Kenjiro Ara Meets Frank Ramsey: Unbalanced Growth in A Neoclassical Optimal Growth Model
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構造

Kenjiro Ara Meets Frank Ramsey:
Unbalanced Growth in A Neoclassical Optimal Growth Model
Harutaka Takahashi1

Abstract
We will study the unbalanced growth in a neoclassical two-sector optimal growth model withsector specific technical progress, which is an optimal growth version of Ara (1969), and will
demonstrate the existence and the saddle-point stability of the optimal steady state (OSS) in theefficiency-units. By so doing the efficiency-units OSS paths will exhibit the unbalanced growth interms of the original-units; each sector will grow at its own growth rate. Furthermore the growthrate of the aggregated output (GDP) will converges to the one of the sector with the higher
technical progress.
JEL Classification:O14,O21,O24,O41

１．INTRODUCTION
We have witnessed a recent resurgence of an interest on growth and structural change. In fact,
the industry-based empirical studies across countries clearly have shown that growth in an
individual industry's per capita capital stock and output grow at industry's own growth rate,
which is closely related to its technical progress measured by total factor productivity (TFP) of
the industry. For example, per capita capital stock and output of the agriculture industry grow at
5% per annum along its own steady-state, whereas they grow at 10% annually in the manufacturingindustry, also paralleling the industry's steady state. Syverson (2011) has recently reviewed thesearguments discussed above. Let us refer this phenomenon as “unbalanced growth among
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2

研 究 所 年 報

industries.”
The attempt to understand this phenomenon has generated a strong theoretical demand forconstructing a multi-sector growth model, yet very little progress has been made so far with the
only exception of Baumol (1967) and Ara (1969). Baumol (1967) has set up an unbalanced growthmodel and demonstrated that if resources are shifting from progressive sector like electricalmachinery sector towards sectors where productivity is growing relatively slowly like service
sector, the aggregate productivity growth rate will slow down. Based on this theoreticalobservation, he has concluded a very pessimistic result that costs of service industry; costs ofeducation, fine arts and government services will increase forever. This phenomenon is often
referred to as “Baumol’s cost disease.” On the other hand, Ara (1969) set up the Uzawa-type twosector growth model with the Cobb-Douglas production functions with constant-returns to scale
and proved that each sector’s growth rate will eventually converge to the growth rate determinedby the sector-specific technical progress.
Recent exceptions are Echevarria (1997), Kongsamut, Rebelo and Xie (2001), Acemoglu andGuerrieri (2008) and Ngai and Pissariadis (2011). Setting up an optimal growth model with threesectors: primary, manufacturing and service, Echevarria (1997) has applied a numerical analysis
to solve the model. Kongsamut, Rebelo and Xie (2001) have constructed the similar model to theone of Echevarria (1997), while they have investigated the model under a much stronger
assumption than her: each sector produces goods with the same technology. In other words, theyalso assume a one-commodity economy. On the other hand, Acemoglu and Guerrieri (2008) havestudied the model with two physically differentiated intermediate-goods sectors and single
final-good sector. Note that the last two models will share a common character: one final good
economy except Echevarria (1997). Ngai and Pissariadis (2011) has set up the multi-sector
optimal growth model with the capital good and demonstrated that contrast to Baumol’s claim,
the economy’s growth rate is not on an indefinitely declining trend.
It is important to note that all the analytical models mentioned above share a common defect;
since they assume the same production functions among sectors, except sector specificexogenous technical progress, their models would be identified as one-commodity economy.
Because of this property, they could aggregate sector’s output even in a transition process.
Contrast to their model, we will study a two-commodity economy in this paper. We assume thateach good is produced with a different technology; consumption-goods and capital-goods arecompletely differentiated physical characters. As I will demonstrate later, this feature of themodel will make the characteristics of the model far more complicated.

Kenjiro Ara Meets Frank Ramsey: Unbalanced Growth in A Neoclassical Optimal Growth Model

3

In Section 1, I will set up a similar two-sector optimal growth model with the general
neo-classical production functions to Ara’s two-sector growth model which will be discussed inAppendix A, in which each industry exhibits the Hicks-neutral technical progress with anindustry specific rate. Then I will study the model under the optimal growth setting, which showsa sharp contrast to Ara’s analysis. In Section 2, I will rewrite the original model into a per capita
efficiency unit model. Then, I will transform the efficiency unit model into a reduced form model,
after which the method developed by Baierl, Nishimura and Yano (1998) will be applied to show
the existence of unbalanced growth. In Section 3, the saddle-point stability of unbalanced growth
will be also demonstrated.

２．MODEL AND ASSUMPTIONS
We will begin with listing up the notation:r: a subjective rate of discount,C (t ) c(t ) Y (t ) K (t ) : the total capital good at t,

K (0) : the initial total capital good,Ki (t ) 

: the tth period capital stock of the ith sector,: the total good consumed at t ,: C (t ) / L(t ) ,: the t th period capital output of the capital good sector,

L(t ) : the total labor input at t,

L(0) 

: the initial total labor input,

Li (t ) 

: the tth labor input of the ith sector,: the depreciation rate,

4

研 究 所 年 報

Ai (t ): the Hicks neutral technical-progress of the ith sector,

where i  0 and i  1 indicate the consumption good sector and the capital good sector
respectively.
We will make the following two assumptions on the model:
Assumption 1.
1) The utility function u(・) is defined on�  as the following:u (C (t ))  c(t ) C (t ) / L(t ) ( 0 for t  0).
2)

L(t ) (1  g )t L(0) , where g is a rate of population growth.

