An Unbalanced Multi-sector Growth Model with Constant Returns
共同研究 1

1

戦後日本経済における地域産業構造の変化と地方財政運営

An Unbalanced Multi-sector Growth Model with Constant Returns:
A Turnpike Approach＊

Harutaka Takahashi
Abstract

Recent industry-based empirical studies among countries exhibit that individual industryʼs
per-capita capital stock and output grow along its own steady state at industryʼs own growthrate, which is highly correlated to industryʼs technical progress measured by the total factor
productivity of the industry. Let us refer to this phenomenon as “unbalanced growth amongindustries”. Since it is totally based on the highly aggregated macro-production functions,“New Growth Theory” cannot explain the unbalanced growth phenomenon. On the other hand,
although Turnpike Theory is based on the multi-sector models, it demonstrates that all sectors
have a common growth rate and each sectorʼs per-capita capital stock and output converge to
its own constant ratio. Therefore, Turnpike Theory cannot explain the phenomenon either.
I will set up the multi-sector optimal growth model with a sector speciﬁc Harrod neutral
technical progress and show that each sectorʼs per-capita capital stocks and outputs grow at
its own rate of sectorʼs technical progress by applying the theoretical method developed by
McKenzie and Scheinkman in Turnpike Theory.
JEL Classiﬁcation: O14, O21, O24, O41

1 Introduction
Since the seminal papers by Romer（1986）and Lucas（1988）, we have witnessed a strong
revival of interest in Growth Theory under the name of Endogenous Growth Theory, andespecially, neoclassical optimal growth models have been used as analytical benchmark models,
which have been intensively studied in late 60ʼs. However, these research models have aserious drawback. Since the models are based on the highly aggregated macro-production
function, they cannot explain the important empirical evidence, as I will give a detailed
discussion in the following section. Recent empirical studies at the industry level among＊

The paper is prepared for the International Conference on “Globalization, interdependences andmacroeconomic ﬂuctuations”, held in Paris, June 11-13, 2009. The paper is very preliminay. Please do not quote.

2

研 究 所 年 報

countries provide a clear evidence that individual industryʼs per-capita capital stock and output
grow at industryʼs own growth rate, which is closely related to industryʼs technical progressmeasured by the total factor productivity of the industry. For example, the per-capita capitalstock and output of an agriculture sector grow at 5% per annum along its own steady state, on
the other hand, those of a manufacturing sector grow at 10% per annum along its own steadystate. Let us refer to this phenomenon as “unbalanced growth among industries”. To tackle
the problem, it has raised a strong theoretical demand for constructing a multi-sector growthmodel. In spite of strong needs for such a model, very little study of this type of model has
been done so far.
On the other hand, the optimal growth model with heterogeneous capital goods has been
intensively studied under the title of Turnpike Theory since the early 70ʼs by McKenzie（1976, 1982, 1983 and 1986）and Scheinkman（1976）. The Turnpike Theory shows that any
optimal path asymptotically converges to the corresponding optimal steady state path without
initial stock sensitivity. In other words, the turnpike property implies that the per-capita
capital stock and output of each sector eventually converge to a sector speciﬁc constant ratioat the common growth rate. Therefore, the Turnpike Theory cannot explain the empiricalphenomenon: unbalanced growth among industries, either. An additional point to notice is that
the Turnpike result established in the reduced form model, which has not been fully applied
to a structure model: a neoclassical optimal growth model. One serious obstacle to apply the
results given in the reduced model is that transforming a neoclassical optimal growth model
into the reduced model will not yield a strictly concave reduced form utility function, but justa concave one. In this context, McKenzie（1983）has extended the Turnpike property to thecase in which the reduced form utility function is not strictly concave: that is, it has a ﬂaton the surface in which an optimal steady state is contained. This ﬂat is often referred to as
the Neumann-McKenzie facet.Yano（1990）has studied a neoclassical optimal growth modelwith heterogeneous capital goods in a trade theoretic context. However, in the case of the
Neumann-Mckenzie facet with a positive dimension, Yano made a direct assumption called
the dominant diagonal block condition concerned with the reduced form utility function（seeAraujo and Scheinkman（1978）and McKenzie（1986））. Thus he still did not fully exploit thestructure of the neoclassical optimal growth model, especially the dynamics of the path on the
Neumann-McKenzie facet to obtain the Turnpike property.
The paper is undertaken to fill the gap between the results derived by the theoreticalresearches explained above and the empirical evidence provided by the recent empiricalstudies at the industry level among countries by way of applying the theoretical method
developed in Turnpike Theory. I will ﬁrst set up a multi-sector optimal growth model, where

An Unbalanced Multi-sector Growth Model with Constant Returns

3each sector exhibits the Harrod neutral technical progress with a sector speciﬁc rate. The
presented model will be regarded as a multi-sector optimal growth version of the Solow modelwith the Harrod neutral technical progress.
Secondly, I will rewrite the original model into a per-capita eﬃciency unit model. Then as
the third step, I will transform the efficiency unit model into a reduced form model. Thenthe method developed in Turnpike Theory are ready to be applicable. I will first establish
the Neighborhood Turnpike Theorem demonstrated in McKenzie（1983）. The neighborhood
Turnpike means that any optimal path will be trapped in a neighborhood of the corresponding
optimal steady state path when discount factors are close enough to one and the neighborhood
can be made as small as possible by choosing a discount factor arbitrarily close to one. Then,
I will show the local stability by applying the logic used by Scheinkman（1976）: there exists
a stable manifold that stretches out over todayʼs capital stock plane. To demonstrate boththeorems, the dynamics of the Neumann-McKenzie facet takes an important role, as we will
see later. Combining the Neighborhood Turnpike and the local stability provides the full
turnpike property: any optimal path converges to a corresponding optimal steady state when
the discount factors are close enough to one. For establishing both theorems, we assume thegeneralized capital intensity conditions, which are generalized versions of those of a two-sector
model. The full turnpike property means that each sectorʼs optimal per-capita capital stock andoutput converge to its own steady state path with the rate of technical progress determinedby the sectorʼs total factor productivity.
The paper is organized in the following manner: In Section 2, I will provide a several
empirical facts based on the recent database at the industry level among countries. In Section
3, the model and assumptions are presented and show some existence theorem. In Section
4, the Neumann-McKenzie facet is introduced and the Neighborhood Turnpike Theorem isdemonstrated. The results obtained in Section 4 will be used repeatedly in the proofs of main
theorems. In Section 5, I show the full Turnpike Theorem. Some comments are given in
Section 6.

2 The Empirical Facts
Over the past few years, a great number of eﬀorts have been done to collect and archive theindustry level database among countries. Recently such a database is easily accessed on the
Web: the EU-Klems Growth and Productivity Database ⑴, which covers 28 countries with 71⑴ URL http://www.euklems.net

to analyze
economic
growthin countries.
moreItinteresting
applications
covers28 countries
with71 industries
fromOne1970oftothe
2005.
contains theGDPs andisth
to theLet us assume
the offollowingproductionof theindustrythetotalindustries.
factor productivity(TFPs)
industries.
Growth function
accounting
has ibeenused4

研 究 所 年 報a country.
toinanalyze
economic growth in countries. One of the more interesting applications is
industries from 1970 to 2005. It contains the GDPs and the total factor productivity（TFPs）of

tothe industries.
Let us assume
the usedfollowingproduction
functionof countries.
the ith industry
industries.
Growth accounting
has been
to analyze
economic
growth in
One ofYi (t) = F i (K1i (t), K2i (t), · · ··, Kni (t), Ait Li (t)),
the more interesting applications is to the industries. Let us assume the following production

in a country.

function of thethith industry in a country.

where Yi : tperiod capital goods output of the ith industry, Kji : ith capital goods

Yi (t) = F i (Kth1i (t), K2i (t),i · · ··,Kni (t), Ait Li (t)),
used in the jth industry
in the t period, At : tth period
labor-argumented technicalwhere Y : t th period capital goods output of the i th industry, K : i th capital goods used in theiprogress,
Li (t) :capitaltth period
labor input
ithji industry.
θj stands
forwhere
Yi : andtth period
goods
outputof theofiththeindustry,
Kji : ithIfcapital
goods
th
th
i
th

j industry in the t period, A t : t period labor-argumented technicalprogress,th
theof thederivefollowing
periodlaborof period,
thefactor,
i th industry.
θjmay
standsfor thethe
factorshare
ofrelation
the jthandfactor
tthth industry
usedinLi（t）
the :jsharein jinput
theinputtth
Aitthen: tthIfwe
periodlabor-argumented
technical-

input factor, then we maythderive the following relation concerned with the ith industry;

concernedandwiththe : i tthindustry;
progress,
Li (t)
period labor input of the ith industry. If θj stands for
¶
µ
·
·
·
Pn
Kji
Yi
Li
th·
−
θji we+ θ Lithe factor share of the j inputYi factor,
then
the following relation
j=1
Kji may0i derive
Ai
=
.
Ai
θ0i
concerned with the ith industry;
¶
µ
·
·
·
Pnto caluculate
KTFPs
Based
equation,
weableable
TFPs
the 20 industries
Y are
Lthe
ji
i
i
Based
on on
thisthisequation,
we· aretocaluculateof
20 of
industries
of a countryof⑵. a−
j=1 θji Kji + θ0i Li
Yi
Ai.
Figures 12show the relationship =between the per-capita U.S. GDP average
growth rate and thecountry . Figures 1 showAithe relationship between
the per-capita U.S. GDP average
θ0i
U.S. TFP average growth rate at the industry level from 1970 to 2005. In Figures 2, those of

growth
rate
and
U.S.
TFP
averagerate ﬁgures,
atTFPsthe industry
level
fromare
1970
to
Based
onecvonomy
this theequation,
we areable
toof 45-degree
the 20industries
oftheJapaneseare exhibited.
Note growth
thatcaluculate
in both
the
lines
alsoa
drawn. If an industry were on the 45-degree line, it would imply that the industryʼs per-capita

1
URL2 .http://www.euklems.net
country
Figures 1 show the relationship between the per-capita U.S. GDP average

GDP2 The
would
grow
at its TFP
rate. Observing Figures 1 and 2, we may conclude that in
U.S.
20 industries
aregrowthfollowings:

growthrate andtheofU.S.
TFP average
growthat the line.
industrylevelsomefromindustries
1970 toboth countries,
most
the industries
lie aroundtherate45-degrees
Although
1:TOTAL INDUSTRIES 2 :AGRICULTURE, HUNTING, FORESTRY AND FISHING

lie far above or below the 45-degree line.
1
URL http://www.euklems.net
2

The U.S. 20 industries are followings:

7

1:TOTAL INDUSTRIES 2 :AGRICULTURE, HUNTING, FORESTRY AND FISHING⑵ The U.S. 20 industries are followings:

7
1: TOTAL INDUSTRIES 2: AGRICULTURE, HUNTING,
FORESTRY AND FISHING
3: MINING AND QUARRYING 4: TOTAL MANUFACTURING
5: FOOD, BEVERAGES AND TOBACCO 6: TEXTILES, TEXTILE , LEATHER AND FOOTWEAR
7: WOOD AND OF WOOD AND CORK 8: PULP, PAPER, PAPER , PRINTING AND PUBLISHING
9: CHEMICAL, RUBBER, PLASTICS AND FUEL 10: OTHER NON-METALLIC MINERAL
11: BASIC METALS AND FABRICATEDMETAL

12: MACHINERY, NEC 13: ELECTRICAL AND OPTICAL EQUIPMENT
14: TRANSPORT EQUIPMENT 15:MANUFACTURING NEC; RECYCLING
16: ELECTRICITY, GAS AND WATER SUPPLY
17: CONSTRUCTION 18: WHOLESALE AND RETAIL TRADE
19: HOTELS AND RESTAURANTS
20: TRANSPORT AND STORAGE AND COMMUNICATIONFor Japanese Economy, three more extra industries are added.

An Unbalanced Multi-sector Growth Model with Constant Returns

5

Figure 1: U.S. Economy, 1970-2005 Source: EU-KLEMS DATABASE

Figure 2: Japanese Economy, 1970-2005 Source: EU-KLEMS DATABASE

We may summarize these facts as follows:
1） Each industrial sector has its own steady state with the sector speciﬁc growth rate.
2） The steady state level and its growth rate are highly related to its own TFP.
These facts cannot be explained by the new growth theory totally based on the macro
production function. Thus we need to set up an industry based multi-sector growth model. On
the other hand, the turnpike theory are established based on the multi-sector model. However
it has a drawback, too. The turnpike theory means that each industrial sector with diﬀerent
initial stocks will eventually converge to its own optimal steady state with the common

From the above discussion, it is an urgent task to set up a multi-sector optimal
growth model with technical progress and demonstrate that each sector’s per-capita
6 From the above discussion, it is研an究urgent
所 年task報 to set up a multi-sector optimal
capital and output will grow at the sectoral specic growth rate determined by thebalanced growth rate. In other words, each industryʼs per capita stock will converges to a

growth model with technical progress and demonstrate that each sector’s per-capita
sector’s
TFP. ratio. Thus the turnpike theory cannot explain the facts that each industryʼscertain constant
From the above discussion, it is an urgent task to set up a multi-sector optimal
capitalandstock
outputwill
the sectoral
specic
growth rate
determinedby theper-capita
grows
at grow
its ownatgrowthrate, which
is determinedby thesectoral TFP.

growthmodel
with technicalprogressand demonstrate
thatindustry
each sector’sOECD（2003）
theproductivity
growth at thelevel inper-capita
detail and3
TheTFP.
Model also
andstudied
Assumption
sector’s
From
the
above
discussion,itisanurgent
task
to
set
up
a
multi-sector
optimalreported the following results, which are consistent with our observations discussed above.
capital and output will grow at the sectoral specic growth rate determined by the・ FromA largeto overall
growthfrom productivity
theiscontribution
above
discussion,
it isproductivity
anandurgenttask
topatterns
setacomesmulti-sector
optimal
Our
model
awith
discrete-timeandmulti-sector
versionof up
thestandard
growthmodel
technical
progressdemonstrate
that
each
sector’sneoclassicalper-capita
3
The
Model
and
Assumption
sector’s
TFP.within industries, rather than as a result of significant shifts of employmentchanges
growthmodel
withtechnical
progressanddemonstrate
that
each
sector’s per-capita
optimalgrowthmodel
withneutraltechnical
progress:
capitaland
outputwill
growtheat Harrod
the sectoral
specic
growthrate determinedby theacrossindustries.
Our
is a discrete-timeand multi-sectorof the standard neoclassical・ model
TFP
depends
onwillcountry/industry
speciﬁc¶versionµ factors.
∞tcapitaland
outputgrow at theXsectoralspecic
growth rate determined by the
3
The
Model
and
Assumption
sector’s
TFP.
1
From the above discussion, itMax
is an urgent task toC(t)set up a multi-sector optimal growth
1 + r technical progress:
optimal growth model with the Harrod
neutralt=0sector’s
TFP.
model with technical progress and demonstrate that each sectorʼs per-capita capital and output
Our model is a discrete-time and multi-sectorof the standard neoclassical¶version∞ µt
3willTheand Assumption
X
growModel
at the sectoral
speciﬁc growthrate determinedby the sectorʼs TFP.
1
C(t)
Max
optimalwith the Harrod neutral
technical progress:
1+r
3 Thegrowth
Modelmodeland Assumption
t=0 to : ki (0) = k isubject
Our3 model
is a discrete-timeand multi-sectorof the standard neoclassicalThe Model
and Assumption¶version
∞ µ
t
X
1
C(t)
Maxmulti-sector version
Our
model
is a model
discrete-time
the standard neoclassicaloptimalgrowthwith theand
Harrod neutral
1 + r technicalofprogress:
t=0(0) = kof
subject
to : kiversion
i the standard neoclassical optimal
Our model is a discrete-time andmulti-sector¶technicaloptimalgrowthmodel
with theHarrodneutralprogress:∞ µt
growth modelwith
the Harrod
neutraltechnical
progress:X
Yi (t) +Max
(1 − δi )Ki (t)1− Ki (t+ 1) = 0
(1)
C(t)
1
+
r
¶
µ
∞
t=0
X
subject
to : 1ki (0)t= k i
C(t)
Max
1+r
t=0
Yi (t) + (1 − δi )Ki (t) − Ki (t + 1) = 0(1)
subject to : ki (0) = k i
C(t) = F 0 (K10 (t), K20 (t), · · ··, Kn0 (t), A0t L0 (t)),
toi (t): k−i (0)Yi (t) +subject(1 − δi )KKi=(t k+i 1) = 0
C(t) = F 0 (K10 (t), K20 (t), · · ··, Kn0 (t), A0t L0 (t)),
Yi (t) + (1 X−n δi )Ki (t) − Ki (t + 1) =i 0
K2i (t), · · ··, Kni (t), At Li (t)),
Yi (t) = F i (K1i (t),Li (t) = L(t),YiF(t)0 (K
+ (1
−
KK
+(t),
1) A
=0 L
0
n δK
i )K
i (t) ·−i (tn0· ··,C(t) =i=0X
10 (t),
20 (t),t 0 (t)),Li (t)
10= L(t),i=0 K (t), · · ··, K (t), Ai L (t)),
Yi (t) = F i (K1i (t),
2i
ni
t i

n K (t), · · ··, K (t), A0 L (t)),(t),
C(t) = F 0 (K10X
20
n0
t 0
Kij (t) = Ki (t),
10
0
n
j=0
(t),
(t), · · ··, Kni (t),(t), A
Ai0tLLi0(t)),
(t)),
C(t)
(K101iX(t),K
K20
Yi (t)==FF i(K2i (t), · · ··, Kn0t
Kij (t) = Ki (t),wherei = 1, 2, ..., n, t = 0, 1, 2,j=0...., and the notation is as follows:

(2)
(1)
(2)
(1)
(3)
(4)
(1)
(2)
(4)
(3)
(2)
(5)
(2)
(3)
(5)

wherei = 1, 2, ..., n, t = 0, 1, 2, ...., and the notation is as follows:10r=i1,subjective
rateof
discount,r≥g,
i follows:= 1, 2, ...,Yn,t=
0,2, ....,rate
andof
the
notationrwherei= subjective
discount,r ≥ g, is asi (t) = F (K1i (t), K2i (t), · · ··, Kni (t), A Li (t)),t

C（t）
r ∈ R+

C(t) ∈ R+

Y（t）
∈ R+
i

C(t)
Yi (t) ∈∈R
R++
Yii(t)
K(t)∈∈RR++

(3)

= =
total
consumption
produced r≥g,
and consu med at t,
subjective
rategoods
of discount,i totalconsumption(K1i (t),K2i (t), · · ··,goods
Kni (t),produced
Ait Lith(t)), and consu med at(3)t,Yi (t) = F=th
= t period capital10
goods output of the i sector,
th
=
total
consumption
goodsoutput
produced
andithconsu
med at t,= t period capital goodsof thesector,10th= ttthperiod capitalof sector,
the ith sector,=
periodcapitalgoods
stock outputof the ithththModel with Constant Returns
I maintainAn
theUnbalanced
followingMulti-sector
standardGrowth
assumptions
throughout the paper.

