*
By
Shigemi Kagawa
Research Center for Material Cycles and Waste Management,
National Institute for Environmental Studies,
Onokawa 16-2, Tsukuba, Ibaraki, 305-0053, Japan,
Phone: +81-298-502843, Fax: +81-298-502840,
E-mail: kagawa.shigemi@nies.go.jp.
First version: November 2001
Second version: July 2002
This version: September 2002
*This revised paper was prepared for the Society for Environmental Economics and Policy Studies,
Hokkaido, Japan, 28-29 September, 2002, and for the Fourteenth International Conference on Input-Output
Techniques, Montréal, Canada, 10-15 October, 2002.

Abstract
In the long run, we cannot examine the dynamic path of waste accumulation without evaluating a path of capital accumulation (investment decision) at the same time.
This paper explores the dependent relationship between capital and waste accumulation. Through this exploration, I find some important conditions in terms of both balanced growth paths and finally reveal a functional relationship between capital and waste accumulation process.
Keywords: Dynamic input-output models; balanced growth paths; capital accumulation; waste accumulation; Frobenius roots
JEL Classification Numbers: O41, Q00, Q01, Q32
2

1. Introduction
Examining the dynamics of waste accumulation is fundamental to evaluating the relationship between environmental externalities related to waste generations and recycling technology innovations.
1
What are the structural determinants of the dynamic path of waste accumulation?
And what are the economic and environmental impacts of the structural determinants?
In order to analytically answer these questions, I (2001) recently proposed a simple dynamic multi-sector model for waste analysis by following Leontief’s dynamic tradition
(1953, 1970b). The crucial point of the model is that a technical coefficient matrix of intermediate waste was definitely incorporated into the traditional Leontief dynamic system and it was indicated that a dynamic path of waste accumulation largely depends on a recycling technology of the intermediate waste. More concretely, assuming that the number of ordinary marketable commodities, and non-marketable wastes is n ,
2
the continuous dynamic equilibrium for waste distribution can be described by or b it
 n
  j  n a b ij a p jk
 k

1 q b kt
 
1   
 n
  j  n a b ij b p jk
 k

1
 
1    b k
 n
 a b ij f jt j   p 
Total waste outputs at time t
Intermediate waste inputs for production of intermediate goods and services at time t it b
Intermediate waste inputs for production of additive capital stock at time t
Intermediate waste inputs for production of final goods and services at time t
 
1 , 2 ,  , n

Final waste disposal at time t
(1)
 i q it p  j n n
 
 
1 a b ij a p jk q kt p  j n n
 
 
1 a b ij b p jk
 k p  j n


1 a b ij f jt p  f it b
 
1 , 2 ,  , n

(1)’ where q b it
is the physical total output of waste i in time period t ; q  b k
is the fluctuation ratio of physical total output of waste k at time period t ; a b ij
is the technical coefficient that stands for
3

the intermediate input requirement of intermediate waste i absorbed to produce a unit of primary and secondary product j ; a p jk
is the technical coefficient that represents the intermediate input requirement of primary and secondary product j absorbed to produce a unit of primary and secondary product k ; p b jk
is the capital coefficient that denotes the stock of primary and secondary product j accumulated to produce a unit of primary and secondary product k ; f jt p is the final demand of product j in time period t ; and f it b is the physical amount of the final disposal of waste i in time period t . In addition, it should be noted that I specify the relationship between capital and waste accumulation as q b k
  k q k p
 q  b k
  k q  k p or q k p    k
1 q b k
 q  k p    k
1 q  b k
where q k p is the total output of primary and secondary product j at time period t ;  k p is the fluctuation ratio of the total output of primary and secondary product j and
 k
is temporally stable generation factor of waste k .
3 Accordingly, this supports that, in the real world, it is reasonable to assume a linear homogeneous relationship between waste generation and commodity production.
4 Although the relationship, q k b
  k q k p , asserts that the physical total output of waste k is simply subject to the total output of primary and secondary product k , we cannot unfortunately examine the impacts of for example the recycling technology changes on waste fluctuations only from the relationship. In order to do so, we need to justify the differential equation (1) by presuming the linear relationship. The justification consequently provides an important eigenvalue problem in terms of the waste accumulation and the comparison with the Leontief’s eigenvalue problem enables us to mathematically discuss the interrelationship between two balanced paths of capital and waste accumulation.
From the specified relationship, the second term of equation (1) can also be written as j n n
 
1 1 a b ij b p jk
  k
1 q  b k
 j n n
 
1 1 a b ij b p jk
 k p  j n


1 a b ij
I j p (2)
4

where I j p is the investment coefficient of the stock of primary and secondary product j . In other words, the second term of equation (1) describes the intermediate waste requirements directly induced by capital investment. This clarifies that the intermediate wastes are required and recycled in order to produce for example machinery. Although there is the criticism that investment balance of dynamic input-output analysis simply refers to purchase and installation of the capital stock, for the waste analysis, I think that it is crucial to consider more extensive role of the capital investment from a physical aspect of dynamic material balance of non-marketable scraps and wastes.
Equation (1) hence asserts that the total waste output (left-hand side of (1)) coincides with the intermediate waste requirement inputted to produce the intermediate input of primary and secondary products (first term on the right-hand side) plus the intermediate waste requirement inputted to produce capital goods (second term on the right-hand side) plus the intermediate waste requirement inputted to produce final goods and/or services (third term on the right-hand side) plus final waste disposal such as reclamation and incineration (last term).
Subsequently, let us recall that the traditional dynamic Leontief system can be described by
5 q it p  j n


