Capital Accumulation and Income Redistribution between Two Generations

(1)

Capital Accumulation and Income Redistribution between Two Generations

著者（英） TAKAHASHI Shuetsu

journal or

publication title

The Literary, Economic and Juristic Association review

number 81

page range 135‑168

year 1979‑12‑20

URL http://id.nii.ac.jp/1204/00024043/

(2)

Capital Accumulation and Income Redist ribu tion between

Two Generations

^*

Shuetsu Takahashi

1

.

Introduction

Samuelson〔5〕〔7〕and Diamond〔2〕clarify that thelaissez faire 1ong

-

^runequilibriumdoes notachieve the golden

-

rule except a mere accident.That i s , t h e y show there is no Invisible Hand or the perfectcompetition is notlikely t o l e a d to an eficient situation.Suitable government intervention may attain thegolden

-

rule.In fact Samuelson〔7〕,Stein〔8〕〔9〕and Takahashi〔10〕show respective instruments for government to attain the goIden

-

^rule

as thelong

-

run equilibrium.

The first purpose of this paper is to show that t h e l o n g

-

^{r u n}

equilibrium in the normalcase achievesthe golden

-

^rule^{and that}

the1ong

-

runequilibrium i n the abnormalcaseattains thewealth-

* The earlier version of this paper w a s presented at the A n n u a l m e e t-

ing of the Japan Association of Economics and Econometrics held at Tokyo Metropolitan University i n September1979.I w o u l d l i k e t o t h a n k Professors Osamu N i s h i m u r a ( D o s h i s h a U n i v e r s i t y ) a s a discussant and Hiroaki 〇sana(Keio University)as a chairman of the fourth sectionof the AnnualMeeting.Especia11y I a m benefited by valuable a n d critica1 comments of Professor Nishimura. I a m also grateful to Professors Shozaburo Fujino (Hitotsubashi University) and Masayuki Sekine (Tohoku Gakuin University) for their helpfu1 comments o n origina1 Japanease draft.〇f course I a m responsible for any remaining errors.

-

¹³⁵

-

(3)

CapitalAccumulation and Income Redistribution between T w o Generations

capita1 equilibrium, which means that the amount in private wealth is exactlyequalto the stock ef capital. We, as we1l as

( 1 )

Gale〔3〕,define the n o r m a l c a s e a s a case that the golden

-

^rule

stock of capitalislarger than the amount of wealth people wish to hold in thelaissez faire and the abnormalcase otherwise.

Thesecond purpose ofthis paper is to consider the economic meanings of both casesof normaland abnormalmoredeeplyand to clarify the causes of occurrence of the abnormalcace.

Eliminatingthe sources of occurrence of the abnormal case, w e can expect the golden

-

r u l e a s t h e l o n g

-

^runequilibrium. The third purpose is to show the instruments for governmenttoexpel to them and to show that the economy carrying out the instru

-

ments asymptotically converges to the golden

-

^rule. T h i s g i v e s a solution for the stability analysis i n t h e economy carrying out a security program,which has been reserved by Samuelson〔7〕. A n d finally the fourth purpose is to show introducing a socialsecurity program to the economy never abate the utilityleve1of represent

-

ative person in steady

-

^state.

W e construct the framework of our model in section 2 a n d envolve arguments to attain thesecond purpose i n s e c t i o n 3 . M a i n subject in s e c t i o n 4 i s the stability analysis of our modeI.The analysis is intensively related to the first purpose.Finally section5 examinesthe stabilityin the economy introducing the security program in order to achieve optimalgrowth.

(l) Gale shows the1ong

-

run equ出brium i n thelaissez faire can inevita

-

bly achieve the golden

-

^rule^{i n the}normalcase,but his m o d e l i s n o t a neoclassicalgrowth model.

2

-

¹³⁶

-

(4)

2

.

The model

W e state behaviors of consumers and producers in this section.

There are two generations at any time: a youngergeneration which worl【s and an older generation which retires.Thelife

-

^cycle

of each individualconsistsof a working periodand a retiring period.The periods are of equallength.A p a r t o f his income received during his youthis available to his consumption during retirement

. H e

remains no bequests for his descendants.

Letting c^l(t) a n dc2(t) b e per capita realconsumpti

on

of the younger and the older generations respectivelythe utilityof a representative person born in period l i s described

by

(1)

U

=

U

(Cl(の

,

C2(t十1))

where

U

is a strictly quasi

-

concave indicator

.

By foI1owing thelife history of a representative m a n , w e can solve an allocation problem between currentconsumption cl(t) a n d future consumption c。(t十1)

.

This individualworksin period t, f o r which he rceives a wage

,

o(t) . H e retires i n period t十1

.Then

interest income becomeshis only income.Thus he allocateswage in periodt between current andfuture consumption soasto maxi

-

mize his utility(1),given the rate of interest existing on one

-

periodloans from period t tot十1,r(t十1).That is,he maximizes his utilityfunction(l)subject to

(2) c

,

^(t⁾

+

1'

j

^.

j j

⁽

j ?

⁾ ⁼^u

^,

⁽

'

⁾

The necessary conditions for optimality are (2) c

,

⁽

'

⁾

+

^1'

j ; t

⁽

j t

⁾⁼^u

^,

⁽

^'

^:⁾

-

¹³⁷

-

(5)

(3)

普

^{= (}

, ⁺

^{r (}

一 ^器

^.

The corresponding sufficiency conditionisalways satisfied by the assumption of strict quasi

-

concavity on

U

Therefore w e can get

by

( 2 ) a n d ( 3 )

(4) C j(t)=Cl(ω(の

,

「(t十1)) (5) c2(t十 l )=c²(ω(t),r(t十 l ) ) .

0 n the other hand,production is assumed tobecarriedon sub

-

ject to constant returns to scale with respect tolabor

L

(t) and capital K(t) . D efine

y

(t) a s output per workerandk(t)ascapita1 per worker.Then production is given by

(6) .y(t )

= f

( k(t) ) ;

f

> 0 and

f

' < 0

.