K（t）
∈ R+
i

7

= t th period capital stock of the i th sector,

K（0）
∈ R+
= initial
stockwhereof the ig issector,ia rate of population growth andAssumption
1. 1) L(t)= (1 capital+ g)t L(0)thjF（· ）:R+n+1→ R+= production function of the j th sector,

given as 0 < g < 1.
2) Ait =labor(1 +inputai )t Aofi0 thewhere
ai is a rate of labor argumented= t th period
i th sector,

L（t）
iperiodlaborand
input,
L（t）technical progress=of
t th thei thtotal
sectorgiven as 0 < a < 1.
i

K i（t）= i th capital goods used in the j th sectorjthe t period,1 means that the sectoral TFP is measured by the sectoral
2) of inAssumptionth= depreciation rate of the ith capital goods,

δilabor argumented
technical
progress (the Harrod neutoral technical progress), which<1,given as 0<δii= t th periodtisAexternallygiven.labor-argumented technical-progressof the i th sector.

I maintain the following standard assumptions throughout the paper.
Assumption
2. 1) All the goods are produced nonjointly with production functions

Assumption 1. 1） L（t）=（1+g）tL（0）where g is a rate of population growth and given asn+1ii 1, · · · , n)(iA=R+ argumented, homogeneousof degree
one,ofstrictly
0 < g < 1.F2）（1+a i ）tA i0 which
whereareai isdened
a rate onof labortechnical
progressthe it=

th sector and given as 0 < a i < 1.

quasi-concave and continuously diﬀerentiable for positive inputs. 2) Any good2） of Assumption 1 means that the sectoral TFP is measured by the sectoral labor argumented

j progress(j = 0,（the1, · · ·Harrod, n) cannot
betechnicalproducedunless, which
Kij >0 for somei = 1, · · · , n.
technical
neutoralprogress）
is externallygiven.
Assumption 2. 1） All the goods are produced nonjointly with production functions F i（i = 1,

3) Labor must be used directly in each sector. If labor input of some sector is..., n）which are defined on R +n+1, homogeneous of degree one, strictly quasi-concave andzero, diﬀerentiable
then its sector’s
outputinputs.
is zero.
continuously
for positive2）Any good j（ j = 0, 1, ..., n）cannot be produced
unless K ij > 0 for some i = 1, ..., n . 3）Labor must be used directly in each sector. If labor inputiwilltransform the original model into perDividing
allzero,the thenvariablesby Aoutput
t , we isof somesector is
its sectorʼszero.
Dividing all the variables by A i , we will transform the original model into percapitatcapita eﬃciency unit model. Firstly,
let us transform the tth sector’s production

eﬃciency unit model. Firstly, let us transform the t th sectorʼs production function as follows;iifunctionbothas sidesfollows;
dividing
by Adividing（t ）, both sides by At L(t),tL

Yi (t)= Fi
Ait L(t)
Then,
Then,
Then,
Then,

µ

Kni (t) Ait Li (t)K1i (t) K2i (t)
,
,
·
·
··,
,
Ait L(t) Ait L(t)Ait L(t) Ait L(t)¶(i = 1, · · · , n).

12eye (t) = f i (ek1i (t),k2i (t), · ·e· , ekni (t), `i (t)) (i = 1, · · · , n)eyei (t)i = f i (eki1ie(t),k2ie(t),· · · , k e(t),` (t)) (i = 1, · · · , n)yei (t) = f (k1i (t), k2i (t), · · · ni
, kni (t),i `i (t)) (i = 1, · · · , n)(t) e
K1i (t) e
K2i (t)Kni (t)where ye (t) =(t)AYiiL(t), k1i (t) K=1i (t), k2i (t) K=2i (t), · ·e· , ekni (t) K=ni (t),and `i (t) =ee
Ait L(t)Ait L(t)Ait L(t)where yei (t)i = AYiiL(t)
,
k
(t)
=
,
k
(t)
=
,
·
·
·
,
k
(t)
=
`i (t) =tnie
Yi (t) 1ie
K1i (t) 2ie
K2i (t)Kni,and(t)
Ait L(t)Ait L(t)Ait L(t)
t i
where
y
e
(t)
=
,
k
(t)
=
,
k
(t)
=
,
·
·
·
,
k
(t)
=
,and
`i (t) =
i
i
i
i
1i
2i
ni
At L(t)
At L(t)
At L(t)
At L(t)i
At Li (t)Ait LiA(t)i L(t) .
Ait Lti.(t)Ait L(t).
Ait L(t)Applying
the sametransformation
to the consumption
we have also
Applying
theto thesector, sector,
we sector,
havewealso
Applying
thesamesametransformationtransformation
to consumption
the consumption
have also
Applying the same transformation to the consumption sector, we have alsoeec(t) = f 0 (ek10 (t),k20 (t), · ·e· , e
kn0 (t), `0 (t)).
eec(t) = f 0 (ek010e(t),k20e(t),· · · , kn0e(t),` (t)).
ec(t) = f (k10 (t), k20 (t), · · · , kn0 (t),0 `0 (t)).

L(t) also
At L(t)transformation
At L(t) to the consumption
At L(t)Applying the same
sector, weAthave
Ait L(t)ec(t)transformation= f 0 (ek10 (t), ek20· ,ekn0 (t), `0 (t)).
Ait Li (t)Applying
the sameto(t),
the· · consumption
sector, we have also.
Applying
the same transformation
to the consumption
sector, we have also
Ait L(t)ec(t) = f 0 (ek10 (t), e
k20 (t), · · · , e
kn0 (t), `0 (t)).
Applyingthesametransformation
toe theconsumption
sector, we havealso asth e
0 e
Furthermore, we may
alsotransformthetequationec(t) = f0 e(k10(t),k20(t),年· · 報· sector’s, kn0 (t), `accumulation
0 (t)).
8
研
究
所
e
e
e
c(t) = f (k10 (t), k20 (t), · · · , kn0 (t), `0 (t)).
iFurthermore,
we
may
also
the tth sector’s accumulation equation as
followas; dividing both sides
by0 transformAt L(t),eeec(t) = f (k10 (t), k20 (t), · · · th,ek (t), `0 (t)).
Furthermore, we may also transformthe tht n0sector’s accumulation equation asi
Furthermore,
we
may
also
transform
thetsector’s accumulation equation as
followas; dividing both sides by At L(t),thKi (taccumulation
Kit(t)+ 1)
Furthermore, we mayYalso
sectorʼsequation as followas;
i (t) transformi thefollowas;dividing
both
sides
by
A
L(t),
+
(1
−
δ
−
=0
)
th
i
i
t
i
i
i
Furthermore,
we
may
also
transform
thetsector’s
accumulation
equation as
followas;dividing
both
sides
by
A
L(t),
i t L(t)At L(t)
At L(t)tdividing both sides by A A）,tL
Ki (t + 1)
Ki (t)Y（ti (t)+ (1 − δi )−=0followas; dividing both
Aitsides
A
L(t)
L(t)
KitiL(t)Yi (t) by Ait L(t),AitK(t + 1)i (t)Note the following relation:
K
K
Yii(t)(t)(t+
1)+ (1 − δi ) i− i=0
At L(t)+ (1 − δi ) Ai it L(t)− Ai it L(t) = 0
Ait L(t)At L(t)
At L(t)Note the following relation:
Ki+(t)1) Ki (t + 1)
(1i (t)+ ai+)(1(1+−g)K
Ki (t + 1) Yδ−= (1i+ a )(1=+0g)ei ) i (tii=Atrelation:ki (t + 1).
NotetheAg)L(t)]
At L(t)iL(t)i followingit L(t)
Notethethefollowing
relation:
Note
relation:
Afollowing
[(1
+
a
L(t)
)A
][(1
+
i
t
t
(1 + ai )(1 + g)Ki (t + 1)
Ki (t + 1)=
= (1 + ai )(1 + g)eki (t + 1).
i
i
A
[(1
++aai )A
L(t)][(1+ g)L(t)]Notethefollowing
relation:
(1
(t
+
1)
)(1
+
g)K
(t
+
1)
K
t
t
i
i
i
Then weKhave
1) =(1 + ai )(1 + g)Ki (t + 1) = (1 + ai )(1 + g)eki (t + 1).
i (ti + nally,
ki (t
+ 1).
Ai t L(t) = [(1 + ai )Ai it ][(1 + g)L(t)]= (1 + ai )(1 + g)e
At L(t)[(1 + ai )At ][(1 + g)L(t)]Then weKhavenally,(1 + ai )(1 + g)Ki (t + 1)
i (t + 1)=+ (1 − δ )e
= (1 +a(ti )(1
+=
g)e
k
Then
we
have
ﬁnally,ii(t) − (1 + ai )(1 + g)e
y
e
(t)
k
k
+
1)
0.i (t + 1).
Then wehavenally,
i
i
i
i
A
[(1
+
a
L(t)
)A
][(1
+
g)L(t)]
itt
Then we havenally,yei (t) + (1 − δi )eki (t) − (1 + ai )(1 + g)eki (t + 1) = 0.
Then
we
have
nally,In a vector formyeiexpression,(t) + (1 − δ )eki (t) − (1 + ai )(1 + g)eeki (t + 1) = 0.
yei (t) + (1 − δi )ieki (t)− (1 + ai )(1 + g)ki (t+ 1) = 0.
In a vector form expression,
In a vector form expression,yei (t) y)ekik(t)(t) −−(1(1++aig)G
)(1 +
ki1)
(t +
e
e g)+ee++(1(I−−δi∆)k(t= 1)
0 = 0.
In a vector form
expression,
In a vector form expression,e −ee + (I −y∆)k(t)(1
where
G Ganddiagonalmatices:
where
and∆
Δare
are following
following
diagonalmatices:
13 + g)Gk(t + 1) = 0
In aGvector
expression,whereand ∆form
are following
diagonalmatices:e − (1e +⎛⎞+ g)Gek(t⎛1) = 0⎞e + (I − ∆)ek(t)ye + (I − ∆)k(t)
y
− 13
(1⎞
+ g)Gk(t +⎛1) = 0⎛⎞00 ⎟⎜ (1 + ai )
⎟
⎜ δ1
0 ⎟
δ
⎜ (1 + ai )⎟13
e 0 − (1e +⎜e + (I − ∆)k(t)y+ g)Gk(t1) =1 0
⎜
⎟
⎜
⎟
13
...
...
⎟ and ∆ = ⎜
⎟.
G=⎜
.
.
⎜
⎟
⎜
⎟.
..
..
G=⎜
⎟ and ∆ = ⎜
⎟
⎝
⎝
⎠
13⎠
⎜
⎟
⎜
⎟
⎝
⎝ 0
0
(1 + an ) ⎠δn ⎠0(1 + an )0δn

We can also rewrite the objective function in terms of per-capita as follows;

Wecan
can
also
rewriteobjectivefunction
inoftermsof per-capita
We
also
rewritethe theobjectivefunctionin terms
per-capita
as follows; as follows;

Then,
Then,
Then,

C(t)
C(t)
C(t)
e
c(t) = C(t)
=
.
0
t
e
c(t) = At0L(t) = (1 + ai ) t(1 + g)ttA000L(0) .
At L(t)(1 + ai ) (1 + g) A0 L(0)¶¸µ∙∞ µ 1 ¶t∞ ∙ (1 + g)(1 + a ) ¸t
P
P
0
t
t
∞
∞
P
C(t) = P (1 + g)(1 + a0 ) ec(t)1C(t) = t=0ec(t)(1 + r)t=0 1 + r
(1 + r)t=0 1 + r
t=0

Now
model
can canbe rewritten
as theas
following
per-capita
eﬃciencyeﬃciency
unit model:unit
Nowthetheoriginal
original
model
be rewritten
the following
per-capita

Now the original model can be rewritten as the following per-capita eﬃciency unit

model:
model:The Per-capita Eﬃciency Unit Model:
The Per-capita Eﬃciency Unit Model:

Now the original model can be rewritten as the following per-capita eﬃciency unit
model:The Per-capita Eﬃciency Unit Model:
An Unbalanced Multi-sector
Growth Model
Constant(1with+ g)(1+ a0Returns)tc(t) where ρ =Max ρ eThe
Per-capita
Eﬃciency
Unit Model:
The
Per-capita Efficiency
Unit Model:(1 + r),s.t. eki (0) = ki (i = 1, · · · , n),(1 + g)(1 + a0 )t,c(t) where ρ =Max ρ e(1 +(ir)= 1, · · · , n),yei (t) = f i (ek1i (t), e
k2i (t), · · · , ekni (t), `i (t))
k·i (0)(i =(i1,=· · 1,
· ,·n),ye (t) = f i (ek (t), eks.t.(t),e·· ,ek=(t),ki` (t))· · , n),
i

1i

2i

ni

i

i
yei (t) =e
k=
k2i
,e
kni·(t),(t))
1, · · · , n),cf(t)(ef 0 (e(t), ·e
k·20· (t),· · ,`ekin0(t),(i`0=(t)),
1i (t),
10(t),e − (1 + g)Gk(te + 1) = 0,
e + (I − ∆)k(t)
y
i e
e
e
yei (t) = f (k1i (t), k2ie(t), · · · , kni (t), `ei (t)) (i = 1, · · · , n),
e+
y
k(t)
e − (1e k(t + 1) = 0,
14+ g)G
e
c(t)
= (I
f 0−
(e
k∆)
10 (t), k20 (t), · · · , kn0 (t), `0 (t)),
e − (1 + g)Gk(te + 1) = 0,
e + (I − ∆)k(t)ynX14`i (t) = 1,nXee + 1) = 0,i=0− (1 + g)Gk(te + (I − ∆)k(t)y`i (t) = 1,

9

(7)
(7)
(7)
(6)
(8)
(7)
(6)(8)
(8)
(9)
(8)
(9)

i=0

n
X

n
X

`i (t) = 1,

(9)

i=0
e
=n e
kj (t) (j = 1, · · · , n).
kij (t) X

(10)
`
(t)
=
1,
(9)
i
e
kj (t)(j = 1, · · · , n).
(10)kij (t) = e
i=0
i=0
We
thethefollowing
assumption
and prove
basic
property,
Lemma 1; Lemma 1;
n assumption
Wemay
mayaddaddfollowing
and theprove
thebasic property,
X
e
e
(t)
=
k
(t)
(j
=
1,
·
·
·
,
n).
(10)
k
ij
j
Assumptin0<ρ<1.
We may3.add
the following assumption
and prove the basic property, Lemma 1;
n
X
i=0
i=0

Lemma 1. Under（6）（10）
except Eq.（8）are summarized as the social
n 2, Eqs.
Assumptin
3. 0Assumption< ρ < 1.X～ee〜 following
～
(t)
=
k
(t)
(j
=prove1, · · ·the
, n).
(10)
k
We
may
add
the
assumption
andbasic property,
Lemma
1;R+2njproduction function c（t）= T（ y（t）, ijk（t））which
continuously
diﬀerentiable
on theinterior
Assumptin
0 < ρfor< initial
1.i=0 and terminalis capitale and eIf x
z3.standstock
vectors
respectively,
then～～and concave=（y（t）, y（t）, ..., y（t）
）andexceptk（t）=（k（t）, k（t）,summarized..., k（t））.
1
32, Eqs.(6)-(10)
n
1
2
n
Lemma
1. where
Undery（t）
Assumption
Eq.(8)
areas thee and eIf x
z stand for initial and terminal capital stock vectors respectively, then
We
may
add
the
following
assumption
prove
the
basic
property,
Lemma
1;as
e
the
reducedform
utilityfunction
V
(e
x
,
z
)
and
the
feasible
set
D
can
be
denedAssumptin
3.
0
<
ρ
<
1.
Proof.
Lemma
1. Under Assumption 2, Eqs.(6)-(10) except
Eq.(8) are summarized as theesocial production function ec(t) = T (ey(t), k(t))
which is continuously diﬀerenBenhabib
and utility
Nishimura（1979）. V (etheSeereduced
form
functionx, ez) and the feasible set D can be dened asfollows:e～social～ production function e
c
(t)
=
T
(e
y(t),
k(t))
which
is arecontinuously
2n terminal
Lemma
Under
Assumption
2, Eqs.(6)-(10)exceptsummarized
as theIf xtiable
and1.z on
stand
for
and
capitalstock
respectively,
then thediﬀerenreduced
Assumptin
3.the0<ρinitial< 1. R
evectors(t) Eq.(8)= (yinterior
1 (t), y3 (t), · · · , yn (t)) and+ and concave where yfollows:～ ～form utility function V（x , z ）and
the feasible set D can be deﬁned as follows:
e−(t)∆)eandconcave=x(y(t),continuously
y3 (t), · · · , yndiﬀeren(t)) andtiablethe interior
Rz2n
e(Iy
e1]is
Vfunction(e[(1z −k(t)),x+) =socialproductionecT(t)=+ Tg)Ge(eywhere
(t),
which
e =on
k(t)
(k1 (t), k2 (t),·x· ,· e, kn (t)).
Lemma 1. Under Assumption 2, Eqs.(6)-(10) except Eq.(8) are summarized as theee]x· ,,2nzk) (t)).
= T [(1 + g)Gez − (I − ∆)ex, xe = (k1 (t), k2 (t),V· (ek(t)ne(t) = (y1 (t), y3 (t), · · · , yn (t)) andtiable on the interior·R+ and concave where yeandsocial production function ec(t) = T (ey(t), k(t))
which is continuously diﬀerenProof.
ande = (k1 (t), k2 (t), · · · , kn (t)).
k(t)
andProof.
e(t) = (y1 (t), y3 (t), · · · , yn (t)) andtiableon theandinterior
R2nn andnconcave where y
See Benhabib
Nishimura(1979).
e] ≥ 0}D = {(ex, ez) ∈ R++×Rz − (I − ∆)ex, x+ : T [(1 + g)Gen (1979).
ne Benhabib
Seek(t)andNishimurae] ≥ 0}={(e,ez)〜· ∈×R〜+: T [(1 〜+ g)Gez −〜(I − ∆)ex, x=〜 (kD1 (t),k2x(t),
··R
, k+
(t)).
Proof.
n～～〜15, k（t+1）
where x =（x（t）
,
x
（t）
,
...,
x
（
t）
）
,
z
=
（k
（t+1）
, ..., ke（）
and I isean n-dimensional
1
2
n
1
2e
n t+1）e = (ewhere xx1 (t), xe2 (t), · · · , xen (t)), ez = (k1 (t + 1), k2 (t + 1), · · · , kn (t + 1)) andunit
matrix.
See
Benhabib and Nishimura (1979). 15 ee = (ewhere xx1 (t), xe2 (t), · · · , xen (t)), ez = (k1 (t + 1), e
k2 (t + 1), · · · , ekn (t + 1)) andIProof.
isFinally,
an n-dimensionalunit
matrix.problem
the above optimization
will be summarized as the following standard
15
Ireduced
isSee
an Benhabibn-dimensional
unit matrix.
form problem,which
is familiarin the Turnpike Theory:
andoptimization
Nishimura(1979).
Finally,
the above
problem will be summarized as the following stan-

