A Reconciliation of Kaldor’s Stylized Facts and the Kuznets
Process: A Turnpike Approach
by
Harutaka Takahashi
Department of Economics
Meiji Gakuin University
(May, 2013)
Abstract
Kaldor (1961) found that aggregated labor share exhibits constant over time. On the other hand,
Kuznets (1965) observed the shift of population from traditional to modern activities. In other words,
although the macro-based observation exhibits the balanced evolution of aggregated labor share, the
industry-based observation shows the unbalanced evolution of industry labor’s share. We have
witnessed recent theoretical models attempted to reconcile of these contradictory facts. For example,
Echevarria (1997), Kongsamut, Rebelo and Xie (2001), Acemoglu and Guerrieri (2008) and Young
(2010) are among others. I will set up a neoclassical two-sector optimal growth model with technical
progresses based on the model studied by Uzawa (1964) and Ara (1969). I will apply the turnpike
theory developed by Scheinkman and McKenzie instead of Uzawa’s analytical method. I will
demonstrate that 1) there exists an optimal steady state (OSS) where each sector’ output grows at the
sector specific technical progress and 2) the optimal path with a sufficient initial stock will converge
to the OSS. The first result implies that the OSS exhibits Kaldor’s stylized facts, and the second one
implies that the optimal path will feature the Kuznets Process through its transition period.
Keywords: Kaldor’s stylized facts, the Kuznets process, unbalanced growth, neoclassical two-sector
optimal growth model, capital intensity conditions
JEL Classification: O14,O21,O24,O41
1. INTRODUCTION
Since the seminal paper by Kaldor (1961), the neoclassical growth models have
been applied as benchmarks and studied intensively Kaldor’s stylized facts: aggregated
labor’s share exhibits constant over time among others. On the other hand, in Economic
Development Theory, Kuznets (1965) observed the shift of population from the
traditional to the modern sectors, often referred to as the Kuznets process. The attempt
to reconcile those contradictory phenomena has generated a strong theoretical demand
for building the model to explain Kaldor’s stylized facts and the Kuznets process
simultaneously. However, those analytical models set up so far have a serious
drawback: they are based on highly aggregated macro-production functions and cannot
explain the Kuznets process. In facts, recent industry-based empirical studies among
countries clearly revealed 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 manufacturing industry,
also paralleling the industry's steady state. Syverson (2011) has recently reviewed these
arguments discussed above. Let us refer this phenomenon as "unbalanced growth
1
among industries." The attempt to understand this phenomenon has generated a strong
theoretical demand for constructing an unbalanced growth model, yet very little
progress has been made so far with the only exception of Ara (1969). Ara (1969) set up
Uzawa’s two-sector 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 determined by the sector-specific technical progress. Some
recent exceptions are Echevarria (1997), Kongsamut, Rebelo and Xie (2001), and
Acemoglu and Guerrieri (2008). However their models and analytical methods are
different from Ara’s and mine. Setting up an optimal growth model with three sectors:
primary, manufacturing and service, Echevarria (1997) has applied a numerical analysis
to solve the model. Kongsamut, Rebelo and Xie (2001) has constructed the similar
model to the one of Echevarria (1997), while they have investigated the model under a
much stronger assumption than her: each sector produces goods with the same
technology. On the other hand, Acemoglu and Guerrieri (2008) has studied the model
with two intermediate-goods sectors and single final-goods sector. Note that the last two
models will share a common character: consumption goods and capital goods are
identical. Contrastingly, Ara’s and my model presented here exhibits a sharp contrast
with them. Since we assume that each good is produced with a different technology,
2
consumption goods and capital goods are completely different physical characters. As I
will demonstrate later, this feature of the model will make the characteristics of the
model far complicated.
The optimal growth model with heterogeneous capital goods has been studied
intensively since the early 1970' under the title of turnpike theory by Scheinkman
(1976) and McKenzie (1983 and 1984). Turnpike theory shows that any optimal path
converges asymptotically 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 industry eventually converge to an industry-specific constant
ratio. Therefore, turnpike theory too, cannot explain the empirical phenomenon:
unbalanced growth among industries. McKenzie (1998) has articulated this point:
"Almost all the attention to asymptotic convergence has been concentrated on
convergence to balanced paths, although it is not clear that optimal balanced growth
path will exist. This type of path is virtually impossible to believe in, if the model is
disaggregated beyond the division into human capital and physical capital, and new
goods and new methods of production appear from time to time." An additional point is
that the turnpike result established in a reduced form model has not been fully applied to
a structural neoclassical optimal growth model. A serious obstacle in applying the
3
results from the reduced form model is that the transforming of a neoclassical optimal
growth model into a reduced form model will not yield a strictly concave reduced form
utility function, but just a concave one. In this context, McKenzie (1983) has extended
the turnpike property to the case in which the reduced form utility function is not strictly
concave, that is, there is a flat segment on the surface, which contains an optimal steady
state. This flat segment is often referred to as the Neumann-McKenzie facet. Yano
(1990) has studied a neoclassical optimal growth model with heterogeneous capital
goods in a trade theoretic context. However, in case of the Neumann-Mckenzie facet
with a positive dimension, Yano explicitly assumed the "dominant diagonal block
condition" concerned with the reduced form utility function (see Araujo and
Scheinkman (1978) and McKenzie (1986)). Thus, he still did not fully exploit the
structure of the neoclassical optimal growth model, especially the dynamics of the path
on the Neumann-McKenzie facet, to obtain the turnpike property.
By applying the theoretical method developed in turnpike theory, this study seeks
to fill the gap between the results derived by the theoretical research explained above
and the empirical evidence from recent studies at the industry level among countries.
First, I will set up the similar two-sector optimal growth model with the general
neo-classical production functions to Ara’s two-sector growth model which will be
4
discussed in Appendix A, in which each industry exhibits the Hicks-neutral technical
progress with an industry specific rate. Then I will study the model under the optimal
growth setting, which shows the sharp contrast to Ara’s analysis. Second, I will rewrite
the original model into a per capita efficiency unit model. Third, I will transform the
efficiency unit model into a reduced form model, after which the method developed in
turnpike theory will be applied. The first task is to demonstrate the existence of the
optimal steady state path (denoted by OSS for short). In the original unit, the OSS
implies that each sector will grow at its own TFP rate with the constant labor’s share.
Thus the OSS exhibits Kaldor’s stylized facts.
The second task is to show the global stability of the OSS: an optimal path will
converge to the OSS. Then, through the transition period, we observe that labor’s share
will change. Thus the global stability exhibits the Kuznets process. We will show that in
two steps: the neighborhood turnpike theorem and the local stability theorem. The
neighborhood turnpike theorem demonstrated in McKenzie (1983) indicates that any
optimal path will be trapped in a neighborhood of the corresponding optimal steady
state path when discount factors are sufficiently close to 1, and the neighborhood can be
minimized by choosing a discount factor arbitrarily close to 1. I will demonstrate the
local stability theorem by applying the logic used by Scheinkman (1976): a stable
5
manifold extends over today's capital stock plane. As we will see later, the dynamics of
the Neumann-McKenzie facet are important in demonstrating both the theorems.
Combining the neighborhood turnpike and the local stability produces the complete
turnpike property: any optimal path converges to a corresponding optimal steady state
when discount factors are sufficiently close to 1. For establishing both theorems, we
assume the capital intensity conditions. The complete turnpike property means that each
sector's optimal per-capita capital stock and output converge to its own steady state path
with the rate of technical progress determined by the industry's TFP.
The paper is organized in the following manner: In Section 2, I will provide a
several empirical facts, which are related to the two-sector model based on the recent
database at the industry level among countries. In Section 3, the model and assumptions
are presented and show the existence theorem of unbalanced growth. In Section 4, the
Neumann-McKenzie facet is introduced and the Neighborhood Turnpike Theorem is
demonstrated. The results obtained in Section 4 will be used repeatedly in the proofs of
main theorems. In Section 5, I show the global stability of unbalanced growth by
demonstrating the complete turnpike theorem. Some comments are given in Section 6.
2. SOME EMPIRICS
Since there are quite a lot of research works done for Kaldor’s stylized facts and the
6
Kuznets process, for recent examples, Young (2010) and, Anand and Kanbur (1993), I
will present here some additional empirical findings needed for setting up the
two-sector model later. In Takahashi, Mashiyama and Sakagami (2011), we have
empirically examined the Post-war Japanese economy and four other OECD countries
based on the two-sector growth model setting originally investigated by Uzawa (1962,
1963, 1964). Our findings are summarized as Figure 1 through Figure 5 1, where the
results of Japan, France, the US and West Germany are reported. Examining them, we
have found the following facts among others.
1.
The two-sector growth paths of the US and West Germany exhibit a sharp
contrast to those of Japan and France; The per-capita capital stocks and outputs
grow at different growth rates among sectors, but the growth rates of both sectors
converge to zero in the end.
2.
In Japan and France, through the observation period (1955-1995), per capita
capital stocks and outputs grow exponentially at different growth rates among
sectors. In facts, the per capita capital stock of the consumption sector grew
much faster than the investment sector.
3.
1
In other OECD countries than Japan, the capital-intensity reversal cannot be
These figures are drawn based on the data used in Takahashi et al. (2012).
7
observed and the consumption good sector is more capital intensive than the
investment good sector over the observation period (1970-1990).
4.
The per capita output paths of both sectors exhibit a co-integrated movement.
The first finding implies that each sector’s steady state path has a sector specific
positive growth rate. Following Baumol (1967), we may call this phenomenon
“unbalanced growth.” The second one exhibits the standard convergence results as
shown by Uzawa (1962, 1963, 1964). The third one implies that in the long-run, the
consumption sector generally turns out to be more capital intensive than the investment
sector.
Based on the empirical evidence examined above, we may therefore conclude that the
two-sector model that I will set up in Section 2 should satisfy the following three
important characteristics:
F1) the per capita capital stocks of the consumption and investment sectors grow at a
sector specific growth rate, which is determined by a sector’s TFP growth rate.
F2) the per capita output of both sectors grows in a co-integrated manner.
F3) along the steady state path, the consumption good sector is more capital-intensive
than the capital good sector.
In Section 3, I will set up a two-sector growth model which satisfies the above
8
findings.
3. THE MODEL AND THE ASSUMPTIONS
My model is a discrete-time two-sector optimal growth version of the model studied
by Ara (1969) 2 with the sector-specific Hicks-neutral technical progress, which will be
measured as the sector-specific total factor productivity.
The model is as follows:
∞
t
 1 
Max ∑ 
 c(t )
t =0  1 + r 
subject
to : K (0) K=
and L(0) L,
=
Y (t ) + (1 − δ ) K (t ) − K (t + 1) =
0,
(1)
C (t ) = A0 (t ) F 0 ( K 0 (t ), L0 (t )),
(2)
Y (t ) = A1 (t ) F 1 ( K1 (t ), L1 (t )),
(3)
L0 (t ) + L1 (t ) =
L(t ),
(4)
K 0 (t ) + K1 (t ) =
K (t )
(5)
, and the notation is as follows:
r
: a subjective rate of discount,
C (t ) ∈  +
: the total goods consumed at t ,
2
I will discuss Ara (1969) in Appendix A. Galor (1992) has extended a two-sector growth model into an
overlapping-generations two-sector model.
9
c(t ) ∈  +
Y (t ) ∈  +
: C (t ) / L(t ) ∈  + , �
K (t ) ∈  +
: the total capital goods at t,
K (0) ∈  +
: the initial total capital goods,
K i (t ) ∈  +
: the tth period capital stock of the ith sector,
F 0 (⋅) :  2+   +
: a production function of the consumption goods sector,
F 1 (⋅) :  2+   +
: a production function of the capital goods 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,
δ
: the depreciation rate,
Ai (t )
: the tth period Hicks neutral technical-progress of the ith
: the t th period capital output of the capital goods sector,
sector
, where i = 0 and i = 1 indicated the consumption goods sector and the capital goods
sector respectively.
We will make the following assumptions on the model:
Assumption 1.
10
1) The utility function u(・) is defined on  ++ as the following 3:
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.
3) 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 ai < 1 .
The condition 1) of Assumption 1 means that we follow the objective function used
in Uzawa (1964) 4 to place an emphasis on technology-driven structure change. Note
that 3) of Assumption 1 means that the sector specific TFP is measured by the sector
specific output-augmented technical progress (the Hicks-neutral technical progress),
which is externally given.
Assumption 2.
0,1)
1) All the goods are produced non-jointly with the production functions F i (⋅) (i =
which are defined on  2+ , homogeneous of degree one, strictly quasi-concave and
continuously differentiable for positive inputs.
=
K i 0=
or Li 0 .
2) Any good i (i = 0,1) cannot be produced unless
Dividing all the variables by Ai (t ) L(t ) , I will transform the original model into per
-capita efficiency unit model. Firstly, let us transform the capital goods sector’s
3
The objective function implies that the optimal solution will be the golden rule path studied by Phelps (1961), and
Uzawa (1964) has introduced this objective function for the first time.
4 Haque (1970) corrected the results of Uzawa’s general case, where no capital intensities are assumed.
11
production function as follows; dividing both side of Eq.(1.3) by A1 (t ) L(t ) , we have:
 K (t ) L (t ) 
Y (t )
= F1  1 , i  ,
A1 (t ) L(t )
 L(t ) L(t ) 
And similarly, for the consumption goods sector’s production function, we obtain
 K (t ) L (t ) 
C (t )
= F 0  0 , 0 .
A0 (t ) L(t )
 L(t ) L(t ) 
Now let us define the following normalized variables:
y (t )
=
=
k0 (t )
K1 (t )
Y (t )
C (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 )
Normalizing the production functions with respect to Ai (t ) L(T ) (i = 0,1) will yield
1
y (t ) f=
=
(k1 (t ), 1 (t )) and c (t ) f 0 (k0 (t ),  0 (t )).
Remark 1. Let us assume the following Cobb-Douglas production functions:
a1
a2
C (t ) A=
and Y (t ) A1 (t ) K1 (t ) β1 L1 (t ) β2
=
0 (t ) K 0 (t ) L0 (t )
where a1 +=
a 2 1 and β1 +=
β 2 1.
a1
a2
Then, we have c (t ) k=
=
and y (t ) k1 (t ) β1 1 (t ) β2 .
0 (t )  0 (t )
Using the normalized production functions will yield the following new accumulation
equation based on the new production possibility frontier:
y (t ) + (1 − δ )k (t ) − (1 + g )k (t + 1) =
05
5
Note that this accumulation equation is not derived from rewriting the original accumulation equation (1).
12
, where
K (t )
K (t + 1) (1 + g ) K (t + 1)
=
k (t ) and
=
=
(1 + g )k (t + 1).
L(t )
L(t )
(1 + g ) L(t )
I can also rewrite the objective function in terms of per-capita as follows: Substituting
the following relation into the objective function yields:
=
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 .
Finally, it follows that
t
∞
 (1 + a0 )  
c
t
=
r t c (t )
(
)
∑
∑
 (1 + r ) 
=t 0=
t 0