Assumption 2.
1) All the goods are produced with the following Cobb-Douglass production functions with theHicks-neutral technical progress:
12C (t ) A
and Y (t ) A1 (t ) K1 (t ) 1 L1 (t ) 2
0 (t ) K 0 (t ) L0 (t )

 2 1 and 1 
 2 1.
where 1 2) Ai (t ) (1  ai )t Ai (0) (i 0,1) , where ai is a rate of output-augmented (the Hicks-neutral)technical-progress of the i th sector and given as 0  ai  1 .
Note that 2) of Assumption 2 implies that the sector specific TFP is measured by the sector
specific output-augmented technical progress (the Hicks-neutral technical progress), which isexternally given.
Before setting up the model, we will divide all the variables by Ai (t ) L(t ) and will transformthe original variables into per -capita efficiency-units variables. Now let us define the following
normalized variables:～y (t )


k 0 (t )Y (t )C (t )K1 (t )c (t ) , ～, k1 (t ),A1 (t ) L(t )A0 (t ) L(t )L (t )K 0 (t )L0 (t )
L1 (t )
,  1 (t ) ,  0 (t )
.
L(t )
L(t )
L(t )

Firstly, let us transform the both sector’s production functions into the efficiency-units ones asfollows; dividing both sides by A1 (t ) L(t ) , we will yield
01～
y(t ) fc (t ) f (k0 (t ),  0 (t )) k0 (t )1  0 (t )2 and ～(k1 (t ), 1 (t )) k1 (t )1 1 (t )2 .

Kenjiro Ara Meets Frank Ramsey: Unbalanced Growth in A Neoclassical Optimal Growth Model

5

The next step is to derive the efficiency-units production possibility frontier (PPF) as shown inLemma 1 :Lemma 1. The efficiency-units production possibility frontier (the efficiency PPF for  is explicitly calculated as follows :y, kshort) : ～c  T ～

2



～
～
y, k   2 1   k  e k , ～
y 

c T

y    k, ～



 

 

 

 

 

(1)

  is the function obtained by solving, where  k , y   2 1k  (1 2   2 1 )e k , y and e k , y

k1 :the following equation with respect to12  k1 ～y  2 1k  (12   2 1 )k1  .
2

2

Proof.

We will apply the numerical method studied by Baierl, Nishimura and Yano (1998).

Under Assumption 2, let us consider the problem (*) where the time index is dropped forsimplicity :～～(*) Max c  k01  02 s.t. y k11 12 ,  0  1 1 and k0  k1 k.

The profit-maximization of both sectors will yield the following first order conditions :r 1 1 1 2 ． .
 
w  2 k1  2 k2
Solving the above equation with respect to  1 and substituting into  0   1  1 , we have

1 

1 2 k1.
 2 1k  (1 2   2 1 ) k1～

(2)

Furthermore, substituting (2) into y(t )  k1 (t ) 1 1 (t ) 2 yields
The next step is to derive the efficiency-units production possibility
frontier (PPF) as shown in




～
～
y   k  (     )k 2 y   (k , ～y )  2 ,k1 (1 2 ) 2 
2 1
1 2
2 1 1



Lemma 1 :(3)

Lemma 1. The efficiency-units production possibility frontier (the efficiency PPF fory    k  (     )k .
where  k , ～
2 1
1 2
2 1 1
short) : c  T y, k is explicitly calculated as follows :Solving (3) with respect to k1 , we will obtain


 

 





2

The next step is to derivethefrontier (PPF) as shown in  production possibility
y,efficiency-units
y .
(1)

c T
k  2 1  k1k  ge kk,, y～  k, y 
Lemma 1 :

 


 

 

(4)

～
Lemma 1.
Thesubstituting
efficiency-units
frontier(the manipulations,efficiency PPF
Then(2) andproduction(3) into c possibility
after some
weforyield thek 1  2 and, where  k , y   2 1k  (1 2   2 1 )e k , y and e k ,0y 0is the function obtained by solving
short) :following
calculated: as follows :PPF eventuallyc  T yefficiency, k is explicitly
the following equation with respect to k1 :


 

 

 

2 2

    
2
2 211 )k yT
y～

 1
 (
k



k


g
c
,
k
e. kk,,y～
1


y  .
2
12c 1
2
2 1 k1 ～   k ,ky, y   

 

2

Proof.

   (1)

We will apply the numerical method studied by Baierl, Nishimura and Yano (1998).
, where  k , y   2 1k  (1 2   2 1 )e k , y and e k , y is the function obtained by solving
Under Assumption 2, let us consider the problem (*) where the time index is dropped forthe following equation with respect to k1 :simplicity :

 

 



12  k1 y 21k  (12   2 1 )k1  .
2

2

1

 

2

1

2

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研 究 所 年 報■
Remark 1.

Note that the derived efficiency-units PPF is constructed such that the original-units

PPF at each period will be pulled back to the corresponding efficiency-units PPF by discounting witheach sector’s rate of TFP growth ( Ai (t ) ) as depicted by Figure 1, below: In Figure1, four PPF curvesof t-th and (t+1)-th periods are drawn. The original-units PPF and the corresponding efficient-units
PPF curves are drawn. The efficient-units PPF curves at t-th period will be constructed by discounting
back the original-unit PPF at each sector’s TFP growth rate along the each axis. The efficiency-units
PPF at (t+1) th period will be obtained by applying the same procedures to the original-units PPF at(t+1) th period.