Finally, the above optimization problem
be summarized as the following stan15 will
dard reduced form problem, which is familiarin the Turnpike Theory:
dard reduced form problem, which is familiar in the Turnpike Theory:
Reduced Form ModelFinally, the above optimization problem will be summarized as the following stanReduced Form Model
Finally, the above optimization problem will be summarized as the following standard reduced form problem, which is familiar
Theory:P∞ int theeTurnpikeeMaximize
t=0 ρ V (k(t), k(t + 1))
dard reduced form problem, which is familiar in the Turnpike Theory:
ReducedForm Model
10
研 究 所 年 報
Reduced
Model
ReducedForm
Form ModelP∞ t ee + 1))
Maximizeρ V (k(t), k(tee +t=0esubject to (k(t),k(tP∞1)) t∈ De foret ≥ 0 and k(0) = k.
Maximize
ρ
V
(
k(t),
k(t
+
1))
t=0

Also note that any interior optimal path must satisfy the following Euler Equae
e + 1)) ∈ D for t ≥ 0 and k(0)esubject to (k(t),k(t= k.
tions, showing an intertemporal eﬃciency allocation:ee + 1)) ∈ D for t ≥ 0 and k(0)esubject to (k(t),
k(t
= k.
Also
note
that
any
interior
optimalpath
mustsatisfythefollowing
Also noteanyderivativeinterior optimal
path must
EulerEquations,
Equawhere
the that
partialvectors meanthat satisfy the followingEuler

showingan intertemporaleﬃciency
allocation:
Also notethat any interior
optimalpath must satisfy the following Euler Equations, showing an intertemporal eﬃciency allocation:
where the partial(k(tederivativee vectors
e thate + 1)) = 0 for all t ≥ 0− 1), k(t))+ ρVmeank(t(11)
zx (k(t),tions, showing anVintertemporaleﬃciency
allocation:tee + 1)) = [∂V (k(t),ee + 1))/∂ eeeeVwherex (k(t),thek(tpartial derivative vectors k(tmean that k1 (t), · · · , ∂V (k(t), k(t + 1))/∂ kn (t)] ,16
e − 1), k(t))
eee + 1)) = 0 for all t ≥ 0
Vz (ek(t+ρVx (k(t),k(t(11) teeeee ek(te − 1))/∂ee
(
k(t
−
1),
k(t))
=
[∂V
(
k(t
−
1),
k(t))/∂
k11(t),(t),······, ,∂V∂V((k(t),knn(t)](t)]t ,Veeeeeek(t),zVx (k(t), k(t +e1)) = e[∂V (k(t), k(te+ 1))/∂
k
k(t
+
1))/∂
k
e + 1)) = 0 for all t ≥ 0
Vz (k(t − 1), k(t)) + ρVx (k(t), k(t
(11)

t
16
e0 means
ean n =
e −zeroee −t (k(t),− 1), k(t))[∂V (k(t1), k(t))/∂k1vector.
(t), · · · ,“∂Vk(t1))/∂ ekn (t)] of
Vz (k(tanddimensional
columne”eimplies
transposition
16
vectors.
Note
under thediﬀerentiability
all the pricevectorsNotewill
and
0 means
an thatn dimensional
zero
column vector. assumptions,“ t ” implies transpositionof vectors.
and 0 means an n dimensional zero column vector. “ t ” implies transposition ofthat under the differentiability assumptions, all the price vectors will satisfy the following

satisfy the following relations:vectors.
relations: Note that under the diﬀerentiability assumptions, all the price vectors will
satisfy the following relations:q = ∂ec/∂ec = 1,
e e
−q∂T
(e
y1,
, k)/∂ki (i = 1, 2, · · · , n),
pq i == ∂e
c/∂e
c=

ee e
q∂T (e
y(e
(i =
= 1,
1,2,
2,······ ,,n),n).
y, k)/∂, k)/∂ke
ki i (ipwi i == −q∂T

e
qe
c +(e
wi0 == q∂Te − weype,yk)/∂kkw
i(i = 1, 2, · · · , n).

Using these relation, we may deﬁne the pricee vectors of capital goods as（n×1）row vectorc + pey − wthek price vectors of capital goods as (n × 1)
w0 =we qe
Using these relation,
may
dene
～
〜
〜
〜

p =（p1, p2, ..., pn）, the output of capital goods as（n×1）vector y =（y 1, y 2, ..., y n）t, the rental
～

〜

〜

ek =
row asvectorp=(pvector· ·=（w, pn1),, wthe
outputgoodsas（n×1）
as ( n ×vector
1) vector
rate（1× n）
row
thecapital
capital stockk =（k 1,y
1 , p2 , · w
2, ...,wn）and of2,
Using these relation, we may dene the price vectors of capital goods as (n × 1)〜..., k n）t. w0 is a wage
rate. For simlicity we may assume that all the price vectors（p, w, w0）t(ey1 , ye2 , · · · , yen ) , the rental rate as (1 × n) row vector w = (w1 , w2 , · · · , wn ) and thee=row
vector p =p2 , · · · ,pricepn ), vectors
the output
capitalas ( n ×1) q. vector y
are expressedas (pthe1 ,relativeof theofpriceof thegoods
consumption
good

t
e
capital stockas (n ×steady1)vector= (kekρ（denoted,ek ,··· ,ekby. w0 henceforth）
is a wage rate.
simlicity
Definition.
An optimal
state kpath
is an For
optimalpathn ) OSS(ey1 , ye2 , · · · , yen )t , the rental rate as (1 ×1n) 2row
vectorw= (w1 , w2 , · · · , wn ) and the～～～which solves the above optimization problem and kρ = k（t）= k（t+1）for all t ≥ 0.

we may assume that all the price(p, w,we0 ) tare expressed as the relative pricee vectorsecapitalas (n × 1)vector(ek1 ,sectorʼsk2 , · · · production,, kn ) . w0 isit isa wagerate. For to
simlicity
Due tostock
the homogenetyassumptionkof=each
often convenient
express

the price
the consumption
q. the technology matrix as follows:
avectorschosen oftechnology
as ofa technology
matrix. Letgood
us deﬁnewe may assume that all the price vectors (p, w,w0 ) are expressed as the relative pricevectorsof the An
priceoptimalof the consumption
good kq.ρ (denoted by OSS henceforth) is anDenition.
steady state path17
Denition. An optimal steady state path kρ (denoted by OSS henceforth) is an17

Due to the homogenety assumption of each sector’s production, it is often convenient to express a chosen technology as a technology matrix. Let us dene thetechnology matrix as follows:

An Unbalanced Multi-sector Growth Model with Constant Returns

⎛

11

⎞

⎜ a00 · · · a0n ⎟
⎜
⎟ ⎛
⎞
⎜
⎟
⎜ a
⎟
⎜ 10
⎟ ⎜ a00 a0. ⎟
⎟=⎜
⎟
A =⎜
⎜ .
⎟ ⎝
⎠
⎜ ..
⎟
A
a.0 A
⎜
⎟
⎜
⎟
⎝
⎠
an0

where a0i = èi /eyi (i = 0, · · · , n), aij = e
kij /eyj (i = 1, · · · , n; j = 0, 1, · · · , n) and⎞⎛⎜ a11 · · · a1n ⎟
⎟
⎜
⎟
⎜ .
.
⎜
.. ⎟
A = ⎜ ..
⎟.
⎟⎜⎠⎝
an1 · · · ann

It directly
follows
that Assumption
impliesthat· · some· , n, ia=ij 1,> 0
It directly
follows
that Assumption
2 implies2thatfor allj = 0,for1, all..., n,j a=ij >0,01,for..., n and a0i > 0 for all i. We make ﬁrst the following assumption in terms of the technology

for some i = 1, · · · , n and a0i > 0 for all i. We make rst the following assumption

matrix to show the existence theorem.

―in terms 4.of（Viability）
the technology
matrixshow
the existence
Assumption
For a given
r（to
≥ g）
, a chosen
technologytheorem.
matrix A satisﬁesr[I − (rI + ∆)A ]−1 ≥ ΘAssumption 4. (Viability) For a given r (≥ g),a chosen technology matrix A satiswhere Θ is a n×n zero matrix ⑶.

where Θ es
is a n × n zero matrix3 .

By the well known equivalence theorem of the Hawkins-Simon condition and Theorem 4 of―Mckenzie（1960）, Assumption 4 is equivalent to 18
the property that the matrix［I−（rI+Δ）A r］

By the well known equivalence theorem of the Hawkins-Simon condition
and The―has a dominant diagonal that is positive; there exists y ≥ 0 such that［I−（rI+Δ）A r］≥ 0.

We 4need
the following
extraAssumption
assumption. 4 is equivalent to the property that the matrix
oremof Mckenzie(1960),
Assumption 5. 1 >r a0 > max｜ai｜

i = 1,a..., n[I − (rI + ∆)A ] hasdominant diagonal that is positive; there exists y ≥ 0 such

Remark 1 The assumptionmeans that the TFP growth rate of the consumption sector is therthat [I − (rI + ∆)A ]y ≥ 0.

highest among those of sectors. Takahashi, Mashiyama and Sakagami（2004）reported thatin theJapanese
economy,consumption sector has exhibited a higer per-capita
Wepostwarneed thefollowing
extra theassumption.
output growth rate than that of the capital goods sector in a two-sector model. If the TFP

growth rate has a positive correlation with the per-capita sectoral GDP growth rate, this fact
Assumption
5. 1 > a0 > max |ai |
i=1,...,nwill partially support Assumption 5.

Remark 1 The assumption means that the TFP growth rate of the consumption⑶ Let A and Θ be n-dimentional square matrix and n-dimensional zero matrix. Then A≫Θ if aij >0 for all i, j,
A >Θ if aij ≥ 0 for all i, j and aij > 0 for some i, j and A≥Θ if aij ≥ 0 for all i, j.

sector is the highest among those of sectors. Takahashi, Mashiyama and Sakagami(2004) reported that in the postwar Japanese economy, the consumption sector has
exhibited a higer per-capita output growth rate than that of the capital goods sector

12

研 究 所 年 報

McKenzie（1983, 1984）has shown that the existence of an optimal path and OSS in the reduced
form model. Actually we can prove the following existence theorem under Assumptions 1 through 5.
Existence Theorem: Under Assumptions 1 through 5, there exists an optimal steady state path
～

～

～

∞
kρ for ρ∈（0, 1］and an optimal path｛kρ（t）｝from any suﬃcient initial stock vector k（0）⑷.

Proof. We need to show that under Assumptions 1 through 3, all the conditions⑸ in Theorem 1of McKenzie（1983）or in the existence theorem of McKenzie（1984）are satisﬁed. Especially,
Assumption 4 and the additional condition are needed to guarantee the non-emptiness of
the interior of D（Condition 5）in the footnote）as we will demonstrate as follows; fromrt t and yb= =ArA
bwhereb= =(1,(1,Notethat
thatthetheequality
equalityrst
wherexb b= =(c,(c,
bx
by
bx
and y
y).y).Noteof ofthetherst
x
y
x)x)
―
r

the Condition 5）, there is an output vector y ≥ 0 such that［I−（rI+Δ）A ］≥ 0. By a scalarr∧∧∧― employment―condition. Since
r∧
rAcan
b
bwill
elements
will
provide(9);thethe
fullmultiplicationof andy, A
weestablish
x Eq.
= AEq.
y (9);
where
x full=（1,x）t and y =（c,condition.
y）. Note that
the
bx
by
elements
of of
x
and
y
provideemployment
Since
∧

∧

r
; the full employment
equality of the ﬁrst elements of x and Ar y will provide Eq.
r condition.
r （9）rA , it followsb
a submatrix
laborconstraints
satised
and
that
by
is is
a submatrixof ofA, it follows
thethelabor
constraints
arearesatisedforfor
y
and
that
A―A
∧
r
r

Since the labor constraints are satisﬁed for y and that A is a submatrix of A , it follows thatr―∧r y holds.
―thatxholds.
=x = Axry=thatAAy holds.
µ µ 1¶ ¶ª© ©I + [I−∆ − (1 + g)Gρ−1
1
r ªr y−1−1−1−1 ]A
G−1
z
−
ρ
x
=
G
z−ρ x =
]A
I
+
[I−∆
−
(1
+
g)Gρ
y
1+
1+
g g
⎞⎞
⎛⎛
0 0 ⎟⎟(1 (1++a1 )a1 )⎜⎜⎟⎟⎜⎜µ µ 1¶ ¶
⎜
⎟⎟
⎜
−1
.
1
−1
... ..
GI +
I+
I−∆
−
(1
+
g)
⎟⎟
⎜⎜
G
==
I−∆
−
(1
+
g)
⎜
⎟⎟
⎜⎜
1+
1+
g g
⎟⎟
⎜⎝
⎠⎠
⎝
0
(1
+
a
0
(1 + an ) n )
∙
¸¸
¾
∙
¸¸ ¾r
(1 (1
++
r) r)
r
AA y y
(1
+
g)(1
+
a
)
0
(1 + g)(1 + a0 )⎧
⎞⎤ ⎫⎫
⎧
⎡⎡
⎛⎛
⎞⎤
⎪
(1+a
)
⎪
⎪
⎪
⎪⎪
1
⎪
1)
0 ⎟⎥
⎢
⎜(1+a
⎪
⎪
⎟⎥ ⎪
⎪⎪
⎪⎪
0
)
(1+a
⎪
⎪
0
⎢
⎜
⎪
⎪
(1+a0 )
⎢
⎜
⎪
⎪
⎟
⎥
⎪
⎪
¶
µ
⎨
⎬
⎢
⎜
⎪
⎪
⎟
⎥
µ
¶
⎢
⎜
⎟
⎥
⎨
⎬
1
r
⎢
⎜
⎟
⎥
−1
.
1
r
⎢
⎜
⎟
⎥
.
−1
.
I
+
I−∆
−
(1
+
r)
G
=
A
⎢
⎜
⎟
⎥
.
..
I + ⎢I−∆=Gyy⎢ − (1 + r) ⎜ ⎜
⎟A
⎥ ⎪
⎟⎥
1+
⎜
⎪ ⎢⎢
⎪
⎟⎥ ⎪
⎪⎪
⎪⎪
1+
g g
⎪
⎜
⎪
⎟
⎥
⎪
⎪
⎪
⎣
⎠⎦ ⎪
⎪
⎪
⎪
⎪
⎝⎝
⎠⎦
⎪ ⎣
⎪
(1+a
⎪
⎪
n)
⎭
⎪
⎪
(1+a
n)
0
⎩⎩
⎭
0
)
(1+a
0
(1+a0 )¶µµ¶ª
© ©I + [I−∆ − (1 + r)I]A
1 1
r ªr y due to Assumption 5,
−1
G−1
≥
≥
G
y
due to Assumption 5,
I
+
[I−∆
−
(1
+
r)I]A
1+
1+
g g
µ µ 1¶ ¶1
r r y > 0 from Assumption 4,−1
G−1[I−(rI+∆)]AG[I−(rI+∆)]Ay > 0 from Assumption 4,==
1called+⑷ A capital stock x1 is+, （t）,
g g suﬃcient if there is a ﬁnite sequence（k（0）, k（1）, ..., k（T））where x =k（0）（kk（t+1））∈ D and k（T）is expansible. k（T）is expansible if there is k（T+1）such that k（T+1）≫k（T）and（k
（T）

−1
will
be
chosenthat
zassured−x≥
0 where
z)εD.
See
also
Lemma
, Therefore
k
（T+1）
）
∈yD.y
Note
that
the
suﬃciency
willzbe−assuming“Inada-type”
conditionontheproduction
Thereforewillbechosenso sothat
ρ−1ρx by≥0 where(x,(x,
z)εD.
See
also
Lemma
3 3

functions.

throughLemma
7 in
Takahashi(1985).
⑸
McKenzieʼsconditions
arefollowings:(1985).
1）
V（x, z）are deﬁned on a convex set D. 2）There is a η >0 suchthroughLemma
7 in
Takahashithat（x, z）∈ D and｜z｜<ξ< ∞ implies｜z｜<η< ∞. 3）If（x, z）∈ D, then（x, z）∈ D for all x ≥ x and 0 ≤ z ≤ z.
～

～

～

～

Moreover V（x, z）≥V（x, z）. 4）Ther is ζ> 0 such that｜x｜≥ ζimplies for any（x, z）∈ D,｜z｜<λ｜x｜where 0 <
～ ～

― ―
―
λ<1. 5）There is（x, z）
∈ D such that ρz― > x.

