The Invisible Multipliers of Joint-Products
SHIGEMI KAGAWA*
HAJIME INAMURA
YUICHI MORIGUCHI
Abstract.
This paper proposes a hybrid input-output model to
estimate the intermediate requirements embodied in the final disposal
such as reclamation and incineration of non-marketable scraps and
wastes from industries or households. The model is based on a mixed
technology assumption in order to connect the monetary distribution of
the ordinary goods and services with the physical distribution of the
scraps and wastes as joint-products. Moreover, some demand-pulled
invisible multipliers in terms of the scraps and wastes were explored
by performing a numerical simulation. From the invisible multipliers,
we find the paradoxical phenomenon that the reduction in the amount
of the final disposal and the promotion of material recycling decreases
the intermediate demand of the scraps and wastes for material
recycling if there is a lag in the introduction of the appropriate recycling
technology.
Keywords: Joint-products, scraps and wastes, final disposal, mixed
technology
assumption
* 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: [email protected].
Hajime Inamura, Professor, Graduate School of
Information Sciences, Tohoku University, Aoba, Aoba-ku, Sendai, 980-8579, JAPAN, Phone: +81-22-2177497,
Fax: +81-22-2177494, E-mail: [email protected]. Yuichi Moriguchi, Research Center for
Material Cycles and Waste Management, National Institute for Environmental Studies, Onokawa 16-2,
Tsukuba,
Ibaraki,
[email protected].
305-0053,
JAPAN,
Phone:
+81-298-502540,
Fax:
+81-298-502572,
E-mail:
The author gratefully acknowledges the helpful comments of Thijs ten Raa and Erik
Dietzenbacher.
2
1. Introduction
Various economic and environmental problems brought about by material
recycling and final disposal in terms of the by-products and scraps/wastes
jointly produced by industries and households have been emphasized from
both the social and political points of view.
Under this background, the
analytical model based on a scientific methodology, to evaluate the economic
and environmental effects in terms of the by-products and scraps/wastes has
been expected in various fields such as environmental economics and waste
management. In order to conduct it, several obstacles should be surmounted
under a certain hypothesis. The most troublesome obstacle is the treatment of
joint-production.
John von Neumann (1945) and Piero Sraffa (1960) proposed the general
equilibrium
models
with
joint-production
and
mathematically
and
economically discussed the existence of a non-negative balanced solution and
the effects of the changes in the rate of profits and/or exploitation on economic
distribution.
particular.
They greatly contributed to mathematical economics in
However, the applied studies using a similar model are,
unfortunately, very few due to some limitations such as the rigidity of the
model and the actual data availability (see for example Steenge, 1977). On the
other hand, Leontief’s input-output theory (1941, 1953, 1986) gets, we believe,
results in empirical and/or applied economics steadily.
The discussion of the input-output model with joint-production arose
when the Cambridge group proposed that the commodity technology
assumption should be positively employed in the construction of input-output
analysis (Stone, 1961; U. N. Statistical Office, 1968). Gigantes (1970) threw
3
doubt on the single adoption of the commodity technology assumption and
proposed the mixed technology assumption that is composed of the
commodity technology assumption in terms of the primary and secondary
products and the industry technology assumption in terms of the by-products.
After that, Fukui & Senta (1985) mathematically evaluated the differences
among the four technology assumptions (lump-sum method, transfer method,
Stone method, and redefinition method) from the viewpoint of the equilibrium
outputs estimated by the technical coefficient matrix based on the respective
assumptions. Ten Raa et al. (1984, 1988) demonstrated the unsuitability of
the industry technology assumption and/or of the mixed technology
assumption by performing the simple numerical test and proposed the
by-product
technology
assumption,
so-called
CB-mixed
technology
assumption, as a fruitful assumption. In addition, Kop Jansen & ten Raa
(1990) proposed four desirable properties, material balance, financial balance,
scale invariance, and price invariance and mathematically proved that the
assumption satisfying all the properties is only the commodity technology
assumption.
Londero (1990, 1999) also supplemented the proofs.
More
recently, Londero (2001) endogenously treated the demand in terms of the
by-products under the profit-maximizing condition and explored the definition
of by-products from the viewpoint of their marginal cost structure. In this way,
the treatment of the by-products has been discussed from the viewpoints of
not only the input-output foundations but also the micro-economic
foundations.
Reviewing the previous studies mentioned above, we must say that the
practical economic model with joint-production still has not been proposed.
There exist mainly four problems.
The first problem is the separation of
4
primary products, secondary products, and by-products as ten Raa et al.
(1984) pointed out.
The primary products, secondary products, and
by-products that have respective markets should be precisely identified under
the technological nature. And also the scraps and wastes that have no market
prices should be identified at the same time.
This paper starts with the
assumption of the separation. In the next section, we will explain about its
treatments in Japanese statistics as well as our basic framework.
The second is the indeterminate characteristic of the economic system. As
Robert Solow pointed out in the critique of the Walras-Cassel model (see for