The original-units PPF at ( t+1)c (t  1)

Discounting backThe efficiency-units PPF at t
The efficiency-units PPF at (t+1)

The original-units PPF at t
Discounting backy(t  1)

Figure 1. The Efficiency-units PPF

Now let us construct the following optimal growth model in terms of the per-capita
efficiency-units:-The Per-capita Efficiency-unit Optimal Growth ModelMax   t ～c (t )
t 0s.t. k (0)  k ,～y (t ), k (t )) (t 0,1,)c (t ) T(～～y (t )  (1   )k (t )  (1  g )k (t  1) 0 
(t 0,1,).

(5)

Kenjiro Ara Meets Frank Ramsey: Unbalanced Growth in A Neoclassical Optimal Growth Model

7

The objective function in terms of per-capita term is derived as follows: Let us rewrite theconsumption in terms of efficiency-units.
～c (t )
C (t )c (t )c (t )t
A0 (t ) L(t ) (1  a0 ) A0 (0) (1  a0 )t ,where I assume that A0 (0)  1 for simplicity.
Then the infinite discounted sum of consumptions2 will be rewritten in terms of efficiency-units
as follows :t (1  a0 )  ～

c
(
t
)
c (t )t ～ (1  r ) 
t 0
t 0




where



1  a0.
1 r

We will make the following additional assumption here:
Assumption3.

A0 (0)  1 and g  a0  r.

The accumulation equation (5) is constructed based on the efficiency-units PPF as explainedin Remark 3, where

K (t )K (t  1) (1  g ) K (t  1)and
 k (t )
 (1  g )k (t  1).
L (t )L (t )(1  g ) L (t )The following important remark is in order now :Remark 2. It is important to note that the accumulation equation (5) will be directly derived fromrewriting the following efficient-units accumulation equation constructed based on the efficiency-units
PPF by dividing both sides with L(t ) :～Y� (t )  (1   ) K (t )  K (t  1) 
0.

Observing Figure 5, the TFP growth effect will be annihilated by discounting the original-units
variables with each sector’s TFP growth rate, but the efficiency PPF could still expand outward
because of the capital accumulation itself. Indeed, the equation (5) will exhibit this process.
If x and z indicate initial and terminal capital stocks respectively, the reduced form utility
function V ( x, z ) and the feasible set D will be defined as follows:V ( x,
z ) u (T [(1  g ) z  (1   ) x, x])
typeof objectivefunction
was originallyused byused
Uzawa
22 ThisThis
typeof objectivefunction
was originallyby(1964).
Uzawa (1964).

8

研 究 所 年 報andD  ( x, z )  �   �  : T [(1  g ) z  (1   ) x, x]  0
where x,k (t ) andz k (t  1). Note that we eliminate the time index for simplicity.

Finally, the per-capita efficiency-units model will be summarized as the following standard
reduced form model, which have been studied in detail by Scheinkman (1976) and McKenzie(1986).
-The Reduced Form Model(**) Max   tV ( k (t ), k (t  1)) s.t. ( k (t ), k (t  1))  D for t  0 and k (0) k.
t 0Also note that any interior optimal path must satisfy the following Euler equation,
which exhibits an inter-temporal efficiency allocation condition :Vz (k (t  1), k (t ))  Vx (k (t ), k (t  1)) 0 for all t  0(6), where the partial derivatives mean thatV (k (t ), k (t  1))
V (k (t  1), k (t ))Vx (k (t ), k (t  1)) and Vz (k (t  1), k (t )) .
k (t )
k (t )
Note that under the differentiability assumptions, due to the envelope theorem, all the prices
will be obtained as the following relations:
～
～
～
d c
T ( y, k )T ( y, k )q ～ 
1, p 
,wq, and w0 
qc～ p ～y  wk
q

y
k
d c
～

, where we normalize the price of consumption good as 1.
Now we will define the optimal steady state as follows :Definition.



An optimal steady state path (OSS) k is an optimal path which solves the

reduced-form model (**) and satisfies k  
k (t ) k (t  1) for all t  0.

３．UNBALANCED GROWTH
In this section, based on the Cobb-Douglas technology, we will calculate the optimal steadystate numerically.

3．1．Existence of the OSS
Using the accumulation equation and let us defineV ( x, z)  T  (1  g ) z  (1   ) x, x where x  k (t ), z  k (t  1).
Then, the Euler equation will be derived as follows :Kenjiro Ara Meets Frank Ramsey: Unbalanced Growth in A Neoclassical Optimal Growth Model(1  g )9T    T T 0.
  (1   ) ～～ y  1 g   y k (7)Evaluating (7) along the OSS yields,   p  p (1   )  r  
 1 g (8)y   r  .
k1  1 ～
 
p 

(9)

From (2),From (8), p 


 m.
  
 r  (1  g )   (1   )(10)

 p 
from (9) and (10) yields,
 
r



Eliminating ～k1  1m y(11)Furthermore, substituting again (11) into (3) and using the facts that～y ( g   )k , wehave(1 2 ) 2 ( 1m) 12  2  (1 2   2 1 )m( g   )  2  k  


2

(12)



Solving (12) with respect to k , we have proved the following result.
Proposition 1. The OSS is numerically obtained as follows :
1

k

2

(1 2 )( 1m)where m.
1  2  (1 2   2 1 )m( g   )(1  g )   (1   )3．2．Unbalanced growth
Due to Proposition 1 and the accumulation equation (5),output of the capital good sector at the original-units value:～y ( g   )k  holds. Thus the～y (t ) can be expressed as～y (t ) (1  a )t A (0) ～y  .
1
1

This means that the per-capita output of capital good sector grows at its TFP growth rate
along the OSS path.