∞
, where
r=
1 + a0
.
1+ r
Now the original model can be totally rewritten as the per-capita constant
production frontier model as presented below:
-The Per-capita Constant Production Frontier Model∞
Max ∑ ρ t c (t )
t =0
s.t. k (0) = k ,
c (t ) = f 0 (k0 (t ),  0 (t )),
(6)
y (t ) = f 1 (k (t ),  (t )),
1
1
(7)
13
y (t ) + (1 − δ )k (t ) − (1 + g )k (t + 1) =
0,
(8)
 0 (t ) + 1 (t ) =
1,
(9)
k0 (t ) + k1 (t ) =
k (t ).
(10)
I may add the following extra assumption and prove the basic property below:
= 1 and 0 < a0 < r.
Assumption 3. A0 (0)
Remark 2. The value of ρ consists of two parameters; the rate of subjective discount
rate ( r ) and the rate of technical progress in consumption goods sector ( a0 ). Note that
the rate of population growth could be negative. For example, we may consider the case
−0.2, r =
0.2 and a0 =
0.1.
where g =
Lemma 1. Under Assumption 2, Equations (6)-(10) except Equation (8) are
summarized as the social production function
c (t ) = T ( y (t ), k (t ))
which is
continuously differentiable in the interior of  2+ and concave.
Proof.
Solving the following problem (*) we can derive the social production function
c (t ) = T ( y (t ), k (t )) :
 Max f 0 (k0 (t ),  0 (t ))