example Dorfman et al., 1958), the input-output system (or general
equilibrium system) with joint-production will generally become indeterminate
in the sense that the unique equilibrium solution can not be determined
because of the singular and/or non-square framework.1 In order to avoid thus
framework and to get the equilibrium solution, we need an inverse demand
function in terms of joint-products or a certain hypothesis such as the mixed
technology assumption and by-product assumption mentioned above.2
The third is that, in a real economic phenomenon, the supply and demand
of commodities are met by interaction among goods and services which can be
easily expressed in monetary terms, while scraps and wastes cannot be
completely expressed in monetary terms. It is clearly seen from the circular
flow of non-marketable products such as scrap plastic, scrap timber, scrap oil
and so on. These are not generally ordinary marketable goods but free goods
or bads (see Lager, 1998).
Then, the monetary distribution system and
physical distribution system cannot be connected due to the problem of
summation (see for example Forssell & Polenske, 1998). In order to solve it,
the so-called hybrid technical coefficient matrix proposed by Bullard &
5
Herendeen (1975a, 1975b) and by Lehbert (1980) should be employed.
The fourth is the limitations of the demand-driven model with
joint-production. If one hopes to estimate, for example, how much direct and
indirect by-product requirements (recycling materials) will be induced by the
exogenous final bills of the primary and secondary products in the near future
or how much direct and indirect pollutants will be increased and decreased by
controlling the amount of the final disposal of scraps and wastes, the
supply-driven model with joint-production should be proposed (see Ghosh,
1958).
However, in order to conduct it, we must accept the criticism by
Professor Oosterhaven (1988, 1989), the comments by Gruver (1989) and by
Rose & Allison (1989), and the reinterpretation of Ghosh’s model (see
Dietzenbacher, 1997) at the same time.
‘The supply-driven input-output model was originally (Ghosh, 1958)
developed not to model the effects of supply shocks in market economies, but
to describe the functioning of centrally planned economies.’ (Oosterhaven,
1988, p. 214)
This paper does not, unfortunately, provide the answer to the criticism
and comments. The purposes in this paper are as follows. The first is to
propose the demand-driven model and supply-driven model including the
physical circular flow of the scraps and wastes from industries and
households.
The supply-driven model without plausibility is negatively
presented in the appendix A. The second is to explore some invisible factor
multipliers of the non-marketable scraps and wastes by performing a
numerical simulation.
Using our model, readers can quantitatively evaluate the value of the
6
intermediate requirements embodied in the final disposal of the scraps and
wastes from firms or households or the value of the economic and
environmental impacts of the recycling structural changes and final disposal
changes. Thus, multipliers as economic and environmental inventories would
be useful for discussing the significance of the material recycling and waste
management.
2. The Basic Framework
In this section, we will explain about the basic framework. In order to simplify
the explanation, let us define the following notations first.
V p   vijp i, j 1,, n
=make matrix of primary and secondary products
V mb   vijmb i, j 1,, n =make matrix of by-products
V ns   vijns i, j 1,, n =make matrix of scraps and wastes
U p   uijp i, j 1 ,, n
=use matrix of primary and secondary products
U mb   uijmb i, j 1,, n =use matrix of by-products
U ns   uijns i, j 1,, n =use matrix of scraps and wastes
q p   qip i 1,, n
=column vector of commodity outputs of primary and secondary
products
q mb   qimb i 1,, n =column vector of commodity outputs of by-products
q ns   qins i 1,, n =column vector of commodity outputs of scraps and wastes
g p   gip i 1,, n
=column vector of industrial outputs of primary and secondary
products
g mb   gimb i 1,, n =column vector of industrial outputs of by-products
g ns   gins i 1,, n =column vector of industrial outputs of scraps and wastes
7
f p   fi p i 1,, n
=column vector of final bills of primary and secondary products
f mb   fi mb i 1,, n =column vector of final bills of by-products
f ns   fi ns i 1,, n =column vector of final disposal of scraps and wastes
y p   yjp  j 1 ,, n
=row vector of value added of primary and secondary products
y mb   yjmb  j 1,, n =row vector of value added of by-products
y ns   yjns  j 1,, n =row vector of adjustment of scraps and wastes
T = transposition of the matrix and vector.
Here, the superscripts p, mb, and ns represent the primary and secondary,
marketable by-products, and non-marketable scraps and wastes respectively.
As is seen from the above-mentioned notation, the key feature is that the basic
framework was separated into three layers which show the make-use
structures in terms of primary and secondary products, by-products, and
scraps/wastes respectively (see Figure 1 and 2).
Here, one might have
questions about separating their structures. We must answer the questions.
Fortunately, in Japan, we can obtain the input-output table on the
by-products and scraps that Japanese statistics defined in their own way.
Needless to say, the definition is crucial for the separation. According to the
explanation of the Japanese input-output table for 1995, the definition is as
follows. Consider one production process that technologically produces other
goods except the concerned good. Then, in the case that there exist the sectors
which produce the other goods as primary products, we call them “by-products”.