10

研 究 所 年 報～Furthermore, since c





～
～
～
t
 T y , k  holds, it follows that c (t ) (1  a0 ) A0 (0)c . Thus

the per-capita output of consumption good sector also grows at its TFP growth rate.
The above result is summarized as the following corollary:
Corollar y. The optimal consumption and capital output steady state paths, denoted byc(t )  and y (t )  respectively, are growing at its own TFP rate: a0 and a1 respectively.
This result can be illustrated as Figure 6, where each sector’s production function will shift at～T ( y , k  )
 ～
y

～
T ( y , k  )kits own TFP rate. Note that p  . Instead of using thoseand w 
expressions, to investigate the prices, it is more convenient to rewrite the prices in terms of theconventional rental-wage ratio used by Uzawa (1964). Note that the following relations concernedwith the rental-wage ratio, often denoted by  will hold:
 1
f 0 
f 1 
A0 (t )  f 0 k0 (t ),1  k0 (t )

A
(
t
)
f
k
(
t
),1
k
(
t
)



1
0
1
 k0 (t )  k1 (t ) 
w(t )


 (t ) 


f 0
f 1
w0 (t )A0 (t )A1 (t )
 k0 (t ) k1 (t )where k0 (t )
,





k0 (t )k (t )and k1 (t ) 1 .
 0 (t )1 (t )The technology
doesn’t affect!

Figure 2. Properties of the Unbalanced GrowthKenjiro Ara Meets Frank Ramsey: Unbalanced Growth in A Neoclassical Optimal Growth Model

Based on these relations, due to the fact that k is constant along the OSS, 

11is alsoconstant. Since the relative price of capital good: p (t ) is a function of  (t ) , it must be also
constant along the OSS.
Now we can define the Gross Domestic Products on the OSS (the OSS-GDP for short) as
follows:c  (t )  p  y  (t )  A0 (t )c  A1 (t ) p  y .

Since the growth rate of the OSS-GDP is expressed as the weighted average of the growthrate of both sectors :～
A0 (t )ca
The growth rate of the OSS  GDP  ～
 A (t )c  A (t ) p  ～y  0
1
 0

～


A1 (t ) p  ya
 A (t )～c   A (t ) p  ～y   11
 0

Thus the GDP grows at a certain constant rate of growth calculated as the weighted sum of
the growth rates of both sectors. Based on the above relation, we will easily demonstrate thefollowing proposition:
Propositiona1 ) thena1 ) .
Proposition 2.
2. If
If a0a>0 a1(aa10(<a0a
the theOSSOSS
growthrate
will
converge
totoa0a( 0a(1).
− GDP1), then― GDP
growthrate
will
converge,
Note that this proposition exhibits a sharp contrast to the result referred to as “Baumol’s costdisease”: the GDP growth rate will converges to that of the stagnated sector. On the contrary,
Proposition 2 implies that the growth rate of the GDP will converges to that of the non-stagnatedsector. Thus we have proved the anti-Baumol thesis.

４．LOCAL STABILITY
In this section we will show the local stability in the sense of saddle-point stability. By so doing,
we demonstrate that the corollary of Proposition 2 will take place at the limit.

4．1．Capital intensity
Let us first define the capital intensity of a sector.
Definition. (capital intensity) When 1 /  2  1 /  2 (1 /  2  1 /  2 ) holds , we may say thatthe consumption good sector is capital intensive (labor intensive) in comparison with the capital
good sector.
As we have examined in Takahashi, Mashiyama and Sakagami (2011), we have presented firm

12

研 究 所 年 報

evidence such that in any OECD countries the consumption good sector is capital intensive.
However, we will study both cases here.

4．2．Local stabilityBy expanding the Euler equation around k , the following characteristic equation will bederived: *Vxz  2  (Vzz   *Vxx)  Vzx 0

If Vxz  0, then we will yield * 2 

where

*


1 g

,

(Vzz   *Vxx )
Vzx
.


0 ．

Vxz
Vxz

(13)

The local stability will be determined by this equation. Let us solve (13) numerically.
Lemma 23. The characteristic equation (13) has the following two roots : 1  m(1   )  m(1   )( 2   2 )  2m( 2   2 )


,



m( 2   2 ) 1  1     2 1  m(1   )  m(1   )( 2   2 ) In other words,  is a root, then1 is also the root of the characteristic equation (13).

Proof. Linearizing the Euler equation (6) around the OSS yields the following well-known
characteristic equation (13).
By the facts that Vx  (1   )Tz  Tx and Vz 
Tz , we will obtain the followings :Vxx  (1   ) 2 Tzz  (1   )Tzx  (1   )Txz  Txx  (1   ) (1   )Tzz  Tzx    x  (1   )  Txz ,
 z


(14)

Vxz 
Vzx 
Tzz (1   )  Txz ,(15)

andandLiviatan(1972)
has proved
this property
in a generalmodel.
33 Levhari
Levhari
Liviatan(1972)
has proved
this property
in a optimal
generalgrowth
optimalgrowth model.