Producer demands the elements of production so as to maximize his profits;

(7) ω (の=f( k(:t⁾)

-

^k(t)

f

(k(t) ) (8) r(t

) = f

' ( k(l::))

Final l y w e assume that population born at period t

, L

(ll), i s (1十g)t, w h i c h is thelabor force in period t.

3

.

Golden

-

rule and we

a

lth

- ^capitale

^【^luiIibrium

Thissection clarifies the reasons why thelong

-

^runequilibrium in thelaissez faire is not attainable to the golden

-

^rule.

W e described(4)and(5)as each individual's al!ocation between current and future consumption by fo11owing hislife

-

^cycleⁱⁿ^the

previous section.Being obtained from maximizing utility subject to a budget constraint(2),equations(4)and(5)identically satisfy ( 2 ) , t h a t is,

4

(9) c¹(u1(t) ,r(t十1))十^c^。(ω(t r(t十1 1十r(t十 l

-

¹³⁸

-

=ω(t) .

(6)

N o w the planned consumption of each individualis notrealized untiltotalmarket demand is equalto totalmarket supply. The market clearing in periodtrequiresthatconsumption

C

(の plus investment K(t十1)

-

^K⁽^t^{) i s}equalto output f( k(の⁾

L

(の

.

Con

-

sumption

C

(t

:

)in periodtconsists of consumptionc1( w (l1),r(t十1)) L(t) o f a yonger generation which works and consumptionc2(ω ( t

-

^{l ) ,}^r(^{t) )}

^L

^(t

-

^{1 ) o f}^anolder generation,born at period t

- ^1,

which retiresat psriodt.Therefore we have

(10) cl(u1(l),r(t十1))

L

(t )十c2(

,

^o⁽t

-

^1),^r^(t^{) )}

^L

^(t

-

^{1 )}

十K(t十1)

-

^K⁽^t

^{) =}

f( k(t) )

L

(t)

TransposingK(t十1)

-

^K⁽t) f r o m theleft handof (10)to the right hand and dividing both hands

by

L( t ) , w e

ob

tain

(、11

一一・、

^f-・、^,⁽^ω一一^{( j}、-

、

/^.l^{r ( t}' 、一^{十 1}'

- 、

¹

、

^・^十^'

̲

^C²^(ω(^{1 十 9}^t

-

^1)'^「⁽^{t) )}

=

f(11(の)

-

^(1十^g^{) k(}^t^十1)

-

^{l k(}^t).

In perfect competition,the equation of demand forlabor to its supply requires the equation of w a g e u,( t ) t o the marginalprodct of labor.Thenequation(9)can be expressed;

(9)

'

c1(u

,

(t)

,

r(t十 l ) ) 十^C²(u1(t,r(t十1 l 十r(t十1

= f

(

,

k(t) )

-

^k⁽^t⁾

^f

^'(k(^の^).

The system of simultaneous equations,(7),(8),(9)'and(11), describes the dynamicpath i n this economy.We reduce equations

( 9 ) ' a n d ( 1 l ) t o a equation;

(12) k(t十1)

-

(

^{i (}の一 ,^l1。[ω(t

c。(ω(t r(t十 1 1十r(t十1)

-

^{1 ) ,}^r⁽^t

1 十r(t

-

^{l 3 9}

-

)

) -

l 十r(t)

一百 X

=0.

(7)

When the dynamic system of (7),(8)and(12)convergesto the 1ong

-

^ru^:n equilibriuml2), i t must satisfy

( 3 _{1 )} _{一商}g

-

^r^*

¹ 、 ^r ̲

⁽¹^c⁺²^{( u}^{g ) ( 1 十}^*^{, r}

^つ

^r

^つ、 ^ノ ^̲

^ー⁰^'

This equation implies either the familiar golden

-

rule equation, (14) r*

=

f (是

つ ^{= g ,}

^,

or the wealth

-

capitalequilibrium defined by Gale〔3〕l31

,

( l 5 ) 1

.̲ -

(1十g ) ( 1 十 r^c²^{( ω}^*^{, r}

^つ - つ

^一

̲ 一

^ω'

-

^cl 十

^,

^{( w}g^'^{, r}^つ

The wealth

-

capitalequiiibrium asserts that the amount which a retired generation holds in private wealth, (

,

^o'

' -

^c

^,つ

^L(^{l )}^{, i s}

exactly equalto the stock of capital

K ' L

(t十1)which a working younger generation uses.By the way,the golden

-

^rule^path^shown

b y ( 1 4 ) i s a balanced growth path which maximizes the utility.

Now equation(l2)is equivalent to the fo1lowing equations;

(16) k(t十 1 ) = ^e。⁽ω(t 1 十 9

r(t十 1

1 十 r ( t 十 1 ^十

要

^fi(t)

(17) h(t十 1 ) =

?

^h(^t^{) ,}

for w e ca:n make a new variable i n ( 1 2 ) a s ( l 8 ) h(t

) =

k(t)

-

^c^。_1十⁽^u

^'

g ) ( 1 十 r (⁽^t

^-

^{1 )}^.^r⁽の^t)

Then w e can make a transformation of ( l 7 ) t o

〔

^t ^{l 十}^{r ( r )}

1

(17)' h(t十 1 ) = I_r=o

i -

_{l 十}_g ^h⁽⁰⁾^.

Therefore i f t h e l o n g

-

^runequilibrium in the economy becomes (2) W e attach asterisk(ⁱ^t) t o the equilibrium values i n the1ong

-

^run

(8) W e can g e t ( 1 5 ) b e c a u s e l l

S

^f⁼^t

^' ・

^一c^{j *} ^{b y ( 9 )}

6

-

^{l 4 0}

-

(8)

the wealth

-

capitalequilibrium,either r' =f( k

つ

>gor

h

(0)=0, atleast,must hold there.We define here

ii

a s t h e capitalper worker inthe wealth

-

capitalequilibrium and i e a s i n thegolden

-

rule.If the economy is achieving the wealth

-

capitalequilibrium when

h

(0)キ0,it impliesk*=

i

>ko b

e

cause f(

1

ll)くg

. 0 n

the other hand,whenh(0)=0,thelong

-

run equilibriumcoincideswith the wealth

-

capitalequilibrium.But w e cannotstatewhetherk*=kj jkg

or k*

= i ;

<kg.