In the case that there exist no other producible sectors, we call them “scraps”. It
is natural that the by-products should be identified from the viewpoint of the
commodity nature. In fact, it would be possible to make the input-output
table on the by-products and scraps by revising the existing input-output
8
table and by adding the physical engineering data in terms of scraps and
wastes.
The first layers Ⅰ and Ⅳ shown in Figure 1 and 2 describe the make and
use balance in terms of the primary and secondary products, respectively.
The second layers Ⅱ and Ⅴ represent the make and use balance in terms of
the by-products which have a market and can be completely expressed in
monetary terms ($). The third layers Ⅲ and Ⅵ stand for the make and use
balance in terms of the scraps/wastes which have no market and can hardly
be expressed in monetary terms. Accordingly, these layers were expressed in
physical terms (ton).
What are the advantages of these separations? Let us consider the role of
final disposal in a national economy and in an environment. In the previous
way of thinking, the amount of final disposal was just the outcome of
productive activity or consumption. The old-fashioned way of thinking largely
rested on the paradigm of mass-production, mass-consumption and
mass-disposal. Therefore, we did not need to consider the influences of the
shifts in the joint-production and final disposal on the national economy and
environment. The waste management policy today, however, obviously affects
the economic system, the economic growth and the emission of various
pollutants.
If one might evaluate the factor requirements under the
temporally stable production technology containing the recycling technology,
the control of the final disposal definitely affects them in the case that there
exists no illegal disposal.
Of course, in the case that there exists illegal
disposal, the factor requirements embodied in the final disposal may remain
constant or decrease. This is related to the concept of effective demand. By
using the separated framework as shown in Figures 1 and 2, it is possible to
9
estimate the factor requirements embodied in final disposal of scraps/wastes
from industries and households.
Here, let us easily explain about the circular flow under the basic
framework. The important point is that there exist no scraps/wastes without
the (joint-) production of primary and secondary products. This implicitly
assumes that the scraps/wastes from the stock and durable consumption
goods do not exist. Although it is unrealistic, the circular flow in terms of the
scraps/wastes from stock or from durable consumer goods should be
discussed separately.
See Figures 1 and 2.
If the amount of the
(joint-production-related) final disposal increases under the fixed production
technology, what will happen to the amount of the marketable goods and
services? First, the final disposal of the scraps/wastes f ns has to be jointly
produced by the primary and secondary commodity technology and
by-product technology as q p  q mb .
Subsequently, the industries inevitably
produce the products as g p  g mb and jointly produce the scraps/wastes as g ns
at the same time. After that, the scraps/wastes are absorbed by industrial
input technology and the input requirements of the scraps/wastes are
determined as q ns . Since the input requirements have to be jointly produced,
the iteration continues infinitely. This is just a demand-driven model with
physical joint-production. The arrows in Figure 1 and 2 show the circular flow
of the demand-driven system. Considering the order of the system, we can
easily understand the input requirements of the scraps/wastes embodied in
the final bills of the primary and secondary products and by-products.
From the explanation, readers would notice that we looked at it from a
different angle.
10
Com.
Primary and secondary products Ⅰ
Ind.
gp
V p $ 
Ⅱ
 q p T
V
mb
$ 
g mb
by-products
Ⅲ
 q mb T
g ns
V t 
ns
scraps/wastes
Com: commodity
 qns T
Ind: industry
Use → Make
Figure 1. Multi-layer make structure in the demand-driven model
Ind.
Com.
g 
Primary and secondary products
U p $ 
yp
p T
Make → Use
ymb
 g mb T
Make → Use
Ⅳ
by-products
U
mb
Ⅴ
$ 
scraps/wastes
U ns t 
fp
qp
f mb
q mb
f ns
qns
Ⅵ
Use → Make
y ns
 g ns T
Figure 2. Multi-layer use structure in the demand-driven model
11
3. The Invisible Multipliers of Joint-Products
In this section, we shall formulate the demand-driven model with
non-marketable joint-production and drive the invisible factor multipliers
from it. The model is based on both assumptions of a commodity technology
and of an industry technology, in short, mixed technology assumption. Hence
it is assumed that the production levels of the primary and secondary
products within industries are technologically constrained, and the market
shares of the by-products and scraps/wastes from industries and households
are temporally stable.
Now let us recall that V p is the make matrix that represents the industrial
outputs of primary and secondary products and V mb is the make matrix that
represents the industrial outputs of by-products. Since both matrices are
expressed in identical monetary terms ($), the make matrix showing the
industrial outputs of the primary and secondary products and of the
by-products can be easily computed as V p  V mb . However, in this study, since
the outputs of the scraps/wastes from industries and households were
expressed in physical terms (ton), in order to treat the physical circular flow of
the non-marketable goods or bads, it consequently brought about the problem
of summation. Forssell & Polenske (1998) pointed out the similar problem in
handling the free goods in the sense that they have no market prices. In order
to solve it, we need some devices on the foundation of input-output analysis
(see for example Millar & Blair, 1985). We employed the hybrid make-use