Kenjiro Ara Meets Frank Ramsey: Unbalanced Growth in A Neoclassical Optimal Growth Model

Vzz  Tzz ,

13

(16)

kk1
kk1


where







(




)
(




)


1 and 
1 .
2
1
1
2
2
1
1
2
2
1
x
z
where 
 2 1  (1 2   2 1 ) x and (1 2   2 1 )  ～
.
x
z
x
 yy
From (3),From (3),kk1

1

xx


y   (  ) 22 11  y
 22 ( )   xx and kk11and  y2((11 22 )) 2
 y

～



 2
y   (  ) 22 11  ((  )) 2 ～y  22 ( )  zz .
.
2
((
11
 22 )) 2

Then we may show :Then we may show : 2 1




 
x

T
T



Txx
Txzxz  x ,,
xx 
 zz

    1 
11  

T



Txz 
     
 zz ,,
1
2
1


  
2


xz

1



2

1



 2 ))m

 (
(
2 
m

T

Tzzzz 
Txzxz 2  2 ..

T


2
2

(17)
(17)

Some manipulations will yield
Some manipulations will yield





 xx
 22
 22





(
)
m
(









 zz (11 22   22 11 )m ( 22 m
 22 ))m

(18)
(18)

Substituting (17) and (18) into the equations (9) through (11) will yield
Substituting (17) and (18) into the equations (9) through (11) will yield
* 


*
 (1   )m( 2   2 )   2 [1  m(1   )] 
 *V
Vxx
xx   *  (1   ) m( 2   2 )   2 [1  m(1   )] 



V
m

 22 
 22 ))

Vxzxz
m(( 


, and
, and


V
m
Vzxzx m((  22 

 22 ))
.
.
 
(1
)
(
)
[1
(1
)]





V
m
m






2
2
2
xz
V
(1   ) m(   )   [1  m(1   )]
xz

2

2

2

Thus the Euler equation will be eventually rewritten as follows :Thus the Euler equation will be eventually rewritten as follows :m(  2   2 )  (1   ) m(   )   [1  m(1   )]      1 m(  2   2 ) ** 22    **  (1   )m( 22   22 )   22 [1  m(1   )]   0.
     
0.
m(  2   2 )  (1   )m( 2   2 )   2 [1  m(1   )]     1 
 






m
(


)
(1

)
m
(


)

[1
m
(1

)]
2
2
2
2
2
 
 


Thus we obtain the two roots of the equation (13).■
Thus we obtain the two roots of the equation (13).■

Based on Lemma 2, we can show the following proposition :Based on Lemma 2, we can show the following proposition :Proposition 2 . The following two properties hold :Proposition 3 . The following two properties hold :1) If the consumption sector is capital intensive, and   2 g  1 44 , then there exists
1) If the consumption sector is capital intensive, and   2 g  1 , then there exists ',1)
 '' 
00 suchsuch thatthat [[ ',1) implies
implies thatthat 
 11 
 1.
1.

2) If the capital good sector is capital intensive, then there exists2) If the capital good sector is capital intensive, then there exists [  '',1) implies that 0    1.

such that
 ''''  00 such that

Proof. Based on Lemma2, let us define the following functions :From the Penn World Table, we may observe that   0.25 and g  0.04 at most. Therefore this condition could

4

4be justified.
From the Penn World Table, we may observe that δ≅ 0.25 and ｇ≅ 0.25 at most. Therefore this4

condition
could
beTable,justiﬁed.
From
the PennWorld
we may observe that   0.25 and g  0.04 at most. Therefore this condition could

be justified.

14

研 究 所 年 報 [  '',1) implies that 0    1.
Proof. Based on Lemma2, let us define the following functions : ( )  2   1
 2 [1  m(  )(1   )]
 (1   )  (1   )   (1   )m(  )( 2   2 )
m
(
)





 2
2  

where m(  ) (1  g )   (1   )

(14)

.

Differentiating (14) yields, m '(  ) 
 2
. .
 2  2   m ( ) (15)(1  g )0,[(1  g )   (1   )]2(16)

 '(  )  

2

and

m '(  ) The second term of the right hand of (15) is positive.
Also note,





1

  g  2  1  (1   ). .．
 (1)   
 1   2    2  (17)

I will prove the proposition in two cases.
Case 1: The consumption sector is capital intensive.
In this case,  2   2 holds. Then we have the following result

 '(  )  0 and  (1)  0.
However, the sing of  (  ) is undetermined.  (1)  0 comes from the fact that
Thus there existsg  2  1 .

 '  0 such that [  ',1) implies that  1    1.

Case 2: The consumption sector is capital intensive.
Due to the definition, 2  2

holds. From (15) through (17), it follows that

 (  )  0,  '(  )  0 and  (1)  g   . The last result comes from the fact that1
 1.
 2 
1  
 2 
Therefor there exists

 ''  0 such that  [  '',1) implies that 0    1.

This completes the proof. ■Kenjiro Ara Meets Frank Ramsey: Unbalanced Growth in A Neoclassical Optimal Growth Model

15

From Proposition 2, there exists a ρ̂�  0 such that the characteristic equation (13) has one
root with its absolute value less than one and the other characteristic root with its absolute value�ρ̂ ,1) . Thus we have demonstrated the local stability.
greater than one for  [ �ρ̂  0 such that the
Proposition 4. Under the both capital intensity condition 5s, there exists a 
optimal steady state k�ρ̂,1) is locally stable in the sense of the saddle point .
for  [ Corollar y: Each sector’s optimal path in the neighborhood of the optimal steady state kfor  [ ρ̂� ,1)will converge to its own optimal steady state :
～
～
～ ～
(c (t ), y (t ))  (c , y ) as t   .