Next w e examine the economic implications of h(

の.Firsof all,

the first term of the right hand in(16)expresses the wealth in terms of a man born atperiod t十1which a generation born at period t holds⁽⁴⁾ Consider neo

-

classicalproducers who borrow allstocks of capitalfrom the market without their

-

selvesfunds.

Then the first term of the right hand i n ( 1 6 ) i m p l i e s c a p i t a l i n terms of a man born at period t十1which a generation born at periodtlends to the producers,that is,private capitalper worker inperiod t^十1.A s 1l(t十1)

(

⁼

当き ⁱ

^f

^,

^(t)

⁾

is the di?erence be

-

tween capitalper worker k(t十 1 ) a n d private capitalper worker

C2(u1(t r(t十 1

1 十r(t十1)

,

we can interpret1l(t) as socialcapital per worker是'(の in the sense deacribed be1owl5):

There exists socialcapitalper w o r k e r 1二( 0 ) i n thebeginningof history.This capital,cooperating with private capitalandlabor,

(4) In(9)dividing wage minus consumption i n youth,i.e.wealth held by a man born at t , b y 1十g , w e can obatin the first te1m of the right hand i na0.

(5l See Stein(8)and(9)regarding to6uch socialcapita1.

-

^{l 4 1}

-

⁷

(9)

CapitalAccumuIation and Income Redistribution between T w o G,e^lnerations

is used in the productive activity.Aseveryone in this economy bindes o n socialrule such that he cannot consume consumption goodsover the presentvaluesof his income at any period,no one consumes both social k^二(t ) L(t )and its imputed interests r(t) k'(t)

L

(の.Therefore(l十r( t

:

)、)k(t)

L

(ll) becomes socialcapita1

in the next period,that is,1十^「(t) k

'

(t)becomes socialcapital

1 十 9 per worker k二(t十 1 ) i n the next period.

The second imlication of h(t) i n economics is to interpret h(ll、) asliabilities of enterprise,

:

1(t:), i nterms

of

a worker emp1oyed in i t . T h e enterprise in its inauguration borrows il(0)

L

( 0 ) f r o m government or a financialinstitution and employs this as capital stockin the productive activity.That i s , t h i s capitalstock in terms of a worker emp1oyed i n the enterprise is ,1l(0). The enterprise must return the principaland interest(l十r(0))1(0)

L

( 0 ) t o thelender at the end of period 0 . T h e n it renews a bond of d e b t . T h a t i s , i t p a y s ( l 十r(0))1(0)

L

( 0 ) t o thelender at the end of period 0 a n d , a t the same time,borrows(1十r(0)) i1(0)

L

( 0 ) f r o m thelenderat the beginning of periodl

.

It em

-

ploys the newly borrowedliabilities11( l )L( l ) = ( 1 十r(0))11(0)

L

(0) as capitalstock in p e r i o d l a s w e l l a s in period 0. T h i s capital stock in terms of a worker emp1oyed i n the enterprise is 1l ( l ) = 1 十「(0)

.

1l(0). I n a similar way

1 十 9

( l 8 ) 1(t) = 1 十「(t

-

1 )1(t

̲

^{1 )}

1十g

holds at any time after p e r i o d 2 a n d (

'

⁹⁾ ^k⁽^の=

能

⁽

i 器 + '

⁽

^,

⁾

8

-

^{l 4 2}

-

(10)

holds at any periodwhen capitalstock is financedbysaving of a young generation andliabilitiesof the enterprise from government or a financialagency.Thus(l6)and(l7)hold.

The third implication of h(t) i s a claim to government for goods.Government owes a amount of natiOnaldebt,d(0)

L

(

- ^1),

to an older generation in the origin of history.Government attaches to the debt the same interest rete as the interestrate of capital.Then government must return the principaland interest,

(1十r(0))d(0)

L

(

-

1),toanoldergeneration at the end

of period

0.This return is realized by that government inducesa younger generation t o t a k e over(1十r(0))d(0)

L

(

-

^{l )}

. T h a t i s , g

overn

-

ment at the beginning of period1owesthe amountof debt,d(1)

= 1 十「(0)d(0),to a man bom at period 0 . A n d

1 十 9

(20) d(t)

=

1十「_1十(t_g

-

^{1 )}^d(^t

^̲

¹⁾

holds inL the same way at any period2.That a younger generation purchases the debtis indifferentthat the generationlends capital to producers,because bothinterest ratesof the debt and capital are samelo1.Therefore a generation born at period^tlends to pro

-

ducers saving minus purchased debt as capitalat the beginning of period t十1;

K(t₊_1)一

̲ ^f 、 ^商

^C

^,

^(u¹^(t⁽^{商解可}^),^「⁽^t^十1))

̲ -

^d¹⁽ⁱⁱ^{十 l )}^g

- 、

^{/ ( t )}

^L

^'

Expressingthis in terms of a worker emp1oyed in producers,we have

C61 w e regard the debt and capitalas wealth in perfect safety.

-

¹⁴³

-

(11)

(21) k (t十 1 ) =

̲

^d⁽^t^十1)

1十g '

Deflning d(t十 l )=

-

^(1十

^g

⁾1^t(t十l ) here,then we can reduce(20) and(21)to(l7)and(16)respectively.

Samuelson〔6〕,〔7〕,Diamond〔2〕and Stein〔8〕,〔9〕 think,as a behavior of capital accumulation,that saving of a working generation become totalcapitalin the next period. T h e dynamic path of the economy i n their models is in fact described by

(22) k(t十 l )= C

・

^{(ω(t) ,}^「⁽^t^十1))

^.