 s.t.

(*)  y (t ) = f 1 (k1 (t ), 1 (t )),
 (t ) +  (t ) =
1
1
 0
and

k (t ) + k1 (t ) =
k (t ).
 0
14
See in detail Benhabib and Nishimura (1979).■
Note that contrast to the fact that the original production possibility frontier:
c(t ) = T ( y (t ), k (t )) is shifting due to the TFP growth, c (t ) = T ( y (t ), k (t )) does not shift
and is constant.
Remark 3.
As explained in Appendix B, Baierl, Nishimura and Yano (1998) have
solved the above problem when the Cobb-Douglas productions are assumed and derived
the following result:
 α β
c  2 1
=
 ∆ k , y

( )
( )
α2

  k − e k=
, y  T y, k

 

( )
( )
( )
( )
, where ∆ k , y = a 2 β1k + (a1β 2 − a 2 β1 )e k , y and e k , y is the function obtained by
solving the following equation with respect to k1 :
( α1 β 2 )
β2
y [α β k + (α β − α β )k ]β2 .
k=
1
2 1
1 2
2 1 1
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 − δ ) x, x])
and
=
D
{( x, z ) ∈  + ×  + : T [(1 + g ) z − (1 − δ ) x, x] ≥ 0}
, where =
x k (t ) and=
z k (t + 1). Note that I will eliminate the time index for
simplicity.
15
Finally, the per-capita efficiency-unit 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 ∑ r tV (k (t ), k (t + 1))
t =0
s.t.
(k (t ), k (t + 1)) ∈ D for t ≥ 0 and k (0) =
k.
Also note that any interior optimal path must satisfy the following Euler
equation, which shows an intertemporal efficiency allocation condition:
Vz (k (t − 1), k (t )) + rVx (k (t ), k =
(t + 1)) 0 for all t ≥ 0
(11)
, where the partial derivatives mean that
=
Vx (k (t ), k (t + 1))
∂V (k (t ), k (t + 1))
∂V (k (t − 1), k (t ))
=
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:
∂T ( y, k )
∂T ( y, k )
d c
q==
−q
q
qc p y − wk ,
1, p =
, w=
, and w0 =+


∂k
∂y
dc
where we normalize the price of consumption goods as 1.
Definition. An optimal steady state path (OSS) k ρ is an optimal path which solves the
per-capita efficient unit model and k r =
k (t ) =
k (t + 1) for all t ≥ 0.
16
Due to the homogeneity assumption of each sector’s production function, it is often
convenient to express a chosen technology as a technology matrix. Now let us define
the technology matrix as follows:
 a00