ρki (t)ieρ =kiρ (t)ρe, it follows that ρkiρ (t)=kAiρiA==
Remarkshould
noted
that
since
e
e
i
=
k
Remark
2 2It Itshould
be be
noted
that
since
kiρk=
Ait,Ltit follows that ki (t)
t t
i
Ai L
t
t

...
⎜
⎟⎥ A yµ = ¶G−1 I + ⎢⎢I−∆ − (1 + r) ⎜⎟⎥ ⎪t 1i eρ
⎪
rThus
1 −1
+
g=to1,avoid
(1 =
+i ).
ai )From
A0 kinow
for
i
·
·
·
,
n.
the
original
seriesoftheindustry’s
optimal
⎢
⎜
(1+a
on,
furthercomplicationsof
ournortation,
all
the
⎪
⎪
⎟⎥variables⎪
G [I−(rI+∆)]Ay> 0 from Assumption
4,
⎪
⎪
⎣
⎝
⎠⎦ ⎪
⎪
⎪
1+g
⎪
⎪
(1+a
)
n
⎩
⎭
0
ρ
)
(1+a
0
per-capitainstockk(t)
isgrowing
at
the
rate
of
its
own
sector’s
technical
progress,measuredeﬃciency
unit
will
be
denotedwithoutthesimbol”e”unlessotherwisei¶µ© ρ−1 x ≥ 0 where (x, z)εD.
Therefore y will be chosen 1so that−1z−
See also Lemma 3rª
G Multi-sector≥
I + [I−∆ Growth− (1 +Modelr)I]Adue toAssumption 5, 13
An Unbalancedwith yConstant
Returns(1+ai ). From now
on,mentioned.
1+
g to avoid further complications of our nortation, all the variables
i eρ µ+ ai )t7Ain
for i1 =¶1,(1985).
· · · , n. Thus ther original series of the industry’s optimalthrough (1Lemma
0 kiTakahashi−1G
0 fromthe
Assumption=
measured in eﬃciency
unit [I−(rI+∆)]A
will be denotedy >without
simbol ” e”4,unless otherwise
1 ρ+gSupposestockthat kki ρ(t)is aninterior OSS
eﬃciency
a givenρ, thenprogress,
it mustper-capita
is growing
at thein rateof its uite
ownwithsector’s
technical
―
―
― ―
−1 ―
Thereforeywillbechosen
so
that
z
−ρ
x
≥
0
where
（x
,
z
）
εD.
See
also
Lemma
3
throughmentioned.y will be chosen so that z − ρ−1 x ≥ 0 where (x, z)εD. See also Lemma
Therefore3
satisfy(1+a
From
nowequations:on,（1985）
to avoid
i ).theLemma
7 inEuler
Takahashi. further complications of our nortation, all the variableskiρ (t)
ρ ρ
ρ e
i ρ
e
inρ noted
Takahashi(1985).
it follows
followsthat
thatk（t）
k (t)〜k=Ait =i）tA0i〜kρi for
Remarkthrough
2 It should
be 7noted
thatthatsince
kiρ〜kρ=Remark
2LemmaIt shouldbesince
i = Ai Lt , it
i Ak
t=
i（1+a
Supposek is anint eﬃciency
uite
withi ia =”givenρ,then
it mustmeasuredin thateﬃciency
unitinterior
will ρbeOSS
denotedwithoutthesimbole”unlessotherwiseρ〜ρ) industryʼs+ ρVx (kρoptimal, kρ ) = 0.
(12)
Vz (kof, kthe
i = 1, ..., n. Thus the original series
per-capita stock k（t）
isgrowingi
21
satisfy
the Eulerequations:
mentioned.
at the rateof its own
sectorʼs technical progress,（1+a i）. From now on, to avoid further
complications of our nortation, all the variables measured in efficiency unit will be denoted

Due to the abovedenitionofρ ,OSS,
wethepartial derivativeskiρ (t)ρ
ρwill ρexpressρρofi theρeksince
) +in
ρV
(k A
,ik
= follows0.
Vz (k,)uiteit=ekit=Remark2 simbolItshouldbeannoted
that
keﬃciency
Supposethatk“〜ρ ”isunlessinterior
OSS
with a thatgivenkiρ,(t)thenotherwise
mentioned.
without
the
t(12)
i x=
i Amustt LtρSupposethat kinisterms
an interior
OSSvectors:
in eﬃciency uite with a given ρ, then it must satisfy the
Eulerequationsof price
satisfy the Euler equations:21

Euler equations:Due to the above denition of OSS, we will express the partial derivatives of theVz (kρ , kρ ) + ρVx (kρ , kρ ) = 0.
Euler equations in terms of price vectors:
Vx (kρ , kρ ) = pρ (I − ∆) + wρ(12)

Due to the above definition of OSS, we will express the partial derivatives of the Euler

equationsterms
of price
vectors:ρof OSS,ρ
Due tointhe
abovedenition
we will expressρ the partial derivatives of theVz (k , k ) = −(1 + g)Gp .

ρkρ ) = pρ (I − ∆) + wρx (k ,vectors:
Euler equations in terms ofVpricewhere I is a n × n unit matrix. Substituting these relations into the Euler equations
Vz (kρ , kρ ) = −(1 + g)Gpρ .
may yield the followings:

where I is a n×n unit matrix. Substituting these relations into the Euler equations may, kρ ) = pρ (I− ∆)relations+ wρ into the Euler equations
Vx (kρSubstituting
where I is a n × n unit matrix.
these
yield the followings:

may yield the followings: Vz (kρ , kρ ) = −(1 + g)Gpρρ.
ρ[wρ + pρ (I − ∆)] − (1 + g)Gp = 0(13)

andfurther
ﬁnallyyield:
where
Ifurtheris a ncalculation× n unit will
matrix.
Substituting
andcalculationwillnallyyield: these relations into the Euler equations
22
µ
¸ ρ=0
ρ ∙
ρ
(1r+¶g)Gp(13)
may yield the followings:ρ[w ρ + p (I − ∆)] −1+
G = wρ .
p −I + ∆+1 + a022

Theseclearly
non-arbitrage
conditions
amonggoods
capital
imply
that
These
areareclearly
non-arbitrage
conditions
among capitalandgoods
imply andthat any
capitalρ of returns
good must yield the sameρ[wrate
as the+ pρ (I − ∆)]− (1subjective+ g)Gpρdiscount= 0 rate ρ. Thus the Euler(13)

any capital good must yield the same rate of returns as the subjective discount rate

conditions are the non-arbitrage conditions.

ρ.Because
Thus theEuler
conditions areconditions.
of thedifferentiability
andthethenon-arbitrage
constant
returns
to scale technologies, the well22
iknown proposition proved by Samuelson（1945）will hold: the cost function denoted by C（w0,Because of the diﬀerentiability and the constant returns to scale technologies,wρ）（i = 1, ..., n）is homogeneous of degree one and ∂Ci/∂wj = aij where aij = kij/yj（i = 1, 2, ..., n;well-known
proposition
proved bySamuelson(1945)
willa hold:cost functionjthe= 0, 1,..., n）. Due tothe cost minimization
conditionand this
property,
unique thetechnology
matrixAρ is chosen along
the OSSkρ. Also note that due to Assumption 3, for a given ρ∈（0,i 1］, theiρdenoted by C (w0 , w ) (i = 1, · · · , n) is homogeneous of degree one and ∂C /∂wj =aij where aij = kij /yj (i = 1, 2, · · · , n; j = 0, 1, · · · , n). Due to the cost minimization
condition and this property, a unique technology matrix Aρ is chosen along the OSS

condition and this property, a unique technology matrix Aρ is chosen along the OSS
kρ . Also note that due to Assumption 3, for a given ρ ∈ (0, 1], the uniquely chosenρcorresponding uniquepositive
14
研 price
究 所 vector
年 報 p and positive factor price vectorρρρtechnology
matrix A
alongthek havecorresponding
uniquepositive
price
vectortoρpsatisfy,― OSS
ρand positive factor price vector
ρ andcorresponding
uniquepositive
price
vectorphave
positive factor
factor priceprice vector
vectoruniquelychosenmatrixAρ along
the vector
OSS k pto satisfy,ρρ technology
corresponding
uniquepositive
price
andpositive(w0 , w ).
ρcorresponding
uniquepositive
price
vectorpandpositivefactorprice
vector(w0ρ ,ρ wρ ).
ρ −1ρρ(w0 ,, wwρ ).
).
[I − (rI + ∆)A ] ≥ Θ.
(w
0ρ
ρ
(w
,
w
).
Proof. 0We can apply the same argument as the one used in Theorem1 of Burmeister
Furthermore
it follows
thatsameaρ00 >
0 and aρ0. ≫as
0 the
from
Assumption
2. Henceforth,
we use theProof.
We cantheone
used
in Theorem1
Burmeister>
0 and aρ0.
fromAssumption
2. ofHenceforth,
we
Furthermore
it applyfollows
thatsameaρ00argument
Proof.
We
can
apply
the
argumentasÀ
the0 oneone
usedin Theorem1
Theorem1of Burmeister
Burmeister
Proof.
We
can
apply
the
same
argument
as
the
used
in
of
ρ
and
Grahm
(1975).
.
Conbiningthese
symbol“ρ”
toclarifythatvectors
andmatricesareevaluated
along
OSS
k
Proof. We can apply the same argument as the one used in Theorem1 of Burmeisterand(1975).
useGrahm
the”ρ”
to clarify
that vectors
and matrices are evaluated along OSS kρ .
results,
thesymbol
following
importantproperty
will be established:
and
Grahm
(1975).
andFromGrahm
this(1975).
lemma, along OSS with ρ, the nonsingular technology matrix Aρ is
and Grahm(1975).
⑹
Lemma
When
ρ∈（0,along1］, there
existsuniqueOSS（kρ ≫ 0）with the matrix
corresponding
From2. thislemma,
OSS
with aρ,the nonsingular
technologyAρ ρ isConbiningtheseresults,
thefollowing
importantproperty
will
be established:
From
this
lemma,
along
OSS
with
ρ,
the
nonsingular
technology
matrix
Aρ is
is
From
this
along
OSS
with
ρ, thenonsingularmatrix A
chosenandthelemma,
cost-minimization
conditionandthe full-employment
conditionwill
uniquepositive
vectorpρ andpositive
price
vector（wρ0,technology
wρ）.
From
thispricelemma,
along
OSS
withfactor
ρ, thenonsingular
technology
matrix Aρ ischosen and the cost-minimization condition and the full-employment condition will
cantheapply
the same argument
as the one
usedin Theorem1 ρof Burmeister
and
Proof.
We
chosenandcost-minimization
conditionandthe full-employment
full-employment
conditionwillchosenandconditionthe0)6 with will
the
Lemma
2. theWhenρ ∈ (0, 1] , there
existsanda uniqueOSS (k À conditionbeexpressed
as cost-minimization
follows:
chosenandthecost-minimization
condition
andthefull-employment
condition
will
Grahm
（1975）. as follows:beexpressed
6
Let x and yasbefollows:
n-dimensional vectors. Then x À y if xi > yi for all i, x > y if xi ≥ yi for all ibe expressed
expressedbeas follows:
From this lemma,along OSS with ρ, the nonsingular technology matrix Aρ is chosen andbe expressed as follows:ρ)=, wi.ρ )Aρ ,piρ≥and at least one j, x > yi and x ≥ y(1,
if x
yi (w
for0allthe cost-minimizationi condition
and the full-employment
condition
will be expressed as follows:ρρρ,ρ
(1, p )ρ = (w0 ,ρ wρ )Aρρ ) = (w0ρ , wρ )Aρ ,(1,p(1, pρ) = (w0ρ, w ρ)Aρ,
230 , w )A ,(1, p ) = (w
and
and
and
and
and
(1, kρ )t = Aρ (cρ , yρ )t .
and
ρ, yρ )ρt .t(1, kρ )ρt t= Aρ (c=AAρρ (c(cρρ ,, yyρ ))t ..
(1, kkρ ))t =
(1,
ρ ρ
t
ρ
ρρ t
If A has the inverse matrix(1,Bk , )then= Asolving(c , yρ )the. above conditions respectivelyρρB ,ρ then solving the above conditions respectively
If ρA ρhas the inverse matrix
ρ
If A
has
the inverse
matrixmatrix
B , thenBsolving
above conditions
ρ has
the inverse
inversethenthesolving
the above
aboverespectively
conditionsyields,
respectively
If A
A
has
the
matrixBρ ,, thensolvingtheconditions
respectively
Ifyields,matrix Bρ , then¶ solving the above conditions respectively
If Aρ has the inverse µyields,aρ0.
1 ρ ρ¶
aρ0.
yields,
ρ
ρµ ρ
ρ
ρ −1
yields,ρ
p = w µa − 1 ρ a.0 a0. ¶ +a ρ = w (b ) +aρρ
yields,
ρ
ρ
0.00
ρ
ρ −1
ρ
a
aρ0.00pρ ρ= wρ ρ µaρ ρ− aρ1100aρ.0ρaρ0.ρ ¶+ aa)ρ −1+ aaρρ
0. = w (b
0.
ρ
¶
ρ
ρ
0.
0.
ρ = wρ µaρ −a00
ρ
ρ
−1
p
+
a
a
=
w
(b
)
+
ρ
a
a
ρ = w (b )ρρ
.0 a0. + a00ρ
00ρ
pρ = w ρ aρ − a1ρρ a.0
+ aa
0.
ρ
ρ
a
0.
0.
ρ
ρ
−1
00 a a00
a00
andp = w a − a00+ a00
ρρ = w (b )ρ
.0 0. + 00
and
a
a
a
00 ¶
00ρ
00 ρ
µ
and
1 ρ ρ¶a.0aandρ t µ ρ
ρ t
ρ −1
ρ t
and
(k ) = µa − 1 ρ a.0 a0. ¶ (y ) +aρρ = (b ) (y ) +aρρ.0
and
ρ
ρ
.000
.000
ρ −1
ρ
ρ
a
(kρ )ρt t= µ(yρ )ρt t+ aaaρ ρ− aρ1100aρ.0ρaρ0.ρ ¶(yρ )ρt t+ aa
ρ .0= (b )
ρ .0
ρ −1
ρ aρ ¶ (yρ.0ρ )t = µaρ −a00
t +a00ρ.0(ka)ρ = (bρ )−1 (yρ )t +aa
ρ
ρ
.0
0.
00
(kρ)t = aρ − aρ1ρ a.0(y ρ)t + aaρ.0 = (bρ)−1(y ρ)t + aaρρ.0
ρ a0.
ρ
00dened
00 = (b ) (y ) + a00
where bρ is a (ksubmatrix(y follows:) = a of−Ba00) + a0000
ρ a.0 a0. as
ρ
a
aρ00
ρ
ρa00
ρ
00
⎞
⎛
ρ
where
b
is
a
submatrixofBdenedasfollows:
whereof Bof deﬁned
as follows:
wherebbbρρisisisa aasubmatrix
submatrixBρρ deneddenedas follows:
follows:⎛ ρ
wheresubmatrix
of B
as
ρ⎞
ρ
ρ
⎜
⎞
⎛ ρb00 bρ 0· ⎟where b is a submatrix of Bρ dened⎞⎛
ρ −1as follows:⎟
B = (A ) =⎜⎜
b
b
⎟
⎞.
⎛
ρ
ρ
00
0·
⎝ bρ bρ ⎟⎠
⎟
⎜
ρ
00
0·
ρ
Bρ ρ= (Aρ )ρ−1−1= ⎜
b
b00
⎟
b0·
ρ⎠
⎟. .
⎜b
·0 b
Bρ =
= (A
(Aρ ))−1 =
=⎝⎜
⎟
⎜
bρρ00
⎟.
⎜
B
⎠
⎝
ρ
0·
⎟
⎜b·0ρρ b ρ ⎠Bρ = (Aρ )−1 = ⎝b·0 bbρ ⎠ .
⎝b·0ρρ
24 b·0 b
24ρ
And the nonsingularity of b comes from the following
observation: From Murata（1977）, bρ =24
24
24 （1960）, it also follows that det Aρ = aρ00det［aρ−（1/aρ00）aρ·0aρ0·］−1. Furthermore, by Gantmacher［aρ·0aρ0·］. Since Aρ is non-singular, the result follows.
⑹ Let x and y be n-dimensional vectors. Then x ≫ y if xi >yi for all i, x >y if xi ≥ yi for all i and at least one j,
xi >yi and x≥ y if xi ≥ yi for all i.

also use “ ∗ ”Whento denote the elements
and variables
are evaluatedat k∗ . Generalized∗Denition.
satisesthe
the GeneralizedDenition. Whenρρ==1,1,thethechosenchosentechnology
technologymatrix
matrixAA∗ satises(1977), bρ =that det Aρ
the GeneralizedDenition.
When ρ = GCI
1, the-Ichosen
technology
matrix Aa∗ satises
Capitalcondition,
ifif there
ofof positive
Capital Intensity
Intensity
GCI -IMulti-sector
condition,
there
existsa set
setReturns
positive number
number
An Unbalanced
GrowthModelexists
withConstant
15
From
arethatconcentrated on the OSS with ρ = 1 denoted by k＊. We will also use(d , ·now· · , don,) we
such

From no“＊” to denote the elements and variables are evaluated at k＊.