framework to connect the monetary circular flow of the primary and secondary
products, and of the by-products, and the physical circular flow of the
scraps/wastes (see for example Kagawa & Inamura, 2001).
12
Subsequently, let us formulate the demand-driven model based on the
hybrid make-use framework. Considering the balance of the use structure of
the primary and secondary products and of the by-products (see the layers Ⅳ
and Ⅴ in Figure 2), we have
B p g p  g mb   f p  q p
(1)
B mb  g p  g mb   f mb  q mb
(2)
where B p and B mb are the input coefficient matrices of the primary and
secondary and of the by-products respectively. The respective matrices can be
obtained by the equations Bijp  uijp  g jp  g jmb  and Bijmb  uijmb  g jp  g jmb  where
 Bijp i, j 1 , ,n and  Bijmb i, j 1 , ,n are the (ij)-th elements of B p and B mb . On the other
hand, the material use balance of the scraps/wastes can be written as
B ns  g p  g mb   f ns  q ns
(3)
where B ns is a hybrid input coefficient matrix which can be obtained by
computing Bijns  uijns  g jp  g jmb  where  Bijns i, j 1 , ,n is the (ij)-th element of B ns (see
the layer Ⅵ in Figure 2). f b would mainly be composed of the final disposals
such as waste reclamation, waste incineration, and exports. The hybrid input
coefficient matrix B ns plays a crucial role as the interface between the
monetary distribution and physical distribution in our framework. Although,
in the traditional monetary input-output system, it was very difficult to
estimate the factor requirements such as energy, materials, by-products,
scraps/wastes embodied in the final disposal the hybrid use system as shown
in equations (1)～(3) enable us to do it easily.
It is also well known that the mixed balance of the make structures of the
primary and secondary products and of the by-products can be written as the
13
following relationship (see Gigantes, 1970).
g p  g mb  C-p I n  D mb  i   D mb q p  q mb   R pmb q p  q mb 
T
(4)
C -p is the inverse of the output coefficient matrix of the primary and secondary
products C p can be obtained by the equation Cijp  vjip gip (see the layer Ⅰ in
Figure 1). D mb is the market share matrix of the by-products which can be
obtained by the equation Dijmb  vijmb  qjp  qjmb  (see the layer Ⅱ in Figure 1). I n
denotes the n-dimensional identity matrix and i describes the n-dimensional
column vector whose elements are one.
^ stands for the matrix
diagonalization. We replaced the mixed market share matrix by R pmb .
The balance of the make structures of the scraps/wastes can also be
written as
g ns  D ns q p  q mb 
(5)
where D ns represents the hybrid market share matrix of the scraps/wastes
revised in order to avoid the problem of summation (see the layer Ⅲ in Figure
1).
Needless to say, the matrix can be computed by the equation
Dijns  vijns  qjp  qjmb  .
Moreover, considering the relationship between q p , q mb and q ns , we have
q ns  D ns  i q p  q mb   q p  q mb 
T
(6)
where  is the diagonal matrix whose elements are the column sum of D ns .
As is seen from equation (6) and from the Figure 1 and 2,  clearly becomes a
singular matrix and does not have the unique inverse matrix in general. This
will consequently become the essential problem in our model. Therefore, the
inverse matrix was adjusted in order to overcome the problem. Let us define
ii and  ii as the diagonal elements of  and of the adjusted inverse matrix
14
01 respectively. Then, mathematically speaking, it was assumed that
 ii  1 ii if ii  0
 i  1,2,  , n  ,
(7)
and
 ii  0 if ii  0  i  1,2,  , n  .
(8)
The mathematical operation means that the primary and secondary products,
and by-products induced by the exogenous final demand are determined in
fixed market share with the intermediate scraps/wastes, while they are not
produced in the case that the market share of certain scraps/wastes does not
exist at all. However, even if the market share of certain scraps/wastes does
not exist, the firms would freely produce the primary and secondary products
in the real economy. The mathematical operation seems like a strange one in
a sense. Here let us recall that the essential problem discussed is to clearly
connect the circular flow of the primary and secondary products, and
by-products with that of the scraps and wastes. Of course, from equations (1),
(2) and (4), the factor requirements embodied in the exogenous final bills of the
primary and secondary products, and of the by-products can be estimated as
q p  q mb  I n  B p  B mb R pmb f p  f mb  .
1
It is, however, impossible to estimate
the scrap/waste embodied the exogenous final demand or to estimate the
primary and secondary products embodied in the exogenous final disposal.
The monetary input-output balances in equations (1) and (2) and the physical
input-output balance in equation (3) should be connected in order to attain it.
The connection inevitably needs the strong assumption as shown in equations
(7) and (8).
We shall consider the connection and formulate the
(scraps/wastes related-) monetary circular flow and its the physical circular
flow.
15
From (4) and (6), the relationship between q ns and g p  g mb can be written
as
g p  g mb  R pmb 01 q ns .
(9)
From the material balance in terms of the primary and secondary products,
by-products, and scraps/wastes, we have the two following relationships.
q ns  B ns g p  g mb   f ns  B ns R pmb q p  q mb   f ns
(10)
q p  q mb  B p  B mb g p  g mb   f p  f mb  B p  B mb R pmb10q ns  f p  f mb
(11)
Hence, substituting equation (10) into (11), the production formula of the
primary and secondary products can be finally obtained as
q p  q mb  B p  B mb R pmb10q ns  f p  f mb
 B p  B mb R pmb 10 B ns R pmb q p  q mb   f ns   f p  f mb
 B p  B mb R pmb 01 B ns R pmb q p  q mb   B p  B mb R pmb 01 f ns  f p  f mb
p
p
mb
pmb 1 ns
p
B