1 a ij p q p jt
 j n


1 b ij p j p  f it p
 
1 , 2 ,  , n

. (3)
As can be seen in both equations (1) and (3), it should be emphasized that the fundamental determinants of the dynamic path of waste accumulation are the technical coefficient matrix of primary and secondary products and of intermediate waste, and capital coefficient matrix and
5

that we cannot explore the dynamic path of waste accumulation without evaluating the path of capital accumulation (investment decision) at the same time.
In the past, although there were many discussions of the dynamic Leontief system, for instance, on the relationship between time lag of output and dynamic stability,
6
the relative stability and instability of the dynamic system,
7
the singularity of the capital coefficient matrix,
8 generalization of the dynamic Leontief system,
9
and numerical solution methods,
10
I do not consider these problems in detail in this paper in order to examine basic properties of both dynamic systems. Of course, it is needless to say that these discussions are fundamental and non-ignorable. Nevertheless, from the basic model described above, I found some important conditions in terms of the mutual relationship between capital and waste accumulation and revealed a functional relationship between them.
In the next section, the structural determinants of the dynamic path of waste accumulation are mentioned in more detail. Section 3 provides some important conditions of capital and waste accumulation. Section 4 formulates the functional relationship between capital and waste accumulation. Finally, section 5 is the conclusion.
2. Structural Determinants of Waste Accumulation
For simplicity, let us transform equations (1) and (3) into the following algebraic forms. q t b 
A b
A p  
1 q t b 
A b
B p  
1  b 
A b f t p  f t b 　 (4)
6

q t p 
A p q t p 
B p  p  f t p
(5)
Here, q t b
, q b
, f t p
, f t b
, q t p
, and q p
represent the column vectors whose elements are q it b
,  i b
, f it p , f it b 　 , q it p
, and q  i p
, respectively. Also, A p
, A b , and B p
stand for the matrix whose elements are the technical coefficient of primary and secondary products a ij p , technical coefficient of intermediate waste a b ij
, and capital coefficient ij p b , respectively. In this paper, it is also assumed that A p
, A b , and B p
are non-singular and these ranks are full. Although, in fact, this assumption may be unrealistic, by doing so, the later theoretical discussion is made clear. Furthermore, in order to keep the generality of the model, let us suppose that

denotes not the diagonal matrix with
 i
but the parametric matrix with
 ij
as mentioned in notation four. It should also be noted that bold notation stands for vector or matrix and italic notation represents a scalar or subscript/superscript. We shall proceed to the following task in order to understand the fundamental relationship between both dynamic paths.
As is well known, the closed dynamic Leontief system can be formulated as q t p 
A p q t p 
B p  p
. (6)
We can also formulate the closed dynamic system of equation (3) as q t b

A b A p
 
1 q t b

A b B p
 
1 b . (7)
7

Although, in general, one may be interested in the accumulated quantity of the final waste disposal, equation (7) lacks the fixed term f t b showing the quantity and does not definitely describe the dynamics of the final waste disposal. At the same time, equation (7) also lacks the fixed term A b f t p
showing the intermediate wastes required to produce final demand and does not describe the accumulated quantity of them. Consequently, equation (7) provides only the accumulated quantity of intermediate wastes required to produce intermediate goods and capital goods. If one wants to completely trace the multi-sector dynamics of intermediate wastes and of final wastes, it is convenient to transform equation (4) into q t b

A b A p
 
1 q t b

A b B p
 
1  b
  q t b
  q t b (8) where

and

represent diagonal matrices with scale parameters
  i

1 ,  , n

and
 i
 i

1 ,  , n

, respectively and it holds that A b f t p
  q t b and f t b
  q t b . Then, we must investigate the basic properties of the dynamic system b 　 
D

1 q t b where
D


I n

A b A p  
1
   


1
A b B p  
1 . Here I n
represents the n -dimensional identity matrix.
This clarifies that the intermediate waste i required and/or generated in order to produce final goods at time t , j n


1 a b ij f jt p , amounts to
 i

100 % of total waste generation at time t and the final waste i at time t , f it b , amounts to
 i

100 % of that. Readers can understand that both parameters should be smaller than one in real world.
The closed dynamic systems play a crucial role in determining the general solutions of equations (4) and (5). Especially, the eigenvalues
 i
 i

1 ,  , n

and eigenvectors u i
 i

1 ,  , n

of the matrix, E


I n

A p


1
B p , affect the particular solution of capital
8

accumulation, while the eigenvalues
 i
 i

1 ,  , n
 and eigenvectors v i
 i

1 ,  , n

of the matrix, D


I n

A b A p
 
1
   


1
A b B p
 
1 , affect the general solution of waste accumulation. The important point is that if matrices D and E are indecomposable and non-negative, there always exist positive dominant roots
  
and
  
corresponding to respective positive eigenvectors from Frobenius Theorem.
11
We often call them F
ROBENIUS roots. In addition, it is well known that if the absolute values of the other eigenvalues are smaller than the F
ROBENIUS
roots
  
and
  
, from the relative stability theorem, the general solutions approach balanced solutions as the time period goes to infinity.
12
Assuming that the F
ROBENIUS
roots are

1
and

1
, respectively, these stability conditions can be easily formulated as inequalities,
  
1
  i
 i

2 ,  ,n

and
  
1
  i
 i

2 ,  ,n

.
With the same postulates, let us consider the mathematical characteristics of both models as shown in equations (6) and (8). From the eigenvalues and eigenvectors, we have the following relationships. p