A=
a
 10
=
a00
,where
a01 


a11 
0
k0
1
k1
=
, a01 =
, a10
and
=
a11
.
y
y
c
c
r
Assumption 4 (Viability). For a given r (≥ g ) , a chosen technology coefficient a11
r
satisfy 1 − (r + δ ) a11 > 0.
4. THE UNBALANCED GROWTH
4.1. Existence of the Unbalanced Growth
McKenzie (1983, 1984) has demonstrated the existence theorem for both optimal
and optimal steady state paths in the reduced form model. Applying a same logic as that
of McKenzie’s, we will prove the following existence theorem under our Assumptions 1
through 5.
McKenzie’s Existence Theorem: Under Assumptions 1 through 5, there exists an
optimal steady state path (OSS for short) k ρ for ρ ∈ (0,1] and an optimal path
{k
ρ
(t )} from any sufficient initial stock k (0) 6. Simultaneously, the normalized optimal
∞
A capital stock x is “sufficient” if there is a finite sequence (k(0),k(1),…,k(T)) such that x=k(0), (k(t), k(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),k(T+1)) ∈ D.
6
17
ρρ
consumption and capital output steady state paths, denoted by c and y respectively,
are determined.
Proof. We need to demonstrate that under Assumptions 1 through 3, all the conditions
listed below in Theorem 1 of McKenzie (1983) (see also McKenzie (1984)) will be
satisfied.
McKenzie’s Conditions:
1) V ( x, z ) is defined on a closed convex set D .
2) There is a η > 0 such that ( ξ, 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) There is ς > 0 such that x > ς impλies for any ( x, y ) ∈ D, z < λ x (0 < λ < 1).
5) There is ( x, z ) ∈ D such that ρ z > x.
It is straightforward to show that Assumptions 1 through 3 satisfy McKenzie’s
Conditions 1) through 4). We will show that Assumptions 4 and 5 satisfies Condition 5)
r
as follows: let us take x such that x = a11 y . Then,
z − r −1 x=
=
1  1− δ
y+
x − r −1 x
(1 + g )
(1 + g )
( )
1  
(1 + g )  r  
a11  y ,
1 + (1 − δ ) −
(1 + g )  
r 

18
=
>
=
=
( )
 r
1  
(1 + g )
(1 + r )  a11
1 + (1 − δ ) −
(1 + g )  
(1 + a 0 )

{
 
y,

( )} y ( from Assumption3),
r
1
1 + [ (1 − δ ) − (1 + r ) ] a11
(1 + g )
{1 + [(1 − δ ) − (1 + r )]( a )} y
(1 + g )
1
r
11
{
( )} y > 0. (by Assumption 4)
r
1
[1 − (r + δ )] a11
(1 + g )
Thus y will be chosen so that z − ρ −1 x ≥ 0 where ( x, z ) ∈ D . This completes the proof.■
4.2. Properties of the OSS
The first important proposition is that the each sector grows at its TFP rate, which is
summarized as the following corollary.
Corollary: The optimal consumption and capital output steady state paths, denoted by
c(t ) ρρ
and y (t ) respectively, are growing at its own TFP rate: a0 and a1 respectively.
y ρ
=
Proof: Note that since
ρ
Similarly,
c
=
ρ
Y ρρ
(t )
y (t )
=
, it follows that y ρ (t )= (1 + a1 )t A1 (0) y .
A1 (t ) L(t ) A1 (t )
ρ
C ρρ
(t )
c (t )
holds. It also follows that c ρ (t )= (1 + a0 )t A0 (0)c .
=
A1 (t ) L(t ) A1 (t )
Hence, the original series of the sector’s optimal per-capita output is growing at the rate
of its own sector’s TFP growth rate ai (i = 0,1). ■
This result can be illustrated by Figure 5, where each sector’s production function
19
ρρ
ρρ
ρρ
∂T ( y , k )
∂T ( y , k )
will shift at its own TFP rate. Note that p =
.
−
and w =
∂k
∂ y
Instead of using those expressions, to investigate the prices, it is more convenient to
rewrite the prices in terms of the conventional expressions used by Uzawa (1964). Note
that the following relation concerned with the rental- wage ratio, often denoted by ω will
hold:

 1
∂f 0 
∂f 1 
A0 (t )  f 0 k0 (t ),1 − k0 (t )
−
A
t
f
k
t
k
t
(
)
(
),1
(
)



1
0
1
∂ k0 (t ) 
∂ k1 (t ) 
w(t )


=
=
w (t ) =
∂f 0
∂f 1
w0 (t )
A0 (t )
A1 (t )
∂ k0 (t )
∂ k1 (t )
(
where k0 (t )
,=
)
(
)
k0 (t )
k1 (t )
=
and k1 (t )
.
 0 (t )
1 (t )
Based on those relations, the fact that k ρ is constant along the OSS implies that ω ρ
is also constant. Since the relative price of capital goods: p (t ) is a function of ω (t ) , it
must be also constant along the OSS.
Now I can define the OSS-GDP (Gross Domestic Products on the OSS) as follows:
ρρ
c ρρρρ
(t ) + p y (t ) = A0 (t )c + A1 (t ) p y .
Suppose that both sector’s TFP growth paths are co-integrated 7, which is obtained from
the fact F3) by our observations of the data, then the GDP grows at a certain constant
rate of growth calculated as the combination of the growth rates of both sectors.
7
On
Consider two series A0 (t ) and A1 (t ) , and a liner relationship exists between them, it is said that A0 (t ) and A1 (t )
are co-integrated and they share a common trend.
20

wρ 
the other hand, since the OSS rental-wage rate  w ρ = ρ  and the OSS capital-labor
w0 

ratio ( k ρ ) are constant, the labor’s income share can be expressed as follows:
w0r (t )
[ A0 (t ) + A1 (t )]w0r
(1 + γ ) A1 (t ) w0r
=
=
,
r
r
r
r
r  