 f p 
L

B
R

f  L
f mb 


0

1
(12)
2
where
Lp  I n  B p  B mb R pmb 01 B ns R pmb   I n  Q p  .
1
1
(13)
The first term  1 on the right-hand side of equation (12) presents the primary
secondary products and by-products embodied in, for example, the waste
incineration and the waste reclamation expressed in physical base.
The
second term  2 presents the factor requirements embodied in the
accompanying final bills of the primary and secondary products and of the
by-products in monetary base.
The first term definitely asserts that the
industrial scraps/wastes incinerated and/or reclaimed are jointly produced
16
by the firms, and the factor requirements such as energy and materials are
directly and indirectly required in its joint-productive processes. Hence, if one
might consider the augmented input-output model, the income distributions
directly and indirectly induced by the activity such as the final disposal of the
scraps/wastes can be easily estimated.
Subsequently, let us consider Neumann’s series of equation (12). Then,
the direct and indirect input requirements of the primary and secondary
products can be decomposed as
q p  q mb  q p  q mb 
direct
 q p  q mb 
indirect
p
mb
1 ns
p
mb
p
1 ns
p
p
 pmb 
 f
 pQ
 p Bmb
Rpmb
f p 

 
B
B
 f

f mb
0 f
0 f
R


L
B


L
Q


 1direct
1indirect
 2direct
(14)
 2indirect
where the first term  1direct on the right-hand side of equation (14) is the direct
factor requirements needed in order to jointly produce the (final disposal
related-) scraps and wastes; the second term  2direct the direct factor
requirements of the primary and secondary products; the third term  1indirect the
indirect factor requirements needed in order to jointly produce the (final
disposal related-) scraps and wastes; and the fourth term  2indirect is the indirect
factor requirements of the primary and secondary products. It should be
noted that Lp  I n  Q p  Q p    . Thus decomposition enables us to pursue
2
the economic and environmental impacts of the level and pattern shifts in the
exogenous final disposal  f ns from the viewpoint of production process.
Similarly, substituting equation (11) into (10) yields
q ns  B ns R pmb q p  q mb   f ns
 B ns R pmb B p  B mb R pmb 10 q ns  f p  f mb   f ns
 B ns R pmb B p  B mb R pmb 10 q ns  B ns R pmb f p  f mb   f ns
17
ns
ns ns
 f p
 
L
B ns
Rpmb

f mb
f


L
3
(15)
4
where
Lns  I n  B ns R pmb B p  B mb R pmb01   I n  Qns  .
1
1
(16)
The first term  3 on the right-hand side of equation (14) presents the
scraps/wastes embodied in the exogenous final demand. The second term  4
presents the scraps/wastes embodied in, for example, the waste incineration
and the waste reclamation.
The second term asserts that the industrial
scraps/wastes incinerated and/or reclaimed are joint-produced by the firms,
and the scraps/wastes are also directly and indirectly utilized in its
joint-productive processes. Considering the expansion of (16), we have
q ns  q ns 
direct
 q ns 
indirect
ns
pmb
mb
ns
ns
pmb
mb
ns
ns ns
 p f
 ns ns
 p  f
 
B
R
Q
B
R
Q
f


f 
  f  L

f
L
 3direct
 3indirect
 4direct
(17)
 4indirect
where the first term  3direct on the right-hand side of equation (17) is the direct
scrap/wastes requirements needed in order to produce the final bills of the
primary
and
secondary
products;
the
second
term  4direct the
direct
scrap/wastes requirements; the third term  3indirect the indirect scrap/wastes
requirements needed in order to produce the final bills of the primary and
secondary products; and the fourth term  4indirect is the indirect scrap/wastes
requirements
needed
to
produce
the
final
disposal.
Note
that
Lns  I n  Qns  Qns    .
2
Since the traditional mixed technology model has not clearly been
separated into the circular flows induced by the final bills of ordinary goods
and by the final disposal of joint-products, it was difficult to identify the factor
18
requirements embodied in their respective final activities. The main feature in
our model is that the input-output structure was clearly decomposed in terms
of the marketable goods containing by-products and the scraps/wastes as
shown in Figures 1 and 2. Therefore, it is possible to estimate factor
requirements of the ordinary goods and/or services embodied in the final
disposal of the scraps and wastes that have no market prices.
Here let us mention about the limitations of the demand-driven model.
We must recognize that it is impossible to completely cover all the circular flow
of scraps and wastes by means of only the demand-driven model. For example,
let us consider the material flow of a disused car. The disused car from the
final consumer such as households and industries, clearly, directly and
indirectly flows in the direction of supplier that produces the intermediate
goods such as for example recycled plastics, recycled irons and so on. If one
hopes to know how much energy and materials are directly and indirectly
induced by the material recycling of the disused car from the household sector,
the planned economy in the supply side should be supposed (see appendix A
for the formulation).
In the next section, the simple numerical simulation using the
demand-driven model is performed.
4. The Simple Numerical Simulation
Let us adduce the simple numerical example as shown in Figure 3.3 From the
figure, we shall consider the inter-industry relations between the wood
industry and the pulp & paper industry.
19
Figure 3 illustrates that the pulp & paper industry produces 100 dollars of
paper products by using 200 dollars of the wood products as the raw materials,
and by using 270 dollars of the marketable wood-chips and 30 tons of the
non-marketable wood-chips as the fuels. It also shows that 250 dollars of
scrap papers and 20 tons are jointly produced at the same time. On the other
hand, the wood industry produces 200 dollars of the wood products by
utilizing 100 dollars of the paper, 250 dollars of the marketable scrap papers,
and 20 tons of the non-marketable scrap papers as the raw materials. One
might notice that the industries also use their own products as the primary
products, by-products, and scraps/wastes.
The final disposal sector such as waste reclamation and waste
incineration absorbs 100 tons of the non-marketable wood chips and 40 tons
of the non-marketable scrap papers. They were not immediately utilized by
industries due to the poor quality. 400 dollars of the wood products, 500 of
paper products, and 20 of the scrap papers are consumed in order to fill the
final demand such as private consumption, public consumption, and exports.
Thus inter-industry relations can be described as the make-use tables in
terms
of
the
primary
and
secondary
products,
by-products,
and
scraps/wastes (see appendix B).
From the numerical example, the input coefficient matrices in terms of the
primary and secondary products, by-products, and scraps/wastes and the
output coefficient matrices can be obtained. Applying the input coefficient
matrices and output coefficient matrices to the demand-driven models with
joint-production yields
q p  q mb  L p B p  B mb R pmb 10 f ns  L p  f p  f mb 
20
4.867 4.700 100 1.615 3.166  100 1 0 400 0.179 0.318 400
(18.a




2.333 7.300   40  1.691 3.359  40  0 1  520 0.184 0.342 520

  
 
 
 1direct
 2indirect
1indirect
 2direct
)
6.482 7.866  100 1.179 0.318 400
,


4.024 10.659  40  0.184 1.342  520 



 
1
2
(18.b)
and
q ns  Lns B ns R pmb  f p  f mb   Lns f ns
0.013 0.019 400 0.005 0.01  400 1 0 100 0.156 0.298 100
(19.a




0.013 0.025 520  0.006 0.013 520  0 1   40  0.182 0.365  40 

  
 
 
 4direct
 4indirect
 3direct
 3indirect
)
0.018 0.029 400 1.156 0.298 100
.