(d11 , · · · ,Intensitydnn ) such GCI
that -I condition, if there exists a set of positive number
Capitalalso use “ ∗Definition.
ρ = 1,thatthe chosen technology¯¯ matrix¯¯A＊ satisﬁes the Generalized Capital(d1 , · · · When
, dn ) suchnDenitionnX¯¯aa∗si∗ aa∗s0∗ ¯¯Xaa∗ss∗ss aa∗s0∗s0
d
(
−
)
>
d
−− ∗s0¯¯¯number
f or s（d= ,1,...,· d· ·）, suchn. thatIntensity GCI-I condition,
dss (a∗∗ if−there> a setdi i¯¯¯¯aof∗∗sipositive∗ )exists
aa00∗00 ¯ f or s =1 1, · ·n · , n.
a0s0s aa00∗00
a
0i0i
i6=
s,0
i6=s,0 ¯¯nX¯ a∗si a∗s0 ¯a∗ss a∗s0¯¯ f or s = 1, · · · , n.
∗d ( ∗ − ∗ )matrix>di ∗¯ satises− ∗ theSimilarly,∗¯ Generalized
Generalized Capital
Capital Intensity
Intensity
Similarly,thethestechnology
technology
A asatises
a0s a00 matrixA
a00 the0iCapiti6=s,0∗thenumber
Generalized
Intensity
Similarly,
thetechnology
matrixSimilarly,
technology
matrixA ＊ satisfies
the Generalized
CapitalGCI-IIGCI-IIthecondition,
if thereexistsaAsetsatisesof positive(d, · Capital· · , Intensityd ) such
that

GCI -II condition, if there exists a set of positive number (d11 , · · · , dnn ) such that

(d1 , · ·condition, if there exists a set of positive number（d1, ..., dn）such that∗aa∗s0∗s0
GCI -II condition, if there existsaass∗sssetofpositive−< 0 number (d1 , · · · , dn ) such that−aa∗0s∗0s aa∗00∗00 < 0a∗ss a∗s0
and
−
<0
and
and
a∗0s a∗00
¯¯ ∗
¯
¯¯ ∗
¯
nn
¯¯a ∗ a∗∗ ¯¯¯ X¯¯a ∗ a∗∗ ¯¯¯Xddss¯¯¯ a∗ssss −− a∗s0s0¯¯¯>>
ddi i¯¯¯ a∗sisi −− a∗s0s0¯¯¯ ffors = 1, · · · , n.
¯aa0s∗ aa00∗ ¯¯aa0i∗ aa00∗ ¯ or s = 1, · · · , n.
and
00
00
s,0 ¯ 0i
¯ ∗0s
¯ i6=i6=ns,0¯¯ ass a∗s0 ¯ X¯ a∗si a∗s0 ¯¯¯¯¯ fcapitalds good− sectordi ¯focus− own> s, ands =input
1, · ·s· and, n. its scapital-labor
Consider aaacapitalon∗ sector∗ ¯s, and∗ itson∗ its¯ agood¯its orConsidercapitalown
input
Considercapitalgoods, focus
and
focus
on
own capital
capitalinput s andand its
its
asector
a
a
0s
00
00
0i
i6=s,0SimilaGCI -ratio in all the other sectors. By the deﬁnition the left-hand side of the GCI-I condition means

capital-laborratio inin all
the
other
sectors.
Bydenition
the
side
ofof
capital-laborallthesector
otherofs,
sectors.
By thethedenition
the left-handleft-handsiderightthe
excess ofaratio
thecapital-laborratio
capitalinput
son
foritstheowncapitalgood
sectorConsidercapitalgood
and focus
capitalinput s.s The
andits

hand side collects the absolute values of the discrepancy between the capital-labor ratio of

andcapital-labor ratio in all the other sectors.
25
25 By the denition the left-hand side ofother sectors（i ≠ s, 0）to that of the consumption sector. The GCI-I condition points that thesum of such absolute values still fall short of the the excess of a＊ss/a＊0s over the comparable

25

ratio in the consumption sector, a＊ss/a＊0s. We may give a similar explanation to the GCI-II
condition.
The following lemma can be directly derived from the definition of the both intensity
conditions.
Lemma 3. If the technology matrix A ＊ satisfies the GCI-I（GCI- II condition）condition, its
inverse matrix B＊ has positive（negative）diagonal ellements and negative（positive）offdiagonal elements.
Proof. From the theorem by Jones et al.（1993）, under the Strong GCI-II, its inverse matrix
has negative diagonal and positive oﬀ-diagonal elements. On the other hand, under the Strong
CGI-I, by considering the case where one price falls with all the other prices constant in their
proof, their exact logic can be applicable and the ﬁrst result will be derived.
Due to Lemma 3, we may prove the following important lemma.
Lemma 4. Under the Strong GCI-I（the Strong GCI-II）,［b＊−（（1+g）G+Δ−I）］has a dominantdiagonal that is positive（negative）for rows⑺.

Considecapital-labodiagonal elements. From the accumulation equation y∗ = (1 + n)Gk∗ − (I − ∆)k∗and y∗Due= b∗tok∗Lemma+ b∗.0 , it3,follows
Proof.
under that
the Strong GCI-I (the Strong GCI-II), the inverse∗∗ ∗∗and y = b k + b.0 , it follows thatmatrix B∗ has positive (negative)and( positive) oﬀ∗
g)G + ∆elements− I)]k∗ =−bnegative[b∗ − ((1 +diagonal
16

研 究 所 年 報

.0∗∗+ g)G + ∆equation− I)]k∗ =y∗−b∗[b∗ − ((1diagonal=.0, (1n)Gkmatrix− (I −
Proof.
Due elements.
to Lemma 3,Fromunderthethe accumulation
Strong GCI-I（theStrong GCI-II）
the+inverse
B＊∆)k
has
Due（negative）
to Lemma diagonal
3, −b∗.0 elements< (>)0. andTheirfore
matrix oﬀdiagonal[b∗ − ((1 +g)G + From∆ − I)]
positive
negativethe（positive）
elements.
thehas
∗
∗ ∗
∗
and
y
=
b
k
+
b
,
it
follows
that∗∗.0
Due to Lemma
3, −b< (>)0.
Theirforek＊thematrix[b − ((1 + g)G + ∆ − I)] has
accumulation
equation
y＊.0=（1+n）
Gk＊−（I−Δ）
y＊ = b＊k＊+b＊. it follows thatthe negative(positive)
dominant
diagonal andfor rows.

the negative (positive) dominantdiagonal∗((1the+ g)G+ for∆ −rows.
I)]k∗ = −b[b∗ −call.0 is negative as the n.d.d.
From now on we may
dominant
diagonal
that
From Lemma
now on3,we−bmaycall thedominant
diagonal
thatis negative as the n.d.d.
＊＊
Due<（ > ）0.
Theirfore
matrix ［b−（（1+g）G+Δ−I）］ has the
andtotoalso
call thedominant
diagonal
thattheis positiveas∗ .0∗the p.d.d. for short.
Due
Lemma
3, −b
<
(>)0.
Theirforethematrix[b−((1 + g)G + ∆ − I)] has.0negative（positive）
for thatrows. is positive as the p.d.d. for short.
and also
call the dominant
dominantdiagonal
diagonal
From
nowmatriceson(positive)
we maycall
the dominantdiagonal
thatis negative
as the important
n.d.d. and also
callthenegativedominant
diagonal
for
rows.
where
all
areequations
evaluated
at
k∗8
will
show
followinglemma,
From
the
Euler
(12),
its. We
JacobianJ(k,theρ) isthe dominant diagonal that is positive as the p.d.d. for short.

From
the
(12),
its Jacobian
J(k, ρ) thatis
From
nowEuleron weequations
may
the dominantdiagonal
which
is corresponding
to callLemma
2.5 of Takahashi(1992). is negative as the n.d.d.
From the Euler equations（12）, its Jacobian
（
J
k,
ρ）
is
∗8
where all matrices. We
will show the following important lemma,J(k, areρ) =evaluatedρVxx (k, at
k) k
+ ρV
xz (k, k) + Vzx (k, k) + Vzz (k, k),
and also call the dominant diagonal that is positive as the p.d.d. for short.
Lemma 5. J(k,Supposethe
GCIρ) = thatρV either(k, k) +ofρVk) conditions+ V (k, k)hold.
+ VzzThen(k, k),there exists a
xz (k,
which is corresponding toxxLemma 2.5 ofTakahashizx(1992).
which at ＊k∗ iswhichat thek isEuler
positivescalarequations
ρ such thatfor itsρ ∈Jacobian[ρ, 1], theJ(k,OSSρ)kρisis unique and is a continuous
From
(12),
∗
which
at k5. isSuppose that either of the GCI conditions hold. Then there exists a
Lemma
ρ
∗ ρ,
vector-value. (k∗ , k∗ ) + Vxx (k∗ , k∗ ), k∗ )namely+ Vxz (kk∗ , k=∗ )k(ρ)+ Vzx
J(k, 1)function= Vzz (kofpositivescalarρ suchfor ρρV∈ [ρ, 1],∗k)
the OSS k∗ ρ k)is uniqueandis a continuousJ(k,∗ that7(kxx,(k,k∗ )k)++V (kxz∗(k,, k ) ++VVzx(k(k,, k∗ )++V
Vzz (k,(k∗k),, k∗ )
J(k,
1) ρ)
==
V ρV

A is arean nevaluated× n matrix
diagonal
elements
are negative(positive).
Let there
where Suppose
all matrices
at andk＊⑻its. Wewill show
the following
importantlemma,
whichexist∗
Proof.
If
detJ(k,1)=
60
holds,
thendueρ to the Implicit Function Theorem, the7P
vector-valuefunctionofρ,
namelyk=k(ρ).
Suppose A istoanLemman × n matrix
and its diagonal
elements
are negative (positive). Let there existn
is corresponding
2.5 of.
a positive vector
h such thathi Takahashi| aii |> （1992）
j=1,j6=i hj | aij |, i = 1, 2, · · · , n. Then A is said to have
which at k∗ isPfollows.
show
this
we willn conditions
use=i theinaApositive
Benhabib
and
Lemma
5.
Supposethat
eitherhold.
aresult
positive
vector
hTo
such
that
h
| athehj | afollowing= 1,fact
2,there· · · shown, n.exists
Thenis said scalar
to havei ofii |>GCIj=1,j6
ij |, iThen∗a dominant main diagonalthat
isnegative (positive)
for rows. See McKenzie (1960) and Murata
Proof.
If
detJ(k,1)=
60
holds,
then
due
to
the
Implicit
Function
Theorem,the――ρsuch that mainfor ρ∈［ρ, 1］, theOSSkρ is (positive)
uniqueandisrows.
a continuous
vector-valuefunction
of
∗
∗
∗
∗
∗
∗
∗
∗
aNishimura
dominant
diagonal
thatisnegativefor
See
McKenzie(1960)
and
Murata(1979):
1) = Vzz (k , k ) + Vxz (k , k ) + Vzx (k , k ) + Vxx (k , k )(1977). J(k,ρ,namelykρ = k（ρ）
resultfollows.
To. show this we will use the following fact shown in Benhabib and7(1977).
Suppose A is
an n × n matrix and its diagonal elements are negative (positive). Let there exist＊
due to=the
Implicit Function Theorem, the result follows.
Proof. If detJ（k , 1）≠0 holds,
T1 =then[∂T /∂y]Pn −p,27T2 = [∂T /∂k] = w
Nishimura(1979):
a positive
vectorh such| aBenhabib
2, · Nishimura· · , n. Then
A is said
to havei | aii |> factij |, i = 1,
j=1,j6=i hjinTo
show thiswe willusethatthe hfollowing
shown
and
（1979）
:
zz

xz

zx

xx

27
p ismainan output
vector.
Diﬀerentiating
bothSee
pricevectors
withandresoect
toawheredominant
diagonalpricethat isnegative(positive) for rows.
McKenzie(1960)
MurataT1 = [∂T /∂y] = −p, T2 = [∂T /∂k] = w(1977).
y and k again will yield the following second-order partial derivative matrices:T11 =
where
price
vector.
Diﬀerentiating
both pricewith resoect
to y andto
wherep pis isananoutput
outputprice
vector.
Diﬀerentiating
both vectors
price vectors
with resoectk[−∂p/∂y],again will yieldfollowing second-orderpartialandderivative
T11 =［−∂p/∂y］
27
T12 the= [−∂p/∂k],
T21 = [∂w/∂y]
T22 =matrices:[∂w/∂k].Also
note that , ifandk again ,will
yield
the following
partial11 =Ty［−∂p/∂k］
T21 =［∂w/∂y］
and T22 second-order=［∂w/∂k］. Also
notederivativethat if thematrices:T
matrices are
12 =the matrices are evaluated at k∗ , then from the previous equation,[−∂p/∂y], T12 = [−∂p/∂k], T21 = [∂w/∂y] and T22 = [∂w/∂k].Also note that if

evaluated at k＊, then from the previous equation,[∂p/∂w] = (b∗ )−1
the matrices are evaluated at k∗ , then from the previous equation,and due to the symmetry of the Hessian matrix∗ −1of c(t) = T (y(t), k(t)),[∂p/∂w] = (b )⑺ Suppose A is an n×n matrix and its diagonal elements are negative（positive）. Let there exist a positive
vector h such that hi｜aii｜> Σ j =1, j≠i hj｜aij｜, i= 1, 2, ..., n. Then A is saidt to have a dominant main diagonal thatn[∂p/∂k] = −[∂w/∂y]due
to the forsymmetryof the （1960）
Hessianof c(t)= T (y(t), k(t)),
isandnegative（positive）
rows. See McKenzie
andmatrix
Murata（1977）
.
2

8

2

Weuseusethe following
convention
the partial
matrices:V⑻ We
the following
notationalnotational
convention forthe partial forderivative
matrices: derivative
Vxx =［∂ V
（x,
z）
/∂x ］
, Vxx
xz = =
2

2

2

［∂ V
（x, z）/∂x∂z］andVzz =［∂ V（x, z）/∂z ］. Note that each matrixt is an n×n
2 matrix.
22[∂p/∂k][∂ 2 V(x, z)/∂x ], Vxz = [∂ 2 V(x, z)/∂x∂z]
and=V−[∂w/∂y]zz = [∂ V(x, z)/∂z ]. Note that each matrix is

8
We use the following notational convention for the partial derivative matrices:Vxx =an n × n matrix.
2[∂ 2 V(x, z)/∂x2 ], Vxz = [∂ 2 V(x, z)/∂x∂z] and Vzz = [∂ 2 V(x, z)/∂z ]. Note that each matrix is28

the matrices are evaluated at k∗ , then from the previous equation,
where the suﬃx ”0 ” means a transpose of a matrix. Utilizing these relations, all the[∂p/∂w] = (b∗ )−1∗
partial derivative matrices at k can be expressed in terms of the matrices b∗ and
An Unbalanced Multi-sector Growth Model with Constant Returns

17and due to the symmetry
matrixof c(t)−1of tthe ∗Hessian−1∗ −1 = T (y(t), k(t)),
andto the symmetryc−1t）t））, = −(b∗ )−1 T22 , and T21 =T
as follows:
T11 = (bof∗ )the
THessian= (bof∗ )（T=22T（y(b（)t）, k,（T22 due1222 (b ) matrix

where thesuﬃx ”0 ” means a transpose of a matrix. Utilizingthese relations, all the−[∂w/∂y]−T22 (b∗ )−1 .Substituting Yx = [∂p/∂k](gI + ∆)=and
Yz = tI into Eq.(2.22) of Takahashi

8partialmatricesnotational
at k∗ canconvention
be expressedin partial
terms ofthe matrices
b∗ xx
and=
Wederivative
use the following
for thederivative
matrices:V
where the“' ” meansof asa matrix.
(1985),
thesuffix
Jacobian
will abetranspose
expressed
follows:Utilizing these relations, all the partial
222−1 z)/∂x∂z]∗⎛−1∗ −1each[∂ 2 V(x,
z)/∂x
],T
Vxzat
= ＊[∂∗2)V(x,
and
V(x,
z)/∂z
that
matrix
is =
zz
⎞
⎞
T
T
(b∗ )−1⎛= in(bV∗terms)−1=T[∂) matrices, T12].=Note)T22Tasand T21
bet22expressed
of(b
the
b＊−(b
and
derivative
matrices
22 as follows:
11 = k(b can
22
22 ,follows:−1＊ −1+ ＊∆）−1−. ISubstitutingT（b⎟, and⎜ (1T21+=g)G
⎟
11 T
Tan
（b
）−1
T t（b＊ ）−1 =（b＊ ）T22（b＊ ）−1, T⎜）12T22−T22（bn ×＊nmatrix.
11 =12 = −∗ ∗−1 22⎟⎜⎟.
−T22 (b) , 1).Substituting
∆)
and
Y
=
I
into
Eq.(2.22)of Takahashi
J(k= [(1 + g)GY+x ∆=−(gII, I]+⎜z ⎠⎝⎠⎝Yx =（gI+Δ）and Yz = I into Eq.（2.22）of Takahashi（1985）, the Jacobian will be expressedas
T21 T22
I
28
follows: the Jacobian will be expressed as follows:(1985),⎞ ⎛ will be completed. Substitut⎞⎛ then the proof
If the righthand side is negative denite,⎜ T11 T12 ⎟ ⎜ (1 + g)G + ∆ − I ⎟⎟⎜⎟.
⎜ the Hessian
ing allJ(kthe∗ ,relations
obtained
matrix of the social production1) = [(1 +g)G + ∆before− I, I]into⎠⎝⎠⎝
T21 T22I
function and suppose that the matrix b∗ is nonsingular, then we may yield the fol-

If therighthand
thenthetheproof
proof
will
completed.
SubstitutIf therighthandsidesideisisnegative
negative denite,
deﬁnite, then
will
bebe
completed.
Substitutingalllowing equation:
the relations obtained before into the Hessian matrix of the social production function anding all the relations obtained
before into the Hessian matrix of the social production＊
suppose that the matrix b is nonsingular, then we may yield the following equation:⎛ b∗ is nonsingular,⎞⎛⎞ the folfunction and suppose that the matrix
then we may yield⎜ T11 T12 ⎟ ⎜ (1 + g)G + ∆ − I ⎟
⎜
⎟⎜
⎟
[(1
+
g)G
+
∆
−
I,
I]
lowing equation:⎝⎠⎝⎠
T21 T22I= ((1 + g)G + ∆ − I)T11 ((1 + g)G + ∆ − I)
⎛
⎞⎛

⎞

(1++∆
g)G
++
∆T
−I ⎟
⎜ T+11T T((1
12 ⎟
+⎜
g)G
− I)
+((1
g)G
I)T
12
22 ⎟
⎜21
⎟
⎜
[(1
++
g)G
++
∆∆
−−
I, I]
⎝
⎠⎝
⎠
T
T
I
21
22
= [b∗ − ((1 + g)G + ∆ − I)]2 [(b∗ )−1 ]2 T22 .
= ((1 + g)G + ∆ − I)T11 ((1 + g)G + ∆ − I)

Due to Lemma 4, the matrix［b＊（（1+g）G+Δ−I）］has the negative（positive）d.d. from the∗＊ g)G + ∆ − I)] has the negative (positive)Due
to Lemma
4,itthematrix− ((1b++T((1nonsingular.
+ g)G + ∆T− isI) negative+ T22 deﬁnite and
+((1
+
g)G
∆[b− I)TGCI conditions
andmust
be+nonsingular.
is 12
also
22

21

nonsingular due to the argument of Benhabib and Nishimura（1979,. Furthermore,d.d.
from the GCI
conditions and it must be nonsingular. b∗ is pp68-69）
also nonsingular.
T22= [b∗ − ((1 + g)G + ∆ − I)]2 [(b∗ )−1 ]2 T22 .
the first two matrices are symmetric and therefore all the elements are positive.Thus theabove matrix is negative deﬁnite and the proof29
is completed.