It follows that in terms of the original-units variables, sector’s per-capita capital stock and output
eventually grow at the rate of sector’s TFP growth:y  .
c  (t ) (1  a0 )t ～c and y  (t ) (1  a1 )t ～５．CONCLUTION
We have shown the existence of the unbalanced optimal steady state growth path and, bydemonstrating the saddle-point stability, the optimal path of each sector will converge to its own
unbalanced growth path with its TFP growth rate. Thus in the end, the sector with the higher
TFP growth rate will dominate at the aggregated GDP growth rate. This result shows a sharp
contrast to that of Baumol (1964).
Needless to say, the existence and the global stability of the efficiency OSS under theassumptions of general production functions among sectors should be left for a further research
problem.
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Baumol, W.J. (1967): “Macroeconomics of unbalanced growth: the anatomy of urban crisis, ” American
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Bosworth, B. and J. Triplett (2007): “Service productivity in the United States: Griliches’s service volume
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Echevarria, C. (1997): “Changes in sectoral composition associated with economic growth,” International

wewe
have
discussed
in Section
2, the2, empiricalsupports
the factthe
thatfact
the thatconsumption-goods
sector is capital
55 AsAs
have
discussed
in Section
the empiricalsupports
the consumption-goods
sectorintensive
among
OECD countries.
is capitalintensive
among OECD countries.

16

研 究 所 年 報
Economic Review 38, 431-452.
Kongsamut, P., S. Rebelo and D. Xie (2001): “Beyond balanced growth,” Review of Economic Studies 68,
869-882.
Levhari, D. and N. Liviatan (1972): “On stability in the saddle-point sense,” Journal of Economic Theory 4,
88-93.
McKenzie, L. (1983): “Turnpike theory, discounted utility, and the von Neumann facet,” Journal of
Economic Theory 30, 330-352.
Ngai, R. and Pissarides, C. (2007): “Structural change in a multisector model of growth,” American
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Oulton, N. (2001): “Must the growth rate decline? Baumol’s unbalanced growth revisited,” Oxford
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Scheinkman, J. (1976): “An optimal steady state of n-sector growth model when utility is discounted,”
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Syverson, C. (2011): “What determines productivity?,” Journal of Economic Literature 49:2, 326-365.
Takahashi, H., K. Mashiyama, T. Sakagami (2012): “Does the capital intensity matter?: Evidence from thepostwar Japanese economy and other OECD countries” Macroeconomic Dynamics,16,(Supplement1),103-116.
Triplett, J. and B. Bosworth (2003): “Productivity measurement issues in services industries: “Baumol’s
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Uzawa, H. (1962) : “On a Two-sector Model of Economic Growth,” Review of Economic Studies 14, 40-47.
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Studies 31, 1-24.

Kenjiro Ara Meets Frank Ramsey: Unbalanced Growth in A Neoclassical Optimal Growth Model

17

AppendixA
A
Appendix
Appendix
AppendixAA
Appendix
AAA
Appendix
Appendix
AppendixA
6
Letus
usreview
reviewthe
thetwo-sector
two-sectorgrowth
growthintroduced
introducedby
byAra
Ara(1969)(1969)
interms
termsofofour
ourmodelmodelnotation
notation
6 666 inLet
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usreview
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growthintroduced
byAra
Ara
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continuous-time
Uzawa-type
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hassetsetupup
a acontinuous-time
Uzawa-type
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continuous-time
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Ara(1969)
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usedin themodel.
(1969)
hasCobb-Douglas
set up a continuous-time
Uzawa-type two-sector non-optimal
growthmodel
withArathefollowing
followingproductionfunctions:functions:
growth
model
with
the
Cobb-Douglas
production
growth
model
with
the
following
Cobb-Douglas
production
functions:
growth
model
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the
following
Cobb-Douglas
production
functions:
growth
model
with
the
following
Cobb-Douglas
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functions:
growth
model
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the
following
Cobb-Douglas
production
functions:
growth
model
with
the
following
Cobb-Douglas
production
functions:
growth model with the following Cobb-Douglas production functions:a0CC((t t))AA
K 0(((t1tt)))1111LLL0(((t2tt)))2222
a00eaea0a00 K
C
a
a
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(
t
)
A
e
K

