Shiinichiro Nakamura 1 , Kenichi Nakajima 2 , Yoshie Yoshisawa 2 , Yasushi Kondo 1 , and Kazuyo
Yokoyama 2
1 Waseda University, Graduate School of Economics, 169-8050 Tokyo, Japan
2 Tohoku University, Graduate School of Environmental Studies, 980-8579 Sendai, Japan
Corresponding author: Shinichiro Nakamura, nakashin@waseda.jp
Abstract
A new methodology for converting a monetary IO table (MIOT) into a physical IO table
(PIOT) is presented, that is fully consistent with mass conservation principle. Exploiting the triangular nature of the matrix of input coefficients, which is obtained by rearranging the ordering of sectors according to degrees of fabrication, the material composition matrix is derived, which gives the material composition of products. By use of the model, monetary input-output tables (MIOT) can easily be converted into a physical material flow account (or physical input-output tables, PIOT) of an arbitrary number of materials, and the material composition of a product can be decomposed into its input origin. This can significantly economize on the otherwise prohibitive cost that is associated with the independent compilation of a PIOT (EU Commission 2001). The proposed methodology was applied to
Japanese IO data of more than four hundred industry sectors for the flow of eleven base metals and nine types of plastics including scrap. The explanatory power of the methodology was tested for the weight and material composition of a passenger car. Final destination of plastics by products and final demand categories was also estimated.
Keywords: PIOT, MFA, WIO, plastics, base metals, mass conservation
This paper is concerned with a new methodology for converting a monetary IO table (MIOT) into a physical IO table (PIOT) that is fully consistent with mass conservation principle ， a principle that is fundamental to any environmental study. As Weisz and Duchin (2006) have correctly pointed out, if there is a single price vector (the same price of an input applies to all its users), PIOTs and MIOTs are equivalent except for the change of unit operation. It is obvious that the mass of a product is the mass of its material components. The mass of the material components of a product, however, is not equal to the mass of material inputs because of the occurrence of process waste: in general, some portion of material input does not enter the mass of output, and ends up as process waste. Proper consideration of this fact is fundamental for a PIOT that is consistent with mass conservation. The conventional MIOT is not consistent with the mass conservation principle because of its neglect of the waste flow.
Accordingly, the change of unit operation by use of a single price vector alone cannot obtain a PIOT that is consistent with mass conservation.
This shortcoming of the conventional MIOT is properly coped with by the Waste IO (WIO,
Nakamura and Kondo (2002)) that explicitly takes into account the waste flow and waste management processes. The WIO thus provides an optimal platform for obtaining a PIOT that is consistent with mass conservation. We convert a WIO into a PIOT with material

composition by use of the WIO-MFA methodology (Nakamura and Nakajima (2005)). The
WIO-MFA converts a monetary flow of inputs into a physical flow of materials, where the term “materials” is defined together with the terms “resources” and “products” by use of the triangular nature of the input coefficients matrix (Simpson and Tsukui (1965)) that is obtained by reordering the matrix according to degrees of fabrication.
Central to the WIO-MFA is the material composition matrix that gives the material composition of an output. By use of this matrix, any flow of inputs, monetary or physical, can be converted to its material components in mass. A distinguishing feature of this methodology consists in that it gives a PIOT with material composition, which is not the case with conventional PIOTs and MFAs. Another important feature is that it provides a general definition of “materials” that exclude the occurrence of double counting in MFA, a point which does not seem to have received scrutiny under general theoretical settings.
This methodology is applied to the flow of 11 types of metals and 9 types of plastics in the
Japanese economy at the level of 400 by 400 input-output sectors. While Duchin and Lange
(1998) considered the physical flow of plastics based on IOA and related MFA studies exist
(Patel et al. (1998) and Joosten et al. (2000)), to the base of our knowledge, none has considered the physical flow of metals and plastics simultaneously at this level of resolution.
2.1 The WIO-MFA model
Write A

[ a ij
] for an n
 n matrix of input coefficients, the i th row j th column element of which refers to the input of i per unit of output j . Because MFA is concerned with the flow of inputs with mass, it is necessary to remove from A the elements without mass such as services. Not all the inputs with mass enter (or embodied in) products: some are discarded as process waste, and some are of ancillary nature, and do not enter products. Using the mass filter
 to remove the elements without mass, and the yield matrix
 
[
 ij
] with
  
to accommodate for the discard of inputs in the production process, the part of A ij that constitutes the mass of output is then given by (see Nakamura and Nakajima [2005],
Nakamura, Nakajima, Kondo, and Nagasaka [2007] for details):
A
  
A (1)
The second ingredient of our model is based on the degree of fabrication among resources
( R ), materials ( M ), and products ( P ): before final products reach the consumers, resources have to be first processed into materials, and the materials thus obtained have to be further processed into products. By definition, the increase in the degree of fabrication occurs only in one direction (recycling of waste materials recovered from end-of-life products is consistent with this, because the latter has to be first reduced to materials before reentering the production process). In input-output analysis, this one-directional increase in the degree of fabrication is known to impose a triangular structure on the matrix of input coefficients, which can emerge when the ordering of sectors is appropriately altered to reflect the degree of fabrication.
Divide n outputs into mutually exclusive and nonempty sets of R ( n elements), M (
R n
M elements), and P ( n elements) such that A can be partitioned as follows
P

A



A
PP
A
PM
A
MP
A
MM
A
RP
A
RM
A
PR
A
MR



(2)
A
RR
We introduce the following set of assumptions with regard to the triangular structure:
1. Resources are not produced but given from the environment:
A iR where O is a zero matrix of an appropriate order.
2. Materials are made of resources:
A iM
   