∗ρ has the negative (positive)Dueto thisLemma
4, itthematrix+ g)G+ ∆pρ−, wI)]
From
lemma,
follows
that[ball −the((1
pricevectors, and the technology matrix Aρ

are continuous vecto-value functions of ρ∈［ρ' , 1］.
d.d.
from the GCI conditions and it must be nonsingular. b∗ is also nonsingular. T22

4 The Neumann-McKenzie Facet29

Now we will introduce the Neumann-McKenzie Facet（NMF for short）, which plays animportant role in stability arguments regarding neoclassical growth models as studied in

by L. McKenzie (see especially McKenzie (1983)). The NMF will be dened in the
reduced form model as follows:
18
研 is
究 a所
報 z) capital stock vectorsρFrom
the denition
above, the NMF
set年
of an
(x,
), is deDenition.
The Neumann-McKenzie
FacetofOSS, denoted by F(kρ , kwhich

Takahashi（1985）and Takahashi（1992）and has been intensively studied by L. McKenzie（seearise fromthe as:exact same net benet as that of OSS when it is evaluated by thenedespecially McKenzie（1983））. The NMF will be deﬁned in the reduced form model as follows:ρ
Definition.
Neumann-McKenzie
of an OSS,F
（ksurface, kρ）, isofdeﬁned
as:pricesof OSS.The
Also,
the VMF is theFacet
projectionof adenotedat onbythe
the utilityρρ ρ, kρ ) = {(x,∈ priceD : c vector+ ρpρ z(−p− pρρ, x+ρpρthe
kρ −
k },
F(k
function V thatis ρsupportedbyz)theρp=ρ , c1)
onto(x,pz)-space.
In

where kρis an OSSand pρ isof kρ when
the subjective
discountgoods
rate r is
Takahashi(1985),
I consider
thea supportingcase of thepriceobjectivefunctionwhere n capitalρρwhere k is an OSS and p is a supporting price of kρ when the subjective discountgiven.
as well as pure-consumption goods are also consumable. Here, the capital goods are
From
deﬁnition above, the NMF is a set of（x, z）capital stock vectors which arise fromrate
r isthegiven.
exact samebut
netthebeneﬁtas that of
OSSofwhen
it is evaluatedby the prices of OSS.
Also,
nottheconsumablediscountedsum
the sequenceof pure-consumption
goods
the VMF is the projection of a ﬂat on the surface
30 of the utility function V that is supported
is directly evaluated. Dueρ to thewell-establishedNonsubstitution Theorem, alongρ

by the price vector（−p , ρp , 1）onto the（x, z）-space. In Takahashi（1985）, I consider

caseof the objectivefunction
wherecapital before
goods aswell
pure-consumption goods
the theOSS,a uniquetechnology
matrixAρ ndened
willbeaschosen.
are also consumable. Here, the capital goods are not consumable but the discounted sum of

By exploiting this fact, we will re-characterize the VMF as a more tractable for-the sequence of pure-consumption goods is directly evaluated. Due to the well-establishedNonsubstitution
Theorem,alongmatrix
the OSS,Aaρ unique
technology matrix Aρ
deﬁnedasbeforewill bemulawith the (n+1)
by (n+1)and (n+1)-dimensional
vectors
follows:
chosen.
By exploiting this fact, we will re-characterize the VMF as a more tractable formula with
ρ
is non-singular,(x,z) ∈ F(kvectors, kρ ) ifas andonly if there exists
Lemma
6. WhenAρmatrixthe（n+1）
by（n+1）Aρ and（n+1）-dimensional
follows:∧

Lemma 6. WhenAρ is non-singular,（x, z）∈ F（kρ, kρ）if and only if there exists y≡（c, y）' = 00b ≡ (c, y) = 0 such thatysuch thatb = Aρ y
b
i) x
¶
µ
1
−1
G [b
y + (I − ∆)]b
x
ii) b
z =
1+n
⎛

⎜ 1
⎜
⎜
⎜ 0
⎜
−1
b = (1, x), b
where xz = (1, z),G = ⎜
⎜ .
⎜ ..
⎜
⎜
⎝
0
31

⎛

⎞

0

···

..
.

1
(1+a1 )

···

0

...

0

0

1
(1+an )

⎞

⎟
⎟
⎟
⎟
⎟
⎟,
⎟
⎟
⎟
⎟
⎠

⎜ 0 0 ··· 0 ⎟
⎟
⎜
⎟
⎜
.
⎜ 0 δ
.. ⎟
⎟
⎜
1
⎟ and I is a (n + 1)-dimensional unit matrix.
∆=⎜
⎟
⎜ .
...
⎟
⎜ ..
0
⎟
⎜
⎟
⎜
⎠
⎝
0 · · · 0 δn
Proof. From the denition of the NMF, we have the following supporting relation:

⎜
. . ⎟
1
0 · · 1·(1+a1. . ⎟ ⎟−1 −1 ⎜ 0⎜ 0(1+a)
1 ) 10
⎟, ⎟,
⎜
⎜
(1+a
n)
b
b
where
x
=
(1,
x),
z
=
(1,
z),G
=
b = (1, x), b
where xz = (1, z),G ⎜
=⎜
⎟⎟
.. ..
... ..
⎟⎟
⎜⎜
.
0
.
⎞
⎛
.
0
⎟⎟
⎜⎜
⎟⎟
⎜⎜
⎠⎠
⎝
⎝
⎜ 0 0 ··· 0 ⎟
1 1
⎟
⎜
0
·
·
·
0
0
·
·
·
0
(1+a(1+a
An Unbalanced
Multi-sector Growth Model with Constant Returns
19
n) n)
⎜
.. ⎟
⎟
⎜ 0 δ
.
⎟⎞ ⎞
⎜⎛ ⎛ 1⎟ and I is a (n + 1)-dimensional unit matrix.
∆=⎜
⎟
⎜ .
⎜⎜ ..⎜
0 00 0.·.·. · · ·0·0 ⎟⎟
0
⎟⎟ ⎟
⎜⎜ ⎜
⎟⎟ ⎟
⎜⎜ ⎜
. ⎠.. ⎟⎝⎜ ⎜0 ·0·δ·1 δ10 δ .. ⎟
. ⎟
⎟
⎜0⎜
⎟
n
⎟
⎜
⎟
⎜
andandI isI ais(nunitmatrix.
∆=a+(n1)-dimensional+ 1)-dimensionalunit
matrix.
∆⎜=⎜⎟⎟.. the. . . . . of.. denition⎟⎜⎜
Proof. Fromthe
NMF, we have the following supporting relation:
⎟
.
0
.
0 ⎟
⎟
⎜⎜ .
⎟⎟
⎜⎜
⎠⎠
⎝⎝
0 0· · · · · · 0 0δn δnρ
c + ρp z − pρ x = cρ + ρpρ kρ − pρ kρ
Proof.
From
thethedenitionof the
NMF,
we we
have
thethefollowingsupporting
relation:
Proof.
Fromdenitionof theNMF,
havefollowingsupporting
relation:
Proof. From the deﬁnition of the NMF, we have the following supporting relation:

where c and y correspond to (x, z). Furthermore, from the fact that Vx (kρ , kρ ) =ρ pρ x ρ= cρ +ρρpρ kρρ −ρ pρ kρρ ρ
c +cρpz−+ ρρpz − p x = c + ρp k − p kρρρρ
w + p (I − ∆) and Vz (k , k ) = −(1 + g)Gpρ , the Euler equation implies thatρ
where c and y correspond to（x, z）. Furthermore, from the fact that V（k, kρ）= wρ+pρ（I−Δ）
xρandy correspond
to to(x,Alsoz). z).
Furthermore,
from
the
fact
thatrelations,VxV(kxρ(k, kρρz,)k==c (1+g)Gandy correspond(x,
Furthermore,from)=ρ whereρc ρ−1ρρ ∆)].
ρ the
ρ fact−1 thatwwhere
[ρ
−
(I
−
note
that
from
the
accumulation
and=Vp
（k
,
k
）
=
−
（1+g）
Gp
,
the
Euler
equation
implies
that
w
=
p
［ρ
（1+g）
G−
（I−Δ）
］.
z

ρ（1+g）ρ ρnoteρ thatρ
ρ= relations,
（I−Δ）
and
k =（1/
Also
from
the
accumulation
（1/
）
w
(I
and
(kzρ(k, andkρρ,)kk−(1+ g)Gp, theEuler
impliesthatρ x］
w++p+g))Gpρ−(I−1∆)
−
∆)
and
)ρ =
−(1
+z+=g)Gp
, [y
the
Euler
equation
impliesthat−1
ρG ［y+
zV
(1/(1
[y+(I
−V∆)x]= (1/(1g))G+(I−equation∆)k]. Substituting−1ρ［yρ+（I−Δ）
kρ］. Substituting theses into the above supporting equation, we ﬁnally（1+g））G−1ρ (1+g)G−1wρ w=ρ intop=ρ [ρ
− (I
Also
note
that
from
thetheaccumulation
relations,z =z =p−1[ρ(1+g)G−−(I∆)].
− ∆)].
Also
note
that
from
accumulation
relations,thesesobtain,ρρ aboveρobtain,{[p y − wthex] − [pρ ysupporting− wρ kρ ]} equation,= 0 and cwe= nallycρ . Thus
it nally yields that
ρ ρ
−1 −1
−1 −1ρ ρ(1/(1+ g))G[y+(I−andandkρ k=ρ (1/(1+ g))G[y [y+(I+(I− ∆)k]. Substituting
¶
µ∆)x]
(1/(1
+ g))G[y+(I− ∆)x]= (1/(1+ g))G− ∆)k]. Substituting
1
ρ
−1
ρ
ρ
ρ ρ
ρ ρ
ρ y −ρw x]ρ−ρ[p y ρ −ρ w k ]} = 0.
(c − c ) + ρ c + pρ yG− w{[p(14)
x=c +p y −w k .
1supporting+gthesesintointo
thetheabove
supporting
equation,
we wenallyobtain,thesesabove
equation,nallyobtain,¶ 6=then
µ kρ¶
ρ
ρ x］
ρ−w
ρ≠0,
If｛［p
y−w
−［p
］｝
term turns
outturns
to be out
a vector.
This
If This{[py−wisx]Condition−[pρyyµ−0, thethensecond
the second
term
to be
a
resulti),1wwhich
implies
the production1k ]}−1ρ ρρ thatρ(c0ρ, y, x)ρ shouldρρ ρ ρlie
ρρ on
−1
ρ
ρ
G
(c
−
c
)
+
ρ
{[p
y
−
w
x]
−
[p
y
−
w
k
=
0.
ρ]}
ρ
ρ (14)
ρ (14)
G
(c
−
c
)
+
ρ
{[p
y
−
w
x]
−
[p
y
−
w
k
]}
=
0.
implies that the above equality
never
Therefore it follows that｛［p y−w x］−［p y −1ρ++ g= holds.
{[pρ y − wρ x] − [pρ yρ − w
k1ρg]}
0 and c = cρ . Thus it nally yields that
vector.
Thisthat intheeachabove
equality
never holds.
Therefore
thatasρ ρρ andfrontierof Timplies(y, k)
sector,the chosen
technology
must itbefollows
the same
ρ

ρ

ρ ρ

ρ ρ

w k ］｝= 0 and c = c . Thus it ﬁnally yields thatρ ρ
If {[py −y wx]ρ−yρρ y−ρ w
kρρ]}
6= 6=
0, 0,
then
thethesecondterm
turns
outout
to to
be be
a a
If {[p
−ρ w
x] [p
−ρ[p
−ρ w
kρρ]}
then
secondterm
turns
ρthat in the OSS. In otherOSS
c +words,pρ y −thew x=32ctechnology+ pρ yρ − matrixwρ kρ . Aρ will be chosen. Thus

vector.
This
impliesthatthetheabove
equalityneverholds.
Therefore
it follows
that
vector.
This
impliesthat
above
equalityneverholds.
Therefore
it follows
thaton theNMF,
the exact
matrixas the corresponding OSS is chosen.
This
resultis Conditioni）, same
whichtechnology
implies that（c0, y, x）should lie on the production frontierThis result is Condition i), which implies that (c0 , y, x) should lie on the production
32 32 must be the same as that in the OSS. In
of T（y, k）and in each sector, the chosen technology
In other words, given OSS technology matrix Aρ , the cost minimization and theother words,
technologyAρ thewill bechosen.
Thus on the
NMF,
same
frontierof Tthe(y,OSSk) andin eachmatrix
sector,chosentechnology
must
bethe
theexactsame
asfull-employment
conditions
for
labor
and
capitalgoods
aresatised.
Therefor,thetechnology matrix as the corresponding OSS is chosen. In other words, given OSS technologythat in the
OSS. In other words, the OSS technology matrix Aρ will be chosen. Thus
matrix Aρ, the cost minimization and the full-employment conditions for labor and capitalfollowing equations must hold:goods
satisﬁed.
the technology
following equations
hold:on
theareNMF,
the Therefor,
exact same
matrixmust
as thecorresponding OSS is chosen.
1)qρ = wρ aρ00 + wρ aρρ·0 ,In other words, given OSS technology 0matrix
A , the cost minimization and the2) pρ = wρ aρ0· + wρ aρ ,full-employment conditions for labor and0 capital
goods are satised. Therefor, thefollowing equations must hold:3)1 = aρ00 c + aρ0· y,4) x = aρ·0 c + aρ y1)qρ = w0ρ aρ00 + wρ aρ·0 ,The cost-minimization conditions 1) and2) imply that the same technology as that2) pρ = w0ρ aρ0· + wρ aρ ,of OSS is chosen. 3) and 4) means that,
under the chosen technology, the full employ3)1 = aρ c + aρ y,Note that the dynamics of the NMF is expressed by the accumulation equationof Condition ii). We will rewrite it using the element of the inverse matrix of Aρ asρ究 所
ρ ρ年−1報
ρ
20 Note
研 NMF
follows:rst
notebρ = [aof−the(1/aa0.expressed] . Solvingii) with respect
thatthethatdynamics
by Condition
the accumulation
equation00 )a.0is

The cost-minimization conditions 1）and 2）imply that the same technology as that of OSS is

toofy,Condition
we will obtainii). We will rewrite it using the element of the inverse matrix of Aρ aschosen. 3）and 4）means that, under the chosen technology, the full employment conditions

Note
that
the dynamics
NMF
by theaccumulation
equationρ isρ expressed
ρthe
hold.
It is rst
not diﬃcult
to see4）
asCondition
Conditionii）. withFrom
thesefollows:notethatbρ that
=of
[a3）
−and
(1/a
)a.0beaρ0.summarized]−1 . Solvingii)
respect
00can
y = bρ x + bρ

·0.respectively. Condition ii）are the（n+1）
conditions,
it follows
thatwill（c t）> 0 and y（≫0 forall elementt,of Conditionii). Werewrite
it t）
using
the
of the inverse matrix of Aρ as

to y, we will obtain-dimensional capital accumulation equations and z is determined through this relation.