E

1 q t p with
 i

1
 i
, u i
 i

1 ,  ,n

(9) b 　 
D

1 q t b with
 i

1
 i
, v i
 i

1 ,  ,n

(10)
Considering that the change in the capital coefficient matrix affects both dynamic paths of capital and waste accumulation and then the eigenvalues and eigenvectors of matrix D inevitably are dependent on the eigenvalues and eigenvectors of E , it is convenient to intentionally derive the dependent relationship between equations (9) and (10). Therefore I used a Spectral Mapping Theorem in linear space. Since any q t b

0 can be decomposed as
9

D

1 q t b
 i n


1
 i u i
 i n


1
 i
 i
 i u i
 j n n

 
1
 i
 i
 ij v j j n n
 
 
1
 i
 i
 ij v j
(11) where
 i
 i

1 ,  , n

and
 ij
 i , j

1 ,  , n

represent n arbitrary scalars corresponding to u i
 i

1 ,  , n

and n
2
arbitrary scalars corresponding to v i
 i

1 ,  , n

, respectively, and the another equivalent resolution form can be obtained as
D

1 q t b
 i n


1
 i
 i v i
(12) where
 i
 i

1 ,  , n

represent n arbitrary scalars corresponding to v i
 i

1 ,  , n

, we finally have the following relationship by comparing equation (12) with equation (11).
 j
 i n


1
 i

 j i
 ij
 
1 ,  , n

(13)
This relationship, which can also be written as the abstract notation
 j
 f j
 
1
,

2
,  ,
 n

10

 j

1 ,  , n

, obviously indicates that the expansion ratio of waste accumulation largely depends on the growth ratio of output in the ordinary sense. If we use an appropriate linear mapping F which corresponds to f j
, equation (13) can also be written as
 
F
 
and/or
 
F -1
 
where

and

represent a n -dimensional column vector with
 i
and
 i
, respectively. This relationship enables us to examine the impacts of changes in the technical structure and capital structure on the dynamic path of waste accumulation. We must note that the reverse relationship of equation (13) does not always hold true, in short
 j
 f j

1
 
1
,

2
,  ,
 n
  j

1 ,  , n

. Rather, we should say that there exists the case in which the reverse relationship has no meanings. For example, even if (recycling-) production technology A b changes and the spectral radius of equation (10) shifts from
 i
to
 i

,
 j remains constant, as can be understood from equation (6). Mathematically speaking, this implies that it holds d
  d F -1
  
F -1
 d
 
O where d stands for a differential calculus in terms of time and O describes a n -dimensional zero column vector. Here I want to investigate a dynamical phenomenon such that it holds that d
  d F -1
  
F -1
 d
 
O . As readers easily notice, I implicitly presume the mapping F as a continuous linear mapping.
With this understanding of the dependent relationship (13), the general solution of equation (10) can be obtained as q t b
 j n


1
 j v
 j exp
 j n


1
 j v
 j exp i n


1
 i
 i
 ij t
 j
. (14) where v
 j
denotes the j th eigenvector and can be expressed as component notation v
 j
  v
1 j v
2 j
 v nj

T
. T represents a transpose. Hence k th component of the solution can be written as
11

q kt b
 j n


1
 j v kj exp
 j n


1
 j v kj exp n
 i

1
 i
 i
 ij t
 j
. (15)
3. Some Conditions of Capital and Waste Accumulation
First, I examine mathematical conditions of relative stability of the dynamic path of waste accumulation. According to Takayama (1984, p. 510), the definition of relative stability is as follows.
D EFINITION : Let x
 t

1
 
M
 x
 
be a system of difference equations where M is an n
 n constant matrix. Suppose that x *
 t

1
   t x

0 is a particular solution of this system of difference equations. Let x
ˆ  
be a solution of this system starting from an arbitrary initial vector x
  
0 . Then the balanced growth solution x * t lim
  x
ˆ i x i
*
 
 
 
is said to be relatively stable if
exists such that
   
0 and

is independent of i , where i stands for the i th component.
13
Although the definition focuses on the discrete dynamic linear model, we can also apply it to our continuous dynamic linear model. Subsequently, let us recall that the balanced growth path is relatively stable if and only if the Frobenius root of M ,

1
, is larger than the absolute value of the other eigenvalues,
 i
 
2 ,  , n

.
Applying the proposed dynamic model to the definition and the condition of relative stability, we can similarly get the condition of relative stability for waste accumulation.
12

Therefore, assuming the indecomposability and non-negativity of the matrices D and E , I start with the relative stability condition of waste accumulation such that there exists q kt b t lim
  q kt b *
 t lim
 
 n j

1
 j v kj exp exp
 
 
  if and only if the positive root of D
-1
,
  
 
1
, is smaller than the other eigenvalue,
  i
 i

2 ,  , n

. This is consistent with the inevitable condition,
 
1

1

1
 i
   
1
  i
 i

2,  , n

. Here superscript * means the balanced growth path.
For simplicity, let us consider the special case of n =2, in short a two-sector dynamic model, then the relative growth ratio is q b kt t lim
  q kt b *
 t lim
 



1 v k 1 exp
1 t exp
 

2
1 t v k
 
2 exp


 t lim
 
 
1 v k 1
 
2 v k 2 exp
  
2
 
1
 t
 
 
1 v k 1
  
0
 
1
  
2

. (16)
The important point in equation (16) is that

1
and

2
can be described by the continuous functions,

1
 f
1
 
1
,

2

and

2
 f
2
 
1
,

2

, respectively and, considering the relative stability zone of waste accumulation such that it holds the inequality
0
 