c r (t ) + p r y r (t ) A (t )c + A (t ) p r y

A1 (t ) c + γ p y
0
1


Thus, if both sector’s TFP growth paths are co-integrated, the labor’s income share is
constant as Kaldor (1957) advocated. Therefore, we may conclude that the growth
model studied here will successfully exhibit Kaldor’s facts.
Remark 4.
When the two-sector Cobb-Douglas production function case, as shown in
Appendix B, I can demonstrate that facts easily. By applying the numerical method used
in Baierl et. al (1998) to our model I will obtain the OSS numerically as follows:
1
kr
β2
(α1β 2 )( β1m)
r
where m
.
=
(
)
(
)
(1
)
(1
)
m
g
g
β1 [αα
+
β
−
α
β
+
δ
+
−
r
−
δ
]
2
1 2
2 1
ρ
From the accumulation equation, it follows that y= ( g + δ )k ρ holds. Thus the output
ρ
of the capital good sector at the original unit value: y (t ) can be expressed as
y ρ (t ) =
(1 + a1 )t A1 (0)( g + δ )k ρ .
This means that the output of capital goods sector grows at its TFP rate along the OSS
path.
< Insert Figure 6 here >
5. THE GLOBAL STABILITY
21
5.1 The Euler Equations and Capital Intensity Conditions
Suppose that k ρ is an interior OSS in the efficiency unit with a given ρ , it must also
satisfy the Euler equation:
Vz (k ρρρρ
, k ) + ρVx (k , k ) =
0.
Note that the following relations hold:
Vx (k ρρρρρρρ
, k ) =p (1 − d ) + w , and Vz (k , k ) =−(1 + g ) p .
Substituting above relations into the Euler equation will lead to the following equation:
ρ[ wρρρ
+ p (1 − δ )] − (1 + g ) p =.
0
We will not prove the following important theorem formally, but it has been well
established by applying the non-substitution theorem in a multi-sector model by
Takahashi (1985).
Lemma 2. When ρ ∈ (0,1] , there exists a unique OSS; k ρ >0 with the corresponding
> 0 and w > 0.
unique positive prices p ρρ
From this lemma, along the OSS with ρ , the nonsingular technology matrix Aρ will
be chosen, and the cost-minimization and the full-employment conditions will be
expressed as follows:
(1, p ρρρρρρρρ
) (=
w0 , w1 ) A and (1, k ) A (c , y ).
=
If Aρ has an inverse matrix B ρ , solving above equations yields,
22

 a
a
1
p= w  a −
a a  + = w (b ) +
,
a00
a00

 a00
ρρ
ρρρρρρρ
−1
01
01
11
10 01
11
ρρρ
and

 ρρ
a ρρ
a
1
k ρρρρρ
a10 a01  y + 01 = (b11 ) −1 y + 01 ,
=  a11 − ρρρ
a00
a00
a00


