this preferablestationarystate,

we

havethefollowingJacobianMatrix

$J_{T}\equiv\lfloor_{\alpha\beta_{r}}^{-\emptyset(i_{T})\mathit{0}oe_{r}(\gamma e_{r}+1)}-a\gamma e$, $\phi(i_{T})[\mathit{0}oe_{r}(\mu_{r}+1)+1]-\beta(\mathit{0}oe,+1)a_{J}e_{r}$ $-\phi(00iT$

where $i_{T}$ istheinterestrate at $E_{l}$

Defming

$B_{1}\equiv\emptyset(i_{T})\mathit{0}oe_{r}(\gamma e_{r}+1)-a\gamma e,$ $=-\mathit{0}oe$,$[\gamma-\emptyset(i_{T})(\mathrm{y}_{r}\mathrm{t}1)]$,

$B_{2}\equiv\phi(i_{T})\alpha Je_{r}+\phi(i_{T})\alpha\beta_{r}=\phi(i_{T})\mathit{0}oe_{r}(\beta+\gamma)>0$ ,

$B_{3}\equiv-\det J_{T}$ $=$

\phi (iT)a\beta \mu r

$(\mathit{0}oe_{r} +1)$-I(iT)$\alpha e\beta\gamma e$,$2=fi(i_{T})ap_{r}>0$,

inorderfor the stationary statetobeasymptoticallystable,the following conditionsmustbe satisfied

$B_{1}>0$ and $B_{1}B_{2}-B_{3}>0$,

whichisequivalent to $B_{1}>\gamma/(\beta+\gamma)$

.

If $i_{T}=i,$, then $B_{1}=-trJ$

,

.

The stability condition is the basic model is $trJ_{l}<0$. Since the above condition is rewritten

as

$trJ,$ $<-\gamma/(/l +?)$$<0,$ it is tougher than $trJ_{l}<0$

.

Therefore, the inflation

targetpolicy introduced here tendstomakethesystem unstable rather thanstable.

$B_{3}\equiv-\det J_{T}=\phi(i_{T})a\beta\mu_{r}(\mathit{0}oe_{r}+1)-\phi(i_{T})\alpha^{e}\beta\gamma e,=\phi\angle(i_{T})ap_{r}>0$,

inorderfor the stationary statetobeasymptoticallystable,the following conditionsmustbe satisfied

$B_{1}>0$ and $B_{1}B_{2}-B_{3}>0,$

whichisequivalent to $B_{1}>\gamma/(\beta+\gamma)$

.

lf $i_{T}=i,$_’ then $B_{1}=$ $trJ$

,

.

The stability condition is the basic model is $trJ_{l}$ $<0$. Since the above

condition is rewritten

as

$trJ,$ $<-\gamma’(\beta+\gamma)<0$_, it is tougher than _{$trJ_{l}<0$}

.

Therefore, the inflation

targetpolicy introduced here tendstomakethesystem unstable rather thanstable.

4.ConcludingRemarks

Krugman $(1998, 2000)$ supposes that the Japanese economy has been in the liquidity trap and

proposes

an

“inflation policy” for it to

recover.

Although the liquidity trap is usually considered in

an

IS-LM ffamework, his analysis is based

on

not the IS-LM but

on a

simple intertemporal optimization

model. Since,in additiontohis name, his proposal

was

considered

as

theresultderived from therigorous

microfoundation, it

soon

attracted

a

greatdeal ofattentions in the policy debate. In his model, rational

expectations play

a

key role. Eventhough the deflation prevails atcurrentperiod, peoplebelieve that the inflation will happen in the ffiture thanks to

an

increase in money supply(orgrowth ofmoney supply).

This expectation of inflation decreases the real interest rate, and hence pulls the economy out of the

recession.

180

believe that the IS-LMmodelisstill useful tounderstand how theeconomybehaves in the short-runorin

themedium-run. Incontrast, inthemodel presented inthispaper,naive

or

adaptive expectationsplay

an

importantrole. As Krugman suggests,

an

increase ingrowth ofmoneysupplycouldcreateinflation. But,

it

can

be the

case

in

our

model only if the adjustment speed of expected inflation rate with respect to

actual inflationrate is sufficiently low. Ifitis high, theeconomy will notrecoverfrom the deflationary spiral.

The IS-LMmodel summarizes the interactionsbetween themoney (orbond)market and the goods marketin

a

simple but understandable

way.

Hence,

_even

thosewho

are

not familiar with intertemporal

maximizationmodels

can

understandit very easily. Ifthispapercould provide

some

intuitionsaboutthe currentJapaneseeconomy andpolicy implications,

our

first objective

was

achieved.

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