1
  
2

0
 f
1
 
1
,

2

  f
2
 
1
,

2

, there may not exist the relative stability zone of capital accumulation such that it holds 0
 
1
  
2
. Naturally, we can also consider the reverse phenomenon. If, by using equations (14) and/or (15) the relative stability zone of waste accumulation is mathematically examined and compared with the relative stability zone of capital accumulation, it would be clear whether both dynamic systems are relatively stable together or not. Moreover, using well-known discriminants 0
 
1

1 and 0
 
1

1 , we can
13

judge whether the path of waste and/or capital accumulation approaches the respective balanced growth paths as time goes to infinity. It should be noted that these discriminants have a dependent relationship to each other.
We can get general propositions by considering, not 2-dimentional subspace, but n -dimensional subspace. Let X be n -dimensional subspace shown in equation (13):
X













 n
1
2 

0
 
1
  
2
     n

















 n
1
2 

0
 i n


1
 i
 i 1


1 i   i n


1
 i
 i 2


2 i    n
  i

1
 i
 in

 n i




.
(17)
And let Y be n -dimensional subspace for relative stability condition of capital accumulation:
Y














1
2 n


0
 
1
  
2
     n




. (18)
Then, the following conditions can be easily obtained.
Condition 1.
Assume that matrices D and E are non-negative and indecomposable. Then, the relationship
X

Y
 
, where

stands for an empty set, asserts that the path of capital accumulation is relatively unstable in the case that the path of waste accumulation is relatively stable, and vice versa (see Figure 1).
14

Remark 1.
Especially, the conditions X
 
and Y
 
assert that both accumulation processes are relatively unstable.
[Insert Figure 1 here]
Condition 2.
Assume that the matrices D and E are non-negative and indecomposable. The relationship,
X

Y
    
X ,
 
Y

, asserts that the path of capital accumulation is relatively stable in the case that the path of waste accumulation is relatively stable (see Figure 2).
Remark 2.
When X

Y holds, any points in the feasible zone of waste accumulation are not always contained in the feasible zone of capital accumulation. Hence even if the path of waste accumulation is relatively stable, the path of capital accumulation may be relatively unstable, and vice versa.
[Insert Figure 2 here]
Condition 3.
Assume that the matrices D and E are non-negative and indecomposable. When it holds that
X

Y , the relative stability of the path of waste accumulation guarantees that of the path of capital accumulation at the same time, while the relative stability of the path of capital accumulation does not guarantee that of the path of waste accumulation (see Figure 3).
15

[Insert Figure 3 here]
Condition 4.
Assume that the matrices D and E are non-negative and indecomposable. When it holds that
X

Y , the relative stability of the path of capital accumulation guarantees that of the path of waste accumulation at the same time, while the relative stability of the path of waste accumulation does not guarantee that of the path of capital accumulation (see Figure 4).
[Insert Figure 4 here]
For instance, let us consider two-sector economy as follows.
A p



1
1
3
3
1
1
3
3

 , B p



1
0
0
1

 , A b



5
4
3
3
7
8
3
3


 


20
0
3
14
0
3

 ,
Ρ 


0 .
1
0 0
0
.
1

 ,
 


0 .
2
0 0
0
.
2


Then, from both dynamic systems, we have
D



1
1
.
07
.
00
2 .
47
2 .
57

 and E



2
1
.
00
.
00
1 .
00
2 .
00

 .
Hence, computing eigenvalues and eigenvectors of the inverse matrices D

1
and E

1
yields
16


1

0 .
33 ,

2

1 .
00

 u
1



1
1
.
00
.
00

 , u
2



1
1 .
00
.
00



 , and

1

0 .
28 ,

2

12 .
47

 v
1



0 .
70
0 .
71

 , v
2



0
0 .
93
.
37



 .
Since, from the eigenvalues and eigenvectors, the necessary parametric vectors and matrices,
 
  j
,
 
  ij
, and
 
  j
can be obtained as
 


1
1 .
.
00
00

 ,
 


1
0 .
02
.
54
4
0 .
.
23
60

 , and
 


7
0
.
18
.
12

 , respectively, the linear mapping F


,

,


in equation (13) and the inverse F

1


,

,

 can be finally computed as
F




0 .
59
0 .
16
0 .
08
12 .
52

 and F 1



1
0
.
69
.
02

0
0
.
.
01
08

 , respectively. In this two-sector economy, it is clear from P
ROPOSITION
1 and R
EMARK
1 that both accumulation processes are relatively unstable.
With same propositions, the mutual relationship between the two relative stability conditions was arranged here. However, in this section, I do not definitely approach an
17

interrelationship of both dynamic systems. The important question is how the balanced path of the waste accumulation can be expressed as a function of the balanced path of capital accumulation and what features the function has. In next section, we shall derive the fundamental functional relationship and explore the basic features.
4. The Functional Relationship Between Capital and Waste Accumulation
As readers can easily understand, for the above-mentioned two-sector dynamic model, the balanced growth path of waste accumulation can be obtained as q b* kt
 exp
 k

1 , 2

. The relationship can also be written as q kt b *  exp
 
1

1

11 t

1
 
2

2

21 t

1
  k

1 , 2

by using equation (13). Hence considering that the balanced path of capital accumulation can be written as q kt p *  exp
 
1 t , the balanced path of waste accumulation can also be expressed as q kt b *  exp
 