where b11ρ is the element of the matrix B ρ defined as follows:
 b00ρρ
b01 
−
1


B ρρ
A )
=
(=

.
ρρ
 b10
b11 

From now on, we are concentrated on the OSS with ρ = 1 denoted by k * . We will also
use the symbol “*” to indicate the elements and variables evaluated at k * .
Definition (Capital-intensity Conditions). When a11ρρρρ
/ a01 < a10 / a00 is established, the
consumption goods sector is capital intensive in comparison with the capital goods
sector. For a11ρρρρ
/ a01 > a10 / a00 , the consemption goods sector is labor intensive in
comparison with the capital goods sector.
As I have examined in Section 2 based on Takahashi, Mashiyama and Sakagami
(2011), I have exhibited firm evidence that in any OECD countries the consumption
goods sector is capital intensive. Based on this finding, I will make the following
assumption:
Assumption 6. a11ρρρρ
/ a01 < a10 / a00
.
23
Note that under Assumption 6, it follows that
1
(b ρρρρ
) −1 =
a11 − ρ a10 a01 < 0.
a00
Now we will show the following Lemma 5:
Lemma 5. Under Assumption 5, there exists a positive ρ such that for ρρ
∈ [ ,1], the
OSS k ρ is unique and is a continuous function of ρ , namely k ρ = k ( ρ ).
Proof. From the Euler equation, its Jacobian can be calculated as follows:
J (k , ρρρ
) = Vxx (k , k ) + Vxz (k , k ) + Vzx (k , k ) + Vzz (k , k ).
Evaluating it at k * , where k * is k ρ when ρ = 1 , yields
J (k * ,1) = Vxx (k * , k * ) + Vxz (k * , k * ) + Vzx (k * , k * ) + Vzz (k * , k * ).
Applying the same logic used in Takahashi (2011), I eventually obtain
J (k * ,1) = [b* − ((1 + g )(1 + α1 ) + (δ − 1)]2 [(b* ) −1 ]2 Tzz
= [b* − ( gα1 + δ )]2 [(b* ) −1 ]2 Tzz < 0 (≠ 0)
,where Tzz is an element of the Hessian matrix of the social transformation
function T ( x, z ) and negative. Thus the result follows due to the Implicit
Function Theorem.■
5.2. The Neumann-McKenzie Facet and Turnpikes
Now I will introduce the Neumann-McKenzie Facet (denoted by “NMF” for short),
which plays an important role in stability arguments regarding neoclassical growth
models as studied in Takahashi (1985) and Takahashi (1992), and has been intensively
24
studied in the reduced-form models by McKenzie (1983). The NMF will be defined in
the reduced form model as follows:
Definition. The Neumann-McKenzie Facet of an OSS, denoted by F (k ρρ
, k ) , is defined
as:
{
ρ
}
F (k ρρρρρρρρ
, k ) =( x, z ) ∈ D : c + ρρ
p z − p x =c + p k − p k ,
where k ρ is the OSS and p ρ is the supporting price of the OSS when a subjective
discount rate ρ is given.
Due to this definition, if a path will happen to deviate from the NMF, then a
value-loss will take place, which will be defined as follows:
Definition. The value-loss δ is defined as,
r
δ =(c + r p r z − p r x) − (c + r p r k r − p r k r ), for ( x, z ) ∈ D.
By the definition above, the NMF is a set of capital stock vector ( x, z ) which arise
from the exact same net benefit as that of the OSS when it is evaluated by the prices of
the OSS. Also, the NMF is the projection of a flat segment on the surface of the utility
, ρ p ,1) onto the ( x, z ) -space. In
function V that is supported by the price vector (− p ρρ
the two-sector model, we can examine the NMF diagrammatically. Based on the
assumption that the consumption goods sector is capital intensive, it is possible to draw
a graph on the coordinates ( y (t ), c(t )) , which is often used in trade theory.
25
<Insert Figure 7 here>
ρρ
Here, ( y , c ) is a production vector corresponding to the OSS and is written as a
point of intersection for the labor-constraint line and the capital-constraint line. Also, the
facts that the labor-constraint line intersects the capital-constraint line from the above is
due to the capital intensity assumed above. Note that production specialization occurs at
points A and B. At point A, no capital goods are produced. Thus, the economy is not
sustainable. On the other hand, at point B, no consumption goods are produced. Due to
Assumption 1, c(t ) > 0 for t ≥ 0 . Therefore, points A and B will be excluded from the
NMF. Now, suppose that k (t ) (< 1/ a10ρ ) , which is greater than OSS, {k ρ } were given.
Then, if we leave the capital-constraint line and the price vector as they are, then move
upward along the labor-constraint line, a new point of intersection “E” is obtained. Also,
the corresponding production vector ( y (t ), c(t )) is obtained at point E. Also, by
substituting this value for accumulation equation (1.2), the next period’s capital stock
k (t + 1) can be attained. The capital stock pairs (k (t ), k (t + 1)) obtained in this manner
can be plotted as point E on plane ( x, z ) . By further altering k (t ) and repeatedly
conducting the similar procedure, line AB can be drawn on plane ( x, z ) in the manner
of figure 6. Now, it can be understood that labor-constraint line AB on the production
plane ( y , c) directly corresponds to line AB on plane ( x, z ) . The portions of this line
26
AB excluding the ends are the von Neumann-McKenzie facet as dipicted in Figure 6.
<Insert Figure 8 here>
and p are continuous functions of ρ as shown in Lemma 5, based on the
Since k ρρ
above discussion, we may observe the following:
F ( k ρρρρ
, k ) 1 , F ( k , k ) ⊂ int D and it is a
Lemma 6. The NMF satisfies that dim=
continuous correspondence of ρρ
∈ [ ,1).
dim F ( k ρρρρ
, k ) 1 and F ( k , k ) ⊂ int D
Proof. Based on the observation of Figure 6, =
are straightforward. To show the continuity, we need to demonstrate that F ( k ρρ
, k ) is
not only upper-semi continuous but also lower-semi continuous. Due to the definition
of the facet, the NMF is upper-semi continuous. Therefor we only need to establish that
F ( k ρρ
, k ) ≡ F ( k ( ρ )) is lower-semi continuous at ρρ
∈ [ ,1). Let us arbitrarily choose a
point ( x, z ) ∈ F (k ( ρ )) . Furthermore choose a point ( x ', z ') such that ( x ', z ') ∈ F (k ( ρ ))
and ( x ', z ') ≠ ( x, z ) . Since the facet is a line segment in the two dimensional space, we
can express the point ( x, z ) by a linear combination of two independent vectors
, k ) such that
( x ', z ') and (k ρρ
=
( x, z ) αα
(k ρρ
, k ) + (1 − )( x ', z ')
, where α > 0 is a constant value.
Now let us choose ρ ' close enough to ρ and construct the following sequence:
27
ν
ν
ν
ν
=
( xν , zν ) αα
(k r , k r ) + (1 − )( x , z ),
ν
ν
ν
ν
where ( x , z ) ∈ F (k r , k r ) .
Since the three points expressed by ( xnnρρ
, z ), ( x ', z ') and (k , k ) are collinear due to our
ν
ν
way of construction, thus ( xν , zν ) ∈ F (k r , k r ) for all ν > 0 . Therefore, as ν → ∞,
ν
ν
( x , z ) → ( x ', z ') and ( xν , zν ) → ( x, z ). We can apply the same logic for any ρρ
∈ [ ,1).
This completes the proof. ■
This property will guarantee that if a path will be away from a ε -neighborhood of
the NMF, then there exists a δ such that δ ≥ δ > 0 . δ is referred to as the uniform value
loss .
Based on the above discussion again, it is possible to redefine NMF in a neoclassical
model as follows.
Definition (Characterization of NMF).
F ( k ρρ
, k ) ≡ {( k (t ), k (t + 1)) ∈ D : There exist c(t ) ≥ 0 and y (t ) ≥ 0 such that they satisfy
following conditions (1) through (5): =
(1) 1 w0ρρρρ
a00 + w a10 , (2)
=
p ρρρρρ
w0 a01 + w a11 , (3)
=
k (t ) a01ρρ
c (t ) + a11 y (t ) , (5) k (t + 1)= y (t ) + (1 − δ )k (t ))},
=
1 a00ρρ
c(t ) + a10 y (t ) , (4)
where the consumption good’s price is normalized as 1; also, for simplification, the
population growth rate has been postulated as zero.
Equations (1) and (2) are cost minimization conditions; equations (3) and (4) are
equilibrium conditions for labor and capital goods. Also, (5) is a capital good
28
accumulation equation.
Based on (3) and (4),
y (t ) bk (t ) + b ρ
=
01
(12)
Here, b ρ and b10ρ are defined before as elements of the matrix B ρ . Also, based on
accumulation equation (5) and equation (12), we obtain the following difference
equation:
k (t + 1) = [b ρρ
+ (1 − δ )]k (t ) + b10 .
(13)
ρ
By defining η (t + 1)= k (t ) − k , difference equation (3.2) is rewritten as
η (t + 1) = [b ρ + (1 − δ )]η (t )
(14)
It is clear that the behaviors of the path on the NMF can be obtained by investigating
this differential equation.
Now, by making a suitable selection of units, it is possible to normalize the element
b ρ of the matrix B ρ as follows.
ρρρρρρ
b=
a00 /( a00a11 − a01a10 ) < 1 .
Since b ρ < 0 , it follows that −1 < b ρ + (1 − δ ) < 1 . Therefore, the difference equation (14)
will exhibit stability and the NMF will become a linear stable manifold. This result is
consistent with the one of Uzawa (1964) when the consumption goods sector is capital
intensive.
29
Let us introduce the following definition:
Definition. The NMF is stable if there are no cyclic paths on it.
Thus we have proved that the NMF is stable and we may prove the following turnpike
property, which I leave for McKenzie (1983).
Theorem 1(The Neighborhood Turnpike Theorem): Provided that the NMF is stable.
Then for any ε > 0 , there exists a ρ > 0 such that for ρρ
∈ [ ,1) and the corresponding
ε ( ρ ) , any optimal path k ρ (t ) with a sufficient initial capital stock k (0) eventually lies
in the e − neighborhood of k ρ . Furthermore, as ρ → 1, ε ( ρ ) → 0.
The neighborhood turnpike theorem implies that any optimal path should be trapped
in a neighborhood of the corresponding OSS and the neighborhood can be taken as
small as possible by making ρ sufficiently close to 1.
Note that by expanding the Euler equation around k ρ , we have the following
characteristic equation:
Vxzρρρρ
0.
λ 2 + (Vxx + Vzz )λ + Vzx =
(15)
The local stability will be determined by this equation and the following property
concerned with it is well-known.
Lemma 7.
Provided that Vxzρ ≠ 0 , the characteristic equation (15) has λ as a root,
then it also has
1
ρλ
.
30
Proof. See Levhari and Liviatan (1976).■
Remark 6. As shown in Appendix B, when the two-sector Cobb-Douglas production
function is assumed, then we can apply the calculation method of Baierl, et al. (1998)
and numerically derive the following two roots:

α [1 − m(1 − δ )] + m(1 − δ )(α 2 − β 2 )
λ = 2
,
m(α 2 − β 2 )




m(α 2 − β 2 )

 1 = 1 


 ρλ ρα
−
−
+
−
−
δ
δ
α
β
1
(1
)
(1
)(
)
m
m
[
]