(1十g)(1十r(t十1))

Camparing(22)with simultaneous equations(16)and(l7),we can understand(22)shows the world of h(t) = 0 , t h a t i s , t h e world withoutsocialcapitalorliabilities of the enterprise or national debt.

Finally in this section w e consider the economic meanings of both cases of n o r m a l a n d abnorma1,and clarify the causes of occurrence of the abnormalcase

. W e

define the normalcase a s a case that the goldon

-

rule stock of c a p i t a l i s l a r g e r than the amountof wealthpeople wish to hold i n thelaissez faire,that is,ko>

i

and

also

the abnormalcase askg<

i

. T h e 1 o n g

-

^run

equ出brium becomes the golden

-

^rule^when ^both ^h(0)キ0^and

ko>

i.

; This means b y ( 1 7 ) '

' ^'

^* ^{= l l m} ^Ij^[ ¹⁺^{r ( r )} ^.

h(0) t

-

^t^'⁰ ^1十^g ^{> 0}

We can interpret

立

^{1 十「}⁽^「⁾ as the future value inperiod 0 in r=o 1十g

(7) see C a s s ( 1 〕 10

-

¹⁴⁴

-

(12)

terms of a unit of capitalorliabilities or nationaldebt(7). T h u s the future value ispositive in the normalcasewhen h( 0 ) キ 0 . 0 n the other h a n d , t h e l o n g-^runequ出brium becomes the wealth

-

capitalequilibrium when both h(0)キ0 and kg<

-

¹^ll. T h i s meansby ( l 5 ) , ( 1 6 ) , ( l 7 ) a n d ( 1 7 ) '

lim h(t )= 0 ,

t

-

^u

that is,

(23) lim

_一 f ^I

^{1 十「}⁽^「⁾^{= 0 .}

ar=o 1十g

(23) implies the future value in period 0 in terms of a unit of capita1orliabilitiesor nationaldebt becomes valueless,or inother words,theterms of trade from present to future never become favorable.Such a path can provide the same consumption or utilitylevelfrom a l o w e r initialcapitalstock.The path can afford the opportunity of decreasing the initialcapitalstock a t n o cost in future consumption.And also there is some other path of capita1 accumulationwhich provides a t l e a s t in each period as much consumption or ut出tylevelas this path and provides in some periods more than this path.Therefore capitaloveraccumulation takes place i n this program.The program should be said to be inefficient.

4

.

Stabiiity analysig sang sociaI security

I n previous section w e got into our discussion under a n assump

-

tion such that k(t) converges to either

i

or kg as t becomes i nlinite.We show,in fact,either k(l1)→

i i

ork(t) →k

,

as t→oo in this section.In order to show this, w e show the stability of

-

^l45

-

¹¹

(13)

simultaneous difference equation of ( 1 6 ) a n d ( l 7 ).

W e specify,for simp「icity

of

analysis,theut出tyfunction(1)as (24)

U=βlog

c¹(t)十(l一β)1ogc2(t十1) (0くβ<1).

Then we can derive from(5)

( 5 ) 'c2(t十1)=( l

-

^,lⁱ^)(1十^:^r(^t^十1))^u

^'

^(t⁾

的nsidering(7)and(8)together w i t h ( 5 ) ' w e can express system of ( l 6 ) a n d ( 1 7 ) a s

(25) 是(

-

⁼

^f

^章

^語

⁽^f(是(t))一是(t)f'(1(t)))+

1十

f

(k(t) )

,

^l^(t⁾

1 十 9

(26) h (

̲

⁼

^「 ^j

( t ) ) h ( t ) .

the

Now w e cosider pairs ( k

,1

t ) o f k(t) a n d

h

(t)such as

k

(t十1)

=

k(t) i n ( 2 5 )

.

Such pairsare

(27) h=

jj f

^,(是)

⁽ ,

^k

- t ⁺ ^- ^j

⁽^f⁽

^o - ^1f ^'

^{(k) )}

⁾

Then w e have

(28)

s i g n h =

s1'g n

(

^k

- ? ^-

⁽^f^(表⁾

^-

¹¹

^f

⁽

^k

^{) )}

⁾

Asassuming that wage function ( 7 ) ω(^t)=f( k(t) )

-

^k(t⁾

^f ^'

⁽¹

^t(

^t^{) )}

is a concavefunction,we can depict twopossible patterns which a r e s h o w n b y F i g u r e s l a n d 2. Capitalstock in the wealth

-

capitalequilibrium,石,satisfies k=

t ⁺ ^- ^j

⁽

^f

⁽

ⁱⁱ

⁾

- ^t ^f

^'⁽

^あ

^).

Then w e get from(2S)and F i g u r e s l a n d 2 (那)' s i p i = n'p ( i

- ^{^}

^是).

12

-

¹⁴⁶

-

(14)

Capita1-Accumulation and Income Redisttibution between T w o Generations

11'

FIGURE 1

FIGURE 2

-

^l⁴⁷

-

13

(15)

Thus we obtain

h< 0

f

or 0 <k <

ii

,^, h

= 0

for k

=

1k

,

^, h> 0

f

or

i

>k

.

Differentiating(27),we have

生 = 1

_dh _1十+_f

g 1 _、

_'1

̲

^1一β(_1十_g

̲

^f^{" )}

̲

_1十^h^f

_gノ

^'

、

and

? l

^h^>⁰^{> 0}

because l > 1 一 β (

_i

_十_g

- ^1f

^'') ^for¹^l^l

_、 ^f

_'

⁼

^k

-

^1一β(_{l 十}_gf

-

kf)_l

、

_._,_l^>

0 in both figures.

0 n the other hand,considering pairs(1k, h ) o f k( t ) a n dh (t) such as

;

l(t十 1 ) =h(t) i n ( 2 6 ) , w e get

(2g) 9

- ^f

^(?)

^; ,

⁼⁰

1 十 9

And we have1l¹(t十1)=h(t) i n ( 2 6 ) w h e n either f ⁽¹l1) =gor h=0.