2

2

21 t

1


  kt

1

1 1

1 . This would be important in analytically evaluating the relationship between both dynamic paths. In addition, it should be noted that relative ratio at time t can be written as q kt b * q kt p *  exp
 
2

2

21 t

1


  kt

1

1 1

1

1
. Applying the above-mentioned example to the function, the diagram of the balanced growth path can be depicted as shown in Figure 5.
[Insert Figure 5 here]
Although the above-mentioned are focused on the functional relationships in a certain time profile, we can also derive the functional relationship between both accumulations by using an
18

integral calculus. For simplicity, I did not consider the problem of time lag, as Sargan (1958,
1961) and Leontief (1961a, 1961b) discussed in past years, and also supposed an infinite time sequence from past time to present time. With these premises and the integral calculus, capital and waste accumulation can be described as
Q kt p *  
 t
 q k p

* d

, (19) and
Q b * kt
  t
-
 exp
 
2

2

21
 
1


  k

1

11

1 d

(20) where Q kt p *
and Q kt b *
represent the amount of capital and waste accumulation. From both equations (19) and (20), readers can easily understand that there are dependent relationships between both accumulations.
Subsequently, let us write the relationships. First, solving equations (19) and (20) reads to
Q kt p * 
1

1 q kt p *
, (21) and
Q kt b * 

1

1

11


1

2

2

21 exp
 
2

2

21 t

1


  kt

1

1 1

1 . (22)
19

See the appendix A for proof. In particular, equation (22) asserts that, for the balanced growth paths, the output level of primary and secondary product k at the present time determines the quantity of the accumulation of capital stock k and of waste k . At the same time, this also asserts that there exists a dependent relationship between capital and waste accumulation.
Furthermore, considering relative quantity, since we have the following relationship,
Q kt b *
Q kt p *


2

2


21

1

1

1

1

11 exp
 
2

2

21 t

1
 
  kt

1

11

1

1
, (23) by differentiating equation (23), the time derivative of the relative quantity can be written as d

Q b kt
* dt
Q kt p *




1

 
1

1
1



1 exp
 
2

2

21 t

1
   kt

1

11

1

1
(24) where

1
 
1

1

11
 
2

2

21
. Hence since it holds that

1

0 , exp
 
2

2

21 t

1


0 and
  kt

1

1 1

1

1 
0 , judging a sign of inequality enables us to know functional forms of equation
(23). The inequality

1

1
 
1

0 implies that equation (23) is a monotonic increasing curve, while the inequality

1

1
 
1

0 implies that equation (23) is a monotonic decreasing curve.
Especially, if

1

1
coincides with

1
, equation (23) is a constant with respect to time and amounts to

1

1
 
2

2

21
 
1

1

11

. In general, we have the following distinctions:

1

1
 
1

1

11
 
2

2

21
    n
 n
 n 1
in case of monotonic increasing curve:

1

1
 
1

1

11
 
2

2

21
    n
 n
 n 1
in case of constant with respect to time:

1

1
 
1

1

11
 
2

2

21
    n
 n
 n 1
in case of monotonic decreasing curve. From the distinctions, for the balanced relationship, we can judge whether the relative quantity of waste
20

accumulation converges or not as time goes to infinity.
Furthermore, if it is supposed that policy makers hoped to decrease the physical amount of waste accumulation at time t from Q b kt
*
to
 
Q kt b *

0
  
1

, where

denotes a reduction parameter, we can easily understand that they should have formulated a plan to reduce the amount of capital accumulation from q kt p *
to
 
1

1

1 1
 q kt p *
(see equation (24)).
Since, from the above-mentioned inequalities of the parameter,
 
1

1

11 is a decreasing exponential function and smaller than one at the same time, it is clear that the difference in the planned output of commodity k , that is

 
1

1

1 1

1

 q kt p *
, is obviously negative. This implication is very simple and is that appropriately reducing the amount of capital accumulation leads to the reduction in the waste accumulation.
We can also consider impacts of technological change, especially recycling technology change. From equation (13), let us transform equation (24) into
Q kt b * 
1

1 exp


2 f
2

1
 
1
,

2
 
21

1
 t
   kt

1

1 1

1 . (25)
Hence if

1
and

2
fluctuate due to recycling technology change in intermediate wastes, we have the following effects of recycling technology change.


Q
 b kt
1
*


 
1



1
1 exp


2 f
2

1
 
1
,

2
 
21

1
 t q kt p *

1

11

1

 (26)


Q
 b kt
2
*


 
2



1
1 exp


2 f
2

1
 
1
,

2
 
21

1
 t q kt p *

1

11

1

 (27)
21

Equations (26) and (27) indicate that it may be possible to reduce the amount of waste accumulation by appropriately controlling recycling technology. The question is whether we can determine

1

and

2

such that it holds:
 
Q kt b * 
1
 
1 exp

 
2 f
2

1
  
1
,
 
2
  
21
 
1
 t
   kt

1
  
1
 
0
  
1

(28) where primed letter represents the appropriate technological spectrum. In this case, what is the appropriate linear operator F

:
    
? Answering this question, policy makers can quantitatively evaluate the mutual relationship between capital and waste accumulation process.
Figure 6 shows the graphical concept in our discussion. Readers can understand in Figure 6 that the volume of accumulated waste in concerned time period, Q kt b*
(area OAB at XY surface), decreases to
η 
Q kt b*
(area OA’B’ at XY surface) because of the appropriate technological changes. Although a fundamental question is how we can choose the appropriate technologies from the viewpoint of waste management policy, I could not, unfortunately, find the answer.
Furthermore, if one hopes to perform the dynamic input-output analysis using actual data, equation (4) should be transformed into a discrete type. Following Leontief’s dynamic inverse, the discrete model can be easily obtained (see the appendix B).
[Insert Figure 6 here]
22