0.019 0.038 520 0.182 1.365   40 

 
3
4
(19.b)
The first term  1direct on the right-hand side of equation (18) presents the direct
and indirect factor requirements (wood products and pulp & paper) induced
by the final disposal of the wood-chips and scrap papers. From the first term
of equation (18.a), the direct backward effects of the shifts in the final disposal
of the wood-chips (100 tons) on the commodity outputs of the wood products
and pulp & paper products can be estimated as $486.7 and $233.3
respectively. Similarly, the direct backward effects of the shifts in the final
disposal of the scrap papers (40 tons) can be calculated as $188 and $ 292.
The second term 1indirect in equation (18.a) shows that the shifts in the final
disposal of the wood-chips (100 tons) brought about by a $161.5 output of the
wood products and $169.1 of pulp & paper products as well as 40 tons of
scrap papers yields to an indirect usage of $126.64 of the wood products and
to $134.36 of the pulp & papers.
21
Comparing the factor intensity of the wood chips (10.506 $/ton) with that
of the scrap papers (18.525 $/ton), we can understand that the latter is 1.76
times larger than the former. Namely, the results assert that the economic
value of the scrap papers is higher than that of wood chips. If it is supposed
that both energy intensities are same, the results also indicate that the scrap
paper is an energy-intensive waste in comparison with that of wood chips.
From the first term  1 on the right-hand side of equation (18.b), the direct
and indirect sectoral backward effects of the final disposal of the wood-chips
(100 tons) can be computed as $648.2 of wood products and $402.4 of pulp &
paper respectively.
Similarly, the sectoral backward effects of the final
disposal of the scrap papers (40 tons) are $314.6 of wood products and $426.4
of pulp & paper products. Totally, the final disposal of the wood-chips and
scrap papers induces 1,792 dollars of the wood products and pulp & paper
products. We can easily understand the backward effects of the final demand
from the second term  2 on the right-hand side of equation (18.b).
The first term  3 on the right-hand side of equation (19.b) would be
important for discussing the strategy of waste management.
Figure 4
illustrates the movement of the physical output of the wood-chips and scrap
papers induced by the final demand. The point A in Figure 4 is an equilibrium
output and can be easily calculated as (0.018  400+0.029  520,
0.019  400+0.038  520)=(22.28, 27.36). The total output is 49.64 tons. Since
the final demand vectors of wood products and pulp & paper products are
definitely nonnegative (see appendix B), [B, C] is the feasible interval. When it
holds the inequality q 1ns  q 2ns   0 where  0 represents the tolerated level of
final disposal, all the points in the triangle OBC are feasible.
If one hopes to maximize the final demand in terms of paper & paper
22
products under the constraint q 1ns  q 2ns   0 , the equilibrium point can be
obtained as B (24.15, 25.49). Accordingly, the maximum value of pulp &
paper output can be estimated as 1341.62. In the case, of course, that the
final demand in terms of wood products is zero. Similarly, if one hopes to
maximize the final demand in terms of wood products under the same scrap
and waste constraint, the equilibrium point can be obtained as C (21.49,
28.15) and the maximum value of wood products can be estimated as 740.90.
Considering that the maximum value of pulp & paper is 1.8 times larger than
that of wood products, it can be understood that the scraps and wastes are
more sensitive to the final demand of the wood products rather than to the
final demand of the pulp & paper. Even if the constraint value becomes larger
or smaller than 49.64, there always exists the feasible intervals such as [B’, C’]
and [B”, C”] and we can freely control the final demand within them.
Figure 5 illustrates that the control of the final disposal affects the
induced effects of the intermediate scraps and wastes. In this example, the
equilibrium point is A (127.52, 72.8). Hence, the tolerated level is 200.32 tons.
If the policy makers determine to reduce the amount of final disposal without
the additive strategy and to change from the point A to A’ (see Figure 5), the
intermediate demand of the wood chips and scrap papers also decrease at
same speed in this case. Environmental policy makers hope to eliminate the
various concerned pollutants, for example dioxin and PCB etc., by directly
reducing the amount of the final disposal and by promoting the material
recycling of scraps and wastes at the same time. However, this strategy,
unfortunately, brings about a paradoxical economic phenomenon such as the
decrease in the intermediate demand of the scraps and wastes for the material
recycling.
23
One might consider that it is a strange phenomenon, because there will be
no such paradox to occur in the case that the (recycling-) technology changes
to cover the reduction in the amount of the final disposal instantaneously.
However, in case that it takes a certain time to change the production
technology, we definitely have the above-mentioned paradox. Let us explain
about it geometrically by using Figure 5.
From the second term on the
right-hand side of (19.b), recall that equilibrium point A can be obtained as
1.156 
0.298
 4  100  
 40  


