『社会科学ジャーナル』

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『社会科学ジャーナル』

38 (1998

〕

The Journal of So"ol Sden" 38 ( 1998

〕

Expectations, 1

巴乱

rningdynamics and the stabilization policy in an ov

，巴

rlappinggener

抗

ionsmodel

Taisei Kaizoji

＊

Abstract

The aim of this paper is to explore whether a least squares learning 1s able to lead the dynamics of阻 econormcsystem to perfect fores1gh

七

equilibrium, and whether a stabilization policy c叩 stabilizeunstable le田tsqu町田learnmgdynarmcs in a monetary overlapping generations model. We show tha

七

leastsquares learnmg dynamics are unstable 1f agents have strong wealth e

町

ectswhich dommate intertemporal sub‑ s

出叫

ione

町

ects,and overreactiona'y expect

叫

10ns.Therefore, a le

酪七

squ町田learning1s a source of endogenous learning which drives ec

か

nomic fluctuations, and moreover

，凶

spercep

も

10nsdoesn't vanish m the least squares learnmg dynamics. Then we illustrate that a sta‑ bilization policy (the in ere田eof the grow

七

hra

担

ofmoney supply through lump sum transfers) can s

回

bili

田叩

yunstable learning dy‑ nam1cs when

む

heleast squares learnmg canno

七

leadto the perfect foresight equilibrium.

Key words : adaptive (over re配t1

。

ⁿ^町^y^)e^x^p^e^c

^臼

^tⁱ^oⁿ^{s l}^e^田

sq凹.reslearningstabilization policy

，，阻

dchaos.

・

^T^hⁱ^sⁱ^s

^も

^hêrê^vⁱ^sê^d^vê^r^sⁱôⁿô^f^t^hê^pâ^pê^r^rêâ^dâ^t^t^hê⁷^t^h^Wô^r^l^d^Côⁿ^g^rê^s^sô^fÊ^côⁿô^mê^t^rⁱ^c

Society held in Tokyo, Augus

七

1995.I a皿 gratefulto Richard H. Day, John Duffy and Albe

同

Marcetfor their

田

efuladvice and helpful comments. However the remaining errors are exclusively mine

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2

1 Introduction

Self‑fulfilling expectations

町

eoften viewed

田

along run concept, where agents have already learned completely the law of motion governing the sys‑ tern in which they operate One should expect the agents to make significant forecasting mistakes for qmte some time while they attemp

七七

olearn

七

he dynamic laws of the signals they receive. It is thus quite natural that

七

he first question we shall address is how expect

叫

10nsand learning take place, and more importan

七

ly,whether expectations and learmng will lead towards perfect foresight eqmhbrium If the temporary equilibrium dynamics with learning (leα m

句

d

世間四回）

dont

∞ 町

erge

切

perfectforesight equilibr

叩

n in the long run, then the carrespondmg efficiency losses will appear. The next question we m

田

tconsider is whe

も

herstabilizat10n policies implemented by

出

egovern

皿

en

七七

Oget rid of efficient lasses, can lead ta s

七

abihzingthe unsta回

ble learning dynamics. Although a considerable number of studies have been made on

七

hesequestions over the p出tfew years, there is httle agreement on how the economic sys

七

emis influenced by expectations and learningi .

The

田

m of this paper is to propose an answer to

七

hetwo questions pased above. The basic model we shall use here is the simple 'Samuelsonianmodel of overlapping generations2 . We apply dα

；

ptive (ove^{作目}ction a

句 j 四：

pee t

αtio

田

asan al

七

ernativeexpectations form to perfect foresight and we as‑ sume th

叫

agentsesti

回目

ethe unknown structural parame

七

er,(error correc‑ tion coe

日

cient)by using a least squares leαm

叩

g First, we demonstrate

出品（

i)if agents have strong

凶

ertemporalsubstitution e

古田恒

wh

凶

dam mate wealth e

百

ects,and/or we

比

lyadaptive

目玉

pectations

，加

mpor

町

yequi‑ librium dynamics converge

土

operfect foresight eqmlibrium, and (n) if agerr

回

1 On the町田ntworks of e

：叩

ectations叩 dlearning mechanisrr

叫

seeD

町田

dLm (1992), Grandmont and Laroque (1

回

6,1992),

Gu•田nene

and Woodford (1992), Marcet and Sar‑ gent (1989), Ben踊y阻dBlad (1989), Evans叩 dHonkapohJa (1994), Bullard (1994) and Kaizoji (1995).