5. Concluding Comments
In this paper, in focusing on the balanced growth path of capital and waste accumulation processes, I explored the dependent relationship between both accumulation processes and derived some fundamental conditions. With the conditions, we can judge not only whether both accumulation processes are relatively stable at the same time or not but also how the relative quantity of waste accumulation temporally changes. Furthermore, I revealed a hidden functional relationship between capital and waste accumulation process. The functional relationship is useful for analytically discussing the long-run dependent relationship between waste generations and economic growth.
Although an emphasis is placed on the simple dynamic input-output model, it is natural that more complex dynamic phenomena, such as time gestation of capital stock and temporally distributed activities should be mathematically considered. At the same time, it would be important to examine the impact of price formation on waste accumulation. In order to attain it, we must connect this consideration with previous studies in terms of duality. This is an important and interesting study for environmental and ecological economics.
23

Notes
1.
For static economic-environmental accounts, see pioneering works of Leontief (1970a) and Leontief &
Ford (1972) and elaborated work of Duchin (1990). In the main subject of this paper, I modeled the dynamic waste accumulation model without technological change. Although it seems that this goes against the awareness of the issue, I think that presenting the dynamic model, which definitely treats the input technology of intermediate wastes, is a first important step toward further elaborate studies and empirical works. For the further studies, it would be important to consider the technological change.
And also Shephard’s lesson may provide us with the important economic properties of the cost in terms of intermediate wastes, if we consider the smooth production function with them.
2.
In reality, the number of the ordinary marketable commodities may not coincide with that of the non-marketable wastes. For the sake of simplicity, we defined that both numbers are equal.
3.
Readers may want to know the relation between equation (1) and q b k
  k q k p
, because it seems that, if we use the latter, the total amount of the waste k simply depends only on that of the product k . It is possible to transform the latter into the former. Let us define q b k
  k q k p
as q k b   k
 k q k p 

1
  k

 k q k p
where
 k

represents the allocation parameter of the intermediate waste k and 1
  k

represents that of the final waste k . Then, if we substitute the well-known Leontief’s dynamic balance, q it p  n
 j

1 a ij p q p jt
 j n


1 b ij p p j
 f it p
, into the above-mentioned relationship, the following relationship (a), q k b  n
 j

1
 k
 k a p kj q p jt
 j n


1
 k
 k b kj p p j
  k
 k f kt p  j n


1

1
  k

 k a p kj q p jt
 j n


1

1
  k

 k b p kj p j


1
  k

 k f p kt
, can be written. This clarifies that

1
  k
  k q k p corresponds to the forth term of equation (1). On the other hand, the important problem is to appropriately decompose the intermediate wastes
 k
 k q k p
under the recycling technology a b ij
. The above-mentioned relationship (a) also clarifies that
 k
 k q k p
can be a priori decomposed into the intermediate wastes for production of intermediate goods (the first term of
24

equation (1)), for production of additive capital stock (the second term), and for production of final goods (the third term). Mathematically speaking, it implies that n
 j

1
 i
 i a ij p q p jt
 k n n
 
1 1 a b ik a p kj q p jt
, n
 j

1
 i
 i b ij p p j
 k n n
 
1 1 a b ik b p kj p j
, and
 i
 i f it p  k n


1 a b ik f kt p
and hence arranging the relationship (a) yields
 i q b i
 k n n
 
1 1 a b ik a p kj q p jt
 k n n
 
1 1 a b ik b p kj p j
 k n


1 a b ik f kt p
where
 i q i b
is the intermediate waste outputs and k n n
 
1 1 a b ik a p kj q p jt
 k n n
 
1 1 a b ik b p kj p j
 k n


1 a b ik f kt p
is the intermediate waste inputs. If we can justify this equality, it is possible to explore not only the simple regressive factor
 k
but also more deep structural determinants of the waste fluctuations, for example a b ij
, a p jk
, and b p jk
. Of course, it is needless to say that the generation factor
 k
(or
 kl
mentioned below) plays a crucial role in our dynamic waste accounting model.
4.
Although I assumed the linear homogeneous relationship q b k
  k q k p   b k
  k q
 p k
, the general linear relationship, q k b  l n


1
 kl q l p   b k
 l n


1
 kl
 l p
, can also be assumed. In this case, the later matrix

may not be a diagonal matrix. This supports a possibility such that a scrap k is generated by the production of a non-homogeneous commodity l
 
. Then, equations (1) and (1)’ can also be written as q b it
 j n n n
 
1 k 1 1 a b ij a p jk
 
1 kl q b lt
 j n n n
 
1 k 1 1 a b ij b p jk
 
1 kl
 l b  j n


1 a b ij f p jt
 f it b n
 j

1
 ij q p jt
 j n n

 
1 a b ij a p jk q p kt
 j n n

 
1 a b ij b p jk
 k p  n
 j

1 a b ij f p jt
 f it b


1 , 2 ,  , n

and


1 , 2 ,  , n

, respectively, and in the real world the formula should be justified. The elaborated formula also plays more important role in exploring mathematical properties of the waste accumulation.
5.
Considering that it holds that q b k
  k q k p
and  b k
  k
 p k
, equation (1) reduces to equation (3). The proof can be obtained as q b it
 n n
  j
 