ρρ ρ −1ρ
Substituting
this
into
equationandsolving
itequation
with
to z
follows:rst
thatthebofρaccumulation= [aNMF− (1/a] ii).theSolving
Conditionii)respectwithrespect
00 )a.0 a0. by
Note that
thenotedynamics
theis expressed
accumulationof Conditiony = bρ x + bρii）. We will rewrite it using the element of the inverse·0.matrix of Aρ as follows: ﬁrst note thatyieldsto y, we will obtainbρ =［aρ−（1/aρ00）aρ.0 aρ0.］−1. Solving Condition ii）with respect to y, we will obtain

Substituting this into the accumulation equation ii) and solving
it with respect to zρ
ρ((1/(1+g))b.
z = (1/(1 + g))(bρy+=Ib−ρ x∆)x
+ b−
·0
·0.
yieldsSubstituting this into the accumulation equation ii）and solving it with respect to z yields) andη(t + 1) = (zequation− kρ ), we
therespect
followingDening
η(t) =(x into− kρtheSubstituting
thisaccumulationii) willandnallysolvingobtainit withto z

z = (1/(1 + g))(bρ + I − ∆)x − ((1/(1 + g))bρ·0 .
diﬀerenceyields equations, which show the dynamics of the NMF:
Defining

η（t）=（x−kρ ）and η（t+1）=（z−kρ）, we will finally obtain the followingDening η(t) = (x − kρ ) and η(t + 1) = (z − kρ ), we will nally obtain the followingρ
diﬀerence equations, whichshow+thedynamicsof ∆)xthe NMF:+I−− ((1/(1 + g))bρ·0 .
z = (1/(1g))(bdiﬀerence equations, which show theof the NMF:
1 dynamics)[(bρ )−1 + I − ∆]η(t).
(15)
η(t + 1) = (1+
1)g= (z − kρ ), we will nally obtain the followingDening η(t) = (x − kρ ) and η(t +It is important to notice that the dimension of the NMF could be zero. The following lemmah
It is important
todwhichnoticethatofNMF
could abezero. Theof
independentvectors(h =show
1, · · the·the, ndynamics−dimension1). It isofclear
that
dh shows
reallocation
diﬀerenceequations,
thetheNMF:
will give us an exact order of its dimension.1
ρ −1)[(b ) + I − ∆]η(t).
(15)
η(t + 1) =ρ ( ρ
ρ
ρ
+
g1).
h= nushfollowing
give
an
order
itscleardimension.
Lemma
7.lemmadimamongF（kwill, ksectors.
）
k· · , ,k1
int ofxed labordenefollowing:h = 1,
2, · · · , n andofa
independentvectorsd(hand=Moreover,
1,F（·exactn）−⊂
ItD.theisthat dforshows
a reallocation

Proof. Let us deﬁne d =（d0, d1, ..., dn）t ∈ R+n+1 such that Σin= 0 a0i di = 0 holds. Because the linear

Itlaboris scalar
importantto ρnoticethat thedimensionof the NMF
could
positive
εh , sectors.
xed
among
Moreover,thefollowing:for h =
1, 2,be· · · zero.
, n andThea
1deneρρconstraint7.must
satisﬁed,
cannint
linearly
independent vectors dh（h = 1,
...,
Lemma
dimbeF(k, kη(t
) we
=
and
,)[(b
kρ ) ρ⊂)−1+D
I −. ∆]η(t).
(15)
+ n1)
=exactly( F(k choose
1+g
shows
reallocationof ﬁxedamong sectors. Moreover, deﬁnen−1）. It scalaris lemma
clearε that
following
willdhgive
us aanexact orderof itslabor
dimension.
positive
h,
h
ρ
h
Pn
b
b
y
≡
y
+
ε
d
,
h
n+1
the following:for= 1, 2, ...,nad1positive
Proof.
us hdene=and(d0,that, · ·the· , dscalar)t ∈ εRh,+ ofsuch
0 holds.
ndimension
i =zero.
It is Let
importantto dnoticethethatNMF couldThe
i=0 a0i dbe
ρ
ρ h
ρ
ρ
h ρ) ⊂
Lemma 7. dim F(k , k yn≡andintε Dρ. hb)xbAρ ρF(ky+d, kbhεsatised,
bρ +
bh= ≡
A, y
y
h=
hA
Becauselinearwill
constraintmust
becand exactly choose n linearly
followingthe
lemmagive us an
exactorder of its wedimension.

ρ h
ρρ y
hρ
bρ y
b+h ε=b=h (d≡Aεh A
d that Pn a d = 0 holds.
=
k
hnA
34
Proof. Let us dene dx)At d∈b R+n+1
such
0, d1 , · · · , d+i=0 0i i
Lemma 7. dim F(kρ , kρ ) = n and F(kρ , kρ ) ⊂ int D .
ρ hbρ +bekεh AdBecause the linear constraint= mustsatised,we can exactly choose n linearly
andP
Proof.
Let us dene d = (d0, d1 , · · · , dn )t ∈ Rn+1
such that ni=0 a0i di = 0 holds.
and
+
34
and
−1 h
h
b
zh ≡ constraint
G [by +must(I − ∆)b]
Because the linear
bexsatised,
we can exactly choose n linearly
−1

−1 hρ
b
zh ≡
yy +
xxhρ34]] + εh G−1 [dh + (I − ∆)Aρ dh ]
G [b
[b
+ (I
(I −
− ∆)b
∆)b
= G
−1
ρ
ρ h[dh + (I − ∆)Aρ dh ]bρ−1+[byερh G+ −1(I[I−+∆)bεh G=(Ix−] +
∆)A
]d
= G
k
−1

bρ + εh G [I + (I − ∆)Aρ ]dh= kP
Note that the rst element of the vector Aρ dh is zero due to the fact that ni=0 a0i dhi =PnbAρρishbh will
0Note
for that
all h.theSincethe rstofelementof ktheduerst
of xrst element
the vectordhone,
is zeroto element
the fact thatabe
0i done.
i =

inde

xe
An Unbalanced Multi-sector Growth Model with Constant Returns

21

Note that the ﬁrst element of the vector Aρdh is zero due to the fact that Σni = 0 a0idhi = 0 for
∧

posi

∧

all h. Since the ﬁrst element of kρ is one, the ﬁrst element of x h will be one. So the vectors～∧

x h（h = 1, ..., n）are well defined. Since the first element of kρ is 1, z h is also well defined
～

∧

∧

∧

∧

for all h. due to the fact that yρ ≫ 0 and kρ ≫ 0, εh can be chosen so that yh > 0, xh > 0 and
∧h

∧

∧

∧

z > 0 for all h. From our way of construction, the vectors yh, xh and z h satisfy Lemma 2
and the corresponding vector（xh, zh）also belongs to F（kρ, kρ）for all h. This implies that
there are n linearly independent vectors（xh−kρ, zh−kρ）. Therefore, there are exactly nlinearly independent line segments on the NMF, F（kρ, kρ）. In other words, the NMF has anandn-dimensional facet（ﬂat）containing the OSS,（k , k ）. This completes the proof.
ρρ

Using Lemma 7, we will show Lemma8.
Lemma 8. F（kρ, kρ）is a continuous correspondence of ρ∈［ρ, 1）.
―

Proof. See the Appendix.
We need the following deﬁnition:
Definition. The NMF is stable if there are no cyclic paths on it.
The stability of the NMF takes very important roles in demonstrating the Turnpike
properties as we will see soon. Under the both GCI conditions, we actually show that the NMF

Not

is stable as we will demonstrate in the next section. Due to the continuity of the NMF, if the

0 fostability of the NNF with ρ = 1 would have been proved, McKenzieʼs Neighborhood Turnpike
Theorem could be applicable as shown in Takahashi（1985）and Takahashi（1992）, and ﬁnally
we would demonstrate the following theorem:So tε> 0, there exists aρ > 0 such that for ρ∈［ρ, 1）and the corresponding ε（ρ）, any optimalezh ipath｛kρt ｝∞ with a sufficient initial capital stock k（0）eventually lies in the ε-neighborhoodtheTheorem 1.（Neighborhood Turnpike Theorem）Provided that the VMF is stable. Then for any――of kρ. Furthermore, as ρ→∞ , ε（ρ）→0.

belo

Proof. See the argument of Section 4 of Takahashi（1993）.
The Neighborhood Turnpike Theorem means that any optimal path must be trapped ina neighborhood of the corresponding OSS and the neighborhood can be taken as small as
possible by making ρ close enough to one.

5 Turnpike Theorem
The full Turnpike Theorem is described as the following theorem:
Full Turnpike Theorem There is aρ > 0 close enoug to 1 such that for any ρ∈［ρ, 1）, an――

optimal path kρ（t）with the suﬃcient initial capital stock will asymptotically converge to the
optimal steady state kρ.

chos

vect

22

研 究 所 年 報

As we have shown, under Assumption 7, the dimension of the VMF is n. We will keep thisassumption henceforth. On the other hand, the dynamics of the VMF is expressed by theindustry’s TFP. Thus our original purpose will be accomplished by showing the Fulln-dimensional linear diﬀerence equation（8）. To show the full turnpike theorem we need tostrengthen
the generalized capitl intensity conditionds, GCI-I and GCI-II.
Turnpike
Theorem.
Remark 3 the first to be noted that in the efficiency unit term, the full turnpike means thateach
sectorʼs
path
converges
the optimal
steadyofstate.
original terms of series,
We usethe optimal
followingproperty
to toshow
the stability
the InVMF.
any industryʼs per-capita capital stock and output grow at the rate of industryʼs TFP. Thus
our original purpose will be accomplished by showing the Full Turnpike Theorem.

Lemma 9. Let us consider the following diﬀerence equation system with the equiWe use the following property to show the stability of the VMF.

= 0,
libriumxe us
Lemma
9. Letconsider the following diﬀerence equation system with the equilibrium xe = 0,x(t + 1) = (C + I)x(t),where x（ t）∈R n and
n C is an n×n matrix. If C has the negative d.d. for rows, C+I is awhere x(t) ∈ R and C is an n × n matrix. If C has the negative d.d. for rows,

contraction for x（t）≠0 with the maximum norm ｜｜｜·｜, i.e., and the equation system is globally

C + I is a contraction
forLiapunov
x(t) 6= 0function
with themaximum
norm k｜｜and as｜theasymptoticallystable and theis V（x）=｜｜x｜｜, where｜··｜k,, isi.e.,deﬁned｜x｜｜= maxi ci｜x｜and
ci is a given set of positive numbers. Furthermore, if C has the positive d.d.
iequation system is globally asymptotically stable and the Liapunov function isfor rows, C+I exhibits total explosiveness for x（t）≠0.

V(x)= ﬁrstkxk,part
wherek · kfromis dened
asinkxk= max
xi | and. On
ci isthea givensetif Ci ci | pp.27-29）
Proof.
Thecomes
the result
Neuman（1961,
contrary,

has the negative q.d.d. for rows, C+I has eigenvalues with their absolute values greater thanof positive numbers. Furthermore, if C has the positive d.d. for rows, C + I
Lemma
10. Under
thefact
negatived.d., then-dimensional
F(kρhave, kρ )
one. This comesfrom thethat if C(positive)
has the positive
d.d. forrows, then itsNMF,
eigenvalues
exhibitsreal
total
for isx(t)6= 0. any path will diverge from equilibrium.
a positive
part.explosiveness
Thus the system
explosive;
where ρ ∈ [ρ, 1) turns out to be a linear stable (unstable) manifold.
We may use the second property later. We will ﬁrst prove the following theorem:

Proof.
rst partcomes （positive）
from theρ result
Neuman (1961,pp.27-29).
On theρ∈LemmaThe
10. Under
the ρnegative
d.d., theinn-dimensional
NMF, F（kρ, kρ）where
+
I
−
∆
=
[b
+
(I
−
∆)
−
(1
+
g)G]
+ (1 + n)G, it follows
Proof.
Becauseb―［ρ, 1）turns out to be a linear stable（unstable）manifold.

contrary, if C hasρ theρ negative
q.d.d. for rows, ρC + I has eigenvalues with ρtheirthat+ g)G[b+=［bI −ρ∆]= (1/(1+ g))G[b+G,(Iit−follows∆) −that
(1 +
n)G]
+G［b
I. Dening
Proof.(1/(1Becauseb +I−Δ
+
（I−Δ）
−
（1+g）
G］
+
（1+n）
（1/
（1+g）
+I−Δ］

ρ
absolute
greaterthan−（1+n）
one. This
from
the
fact）
that
if
C has−（1+g）
the positive=（1/（values
1 +g）
）
G
［b
+
（I−Δ）
G］
+I.comes
Deﬁning
C =（1/
（1+g）
G
［bρ+
（I−Δ）
G］
, Eq.
（8）
C = (1/(1 + g))G[bρ + (I − ∆) − (1 + g)G], Eq.(8) can be rewritten as:

can be rewritten as:

d.d. for rows, then its eigenvalues have a positive real part. Thus the system isη(t + 1) = (C + I)η(t).
explosive; any path will diverge from equilibrium.
Note again that

η（t）=（x−kρ）and η（t+1）=（z−kρ）. Thus applying Lemma 4, under the

Note
again
η(t) =property(x − kρ )later.
and η(t1) rst= (zprove− kρthe). ThusapplyingLemma
We mayusethatthe second
We +
will
following
theorem:negative d.d.（the positive d.d.）, any path on NMF will converge to（diverge from）the OSS.

4, From
underthisthelemma,
negative
d.d.the(thepositive
any thepathonstability
NMF willtounderStrong
GCI-II d.d.),condition,
local
and converge
the stability
38of the NMF hold simultaneously. The stabilityof the NMF implies that the Neighborhood(diverge from) the OSS.
turnpike holds. Thus combining both results, the following full Turnpike theorem will be

From this lemma, under the Strong GCI-II condition, the local stability and theestablished.
stability of the NMF hold simultaneously. The stability of the NMF implies that
the Neighborhood turnpike holds. Thus combining both results, the following fullto utilize the following well-known lemma by Levhari and Liviatan (1972):
OSS. Thus if the local stability holds in the neighborhood of OSS, then the optimalρ6= here0, if thetheNMF
following
characteristic
equation,Lemma
11. jump
Providedthat
detmanifold,Vxzpath muston thestableitself, andwill converge
toAn Unbalanced Multi-sector Growth Model with Constant Returns

23

thegiven
OSS. byThus
the full Turnpike
theoremwill beestablished.
expandingthe Eulerequationaroundthe OSS, has λ as a root, thenCorollary. Under the Strong GCI-II condition, the full Turnpike Theorem will be established.

Proof.
Toachievethe fulltoTurnpike
we needunder
to combin
the Neighborhood
Turnpike
thehasother
hand,show thetheorem,
local stability
the GCI-I
condition, we
need
itOn
also
1/(ρλ),
Theorem and the local stability of the OSS. The Neighborhood Turnpike Theorem means thatto utilize the following well-known lemma by Levhari and Liviatan (1972):

any optimal path should be trapped in the neighborhood of the OSS. Thus if the local stability

holds in the neighborhood of OSS, then the optimal path must jump on the stable manifold,ρ
6= 0, if the following characteristic equation,
Lemma 11. Provided that det Vxz

here the NMF itself, and¯ will converge to the OSS. Thus the¯ full Turnpike theorem will be¯Vρ λ2 + (Vρ + Vρ )λ + Vρ ¯ = 0.

xz Euler equationxx
zz
zx OSS, has λ as a root, thenestablished.
given by expanding thearound the(16)

On the other hand, to show the local stability under the GCI-I condition, we need to utilize

it also has 1/(ρλ),
Proof.
See
Levhariand lemma
Liviatan(1972).
the following
well-known
by Levhariand Liviatan（1972）:

Lemma 11. Provided that det Vρxz≠0, if the following characteristic equation, given by expanding（ρλ）,
the Euler equation around the OSS, has λ as a root, then it also has 1/

Lemma 12. Under the GCI-I
condition, the OSS satises
the local stability.
¯
¯
ρ 2
ρ
ρ
ρ ¯
¯Vxz
λ + (Vxx+ Vzz)λ + Vzx
= 0.

(16)

ρ
Proof.
All Levhari
we needshow
is thatdet Vxz
6= 0 under the GCI-I condition due
Proof. Seeand toLiviatan（1972）.

Proof. See Levhari and Liviatan (1972).

Lemma 12. Under the GCI-I condition, the OSS satisﬁes the local stability.

to Lemma
11. From the fact that V (kρ , kρ ) = pρ (I − ∆) + wρ (see Benhabib and
Proof. All we need to show is that det Vxρ ≠0 under the GCI-I condition due to Lemma 11.
xz

From the12.
fact Under
that V（k,k）= p （I−Δ）+w （seeBenhabib
Nishimura（1985）
for a twoLemma
GCI-I
condition,
the OSSsatises
thelocalshow
stability.
Nishimura(1985)
forx athetwo-sector
case)
and
Lemma
4,andwe
may
that
ρ

ρ

ρ

ρ

sector case）and Lemma 4, we may show that

Proof.

ρAll we need toρ show is ρthat
the
GCI-I condition dueρ−1 detρ Vxz 6= 0 underρ −1

Vxz = −(b ) [b + (I − ∆)]T22 (b )ρρρρto Lemma
thefact(see Benhabib andρ 11. 2 Fromρρ
2 that Vx (k , k ) = p (I − ∆) + w

where T 22 =［∂ T（y , k ）/∂k ］. Then,where Tρ22 = [∂ 2 T (yρ , kρ )/∂k2 ]. Then,

Nishimura (1985) det
for V
a two-sector
case)
Lemma
may
ρ
ρ −1andρ= −(det(b) )2 det[b+ (I4,−we∆)]detshowTρ that
22

xz

40

−1 ρ
Since Tρ22 is negative-definite,
it isρ )non-singular.
［bρ+（I−Δ）］has a quasiVρ = −(b[b + (I −Furthermore,∆)]Tρ (bρ )−122xz
Since Tρ22 is negative-denite,
it is non-singular. Furthermore,[bρ + (I − ∆)]ρhasdominant diagonal that is positive under the GCI-I condition, it also nonsingular. Thus V xz isρ
2 the isGCI-I
the
other
hand,
condition
impliesthat thecondition,
VMF is explosive.
Thisanon-singular.
quasi-dominantthat
positive
under
the GCI-I
it also nonwhereTρ22 = On[∂ 2 T(ydiagonal, kρ )/∂k]. Then,means that there are n characteristic roots with absolute value greater than one. Applyingρ
singular. Thus Vxz
is non-singular. On 40
the other hand, the GCI-I condition implies

Lemma 11, this also implies that there are n characteristic roots with its absolute value lessthan one.
the OSSsatisﬁes theThis
local stability.
that
the SoVMF
is explosive.
means that there are n characteristic roots withThus we have established the following theorem:

absolute value greater than one. Applying Lemma
11, this also implies that there areρTheorem 2 Under the both GCI conditions, the OSS k exhibits the full Turnpike Theorem.

Under the roots
GCI-II withcondition,
the full Turnpike
will Sobe theestablished
due to thethenProof.
characteristic
its absolutevalue lessTheorem
than one.
OSS satises
above corollary. On the other hand, under the GCI-I condition, from Lemma 12, the OSS

local stability.

will exhibit the local stability. Since any path on the VMF is totally unstabl, the NMF is“stable”
Neighborhood
Theoremhold. Combining both results again, the fullThusandwethe
haveestablishedTurnpike
the following
theorem:

Theorem 2 Under the both GCI conditions, the OSS kρ exhibits the full Turnpike

24

研 究 所 年 報

Turnpike Theorem is also established. This completes the proof.

6 ConcludingRemarks
We have demonstrated the Turnpike properties under the two types of the generalized
capital intensity conditions. As I mentioned before, the full turnpike means that each sectorʼ
s per-capita capital stock and output converge to the industryʼs own steady state paths
respectively. For example, the per-capita capital stock of the agriculture sector grows at its
own rate of technical progress along its steady state, but another sector, say a manufacturingsector also grows at its own rate of technical progress along its own steady state. The similar
explanation can be applicable to other sectors. Thus our theoretical result derived above isconsistent with the evidence obtained in recent empirical research.