Next w e ascertain a n existence of pairs such that both 1l1(t十1)

= k

(t)andh(t十 1 ) =h(t).Pairs such ask(t十1)=k(t)are represented b y ( 2 7 ) a n d pairs such as h (t十 1 ) = 1l(t) a r e expressed by(29).

Then w e can depictFigures3

-

5 , b y which w e can ascertain pairs ( k

,

h) t h a t s a t i s f y b o t h ( 2 7 ) a n d ( 2 9 ) . F i g u r e s 3 a n d 4 c o r r e s p o n d to F i g u r e 1 , i n whichh< 0 f o r 0 <k くk, h

= 0

for k

=

k and h> 0 for k>k

t

And Figure5corresponds t o F i g u r e 2 , i n whichalways

;

t> 0 for any k> 0 ,

Now w e discuss the stab出ty of the system.First of a11,we sketch the movement of the system which is shown b y ( 2 5 ) a n d ( 2 6 ) . W e

14

-

¹⁴⁸

-

(16)

get k(t十 l ) >k(t) i n position above the curve which

is

shownby (27),because

and (25)

一

^>

^# ⁽

^{i (}

^'

⁾

^一 ^常

⁽^f ^(是(

^'

^))一

一

^f^'(是(t

^型

k (t十 1 )=

? ( f

⁽¹k(t) )

-

^k(^t⁾^f⁽^k⁽^t^{) )}

)

十 1 十

f

( i ( t ) ) h (^,l¹

).

1十g

Similarly w e get k(t十 1 )く1i(t) i n position be1ow the curve which is shown by(27). A n d w e have 1t(t十1)

( = -

^x

h

(t)

)

> 1ⁱ(t)because of

f

' (k(t) ) >g when both h(t) > 0 and k(t) <ko.Similarly w e haveh(t十 1 )く h(t) w h e both h(t) > 0 andk(t) >ko, h

(

t十 1 )く h(t) w h e n both h(t) < 0 and k(t) <kg,

and 1ⁱ(t十 1 ) >h(t

、

^whe^:n ^both ^h⁽^t) < 0 a n dk (t) >ko

.Thus

wecan draw thearrows in Figures3

-

5 w h i c h indicate the direction of movement of

k

( t ) a n d

h

(t)

. W e

show them in Figures6

-

⁸^.

;

t''く0 in the1ong

-

^runequilibrium(「, 1 「)of the system of (25)and(26)holds only in Figure6. h''く0impliesh (t) く 0 a n d h( 0 )く0 because

(26) h (t十 1 ) = 1十f

常

^l^{) )}^h⁽^t⁾

and

1十f (k1 (t) > 0 1 十 9

W e can obtain h(t) く 0 only when w e interpret h(t)as a claim to government for goods among three implications in eco!lomics i n t h e previous section

. B y

the w a y , i t becomes clear from

-

¹⁴⁹

-

^{l 5}

(17)

hl1十l1

-

^h^l^t⁾

= li

il

FIGURE 4

-^{l 5 0}

-

(18)

F:CURE 5

-

^{l 5 l}

-

17

(19)

18

CapitalAccumulation and Income Redistribution between Two Generations

FIGuRE 7

FICllR E 8

-

¹⁵²

^-

(20)

F i g u r e 6 t h a t this equilibrium(1「 , 1 P)is a saddle point,as indicated by thearrows pointing the directionof movement.This means that the1ong

-

^runequilibrium becomes the golden

-

^rule

only when thepath in this economy is on the boldline through the equ出brium. Govemment must specify the initially issued amountsof nationa1 debtd(0)which must be chosen for initia1 capitalk( 0 ) i n order to attain the

golden -

^rule

.

And when h( 0 )=0 in Figure6,the path movesonly on the k

-

^axis^as ^h⁽^t⁾

^{= 0}

andconverges to the point(

ⁱ - ^{, 0 )}

⁽⁸⁾

^.

W e show in this paragraph thatthe equilibrium point(^、k, 0 ) i s stable wben k( 0 ) > 0 and h( 0 ) > 0

. A t

first,we point t h a t (k( t

), h

(t))≧0 necessarily holds for any period because

of

( 2 5 ) a n d ( 2 6 ) w h e n (1k( 0 ) , h( 0 ) ) > 0 , t h a t i s , (k(t) , h(t) ) i s bounded

f

rom be1ow when( k(0) h( 0 ) ) > 0

.Secondly,when

11o>k(t) > 0 and h(t) > 0 , w e have k(t十 1 ) >k(t) a n d 1t(t十 1 ) >h( t) f r o m Figure6.As 0 >ko

- ^1一β(

_1十

_g f

(ko)

-

^k^o

^f

^(k^o^{) ) =}^h

^{= l i m}

_t

-

^h(^t^)by(27)

and Figure 6 w h e nk(1 t) →ko,both1ll(t)→o o a n dk(t) → ignever hold ast

-

^oo.Thusthe path i n t h e domain of ko>k( t) > 0 and h(t) > 0 necessarily thrust into the domain of (k(t), h(t) ) >

(

,

k^o,0). Thirdly 1ll(t十 1 ) <h(t) h o l d s because of ( 2 6 ) w h e n ( k( t ) , h(t) ) > (ko, 0 ) , t h a t is h(t) becomes monotonica1ly decreasing sequence when (.1i(t), h( t ) ) > (1to,,0).Becauseh(t) is bounded from below as described above,1l(t)converges to zero.Fourthly,when(k(t),h( t) ) > (

,

ko, 0 ), k( t十 l ) >k( t ) h o l d s

(8) The stabi「ity of the system w i t hh( 0 ) = 0 hasbeen already analyzed by D i a m o n d 〔 2 〕 a n d S t e i n 〔 8 ) , 〔 9 ) , They only discussed the movement of the path on the

,

t

-

axis from our viewpoint.

-

¹⁵³

-

^{l 9}

(21)

in theleft position

of

the k(t十 1 ) =k(t) c u r v e u n

u

l k(t十 1 ) intersects the k(t十 1 ) =k(t) curve,where(27)expresses the k(t十 1 )=k(t) c u r v e . 0 n the other hand, k( t十1)く1k(t)holds i n the right positio:nof the curve until k(t十1)interseGts the curve.