1 a b ij a p jk
 k

1 q b kt
 n
 j

1 a b ij q p jt
 n n
  j
 
1 a b ij a
 j n n
 
1 1 a b ij b p jk
 k

1  b k
 j n


1 a ij b f p jt
 p jk
 k

1 q b kt f it b
 n n
  j
 
1 a b ij b p jk
 k

1  b k
 j n


1 a b ij f p jt
25

 q it p  j n


1 a ij p q p jt
 j n


1 b ij p j p  f it p .
Here it should be noted that q b it
 f b it
 j n


1 a b ij q p jt
.
6.
See Sargan (1958, 1961) and Leontief (1961a, 1961b) for the discussion of the relationship between time lag of output and dynamic stability.
7.
See for example Morishima (1958), Solow (1959), Jorgenson (1960, 1961), and Tokoyama & Murakami
(1972) for the relative stability of the dynamic input-output model.
8.
See Kendrick (1972), Luenberger & Arbel (1977), and Meyer (1982).
9.
See Brody (1974), Johansen (1978) and
○
,Aberg & Persson (1981) for the time lag model of the capital equipment. Also, see ten Raa (1986a) for the temporally distributed activity model.
10.
See for example Almon (1963), Duchin & Szyld (1985), and ten Raa (1986b). More recently, Kurz &
Salvadori (2000) developed a dynamic input-output model which outlines AK model, while Los (2001) proposed an analytic dynamic input-output model with some endogenous growth properties for example spillover effects.
11.
I hope to discuss a negativity and decomposability of the matrix D in another paper.
12.
More correctly, as is mentioned later, we need the mathematical conditions
 is well known that if the conditions
 
1 and


1 and


1 hold, Euclidean norm between the

1 . It particular solution and balanced solution may be diverge as time goes to infinity. Furthermore, the equalities
 
1 and
 
1 implies that the linear mappings of both matrices D -1 and E -1 have fixed points.
13.
Since I do not intend to completely explain in this paper, see chapter 6 of Talayama (1984).
26

References
○
,Aberg, M. and Persson, H. (1981) A note on a closed input-output model with finite life-times and gestation aaa lags, Journal of Economic Theory , 24, pp. 446-452.
Almon, C. (1963) Numerical solution of a modified Leontief dynamic system for consistent forecasting or aaa indicative planning, Econometrica , 31, pp. 665-678.
Brody, A. (1974) Proportions, Prices, and Planning (Amsterdam, North-Holland).
Duchin, F. and Szyld, D. B. (1985) A dynamic input-output model with assured positive output, aaa Metroeconomica , 37, pp. 269-282.
Duchin, F. (1990) The conversion of biological materials and wastes to useful products, Structural Change aaa and Economic Dynamics , 1, pp. 243-262.
Johansen, L. (1978) On the theory of dynamic input-output models with different time profiles of c apital aaa construction and finite life-time of capital equipment, Journal of Economic Theory , 19, pp. 513-533.
Jorgenson, D. W. (1960) A dual stability theorem, Econometrica , 28, pp. 892-899.
Jorgenson, D. W. (1961) Stability of a dynamic input-output system, Review of Economic Studies , 28, pp. aaa 105-116.
Kagawa, S. (2001) A simple dynamic input-output model for waste analysis, discussion paper, National aaa Institute for Environmental Studies (NIES), Japan.
Kendrick, D. (1972) On the Leontief dynamic inverse, Quarterly Journal of Economics , 86, pp. 693-696.
Kurz, H. D. and Salvadori, N. (2000) The dynamic Leontief model and the theory of endogenous growth, aaa Economic Systems Research , 12, pp. 255-265.
Leontief, W. W. (1953) Dynamic analysis, in: W. W. Leontief & others (eds) Studies in the Structure of the aaa American Economy (New York, Oxford University Press).
Leontief, W. W. (1961a) Lags and the stability of dynamic systems, Econometrica , 29, pp. 659-669.
27

Leontief, W. W. (1961b) Lags and the stability of dynamic systems: a rejoinder, Econometrica , 29, pp. aaa 674-675.
Leontief, W. W. (1970a) Environmental repercussions and the economic structure: an input-output approach, aaa The Review of Economics and Statistics , 52, pp. 262-271.
Leontief, W. W. (1970b) Dynamic inverse, in A. P. Carter and A. Br
/
,ody (eds) Contributions to aaa Input-Output Analysis (Amsterdam, North Holland).
Leontief, W. W. (1970a) Lags and the stability of dynamic systems, Econometrica , 29, pp. 659-669.
Leontief, W. W. and Ford, D (1972) Air pollution and the economic structure: empirical results of aaa input-output computations, in A. Br
/
,ody and A. P. Carter (eds) Input-Output Techniques (Amsterdam, aaa North Holland).
Los, B. (2001) Endogenous growth and structural change in a dynamic input-output model, Economic aaa Systems Research , 13, pp. 3-34.
Luenberger, D. G. and Arbel, A. (1977) Notes and comments singular dynamic Leontief systems, aaa Econometrica , 45, pp. 991-995.
Meyer, U. (1982) Why singularity of dynamic Leontief systems does not matter, in H. D. Kurz, E. aaa Dietzenbacher and C. Lager (eds) Input-Output Analysis Volume Ⅰ (Northampton, Edward Elger aaa Publishing).
Morishima, M. (1958) Prices, interest and profits in a dynamic Leontief system, Econometrica , 26, pp. aaa 358-380.
Sargan, J. D. (1958) The instability of the Leontief dynamic model, Econometrica , 26, pp. 381-392.
Sargan, J. D. (1961) Lags and the stability of dynamic systems: a reply, Econometrica , 29, pp. 670-673.
Shephard, R. W. (1970) Theory of cost and production functions (Princeton, Princeton University Press).
Solow, R. M. (1959) Competitive valuation in a dynamic input-output system, Econometrica , 27, pp. 30-53.
28