2
2
2



, where m =
ρ
1 − ρ (1 − δ )
. Thus we can easily confirm Lemma 7 when the
Cobb-Douglas productions are assumed.
Since the NMF is stable, it implies that the one characteristic root along the NMF has
an absolute value less than 1. In the two-sector model, V12ρ is calculated as follows (see
in detail Benhabib and Nishimura (1985)).
 ∂w 
=
−((b ) −1 ) 2 
Vxzρρρ
 b + (1 − δ ) 
 ∂k 
(16)
Based on this, Equation (16) does not become zero other than in b ρ =−(1 − δ ) . By
assuming that b ρ ≠ −(1 − δ ) , we can apply the Lemma and obtain that the both
characteristics of Equation (15) have absolute values less than 1. This implies that the
ρ
OSS k is locally stable; any path near the OSS will converge to the OSS. Thus we
have proved the following local stability concerned with the OSS:
31
Theorem 2 (the Local Stability). The OSS k r for r ∈ [ r ,1) satisfies the local stability.
Proof. Suppose that the optimal path k ρ would satisfy the neighborhood
turnpike theorem; for any ε > 0 , there exists a ρ such that for ρρ
∈ [ ,1) and the
corresponding ε ( ρ ) , any optimal path k ρ (t ) with a sufficient initial capital stock
k (0) = k eventually lies in the e − neighborhood of k ρ and ρ → 1, ε ( ρ ) → 0. By
choosing ρ close enough to 1 such that the local stability will also hold. ■
We have now proved the following complete turnpike theorem:
The Complete Turnpike Theorem: There is a ρ > 0 �close enough to 1 such that for
any ρρ
∈ [ ,1) � , an optimal path k ρ (t ) � with the sufficient initial capital stock
ρ
k (0) = k �will asymptotically converge to the optimal steady state k �.
Converting all variables into the original units, we have proved the following
corollary:
Corollary: Each sector’s optimal path will converge to its own optimal steady state:
ρρρρ
(c (t ), y (t )) → (c , y ) as t → ∞ . It follows that in original model of series, sector’s
per-capita capital stock and output grow at the rate of sector’s TFP growth:
ρρ
c ρρ
(t ) =
(1 + a 0 )t c and y (t ) =
(1 + a1 )t y .
Note that our original purpose, stated below as the proposition, have accomplished by
demonstrating the complete turnpike theorem.
32
Let us consider the transition process of the optimal path. Since each sector’s optimal
path eventually converges to the optimal steady state path which grows at the growth
rate of its own technical progress and the labor’s share is constant. It is not difficult to
see that during the transition period, the labor’s share will change. For example, if the
initial point is near Point B of the NMF. Then the optimal per-capita capital will
increase along the NMF, which is the stable manifold. Thus the labor’s share will
change during the transition period. In other words, we observe the labor shifting
between sectors as Kuznets (1965) advocated.
Proposition (Global stability of the unbalanced growth).
Under our assumptions, each sector’s optimal path with a sufficient initial stock will
converge to its own optimal steady state with a sector-specific TFP growth rate.
Furthermore the transition process of the optimal path will exhibit the Kuznets process.
6. CONCLUTION
We have shown the existence of the unbalanced optimal steady state growth path
and, by demonstrating turnpikes, the global stability of the unbalanced growth path in
the neo-classical two-sector optimal growth model. By so doing, I have shown that the
model presented here uniformly exhibit not only Kaldor’s facts but also the Kuznets
process.
33
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1955-1973 and the Era of High-speed Growth end after the Oil-shock?: Measuring
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16, Supplement S1, 103-116.
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Economic Studies 14, 40-47.
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Economic Studies 19, 105-118.
Uzawa, H. (1964) "Optimal Growth in a Two-Sector Model of Capital Accumulation, "
Review of Economic Studies 31,1-24.
Yano, M. (1990) "Von Neumann facets and the dynamic stability of perfect foresight
equilibrium paths in Neo-classical trade models," Journal of Economics 51, 27-69.
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stable?, "Journal of Macroeconomics 32, 90-102.
35
APPENDIX A
Let us discuss the two-sector growth model presented by Ara (1969) 8 in terms of my
model notation used here. Ara (1969) has set up the continuous Uzawa-type two-sector
non-optimal growth model with the following Cobb-Douglas production functions:
C (t ) = A0 e a0 K 0 (t )a1 L0 (t )a 2



a1
β1
β2
Y (t ) = A1e K1 (t ) L1 (t )
He also made the following two extra assumptions other than mine:
(A) Economic agents have a static expectation.
(B) Labor income is only consumed and rental income is solely saved.
Let us define G ( x) = (1/ x)(dx / dt ) and the long-run equilibrium will be defined as the
economic situation in which G=
( K1 (t )) G=
( K 2 (t )) G=
( K (t )) w* holds, where w*
stands for the long-run equilibrium profit rate, which is stable. Then his model in the
long-run will be finally characterized by the following seven equations:
a0
(i ) G (C (t=
))
*
(ii ) w=
a0
a2
a2
+ n,
+ n,
(iii ) G ( K (t=
))
a0
a2
+ n,
(iv) G ( L(t )) = n,
8
Since Ara (1969) was written only in Japanese, very little attention to the paper has been made.
36
(v) G (Y (t )) =+
a1 β1
a0
a2
+ n,
a a 
(vi ) G (=
p (t )) a1  1 − 0  ,
 β2 a 2 
a
(vii ) G ( w0 (t ))= a1 + β1 0 .
a2
We easily understand that his model is the unbalanced growth model, because
a0
a
≠ G (Y (t )) − G ( L1 (t )) =
G (C (t )) − G ( L0 (t )) =
a1 + β1 0
a2
a2
holds. Thus each sector’s
per-capita output grows at the sector-specific growth rate. However, note that his results
show a sharp contrast to mine 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, I have shown that the long-run per capita capital total stocks are constant. On
a0
and the long-run per capita
the contrary, from (iii) and (iv), G ( K (t )) − G ( L(t )) =
a2
total capital is 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 and assumptions.
37
APPENDIX B
In this appendix, I will derive the important results in the paper, when the
Cobb-Douglas productions are assumed by applying the method developed by Baierl,
Nishimura and Yano (1998).
B1.
( )
The derivation of the social production function: c = T y, k
We have to solve the following problem (*):
 Max c = k0 (t )a1  0 (t )a 2


 s.t.