Accordingly

k

(t十 1 ) w i 1 l b e , a t l a s t , o n the curve i:n bothcases.

A s t h e steep of the curve is positive andh(t十 1 ) <h(t) i n (k(t) , h(t) ) > (kl l

, 0 ) , a

point on the curve,as soon as the point is on the curve,thrust into the right position of the curve.In the position k(t十 1 ) <k(t) h o l d s until k( t 十 l ) i n t e r s e c t s the curve. And again 1t( t 十 1 ) w i l l b e on the curve.Afterthis, 1t(t)repeatsthe similarity.After all, k(t)converges to

i

. T h u s l i m ,lt(t

) = i

^and

l

̲

1

^要 h ( t ) = 0 , t h a t is,the s e q u e n c e ( i ( t ) , 1t(t))oonverges to the p o i n t (

1i,

0).

I n F i g u r e 7 t h e l o n g

-

run equilibrium(

i

, 0 ) is a saddle point.

But as alwaysh( t

) = 0

when h( 0 )=0 , t h e path moves only on thek

-

axis andconverges to the point

(i

k

,0).

Summing up above,we obtain the fo1lowing results:the wealth

-

capitalequilibrium(

i

, 0 ) i s stable when both

;

^l(0)j j0 and

-

^k^>^ko

hold,the golden

-

rule equilibrium(ko

, ;

^t

つ

^is^a saddle^pointwhen

b o t h 1¹( 0 ) < 0 a n d k>

?

h o l d , a n d the path when h( 0 )=0 converge to the wealth

-

capitalequilibrium even though the normal case ko >

-

^k^.

The remainning problem is whether the golden

-

rule equilibrium (kg,1

t つ

ⁱⁿ^Fig^u^re^s7 a n d 8 i s stable or not

.

The golden

-

^rule

equilibriumpointis on thepositive orthant in bothfigures.If initialconditions are k( 0 ) > 0 and h( 0 ) < 0 , t h e path in this

20

-

¹⁵⁴

-

(22)

economy never converges to t h e p o i n t (1l・1o

,1

i'')because ¹ll( t) < 0 always h o l d s b y ( 2 6 ). Accordingly thepossibilityof convergence to the golden

-

^ruleis only remained by(k( 0 ),h( 0 ) ) > 0 o r (k(0)

,

h(0))

e R

11l,.In fact w e can state the globalstabilityof thegolden

-

r u I e (.11o

,1

^t

つ ^by

applying the following theorem to the system

of

( 2 5 ) a n d ( 2 6 ) .(9)

THEOREM Constlder an autono mou s dj

f

erencel syste

,

n

' '

(t十 1 ) =

f

('t'(t),u

'

( t) ) ω (t十 l ) =g( v (t),u

'

⁽t) )

ωh

-

e ( u , ω ) む o a the p^a^配'a've orthaat. m^ef^mctionsf a ㎡

g

a r e assum e dto1be ofclassCl o

,

^t ^R

^{i .}

^Furthêrn^lô^re^sû

pp

ose there e a sなa

-

^gae eguiZ^t^'^6rt^'^{ω n 如}^'^{n t (}

^-

^v,ⁱ^)eR

^;

^s ^c^a^t^that^{i = f}⁽^t

^{i ,}

^一ω)

a nd

-

^u¹⁼^g⁽

i;

^u

- ,

^{) .}^Thenⁱf t hefol1o

,

ot

,

ⁱgcond^t

,

ltio n s a r e satis

f

ie d

,

the

eq

tnllib r iumpo it lt (^v

- ^, ^, -

^o^{) isstable} ^加 ^{the g}^t^obal^:

(A) 1

-

⁽^fi,g^u

^, - f

u

, g, ,

)v u11fig >・

o

(B) 1₍

_f -

_。g,,,⁽^{f ,}

_-

^u_f

^lf

_u_{g .}^十_。₎

^g, _, - _,

_ω_, ₁

^1g _f

_g^l⁾

_l

_>^十_o

(0) 1 十 (f ,lvげ十g

̲

^u

^,

¹^g

¹

⁾^十

(f^vgu,

-

fu

,g,,

)v u

,

^1f^gⁱ^>⁰

f^o^r ^a11^(v,

,

^o^{) 加}

R

i;

for a

u

(υ,ω・ ) 加R

;

^;

for a li(・t

, ,

u

,

^{) 加}

^R 1

.

W h e n (k( 0 )

, h

( 0 ) )

' e R

j

, (k

(t) ,h(t) ) liRj necessarily holds b y ( 2 5 ) a n d ( 2 6 ) . T h e r e exists a unique equ出briumpointonthe

(9) See T a k a h a s h 〔 1 1 ) w i t h regard to proof of Theorem.

-

¹⁵⁵

-

²¹

(23)

psitive orthant in F i g u r e s 7 a n d 8 . T h u s w e can state theg1obal stability of the difference system of (25)and(26)if three conditions are satisfied.Theconditions(A)

-

( C ) f o r the g1obalstab出ty of

( 2 5 ) a n d ( 2 6 ) a r e expressed by

(30) 1

-

_{1 十 9}^1一β(

- ^,

^l^,^f

^つ ^士

^{> 0}

^f

^{o r a l}

^Z

⁽

¹

^k

^,

^h)

^{i n R i}

^;

(31)

-

_1十^f^″_g^.¹^l

, 去

^{> 0} ^f^o^r

^a11

⁽

^k

^.

' ^,

⁾ ^{i n}^Rⁱ^;

(32) 2十

(

²

?

⁽

-

^?^″⁾^十

, : j

^{g l}

ⁱ

⁾

^)去

^{> 0}

fora li( k

,

h) ^加

R

';l

,

where

K

= ?

⁽

^f

⁽

^,

^k⁾

^-

^,lf ⁽^{k) )}^十

?

^h^{> 0}

forl a11 (k h) in

R

j

.