Takayama, A. (1984) Mathematical Economics: second edition (Cambridge, Cambridge University Press). ten Raa, T. (1986a) Dynamic input-output analysis with distributed activities, Review of Economics and aaa Statistics , 68, pp. 300-310. ten Raa, T. (1986b) Applied dynamic input-output with distributed activities, European Economic Review , 30, aaa pp. 805-831.
Tokoyama, K. and Murakami, Y. (1972) Relative stability in two types of dynamic Leontief models, aaa International Economic Review , 13, pp. 408-415.
29

Appendix A.
Equations (23) and (24) can be proved as
Q kt p *   t
  exp
 
1 d
 
1

1
 exp
1
 t

 

1

1 q kt p *
, and
Q b * kt
  t
-
 exp
 
2

2

21
 
1


  k

1

11

1 d



2


1
2

21 exp
 
2

2

21 t

1


  kt

1

11

1
 
 t

1

11
 
2

2

21 exp
 
2

2

21
 
1
 q  k p

*
  k

1

11

1

1 d



2


1
2

21 exp
 
2

2

21 t

1

 q kt p

1

11

1
 
 t


1

1

11

2

2

21 exp
 
2

2

21
 
1
   k

1

11

1 d



2


2
1

21 exp
 
2

2

21 t

1


  kt

1

1 1

1



1
2


2
1


11
21

Q kt b *


1

1

11

1
 
2

2

21 exp
 
2

2

21 t

1


  kt

1

11

1 .
30

Appendix B.
In this appendix, we shall derive a discrete solution from the proposed dynamic input-output model. The basic dynamic balance without dropping the time subscript in terms of technical coefficient matrix and of capital coefficient matrix, that is the dynamic system with technological progress, can be rewritten as q t b

A t b A t p  t
-1 q t b

A t b B t p

1
 t
-1

1 q t b

1

A t b B t p

1
 t
-1 q t b

A t b f t p  f t b . (B-1)
Arranging equation (B-1), we have q t b

L t
1 A t b B t p

1
 t
1

1 q t b

1

L t
1 A t b f t p 
L -1 t f t b (B-2) where L 1 t


I n

A t b A t p  t
1

A t b B t p

1
 t
1


1
. Hence, regarding equation (B-2) as a backward lag-type model and considering a finite time sequence from – m to zero, the dynamic equilibrium solution can be finally obtained as
 q





 q q q
 b
0 m b

2 b

1















 L


1 m
A b
 m


L 1 m
A b
 m t

2
 

P t m

1

L


1
2
A b

2
O
L 1 m
A b
 m t


1

P t
 m

1

L


1
2
A b

2
P

1
L 1
1
A b

1
L
L
1 m
A b
 m t
0

 
P t m

1

1
2
A b

2
P

1
P
0
L 1
1
A b

1
P
0
L

1
0









 f





 f f
 p m

 p
2
 p
1 f
0 p







31









 L 1 m
A b
 m


L 1 m
A b
 m
 t


2

Q
 m

1 t
L 1
2
A b

2
O
L 1 m
A b
 m
 t
 

1

Q m

1 t
L


1
2
A b

2
Q

1
L 1
1
A b

1
L
L
1 m
A b
 m
 t
 
0

Q m

1 t
1
2
A b

2
Q

1
Q
0
L 1
1
A b

1
Q
0
L

1
0









 f





 f f b m
 b

2 b

1 f
0 b







(B-3) where P t

B t p  t
1 L t
1 A t b and Q t

B t p  t
1 L t
1 . The first term on the right-hand side of equation
(B-3) represents the temporal fluctuation in the outputs of the wastes embedded in the final bills of primary and secondary products. The second term represents the temporal fluctuation in the outputs of the wastes embedded in the final disposal. Accordingly, by estimating the respective terms, we can explain about not only the ordinary growth trajectory of primary and secondary products but also that of wastes. In this case, we no longer need the inverse of B p .
32

Figure
X
Figure 1.
Y
X
Overlapping zone
Figure 2.
Y
X
Overlapping zone
Figure 3.
X
Y
Overlapping zone
Figure 4.
33
Y

Time
18
16
14
12
10
8
6
0
500
4
2
400
XZ surface
300 q t p
200
100
0 0
Figure 5.
50
YZ surface
XY surface
100 q t b
150
200
Time
18
16
14
12
10
8
6
0
500
4
2
400
XZ surface
300 q t p
200
100
F


,

,


F


 
,
F
 
,
 
 F
B ’
0
O
0
B
YZ surface
XY surface
Area OAB=
Q b * kt
50
A ’
Area OA’B’= 
A
100 q t b

Q b kt
*
150
Figure 6.
200
34

Figure Captions
Figure 1. Illustration of the condition 1
Figure 2. Illustration of the condition 2
Figure 3. Illustration of the condition 3
Figure 4. Illustration of the condition 4
Figure 5. Diagram of balanced growth path in our example
Figure 6. Changes in balanced growth path
35