C

(
t
)
A
e
K
(
)0(t )L1 1L(t0)0(t ) 2 2

0
0
(
t
)
A
e
K

0
0 0K0t0(
C
C(((ttt)))0A
AA0000eeea0 K
K000((ttt)))10LLL0000(((ttt))) 2
 C a
K1(((t1tt)))1111LLL1(((t2tt)))2222
 YYY(((ttt)))AAaA1 e1eaaea1aa111K
K(YYt ()((tteK(1e)t))A
A
K
L(Lt11()(1(ttt)))222
1 11
1a
1111L
1
1
YY AAee 1 KKt11()(1(ttt)))LY (t )  A111e K111(t ) L111(t )Healsoalsomademadethethefollowing
followingtwo
twoextraextraassumptions;assumptions;(A)(A)andand(B)(B)otherotherthanthanours:
ours:
He
He
also
made
the
following
two
extra
assumptions;
(A)
and
(B)
other
than
ours:
HeHe
also
made
the
following
two
extra
assumptions;
(A)
and
(B)
other
than
ours:
He
also
made
the
following
two
extra
assumptions;
(A)
and
(B)
other
than
ours:
He
also
made
the
following
two
extra
assumptions;
(A)
and
(B)
other
than
ours:
also
made
the
following
two
extra
assumptions;
(A)
and
(B)
other
than
ours:
He
also
made thefollowing
extraexpectation.
assumptions; (A) and (B) other than ours:(A)
Economic
agentshavetwoastatic
static(A)
Economic
agentshave
expectation.
(A)
Economic
agentshave
staticexpectation.
(A)
Economic
agentshavea static
expectation.
(A)
Economic
agentshaveaaaastatic
staticexpectation.
(A)
Economic
agentshave
staticexpectation.
(A)
Economic
agentshaveaexpectation.
(A)agentsishave
aconsumed
static expectation.
(B)Economic
Laborincome
incomeonlyconsumed
andrental
rentalincome
incomeisissolely
solelysaved.
saved.
(B)
Labor
is
only
and
(B)
Labor
incomeisonlyonly
consumedandrentalincomeissolelysolelysaved.
(B)
Labor
incomeis
onlyconsumedandrentalincomeis issolelysaved.
(B)
Labor
incomeisconsumedandrentalincomeissaved.
(B)Labor
Laborincome
incomeis
isonlyonlyconsumed
consumedandrentalincomeissolelysolelysaved.
(B)andrentalincomesaved.
(B)
Labor
incomeis
onlyconsumedandrentalincomeissolelysaved.
Let usus definedefine GG((xx))(1/and thethe long-run
long-run equilibrium
equilibrium will
will bebe defined
defined asas thethe(1/xxx)()(dx
dx///dtdt))) and
Let
Let
us
define G
andthelong-runequilibriumwillbedefined
as
the
G
(1/
dx
dt
Let
usus
definethelong-runequilibrium
will
bebe
defined
asas
the
Let
us
defineandthelong-runequilibriumwillbedefinedasthe(Gx(()(x(xxx))))
(1/x)(xxx)(
dx
/ dt
)dt
(1/
)()()(dx
dx
dt
Let
us
defineGG
andthelong-runequilibriumwillbedefinedastheLet
definethelong-runequilibriumwill
defined
the
(1/
(1/
dx
) and
////dt)))and*as*Letusdefineandthelong-runequilibriumwillbedefined
the
G
(
x
)

(1/
x
)(
dx
dt
w*** stands
economicsituation
situationininwhich
which GG
holds,where
where*w
stands(K
K1(((ttt))
)) G
G
(K
K 2(((ttt)))) GG
K(((ttt)))) w
w*** holds,
economic

(1(K
(

K
economicsituation
inwhichwhich G
holds,
where
stands
G

G
(
K
G

((K
ww
economicsituation
inin
which
where
www***stands
economicsituationinholds,
where
stands(G
K(Kt11())
(GK
(2t2())
(G
K
tK())

(
(1(ttt))))))))GGG(K))))))G
G

(((K
K((ttt))))))))w*w
www***holds,
2((ttt))2K
economicsituationinwhichG
holds,
where
standseconomicsituationwhichholds,
where
stands
(
K
(

G

(
K
(
K
G

(
1
2
1
2
w
economicsituationinwhichholds,
where
standsGprofit( K1 (trate,)) whichG( K 2 (ist ))stable.
G( KThen(t )) hiswmodel in the long-run will beforthethelong-run
long-runequilibrium
equilibriumprofit
for
rate,
which
is
stable.
Then
his
model
in
the
long-run
will
be
for
the
long-runequilibrium
profitrate,
whichisstable.
stable.
Then
his
model
inthe
the
long-runwill
beforforthelong-runequilibrium
profit
rate,
which
is is
stable.
Then
his
model
inin
the
long-run
will
bebe
for
the
long-runequilibrium
profit
rate,
which
is
Then
his
model
in
long-run
will
be
for
the
long-runequilibrium
profit
rate,
which
is
stable.
Then
his
model
in
the
long-run
will
be
the
long-runequilibrium
profitrate,
whichstable.
Then
his
model
the
long-run
will
for
the
long-runequilibrium
profit
rate,
which
is
stable.
Then
his
model
in
the
long-runwill
beeventuallycharacterizedbythefollowingsevenequations:
eventuallycharacterizedbythefollowingsevenequations:
eventuallycharacterized
bythe
the
followingsevenequations:
eventuallycharacterizedbyby
thefollowingsevenequations:
eventuallycharacterizedbyfollowingsevenequations:
eventuallycharacterizedbythefollowingsevenequations:
eventuallycharacterizedthefollowingsevenequations:
eventuallycharacterizedbythefollowingsevenequations:a( i ) (i ) G (C (ta aaa0000 nn, ,i)i))G
t))
))
G((C
t

)))) 0aaaα
0CtC
n2 ,nn, ,(i )((i(G
(G
(C

))
0 
((t(

G(((C
t

))  0 2
nnn,,,(((iii)))G))
2
G
CC(((tt
))
22 2

22

a0 2
*
** a aaw(ii))*)w
 n,
*
000 
aa0
(ii)