I t is clear t h a t ( 3 1 ) h o l d s fora1l(k(t) , 11( t) ) i n

R

:11.

. A n d

(33) ?(k)

-

¹

^f

^{' ( k}^))/^k^>^一

要

f⁽

f

(k

)一

ll

f

'( k) )=

-

¹

^f

^''(^k⁾

holds by the concavity of wage f u n c t i o n ( 7 ) . F i n a l l y ( 3 2 ) h o l d s when the elasticity of the marginalproduct of c a p i t a l (

-

¹^f^''

^1f

^')

is sma1ler than2.

Furthermore,aslong as w e suppose k( 0 ) >h( 0)l≧0,we have k(t十1)

-

^h(^t^{十 1 ) =}

? ^U

^(,l¹⁽^t^{) )}

-

^k(t、)

^f

^'⁽^{k ( t}^、)^{) ) > 0}

(

,

⁺

,

⁾⁼

'

⁺

ifi '

^{) )}

- ^。

^・

Thus w e get k(t十 1 ) >h(t十 1 ) ≧ 0 . T h i s meams socialcapita1 or capitalfinanced by theliabilities of the enterprise never exceeds

22

-

¹⁵⁶

-

(24)

totalcapitalin this economy.

W e summarize thissection. W h e n there exists socialcapita1or theliabilities of the enterprise in this economy,the golden

-

^rule

equilibrium is globa1ly stable in the normalcase and thewealth

-

capitalequilibrium is globally stable intheabnormalcase. When there exists the claim to government for goods, the golden

-

^rule

only exists in the abnormal case and becomes a saddlepoint.

When none of them exists,the path in this economy converges to the wealth

-

capitalequilibrium in both n o r m a l a n d abnormal cases.

5.Stabiiity analysis with social Eecurity

The previous section shows that the path in the economy converges to the golden

-

^ruleequilibrium when both i,o、>1

1

l1 and h( 0 ) > 0 hold and to the wealth

-

capitalequilibrium when h( 0 )

=0 or both h( 0 ) > 0 and ko く

i

hold

.

The optimality of dynamic a11ocation requires that the path converges to the golden

-

^rule

instead of the wealth

-

capitalequ出brium.This section showsthat government can convert the wealth

-

capitalequilibrium to the golden

-

rule equilibrium when it adds a socia1 security program between a younger generation and an older generation as its instrument to the system described in the previous section. Inthis connection such a socialsecurity system is not fully

-

^founded^but

pay

-

^as

-

^you

-

gosystem. Furthermore this section examines intro

-

ducing pay

-

^as-you-^gosocia1 security program to the economy never decreases the utilitylevelof each generation in steady

-

^state.

N o w the socialsecurity systemlevies taxes of Tl( > 0 ) o n each

-

¹⁵⁷

-

²³

(25)

worker and pays benefits of T a( > 0 ) t o eachretiredperson.Then ( 1 2 ) i n section3is rewrtten as

(34) k(t十 l )

-

^c^。⁽^ω⁽^t

1 十 9

r(t十 l 1十1l

-

⁽^t^十1

- ii 「

⁾

^(, '

⁽

^'

⁾

-

T ,

-

^一 ^十

1十g

c。(ω' (t l 十 9

-

¹ ^r⁽^t

1 十 r ( l

= 0 .

There are fuliy founded and pay

-

^as

-

^you

-

^go^social ^security

systems. The fully

-

fouded socialsecurity system is a system in which governmentlevies taxes of

T

l o n each younger person, uses these as capitalin the nextperiod and pays the principal and interest of

T ,,

^to him

. T h e n T

:t= ( 1 十r( t十 1 ) )

T ,

holds.

Considering this equation,(34)becomes (35) 1'( t 十 1 )

-

c c。lω(t

1十g ω(t 1 十 9

r(t十1 1十r(t十 l

-

¹ ^r⁽^t

1十r(t

- ' j j j

⁾

( , '

⁽

,

⁾

)

^{= 0 .}

( 3 5 ) i s the same a s ( 1 6 ) , w h i c h is equivalent t o ( 1 6 ) a n d ( l 7 ) that imply no socialsecurity system. This means that the path in fully

-

^foudedsocialsecurity is the same as the path without socialsecurity a s l o n g as fully

-

^foundedsystem does not vary the initialvaluesof 是 ( 0 ) a n d 1l( 0 ).

The path in pay

-

^as

-

^you

-

^gosocialsecurity system is different from the path in fully

-

founded socialsecurity. Pay

-

^as

-

^you

-

^go

security system in which governmentlevies taxes of T^t upon each younger person and pays benefits of T² to each retired person in this period,that is,governmentlevies ' 「l

L

( t

、 ^,

^s^sum

totalon a younger generation and transfers allamounts⁽l,f

T , L

(t)

24

-

¹⁵⁸

-

(26)

CapitalAccumulation andlncome Redistribution between T w o Generations

to a n older generation.Asapopulation of a n older generation

1 ' l '

f

'

is

-

^times^as^manyas a popuation o a younger generation,

l 十 9

a retired person can receive(1十g) t i m e s as much asayounger person pays.Thus T²=(1十g)

T , holds.