(*)  y (t ) = k1 (t ) β1 1 (t ) β2 ,
 (t ) +  (t ) =
1
1
 0
and

k (t ).
k0 (t ) + k1 (t ) =
The profit-maximization of both sectors will yield the following first order conditions:
r α11 β1 2
.
= =
w α 2 k1 β 2 k2
Solving the above equation with respect to 1 and substituting into  0 (t ) + 1 (t ) =
1 , we
have
1 =
α1β 2 k1
α 2 β1k + (α1β 2 − α 2 β1 )k1
(a1)
Substituting (a1) into y (t ) = k1 (t ) β1 1 (t ) β2 yields
y [α β k + (α β − α β )k ]β2
2
k1 (α1β 2 ) β=
2 1
1 2
2 1 1
(a2)
( )
Let us define ∆ k , y ≡ α 2 β1k + (α1β 2 − α 2 β1 )k1. Solving (a2) with respect to k1 , we
may finally obtain
( )
k1 = g k , y
(a3)
1
Substituting (a1) and (a2) into c = k0 (t )αα
 0 (t ) 2 and after some manipulations, finally
38
we have the following social production function:
 α β

c  2 1
=
 ∆ k , y

α2




k − g ( k , y ) ) .
(
( )
B2.
(a4)
Derivation of the OSS: k ρ
Using the accumulation equation and let us define
V ( x, z =
) T [ (1 + g ) z − (1 − δ ) x, x ] where x= kt , z= kt +1.
Then, the Euler equation can be expressed as follows:
(1 + g )
∂T  ρ  
∂T ∂T 
+
−(1 − δ )
+
0.

=

∂ y  1 + g  
∂ y ∂k 
(a5)
Evaluating (a5) along the OSS will yields,
 r  r
r
pr

  p (1 − δ ) + r  =
+
1
g


(a6)
From (a1),
r
r  r 
k1r = β1 y  r 
p 
(a7)
From (a6),
 pr 
r
=
≡m
 r 
 r  (1 + g ) − r (1 − δ )
(a8)
 pr 
Eliminating  r  from (a8) and (a7) yields,
r 
ρ
k1ρ = β1m y
(a9)
ρ
Furthermore, substituting again (a9) into (a2) and using the facts that y= ( g + δ )k ρ ,
we have
(α1β 2 ) β2 ( β1m)= β1β2 [αα
2 + ( 1 β 2 − α 2 β1 ) m( g + δ ) ]
β2
(k )
ρ β2
(a10)
Solving (a10) with respect to k ρ , we have the result.
39
B3: The Local Stability of OSS
Linearizing the Euler equation around the OSS yields the following well-known
characteristic equation:
r *Vxzr λ 2 + (Vzzr + r *Vxxr=
)λ + Vzxr 0
r*
=
where
r
1+ g
,
or
(Vzzρρρ
+ ρ *Vxx )
V
ρλ +
λ + zx =
0
ρρ
Vxz
Vxz
*
2
, and all matrices are evaluated at OSS.
By the facts that Vx =−(1 − d )Tz + Tx and Vz =Tz ,
= (1 − δ ) 2 Tzz − (1 − δ )Tzx − (1 − δ )Txz + Txx
Vxxρρρρρ
 ∆ ρx

ρρρ


= (1 − δ ) (1 − δ )Tzz − Tzx  +  ρ − (1 − δ )  Txz ,
 ∆z

(a11)
Vxzρρρρ
Vzx =
=
−Tzz (1 − δ ) + Txz ,
(a12)
Vzzρρ
= Tzz ,
(a13)
∂k1r
∂k1r
r
where ∆=
a 2 β1 + (a1β 2 − a 2 β1 )
and ∆=
(a1β 2 − a 2 β1 )
.
x
z
∂x
∂ y
r
From (a3),
ρ
ρ
y β (∆ ρρ
) β2 −1 ∆ x
∂k1ρρρρρ
∂k1
(∆ ) β2 + y β 2 (∆ ) β2 −1 ∆ z
2
= =
and
.
∂x
(a1β 2 ) β2
(a1β 2 ) β2
∂ y
Then we may show the following relations:
−α 2 −1
ρ
ρ
ρρρρρρ
x
1
xz
z
xx
xz
zz
xz
ρ
 ∆ 
∆
( 2 − β 2 )m
αα
T =
− 
∆ , T =
T
, T =
−T

β1  α 2 β1 
∆z
α2
(a14)
Some manipulations will yield
∆ ρx
αα
2
2
=
=
ρ
∆ z (α1β 2 − α 2 β1 )m −(α 2 − β 2 )m
(a15)
Substituting (a14) and (a15) into the equations (a11) through (a13) will yield
40
ρ *Vxxρ
ρ
Vxz
 (1 − δ )m(α 2 − β 2 ) + α 2 [1 − m(1 − δ )] 
= ρ* 

m( β 2 − α 2 )


, and
Vzxρ
m( β 2 − α 2 )
.
=
ρ
Vxz (1 − δ )m(α 2 − β 2 ) + α 2 [1 − m(1 − δ )]
Thus the Euler equation will be eventually rewritten as follows:

 (1 − δ )m(α 2 − β 2 ) + α 2 [1 − m(1 − δ )]  
 
m( β 2 − α 2 )
0.
+
 +1 =
m( β 2 − α 2 )

  (1 − δ )m(α 2 − β 2 ) + α 2 [1 − m(1 − δ )]  
ρ *λ 2 +  ρ * 

Thus we can easily obtain the two roots of the above equation as reported in Remark 6.
41
Figure1.a: Two-sector Capital Intensities in Japan
(Million Yen in 85 prices/1000 persons*year)
Figure1.b: Two-sector Output in Japan (log scale)
Figure2.a: Two-sector capital intensities
in France(Million FF/persons*year)
42
Figure2.b: Two-sector per-capita outputs
in France (log scale)
Figure3.a: Two-sector capital Intensities in US
(Million US$ in 90 prices/1000 persons*year)
Figure3.b: Two-sector per-capita-outputs
in US (log scale)
43
Figure4.a: Two-sector capital intensities in West Germany
(Million DM/persons*year)
Figure4.b: Two-sector per-capita outputs
in West Germany (log scale)
2.00
1.90
1.80
1.70
1.60
Canada
1.50
France
1.40
US
1.30
Germany
1.20
1.10
1.00
1970
1975
1980
1985
1990
Figure 5: Capital-intensity Ratios in OECD Countries
44
Figure 6: Properties of the Unbalanced Growt
c(t )
A
Labor − const.
(p ρ ,1)
c (t )
(p ρ ,1)
E
k (t ) − const.
c
ρ
k ρ − const.
y (t )
yρ
B
y (t )
Figure 7: Derivation of the NMF in the Two-sector Model
45
z
D
NMF
E
k (t + 1)
k
45o
z= (1 − δ )x
A
ρ
B
[b ρ + (1 − δ )]
kρ
k (t )
x
Figure 8: The NMF of the Two-sector Model
46