* 0a
w
*
0
n,nnnnn,,,,,(ii((()(iiii(iiiiiiw
w
))

a

α
*
0 2
w


2


w
n,
(ii ))w


2

22 2
222
a
iii)))G
G((KK(((ttt))(iii))a0aaaaa0000 nnn,,,
iii
G
()iii
))
0
0
(iii)
G
K((((Kn,nnn,,,
(iii
(G

iii
G
KKtK())((((iii)))G((ttt))))
a2
02222n,(iii ) G ( K (t ))))α2n22,2
iv
G
L
t
(
)
(
(
))

iv
G
)G
iv
G
t))
(tLL))))n,nnn,,,G
L(((((LL(iv((((iv
)(iv
(G
iv
((t(t)))))))G(iv)
G(iv( LL(((ttt))))))nnn,,,aa aa0 0)G
G((Y
(Y(((ttt))
a1 0a1aa0 0 nnn,,,((vv())(v((vv))v)))GG))))

0
0
(G
Y
(YYtY())
an,n,G(t ))
aaaa111 11aα1a
0 2
(((vvvG((((Y
2nn,,)G
Y ((ttt))))))

a11 1111
2  n,1



1 2 2222aaaa111a2aaaa000 ,


a
(
vi
)
G
(

p
(
t
))
a
0 a0 ,
1 
vi
)G
G
((
vi
G
t))
))a aaa1111aaa1
0,
1 a
(vi((((vi
)(vi
G
(G

p((
tppp())vi(((tt(tt))))(vi)
2,,,,,0 ((a0)))))Gppp
22122(viG((t )))) 1aaa1111

2α

2222
222  2 a2a220(viivii)))GG((ww0(((ttt)))) aaa000 ...
(vii
aaa11
(vii
G

0
(vii
)vii
(G
(w0t0())
11101a1aa.0G
w((((0w
a1aaa
))G
))
))
G
ww
tt))
0((
1 1
(vii)))
((((vii)t0 ..2.
vii ) G ( w000(t )) a111 111α22222.

222

We maymay easily
easily notice
notice thatthat his
his model
model isis an
an unbalanced
unbalanced growth
growth model,
model, because
becauseWe
We
mayeasilynoticethat
hismodelmodel
an growth
unbalanced
growthmodel, because
becauseWe
mayeasilynoticethat hismodel
anunbalanced
model,
becausemay
easilynotice
that
his
is is
growthmodel,
Wemay
easilynotice
that
his
model
isisisanananunbalanced
growthmodel,
Wemay
easilynotice
that
hisis
model
anunbalanced
unbalanced
growthmodel,because
becauseWe
mayeasilynotice
that
his
model
unbalanced
growthmodel,
becauseaaWe
may
easilynotice
that
his
model
is
an
unbalanced
growthmodel,
because0 holds. Therefore each sector’sa
aaaa0000 GG(Y
G((CC(((ttt))))G
G ( L0(((ttt))(Y(((ttt))))G
G((LLL1(((ttt)) a aaa000holds.
0aholds.
Therefore
each
sector’sholds.
Therefore
each
sector’sG
aaa11
Therefore
each
sector’sholds.
Therefore
each
sector’s
G
G
))))

G
G(((Y
G
))))

0 holds.
GG
(G
C
(C
tC())
))))
G(G
L((0((L(L(LLL0t0())
0 2G(G
Y(Y(YtY())
))))
G(G
L((1((LL
t11())a111110a1aa0
G((CC((ttt))
))
G
G
))

G
((ttt))))
GG(1(ttt))
))

aa
a0
0((ttt))holds.
Therefore
each
sector’s
Therefore
each
sector’s
1a
1
0
0
))


L
))
(

2 holds.

Therefore
each
sector’s
α
1
1
1
1
1
1


2
2
G (C (t ))  G ( L000(t )) α


G
(
Y
(
t
))

G
(
L
(
t
))

a


  221  22
1
1
22

222

22

222

6
Ara(1969)(1969)waswaswritten
writtenonly
onlyininJapanese,
Japanese,very
verylittle
littleattention
attentiontotothe
thepaperpaperhas
hasbeenbeenmade.
made.
6 SinceAra
6 6 Since
Since
Ara(1969)
waswritten
only
inJapanese,
Japanese,verylittleattention
tothe
thepaper
paper
has
been
made.
Ara
(1969)
waswrittenonly
inJapanese,verylittleattention
to to
the
paper
been
made.
Ara
(1969)
waswritten
only
in
very
littleattention
to
has
been
made.
66 Since
Ara(1969)
waswrittenonly
inJapanese,verylittleattention
tohas
the
paper
has
been made.
66 Since
Since
Ara(1969)
waswritten
only
inJapanese,
Japanese,verylittleattention
tothe
the
paper
has
been
made.
Since
Ara
(1969)
waswritten
only
in
very
littleattention
paper
has
been
made.
6 SinceAra(1969)
waswrittenonly
inJapanese,verylittleattention
to
the
paper
has
been
made.

18

研 究 所 年 報

per-capita output grows at the sector-specific growth rate. However, note that his results show a
sharp contrast to the one we obtained here in three points; firstly, we have proved that in his
terms,G (C (t ))  G ( L0 (t )) 
a0  G (Y (t ))  G ( L1 (t )) a1 .
Secondly, we have shown that the long-run per capita capital total stocks are constant. On thea0 and the long-run per capita total capital
contrary, from (iii) and (iv), G ( K (t ))  G ( L(t )) α22is growing. Thirdly, in his model, all the prices grow as shown by (vi) and (vii). In our model
those are constant. Of course these differences come from his special model setting andassumptions.