References
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Benhabib, J. and K. Nishimura, 1981, “Stability of equilibrium in dynamic models of capital theory,”
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Benhabib, J. and K. Nishimura, 1985, “Competitive equilibrium cycles,” Journal of Economic Theory 35,
284-306.
Benhabib, J. and A. Rustichini, 1990, “Equilibrium cycling with small discounting,” Journal of Economic
Theory 52, 423-432.
Burmeister, E. and A. Dobell, 1970. Mathematical Theory of Economic Growth（Macmillan, London）.
Burmeister, E. and D. Grahm, 1975, “Price expectations and global stability in economic systems,”
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39, 88-93.
Jones, R., S. Marjit and T. Mitra, 1993, “The Stolper-Samuelson theorem: Links to dominant diagonals,”
in: R. Becker, M. Boldrin, R. Jones and W. Thomson, eds., General Equilibrium, Growth and Trade
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88-93.
Lucas, R., 1988, “A Mechanics of Economic Development,” Journal of Monetary Economics 22, 3-42.
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Suppes, eds., Mathematical Methods in the Social Sciences,（Stanford University Press）.
McKenzie, L., 1983. “Turnpike theory, discounted utility, and the von Neumann facet,” Journal of Economic
Theory 30, 330-352.
McKenzie, L., 1984, “Optimal economic growth and turnpike theorems,” in: K. Arrow and M. Intriligator,eds., Handbook of Mathematical Economics Vol.3,（North-Holland, New York）.

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25

Murata, Y., 1977. Mathematics for Stability and Optimization of economic Systems（Academic Press,
New York）.
Neuman, P., 1961. Approaches to stability analysis. Economica 28, 12-29.
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Samuelson, P., 1945, Foundations of Economic Analyssis（Harvard University Press）.
Scheinkman, J., 1976, “An optimal steady state of n-sector growth model when utility is discounted,” Journalof Economic Theory 12, 11-20.
Srinivasan, T., 1964, “Optimal savings in a two-sector model of growth,” Econometrica 32, 358-373.
Takahashi, H., 1985, Characterizations of Optimal Programs in Infinite Economies, Ph.D. Dissertation,
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Takahashi, H., 1992, “The von Neumann facet and a global asymptotic stability,” Annals of Operations
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Takahashi, H., K. Mashiyama, T. Sakagami, 2009, “Why did Japan grow so fast during 1955-1973 and27-69.
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APPENDIXAPPENDIX
We will prove here Lemma 8.

Proof.
We willSince
prove thehere VMF
Lemmais8.upper-semi continuous from its denition, all I need to
Proof. Since the VMF is upper-semi continuous from its deﬁnition, all I need to establish isestablish is that F(kρ , kρ ) = F(kρ ) is lower-semi continuous (l.s.c.) at ρ ∈ [ρ, 1).
［ρ, 1）. Under Assumption 7, wethat F（kρ, kρ）= F（kρ）is lower-semi continuous（l.s.c.）at ρ∈―

Under
Assumption
7, we can vectors
choose dnh labor
redistribution vectors dh as follows:
can choosen labor redistribution
as follows:nX
d = (−1, −1, −1, · · · , −1,aρ0i /aρ0h , −1, · · · , −1) (h = 0, 1, · · · , n).
h

i=0

th
This means
producing
sectortransfers
one unit
labor
the htosector.
Thismeansthat
thateacheachproducing
sectortransfers
one ofunitof tolabor
the hthFrom
sec-

Lemma 4, dh is a continuous vector function of ρ in ρ∈［ρ, 1）and that Σni = 0 aρ0idhi = 0―

tor. From Lemma 4, dh is a continuous vector function of ρ in ρ ∈ [ρ,h 1) andfor all h. This means that there are n linearly independent redistribution vector d . Let us

Pn ρ hdenote theseredestribution vector as dh（h = 1, ..., n）. Henceforth, we may use the notation,that
i=0 a0i di = 0 for all h. This means that there are n linearly independenthk（ρ）, d（ρ）
（h = 1, ..., n）and F（k（ρ））for denoting OSS, redistribution vectors and theredistribution vector dh .h Let us denote these redestribution vector as dh (h =VMF respectively. Using d（ρ）and from Lemma 6, we can deﬁne the following n linearly
∧

∧

h
h
h
h
independentvectors for weeachmayh（huse= 1,the..., n）: x（ρ）=k(ρ),
k（ρ）
（ρ）
d（ρ）
= k（ρ）
1,· · · , n).Henceforth,notation,
d+ε
(ρ)
(h =
1, · · ·and
, n)z（ρ）
and F(k(ρ))
hA
―

―

∧

∧

h
h
h
［I+
（I−Δ）A（ρ）］d（ρ）
where εh is chosen so that x（ρ）
≫ 0 and z（ρ）
≫ 0（h = 1, ...,
+εhG−1for denoting OSS, redistribution vectors and the VMF respectively. Using dh (ρ) andn）. Let us arbitrarily choose a point（x, z）∈F（k（ρ））. Furthermore choose（x', z'）∈ F（k（ρ'））

where（x',z'）∈int
D ,（x,≠（x', z'）ρ' ∈［ρ,n1）
is chosenindependent
close enough vectors
to ρ. Now
us
from
Lemma
6, we
canz）
dene
theandfollowing
linearly
forleteach―−1bh (ρ) = k(ρ) + εh A(ρ)dh (ρ) and bh (h = 1, · · · , n) : x
zh (ρ) = k(ρ) + εh G [I + (I −bh (ρ) >> 0 and b∆)A(ρ)]dh (ρ) where εh is chosen so that x
zh (ρ) >> 0 (h = 1, · · · , n).

{(x,
DD
: t0,[αk(ρ+ (x(1−α)k(ρ),αk(ρ)+
H ≡(xF(k(ρ )) where, z z)) ∈∈int(x, z))6=, z ) and ρ ∈ [ρ,1)(1−α)k(ρ)]+
is chosen close enoughα00) + (1−α)k(ρ),αk(ρ ) + (1−α)k(ρ)]+
n H ≡ {(x, z) ∈ D : t0 [αk(ρnXXαto ρ. 0Now
let usj 0dene
the plain0 0
j H isj dened
0 0 0 as follows:
j
0
t
[αx
(ρ
)
+
(1
−
α)x
(ρ),
αz
(ρ
)
+
(1
−
α)z
(ρ)]}
where
th = 1.
h
F(k(ρ ))n where(x , z ) ∈ int D , (x, z) 6= (x , z ) and ρ ∈ [ρ, 1) is chosenn close enough
X
X
h=1
h=1
0
th [αxj (ρ0 ) + (1 − α)xj (ρ),αzj (ρ所) +年(1 報− α)zj (ρ)]} whereth = 1.
α 究
26
研
α
0
0
α
α
α
toWe
ρ. can
Now
let
us
dene
the
plain
H
is
denedasfollows:
≡
{(x,
z)
∈
D
:
t
[αk(ρ
)
+
(1−α)k(ρ),αk(ρ)+(1−α)k(ρ)]+
H
0
,
z
)
between
H
and
the
line
alwaysndanintersection(xh=1
h=1 obtained bydeﬁne the plain Hα is deﬁned as follows:

α α
α
by
We canX
nalways nd an intersectionn
0 0 (x , z ) between H
0 0and the line
α obtainedαXconnecting
(x,
z)
and
(x
,
z
)
unless
(x,
z)
=
(x
,
αpoints00z ). Since (x , z ) is on the
j
0
j
j
0
j
≡ {(x,∈D: t0 [αk(ρ) +(ρ(1−α)k(ρ),αk(ρ) + where(1−α)k(ρ)]+
Hth [αx(ρ ) z)+ (1− α)x(ρ), αz) + (1 − α)z (ρ)]}
th = 1.
0 00 0
α αconnecting, z ) is on theh=1
α points (x, z) and (x , z ) unless (x, z) = (x , z ). Since (xh=1
plain Hn , it can also be expressed as follows:
n
X
X
j nd
0
j
j α ,0 zα ) betweenj Hα and the line obtained by
We canα always
anintersection(x) +be(1 expressed− α)x (ρ),as
αzfollows:(ρ ) + (1 − α)z (ρ)]} whereth = 1.
h [αxplain H ,titcan(ρalso
h=1

α

α

0

α

0

h=1

0 (1−α)k(ρ)]+
z )z)
= and
t0 [αk(ρ(1−α)k(ρ),αk(ρ)+(x , (x,connecting points(x0 , )αz0+) αunless(x, z)α= (x,αz0 ). Since (xα , zα ) is on the
α betweenαWe cananintersection
（x
,
z
）
H
and
the
line obtained
connecting,z)
betweenHandthe line byobtainedbyWe
canalways
alwaysﬁnd
ndanintersection(x0) + (1−α)k(ρ)]+nα(xα , zα ) = tα0 [αk(ρ0 ) + (1−α)k(ρ),αk(ρnααXX, it can also beαz）and（x', z'）unless（x, z）=（x', z'）points. Since（x , z ）isj on the plain H
plain（x,H, itα canj also
as follows:
0 be expressed
0 j (ρ),00 (ρ)]
0
tpoints[αx (ρ(x,
)+
(1and
− α)x
αzj (ρ0 )(x,
+ z)
(1 −
α)z
where(xα , ztααh) =is 1.on thehconnecting
z)
(x
,
z
)
unless
=
(x
,
z
).
Since
n
n
X
X
expressed
as follows:
h=1
h=1 α
α
j 0
j
j 0
j
t
[αx
(ρ
)
+
(1
−
α)x
(ρ),
αz
(ρ
)
+
(1
−
α)z
(ρ)]
where
th = 1.
h
α
α αbe
0
h
h 0
plain
Hh=1
it canalso, z 0) =tα0 [αk(ρ0 )as+ follows:(1−α)k(ρ),αk(ρ+0 )(1−α)k(ρ)]++expressed(1−α)k(ρ)]→ k(ρ) and [αxh) (ρ+ (1 − α)x
where
α, →
0,(x
[αk(ρ
h=1 (ρ), αz (ρ ) +hh 0
where αX
→ 0, [αk(ρ0ρ) +ρ (1−α)k(ρ)] → k(ρ) and [αxh (ρ0 )α + α(1 − α)xn (ρ), αz (ρ ) +X(1 − α)znh (ρ)]→αj (x0α , z ) αfor h =Therefore, z ) convergesto (x, z) as
0j 1, · · ·j , n.
0 (x
α (x
0
j
α
, z ))+=(1t0−[αk(ρ+ (1−α)k(ρ),αk(ρ) +(ρ)](1−α)k(ρ)]+
th [αx (ρα)x )(ρ),αz (ρ ) + (1 − α)zwhere
th = 1.
hρ ρα α(1 − α)z
(ρ)]
→
(x
,
z
)
for
h
=
1,
·
·
·
,
n.
Therefore(x, z ) converges
h=1
h=1 to (x, z) as
α α
α → 0.nAlso note that (x , z ) ∈ int D due to the convexity of D andthe fact that
n
X
X
h
h
0
hj 0
hhα
h 0
α［α
j[αk(ρ
0 +
j
j
0
where
k
（
ρ
'）
（1−α）
k
（
ρ）
］
→k
（
ρ）
and
［α
x
（ρ
'）
+
（1−α）
x
（ρ）
,
α
z
（ρ
'）+
（1−α）
)
+
(1−α)k(ρ)]
→
k(ρ)
and
[αx
(ρ
)
+
(1
−
α)x
(ρ),
αz
(ρ ) +
whereα→0,
α→
0,
α
α
t
[αx
(ρ
)
+
(1
−
α)x
(ρ),
αz
(ρ
)
+
(1
−
α)z
(ρ)]
where
t
=
1.
h
h
α→
0.
Also
note
that
(x
,
z
)
∈
int
D
due
to
the
convexity
of
D
and
the
fact
that
0 0
(x , z］→)（x∈ int
Onh theother
hand, because
the continuityof k(ρ),x(ρ)Also
andnote
z(ρ)
h
ρ
h=1
, zρD.
）for= 1, ...,
n. Therefore（xα, zαof）converges
to（x, z）
ash=1
α→0.
z（ρ）
h
ρ ρ
α α
(10 −0 α)z
(ρ)]
→
(x
,
z
)
for
h
=
1,
·
·
·
,
n.
Therefore(x,z)
converges
to
(x,
z)
as
α
0thetoh thath x(ρ)
h other0(x ,（xz )α∈z→intOn
hand, →
becauseofthe
thecontinuityof k(ρ),
and
that
）∈
int
D
due
convexityofk(ρ)
Dδ αand
fact
int
On
the
) ε+α other(1−α)k(ρ)][αx(ρ0 )（x',+αz(ρz(ρ)
)+
where
0,D.
[αk(ρ
in ρ ∈α,[ρ,
1).
for
any
>the
0 thereexists>and0 suchthat|ρα(1z'）−−∈ρ|α)x< δD.α(ρ),
implies
that
―
hand,
the that
continuity, x（Dρ）
andto（z theρ）convexity
in ρ ∈［ρ1）. forandany
εα >fact
0 thereα →because
0. hAlsoofnote(xα , zofα )k（∈ρ）
int
due
D
that
α ,of
α the
ρ εαρ > 0 there exists δ α > 0 such that
α |ρα
in
ρ
∈
[ρ,
1).
for
any
−
ρ|
<
δ
impliesthat(1 − α)z
(ρ)]
→
(x
,
z
)
for
h
=
1,
·
·
·
,
n.
Therefore(x,z)
converges
to
(x,
z)
as
α
exists δα > 0 such that ｜ρα−ρ｜<δ
implies0 that
α
α
0 0
)
−
[αk(ρ
)
+
(1
−
α)k(ρ)]k<εkk(ρ(x , z ) ∈ int D. On the other hand, because of the continuity of k(ρ), x(ρ) and z(ρ)α → 0. Also note that (xα , zαα ) ∈ int D0 due to the convexityαof D and the fact thatkk(ρ ) − [αk(ρ ) + (1 − α)k(ρ)]k < εin0 ρ 0∈ [ρ, 1). for any εα > 0 there exists δ α > 0 such that |ρα − ρ| < δ α implies thatand= D.
1, · On· · , n,(x
, z for) ∈ hint
the other hand, because of the continuity of k(ρ), x(ρ) and z(ρ)

and for h = 1, ..., n,

and for h = 1, · · · , n, ααα°° 1). αfor any
in ρ ∈ [ρ,
0 thereexists0 )δ+>that <|ραεα− ρ| < δ α impliesthat) − [αk(ρ(1 0−suchα)k(ρ)]kkk(ρ°(k(ρ ), k(ρα ))ε −>[αxh (ρ0 ) + (1 − α)xh (ρ), αzh (ρ0 ) + (1 − α)zh (ρ)]° < εα
°
α
0
α α →
where °= ααρ+ α(1k •(1k−isα)x
theh (ρ),
Euclidean°ρ(k(ρ
), k(ρ
)) −− α)ρ
[αxhαand(ρ0 ) +
αzh (ρ0 )norm.
+ (1α−Furthermoreα)zh (ρ)]° < εas
α
0
where
αρ
α kk(ρ
）ρ and｜•｜｜is thenorm. <
Furthermore
as α→0, εα→0.
47) −｜[αk(ρ) +Euclidean(1 − α)k(ρ)]kεand forρh==1,'+（1−· · · , n,―α ―αa point (xα ,zα ) as follows:0,
εα let→us
0.Now
let usa also
dene
Now
also deﬁne
point（x, z ）as follows:
47
°
°
α
α
h
0
n αzh (ρ0 ) + (1 − α)zh (ρ)]° < εαand for°h= 1,),· k(ρ· · , n,)) − [αx (ρ ) + (1 − α)xh (ρ),(k(ρX(xα ,zα ) = tα0 (k(ρα ), k(ρα )) +tαh (xj (ρα ), zj (ρα ))h=1°°°(k(ρα ), k(ρα )) − [αxh (ρ0 ) + (1 − α)x47h (ρ), αzh (ρ0 ) + (1 − α)zh (ρ)]° < εα
―α ―α
α α
α
α choosing
where（tdeﬁne,z ）.(xByclose enoughto zero,we canwhere(tα00,, ttα11, , ...,· · ...,
· , t·n·）· is, tαnused) is to
used
to（x
dene
,zα ). By αchoosing
α closeenough
to
―α ―α
―
―
―
―
α
α
α
α
α
α
47
make（x , z ）close enough to（x , z ）and（x , z ）∈ int D. Thus（x , z ）is feasible. Since itzero, we can make (xα ,zα ) close enough to (xα , zα ) and (xα ,zα ) ∈ inth D.
Thushh
is also expressed as the linear combination of（n−1）linearly independent vectors x , y and z ,it αfollows（x , z ）∈ F（k（itρis））. Alsoexpressed
note that due
to ourway ofconstruction of
of（x
）,
(x
,zα ) isthat
feasible.
Since
also
as thelinearcombination(n ,−z 1)
―α ―α

―α ―α

α

as α → 0,（xα, zα ）→（xα, zα ）. Now make α converge to zero, then（xα, zα）→（xα, zα）and――linearly independent vectors
xh ,yh and zh , it follows that (xα ,zα ) ∈ F(k(ρα )). Also――（x, z）. Thus（xα, zα）∈ F（k（ρα））converges to（x, z）∈ F（k（ρ））. therefore F（k（ρ））（xα, zα）→
α andα
is l.s.c.thatat ρ∈［ρ. Apply
theconstruction
same arguments
to αany
F（k（
F（k
（, ρ）
note
due
to, 1）
ourway ofof (x,zα ),point
as αof →0,ρ）(x）,zα ) any→ (xzα ).）―of ρ∈［ρ, 1）, it follows that F（k（ρ））is l.s.c. on［ρ, 1）
.
―

―

α

α

Now make α converge to zero, then (x , z ) → (xα , zα ) and (xα , zα ) → (x, z). Thus(xα ,zα ) ∈ F(k(ρα )) converges to (x, z) ∈ F(k(ρ)). therefore F(k(ρ)) is l.s.c. at