Using this relation,we can express(34)as

(36) k(t十 1 ) co(w(t

( 1 十g

r( t 十 1 1十r(t十1

-

¹

常

⁾

(

¹t ( t )

-

十 1

:

十

;

⁽ⁱ^t⁾

⁾ ^{= 0}

^'

This is e(luivalent that (37) 是 ( t 十 1 ) =

1 十 9

+

'1ii

⁾

-

(能)

,

⁽

-

⁼

ⁱⁱ ^「

⁾

-

^,

because we can set in (36)

(39) h(i十 l ) =k (t十 1 )

-

_1十_g

+ ^,

r,

^,

l 十 r ( l 十 l ) '

十

m T

( l^l

-

^十^:

. ^: ^t

⁾

^-

t〇(t

-

¹

1十g

r(t十1 1十r(t十1

c

,

('u

'

⁽^t

r( li 1十r( l l

T

,

一商

r(t十 1 1 十r⁽t十 l

When pay

-

^as

-

^you

-

go socialsecurity program has bee:n enfo

-

rced by government,a representative person regards his payments T

,

as a decrease of his income and present value of benefits

T

2

(=( 1 十g)

T ,

^{) a s}^aⁿ^increaseof his income. Then his budget constraint becomes instead of (2)

(40) c¹(l) 十_{( l 十}C2(_r(tt十 1 )_十1))=

,

^〇^{( t}⁾

̲ ^T

^t^十_1十_r(^l

-

^l_t⁹_{十 1 )}

^̲ ^T

¹

-

^l⁵⁹

-

(27)

Specifying the utility function a s ( 2 4 ) i n this section t o o , w e can derive his future consumption as

(4l) l

j ; ^組

⁾ ⁼^{( 1}

-

ⁱⁱ^{) (}

^f

^(k(^t^、)、)

-

^{k(t )}

f

(k(t、)、))

-

^(1一β)

⁽

¹

^- )

^T^l^.

Substitutin g ( 4 l ) i n t o ( 3 7 ) a n d arranging it with a n aid of (7) a n d ( 8 ) , w e get the system describedasfol1ows;

(37)

' k

(t十1)=

?

⁽

^f

⁽^k(t、)⁾

-

^k(t^、)

f

⁽k(t)) )

̲

^T

_'

^f

_、

_.₁

^1一β ₊

_g^十_商 ^β

、

_/^{十 1 十}

^f

₁₊⁽_g

^'

^k(^の⁾^x

/l( t) .

(38) h( t十 1 )= 1十

f

(k( t ) ) 11( t)

.

l 十g

(37)'implies that t o t a l c a p i t a l i n t h i s economy consists of a n amount which would become private capitalif pay

-

^as

-

^you

-

^go

security program were not enforced,an amount which in fact does not become private capitalbecause of the security program,and an amount which forms capitalas socialcapita1orliabilities of enterprise. Three amountsare shown by the first term,second term and the third term of the right hand of (37)'respectively.

Now consider p a i r s (k, ,h) o f k(t) a n d

;

l(t) s u c h as k(t十1)

=

^k(^t) i n ( 3 7 ) ' ;

十「1 ^l

( ?

^{十 l 十}^f

^量

⁽¹^k⁾

^- ))

Comparing(42)with(27),we see that the socialsecurity program and the increment of payment shift the k(t十 1 )=k(t) c u r v e

26

-

¹⁶⁰

-

(28)

upward.

(43)

and

Differentiating(42),we have

dh

̲

^1十

^g

^f

̲ ^1一β

^{k ,} ^?

^,

^{f '}

- a

f

-

¹

⁺ ^f

⁽^k⁾^一

、

¹

^ii:i ^j ^-

⁽

- ^f

⁾

-

^一

^てi i ^f ⁱ

ⁱ

̲

^h^f^'

、

1+

gノ

(44)

告 l

¹

^r ^,

^>^o^>

? l ^T ^,

⁼^o^・

0:n the other hand,we see f r o m ( 3 8 ) t h a t h(t十 1 )

=

h ( t ) holds when h

= 0 or f

(

k

)=

g.

F i g u r e s 9 a n d 1 0 show above,where言is a solution of (42) withh=0

a . T h e

arrows pointing the direction

of

movement

of

(k(t),h(t) ) i n F i g u r e s 9 a n d 7 a r e same,while the arrows in Figures10 a n d 6 a r e same.Theconditions of theorem described in the previous section being satisfied,the path in the economy converges to thegolden

-

^{rule in}F i g u r e 9 a n d to new wealth

-

capitalequ出brium in Figure10lu1.

Figure 10 shows that pay

-

^as

-

^you

-

go security system only

的 As

i :

is a solution of (42) with h

=

o,

i

satisfies

f =

j ; 語

⁽

^f

⁽ⁱ⁾

-

^{i f}⁽

^f

^{) )}

- ^r, ^(再 ⁺ ふ ⁾

This equation shows the wealth

-

^capitaleq^uillbriumunder pay

-

^as

-

^you

-

go security system.

的 T r a n s p o s i n g therighthandof (37)

'

t o t h e l e f t hand a n d diffentiating it with respect to k(t十 l ) , w e have

K =1

-

^{> 0}

Thus w e can express (37)

'

by the theorem of implicit function as ( a ) ,ll(t十 1 ) =F(

,

t( t),h(t) )

Then the c o n d i t i o n ( B ) f o r convergence described i n the theorem i n th (

r

revioussection becomes

( b )

-

^{> 0}

-

^{1 6 l}

-

27

(29)

CapitalAccumulation and Income Redistribuiion between Two Generations

converts old wealth capital equilibrium to new wealth

-

^capital

equilibrium.Showing that it convertsthe wealth

-

^capita1^equilib

-

rium to the golden

-

^{rule is} ^{Figure 9}^.^The difference between F i g u r e s 9 and l 0 occurs from different amount of payments

T ,

ior or benefitsT² from the security system.Evaluating(42 ) w ith k=ko and T

,

^{= 0}when government has not added pay

-

^as

-

^you

-

^go

security programtothe economy yet,we obtain from Figures 9 and10.

FICllRE 9

(conti m,e dfromt he preol.ou spa gli)which is always satisfied.And w h e n other t w o conditions

( c ) ?(t+1 )

- t- ⁺ j

^(k(^t⁾^f^″(^k⁽^t^{)) )}¹ⁱ⁽^t^{) >}o

( d ) 2基十 2

j

g

⁽

-

^,^t(t)

^f

^″(¹^t(t)^{) )}^1t(^{t )}

L

̲ ^f

^″(¹ⁱ⁽^t^{) )}^h^(t)

^.

¹^l⁽^t) ^、.^{1 >} ^o

' l 十 9

- -

are satisfied,the system of (37)'and(38)converges t o (ko,h

'

、)

.

28

-

^{l 6 2}

-

Capital Accumulation and Income Redistribution between Two Generations