0.182
1. 365 
 f 1ns  a 1  f 2ns  a 2 .
(20)
Then, if the final disposals, f 1ns and f 2ns , decrease to  f 1ns ' and  f 2ns ' , is it
possible for the equilibrium outputs of the wood-chips and scrap papers to
remain constant? The answer is yes. It can be easily proven as
α 4  f 1ns  a 1  f 2ns  a 2
  f 1ns 'a 1   f 2ns 'a 2
 f1ns '  f1ns ,  f 2ns '  f 2ns
  1 f 1ns   a 1  1    2 f 2ns   a 2  2 
(0  1  1, 0   2  1 )
(21)
where  1 and  2 are real numbers in the open set (0, 1). In this way, we can
always choose the real numbers in terms of the respective final disposals.
This also requires that the embodied intensity vectors, a 1 and a 2 , have to
change into a 1  1 and a 2  2 respectively.
This obviously implies that the
embodied intensities go up in same proportion. The changes correspond to
the technological progress to cover the reduction in the final disposal and can
be defined as the progress in the material recycling.
What the policy makers should emphasize is the speed of the changes.
Even if they hope to reduce from f 1ns to  f 1ns ' in the technological progress
24
( a 1  a 1  1 and a 2  a 2  2 ), the reduction in the final disposal f 1ns has
nowhere to go.
Under these circumstances, the further decrease in f 1ns
brings about the decrease in the intermediate demand of the wood-chips and
scrap papers and would have nowhere to go because of the above-mentioned
paradox.
Final disposal (waste reclamation and waste
incineration etc.)
Wood-chip 100TN
$ 600
$130
20TN
Scrap paper 40TN
Paper $100
Scrap paper $250
Wood
Industry
Scrap paper 20TN
Wood product $200
Wood-chip $270
Wood-chip 30TN
Pulp &
Paper
Industry
Scrap paper $20
Wood product $400
$400
$330
40TN
Paper $500
Final demand (private and public consumption,
exports etc.)
Figure 3. Simple example of the inter-industry relations
25
B’
Physical output of scrap papers (ton)
q 2ns 30
B
C’
A
27.36
●
C
B”
20
C”
q1ns  q 2ns   0
●
q 1ns  q 2ns   0  49. 64
10
●
q 1ns  q 2ns   0
0
O
10
20 22.28
30
Physical output of wood-chips (ton)
40
q1ns
Figure 4. Backward effects of the scraps and wastes induced by the final
demand
q 2ns 200
B’
Physical output of scrap papers (ton)
B
160
B”
120
80
●
72.8
●
A
●
A’
40
C
C”
q1ns  q 2ns   1  200.32
●
0
O
40
80
120
C’
160
200
127.52
Physical output of wood-chips (ton)
240
140
q 1ns
Figure 5. Backward effects of the scraps and wastes induced by the final
disposal
26
5. Conclusions
This paper succeeded in connecting the monetary distribution in terms of the
primary and secondary products and that of the by-products with the physical
distribution in terms of the non-marketable scraps and wastes by applying the
appropriate hybrid make-use matrices to the adjusted mixed technology
model. The main feature of our model is that the final disposal was definitely
dealt with into the framework.
Hence the definite dealing enables us to
estimate the intermediate requirements directly and indirectly induced by the
scrap and waste emissions and to estimate the scrap and waste requirements
directly and indirectly required to produce the final bills of the primary and
secondary products and of the by-products and to finally dispose them.
Considering the invisible multipliers in terms of the scrap and waste
requirements, we can interpret the paradoxical phenomenon such that
reducing the amount of the final disposal and promoting the material recycling
of scraps and wastes at the same time, bring about the decrease in the
intermediate demand of the scraps and wastes for the material recycling in
case that it takes a certain time period to introduce the appropriate recycling
technology.
Finally, let us mention about the further studies. First, by performing the
structural decomposition analysis of the demand-driven model, the sources of
the changes in economic system on the factor requirements and employment
can be quantitatively identified as the effects of (1) technological changes of the
primary and secondary products, (2) technological changes of the by-products
and scraps/wastes, (3) final bill shifts of the primary and secondary products,
27
and (4) final disposal shifts of the scraps and wastes (see for example Rose &
Casler, 1996; Dietzenbacher & Los, 1998).
Second, in fact, it would be difficult to completely identify the effects of the
recycling technology changes of the by-products and of the scraps/wastes.
Considering the recycling process of the paper products produced by the
joint-use of the virgin pulps and the pulps from the scrap papers, we can
understand that it is difficult to separate the additive activities required to
recycle the scrap papers from the production process of the paper products in
spite of the existence of the definite recycling activity such as the collection
activity of the scrap papers.
In order to examine the intermediate
requirements induced by the recycling technology, the organic relationship
between the conventional manufacturing technology of the primary and
secondary products and the recycling technology of the by-products and
scraps should be precisely identified by using the engineering production
function.
Naturally, the empirical analysis should be performed.
The
empirical analysis enables us to estimate the effects of the structural changes
in the so-called venous economic system, which cannot be completely
described in monetary terms, on the intermediate requirements containing the
energy-environmental inventories.
We think that the proposed model opens a field of interdisciplinary
research
between
applied
economics
and
resource-environmental
management.
Notes
1. ten Raa (1986) introduced the generalized inverse matrix to solve the singularity of
28
capital coefficient matrix.
2. Londero (2001) employed the by-product assumption to derive the demand function of
the by-products.
3. We used the imaginary values.
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31
Appendix
A.
Formulation
of
the
Supply-Driven
Model
with
Joint-Production
In this appendix, the supply-driven model is revised in order to express the
forward physical circular flow of the scraps and wastes from industries and
households.
Now, let us consider the balance of the use structure of the primary and
secondary products and of the by-products. Then, the two following
relationships like price and cost relations can be written as
q p  q mb T G p  y p  g p T ,
and
q p  q mb T G mb  y mb  g mb T
where G p is the production coefficient matrix of the primary and secondary
products and can be computed by Gijp  uijp  qip  qimb  and G mb is the production
coefficient
matrix
of
the
by-products
and
can
be
obtained
by
Gijmb  uijmb  qip  qimb  .
On the other hand, the similar balance of the scraps/wastes are given as
q p  q mb T G ns  y ns  g ns T
where G ns represents the production coefficient matrix of the primary and
secondary products and can be obtained by Gijns  uijns  qip  qimb  .
In case of
supply-driven model in terms of scraps/wastes, y ns becomes the exogenous
driving force. Moreover, note that y ns is expressed in physical base.
Next, we shall consider the make structure in the supply-driven system.
From the output mechanism of the primary and secondary products and of the
by-products, we have
32
q p  q mb T  g p T H p  g p  g mb T H mb
 g p  g mb  g mb  H p  g p  g mb  H mb
T
T
 g p  g mb  I n  H mb i H p  H mb  g p  g mb  S pmb
T
T
where H p and H mb represent the output coefficient matrices of the primary
and secondary products and of the by-products. They can be obtained by
H ijp  vijp  gip  gimb 
and
H ijmb  vijmb  gip  gimb 
respectively.
Note
that
I n  H mb i H p  H mb  was replaced by S pmb . Moreover, considering the output
mechanism of the scraps/wastes, the following relationships can be
formulated as
q ns T  g p  g mb T H ns ,
g ns T  g p  g mb T H ns i  g p  g mb T  .
Here, H ns is the hybrid output coefficient matrix of the scraps/wastes and can
be obtained by H ijns  vijns  gip  gimb  .  is the diagonal matrix whose elements
are the row sum of H ns . Since the matrix  will clearly become a singular
matrix, the inverse of  was also adjusted as  01 like in equations (7) and (8).
From the above-described equations, the production formula in terms of
the scraps/wastes can be written as
g ns T  q p  q mb T G ns  y ns
 g p  g mb  S pmbG ns  y ns
T
 q p  q mb  G p  G mb S pmbG ns  y p  y mb S pmbG ns  y ns
T
 g ns   01 S pmb G p  G mb S pmbG ns  y p  y mb S pmbG ns  y ns
T
 y p  y mb S pmbG ns Mns  y ns Mns ,
 