2 Benhabib and Day (1982), and G

n

。

û^sc^y^c^lê^sô^c^cû^râ^sⁱⁿ^tê^r^tê^田^p^mâîe^qûⁱ^lⁱ^b^rⁱû^m^p^hêⁿô^mêⁿâ^T^hêⁱ^r^s

^七

û^dⁱê^sdê^môⁿ^,^t^râ^tê^t^hâ

t

persisten

む

econo皿ic

日

uctuationsgenerated by volatile forecasts are indeed compatible with md1v1dual optimization, self‑fulfilling expectations阻 dWi

油田

ianmarket clearing provided

出

atthere are c

叩

italmarket i

皿

perfectio田 .The outcome of these studies is that under increasingly plausible assumptions, endogeno田 fluc凶ationsequilibria with self‑ful

臼

ling expectations do occur in such models, see Boldrin and Woodfords' review [Boldrin and Woodford (1990)].

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Expeo"tion>

，同

mingdynomi" ond

血

e"abil

回

lionpolioy in an °'"lapping gonorntinn> mndd 3

have strong wealth effects which dominate intertemporal substi

七

utione

旺

ec

回，

and/or strongly overreac

も

10naryexpectations, temporary eqmlibrium dynam i

田町

eable

も

obe chaotic Then we illustrate tha

七（

iii)a least square learnin for expectations leads to the perfect foresight equilibrium, provided that

七

he endogenous fluctuations are caused by agen

胎

overreactionaryexpecta

七

10ns, and (1v) on

七

hecontrary

七

heleast squ

町田

learnmgcannot lead

加七

heperfect foresight equilibrium, provided that the endogenous fluctuat10ns are cansed by strong weal

も

heffec

臼.

Finally, we show

七

ha

も（

v)a stabilization policy (an in ere

蹴

of

出

egrowth rate of money supply) can stabilize unstable learnmg dynamics when the least squared learning cannot lead the economic system to the perfect for

^田

ighteqmlibrium.

2 The model

Consider an overlapping generations model where each generation lives two periods The model mvolves one non‑s

七

orablegood and a single asset, money, that is employed for

七

ransferringwealth

仕

omone period to the nex

七

For most of this paper, the money stock, M will be

^田

sumed

七

obe consta

叫

over time Agents hve two periods and

町

e

協同四

1( eqmvalerr

七

ly

七

hereis a single agent) m each generation. The agen

恒

endowmentsof the good a

七

e配hage

T =

1,2, are e1

田

ide2 The representative consumer is assumed to have the separable utility function,

U(ci, c2)

=

logc1

＋占的／（

1 β

，） ^︵ ¹ ^︶

where c1 and c2 denote consumpt10n in the first and second periods of life respectively. As concerns the attitude to risk, this utility funct10n 1s known

描＇

constant

由同市

kave

凶

on

（印刷，

wherethe RRA coefficient is

-v~ （匂） Jv；（匂）＝ β.

A large β

，

which is greater

七

hanunity, means

七

hatan agent have a strong wealth effect which dominates

阻

intertemporalsubsti・^目

印刷

neffect, and on吐1econtrary a smallβ

，

which 1s less than unity, means that an agent have a weak wealth effect3 .

We focus on the

S α

muelson case. Namely, we consider that the consumers hold positive money balances,

（他国

is,technically this

四

nbe ensured by 3 See the chapter 5 of Blanchard^叩 dFischer (1985) for a full accoun

色

ofthe relation be

も

weena wealth effect andβ

『社会科学ジャーナル』