} .
3. Products are made of products and materials:
A
RP

O .
O i { P M R } ,
With these conditions imposed, (2) becomes
A


A
PP
A
MP
O
A
O O
O O


.
O
RM
(3)
It then follows that any physical product can be reduced to elements in M , with its “ M composition” being given by the elements in the corresponding column of the following n
M
 n
P
matrix:
C
MP

A
MP

I

A
PP

A
PP
2 
A
PP
3 


A
MP

I

A
PP
 
1 
(4) where I refers to an identity matrix of order n . The i th row j th column element of this
P matrix, C , gives the amount of material i that is contained in a unit of product j . When
M P j all the materials are measured in kg, then
 i
C
M P j
gives the mass of a unit of j in kg, where the unit of j can be physical or monetary. Note that recycling of waste can easily be introduced into the above framework by counting waste as materials (see Nakamura and
Nakajima (2005) and Nakamura, Nakajima, Kondo, and Nagasaka (2007) for details). By use of C
MP
, the conventional input-output table can easily be converted into the corresponding flow of materials.
2.2 The data
The above methodology has been applied to the flow of base metals and plastics in the
Japanese economy. We consider as materials 11 types of metals consisting of pig iron, ferroalloys, copper, lead, zinc, aluminium, and their scrap, except for ferroalloys, and 9 types of plastics consisting of thermo-setting resins, polyethylene of low density, polyethylene of high density, polystyrene, polypropylene, polyvinylchloride (PVC), high-performance resins, other resins, and plastic scrap. The Japanese IO table for year 2000 is used as a major data source after having extended/modified it by use of detailed information on the production and supply of metals and plastics. With regard to plastics, the plastics products sector of the

original IO table was disaggregated into 10 product types: plastic film and seat (except PVC), vinyl chloride film and sheet, plastic board, pipe, and stick (except PVC), vinyl chloride board, pipe, and stick, plastic foaming products, industrial plastic products, reinforced plastic goods, plastic containers, miscellaneous plastic goods for daily use and table articles, and other plastic goods.
2.3 The results
For a product the unit of which is in monetary term, the column sum of C gives the
MP estimated metal and plastics weight of the product per 1 million yen. For a product for which reliable data are available on the weight, price, and its material composition, the predictive power of the model can be tested by comparing the estimated weight and composition with its real counterparts. A good example for this purpose is a passenger car. Table 1 shows that the estimate weight (in terms of metal and plastics) of a representative passenger compares fairly well with the values reported by car manufacturers. Figure 1 indicates that the model is able to reproduce the composition of iron, aluminium, copper, lead, and PVC fairly well.
Figure 2 gives the estimates of C for some other products.
MP
Weight of metals and plastics (kg)
JAMA 1997
897.8
JAMA 2001
904.5
WIO-MFA
892.5
Table 1: Comparing the (metal and plastics) weight of a passenger car reported by Japan Automobile
Manufactures Association and that estimated by the WIO-Model
Pb
Zn
PVC
JAMA01
JAMA97
WIO-MFA
Cu
Al
Other plastics
Fe
0.0
20.0
40.0
Composition %
60.0
80.0
Figure 1:
Comparing the metal and plastics composition of a passenger car estimated by the WIO-MFA model with those published by the Japanese Association of Automobile Manufacturers (JAMA)

1.000
0.800
0.600
0.400
0.200
0.000
Pig iron
Zinc
Polyethylene (high density)
High-performance resins
Lead scrap
Ferroalloy
Aluminum
Polystyrene
Other resins
Zinc scrap
Products
Copper
Thermo-setting resins
Polypropylene
Iron scrap
Aluminum scrap
Lead
The polyethylene (low density)
Vinyl-chloride resins
Copper scrap
Plastic scrap
Figure 2: Metal and plastics composition of selected products estimated by the model. 1000kg per unit (1000kg for the seven metal products on the left hand side, and 1 million yen for the remaining five products on the right hand side.)
Trucks, buses and other cars
Cellular mobile telephone
Electric audio equipment
Motor vehicle parts and accessories
Soft drinks
Medical service (medical corporations, etc.)
Public construction of rivers, drainages and others
Plastic containers
Household electric appliance (except air conditioner)
Retail trade
Plastic film and seat (except PVC)
Residential construction (non-wooden)
Non-residential construction (non-wooden)
Electric computing equipment (accessory equipment)
Residential construction (wooden)
Miscellaneous plastic goods for daily use and table articles
Passenger motor cars
0 0.02
Household consumption
Government consumption
Public investment
Private investment
Inventory investment
Export
0.04
0.06
0.08
Figure 3: Final destination of plastics by products and final demand categories: major users that consume about 30 persent of doestic plastics production

Finally, Figure 3 gives the final destination of plastics by final demand categories for the major users that consume about 30 percent of annual domestic production. We find that accumulation of plastics in products mostly occur in automobiles and construction (see
Nakamura and Nakajima [2005] for the results for metals).
The compilation of a PIOT has usually been done independent of existing MIOTs, that is, almost from scratch without resorting to available MIOT resources (see Stahmer (1990), for instance). Using the material composition matrix, however a PIOT can easily be obtained from readily available MIOT resources. This can significantly economize on the otherwise prohibitive cost that is associated with the independent compilation of a PIOT (European
Commission 2001). Besides the cost issues, there is also a conceptual issue. The possible lack of consistency with the established framework of MIOTs is pointed out as a drawback of the PIOT based on independent compilation (Weisz and Duchin 2006). It does not apply to our approach because of its use of a consistent IO framework.
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