1
(A1)
2
where
33
Mns  I n   01 S pmb G p  G mb S pmbG ns  .
1
The first term 1 on the right-hand side of equation (A1) presents the scraps
and wastes that are induced by the exogenous supply of the primary and
secondary products and directly and indirectly required in the planned
economy. The second term  2 presents the scraps and wastes induced by the
exogenous supply of the scraps and wastes.
Next, considering what is the supply-driven input-output system in terms
of the primary and secondary products that are induced by the exogenous
final bills of the scraps/wastes, the following relationships can be derived from
equations (16)～(21).
g p  g mb T  q p  q mb T G p  G mb   y p  y mb
 g p  g mb  S pmb G p  G mb   y p  y mb
T
 g ns   01S pmb G p  G mb   y p  y mb
T
 g p  g mb  S pmbG ns  01S pmb G p  G mb   y ns  01S pmb G p  G mb   y p  y mb
T
1
pmb
p
mb
p
p
p
G 
M
y 
,
 yns


G

y mbM
0 S






(A2)
4
3
where
M p  I n  S pmbG ns  01S pmb G p  G mb  .
1
The first term  3 on the right-hand side of equation (A2) represents the
forward linkages of the primary and secondary products induced by the
exogenous supply of the scraps and wastes. The second term  4 describes the
forward linkages of the primary and secondary products induced by the
exogenous supply of the primary and secondary products. Hence the forward
factor requirements induced by the exogenous demand in physical base can
be easily estimated. The analytical model would be useful to examine, for
34
example, the impacts of the changes in the input-output structure of the
scraps and wastes produced by industries and households on the factor
requirements directly and indirectly required in the planned economy.
35
Appendix B. Make-Use Tables for the Numerical Simulation
Table B1. Make-use table of the primary and secondary products in dollars
Pulp &
paperC
WoodC
WoodI
WoodC
Pulp
paperC
WoodI
Pulp
paperI
Pulp &
paperI
F. D.
T. D.
600
200
400
1200
100
400
500
1000
&
1200
0
0
1000
1200
&
1000
V. A.
T. S.
1200
1000
500
400
1200
1000
Notes: The superscripts C and I denote commodity and industry respectively. F. D. and T. D. represent
final demand and total demand respectively. V. A. and T. S. stand for value added and total supply
respectively.
Table B2. Make-use table of the by-products in dollars
Pulp &
paperC
WoodC
WoodC
Pulp
paperC
Pulp &
paperI
F. D.
T. D.
130
270
0
400
250
330
20
600
&
WoodI
Pulp
paperI
WoodI
400
0
400
0
600
600
&
V. A.
T. S.
400
600
20
0
400
600
Notes: The superscripts C and I denote commodity and industry respectively. F. D. and T. D. represent
final demand and total demand respectively. V. A. and T. S. stand for value added and total supply
respectively.
Table B3. Make-use table of the scraps/wastes in ton
Pulp &
paperC
WoodC
WoodI
WoodC
Pulp
paperC
T. D.
20
30
100
150
20
40
40
100
150
0
150
0
100
100
&
R. A.
T. S.
F. D.
&
WoodI
Pulp
paperI
Pulp &
paperI
150
100
110
30
150
100
Notes: The superscripts C and I denote commodity and industry respectively. F. D. and T. D. represent
final disposal and total demand respectively. R. A. and T. S. stand for the row of adjustment and total
supply respectively.
36