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A new approach to modeling waste in physical input–output analysis
Ecological Economics 68 (2009) 2475–2478
Contents lists available at ScienceDirect
Ecological Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e c o n
Commentary
A new approach to modeling waste in physical input–output analysis Yijian Xu ⁎, Tianzhu Zhang Department of Environmental Science and Engineering, Tsinghua University, Beijing 100084, China
a r t i c l e
i n f o
Article history: Received 11 January 2006 Received in revised form 16 December 2008 Accepted 11 April 2009 Available online 23 May 2009 Keywords: Physical input–output tables Input–output analysis Waste Land appropriation
a b s t r a c t It is important to treat waste properly in physical input–output analysis and a series of publications discussed this topic in this journal recently. In this paper, we propose a new approach to deal with physical input– output table (PIOT) measured in a single mass unit, by which the structure of PIOT need not be changed. The new approach yields consistent and reasonable results. It not only is simpler than the existing approaches but also can reflect the physical reality of economic systems. We first discuss and clarify the concept of different kinds of inputs and outputs of economic systems. We then present the details of the new approach. During the process we define a new multiplier, which builds a bridge between the total input and the final demand in PIOT, just like the traditional Leontief inverse in MIOT. We select the three-sector PIOT for Germany 1990 as a case study to show the validity of the new approach. Finally, we prove the equivalence between the new approach, Suh's approach and Dietzenbacher's approach and the equivalence of non-waste part and waste part multiplier of the new approach and Dietzenbacher's approach. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Recently, a debate over how to treat waste properly in physical input– output analysis was initiated in this journal. The debate was triggered by a paper by Hubacek and Giljum (2003), in which the embodied land appropriation by international trade activities of the EU-15 is analyzed using a physical input–output table (PIOT). But their approach causes a problem of inconsistency, i.e., none of the calculated sectoral land use is equal to the actual sectoral land use; even the calculated total land use is not equal to the actual total land use. The cause is that they violate the mass balance during the allocation of waste. In a reply to this paper, Suh (2004) pointed out this inconsistency and afforded three approaches, which yields two different results (the results of Suh's approach 1 and approach 3 are the same). All the three approaches are consistent, because they are all based on the overall balance. However, Suh's approach 2 is somewhat unreasonable, because it assumes that the waste and the usable output of an industry are equally responsible for the factor inputs to the industry, in proportion to their mass. As a reply, Giljum and Hubacek (2004) raised another approach, which has corrected the inconsistency and yields the same result as Suh's approach 2. However, it is also somewhat unreasonable, since the assumption is the same as Suh's approach 2. Then, Dietzenbacher (2005) reviewed all the existing methods and introduced another approach to reconcile Suh's approach 1 and GH approach. Dietzenbacher's approach yields the same result as Suh's approach 1 (from now on, if not specified, Suh's approach refers to Suh's approach 1). ⁎ Corresponding author. Tel.: +86 10 62794144; fax: +86 10 62796956. E-mail address: [email protected] (Y. Xu). 0921-8009/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2009.04.010
Both Suh's approach and Dietzenbacher's approach yield consistent and reasonable results. Because they are all based on the overall balances and assume that the usable output of an industry is responsible for the whole factor inputs to the industry regardless of whether they are actually used to produce usable product or to generate wastes, which means that they do not rely on the volume of final demand, where the wastes are included, but the indirect supplychain effect through input–output relations. However, they regard the waste as negative input so that the structure of PIOT must be changed. Due to the structural change, the sectoral input in the model is smaller than the real sectoral input, which does not reflect the physical reality of an economic system. A new approach is presented in this paper, which solves this problem. Without changing the structure of PIOT, the new approach yields consistent and reasonable results. It not only is simpler than the existing approaches but also can reflect the physical reality of economic systems. 2. Method 2.1. Inputs and outputs of economic systems Before presenting the new approach, we would like to discuss and clarify different types of inputs and outputs of economic systems first. The PIOT given by Hubacek and Giljum (2003) can be summarized as Table 1, which is called PIOT pattern 1 in this paper. The n × n matrix Z denotes the intermediate deliveries of secondary inputs, d the vector of domestic final demand, e the vector of foreign final demand, w the vector of waste, and x the vector of total input/output.
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Y. Xu, T. Zhang / Ecological Economics 68 (2009) 2475–2478
From Eq. (4), we have, w = q̂x, so
Table 1 PIOT pattern 1: measured in single mass unit. Supply
ðI − A − q̂ Þx = d + e:
Use Sectors (1,…, n)
Sectors (1,…, n) Primary material inputs (domestic extraction and imports) Total input Land appropriation
Z r′
Domestic
Final demand Exports
Disposal to nature
Total output
d
e
w
x
M1 = ðI −A −q̂ Þ
x′ s′
Dietzenbacher (2005) argued that Table 1 is entirely correct from a bookkeeping perspective, but waste is not an output but merely an outflow of the production process, thus the structure of PIOT should be changed into another pattern to better reflect the production principle (see CEC/IMF/OECD/UN/WB, 1993), which is shown in Table 2 and is called PIOT pattern 2 in this paper. In this way, the waste is treated as negative input, thus total input and output are changed into x2 accordingly. Both Suh's approach and Dietzenbacher's approach rely more on PIOT pattern 2 than PIOT pattern 1. However, PIOT is related to material flows, which is quite different from monetary input–output table (MIOT). Thus, a problem in PIOT pattern 2 would then be found, if we carefully examine the concept of inputs and outputs of economic systems combined with the view of MFA (Material Flow Analysis). Total input consists of primary input and intermediate input. Total output consists of usable output and waste, and usable output consists of intermediate use and final use (or called final demand). Here, waste refers to the waste that finally disposed to nature, in which recycled waste is not included. In the whole economic system, total input should be equal to total output. In each sector, total sectoral input should be equal to total sectoral output. This is the mass balance principle of physical input–output analysis. Total input equals usable output plus waste numerically, which means that total input is larger than usable output. In PIOT pattern 2, the so-called total output is actually usable output. The so-called total input is neither total input nor primary input, nor intermediate input. It is only equal to usable output numerically, while it is smaller than the real total input. Therefore, PIOT pattern 2 does not reflect the physical reality well, while PIOT pattern 1 do, which calls for a new approach totally based on PIOT pattern 1. 2.2. New approach to modeling waste The new approach presented here is totally based on PIOT pattern 1. The matrix of input coefficients A, the Leontief inverse M and the vector of land appropriation coefficients c are defined by Eqs. (1)–(3) respectively1. A = Z x̂
−1
ð1Þ −1
ð2Þ
:
ð3Þ
M = ðI − AÞ −1
c V= sVx̂
Now, define a new multiplier M1 by Eq. (5).
In order to treat waste properly, the vector of waste coefficient is defined by Eq. (4).
−1
:
The multiplier M1, derived from Eq. (5), has its analytical significance. The typical element (i, j) of the matrix M1 gives the extra output in sector i that is (directly and indirectly) required to generate 1 mt of final demand in sector j. The multiplier M1 builds a bridge between the total input and the final demand in PIOT, just like the traditional Leontief inverse in MIOT. On the basis of the newly defined multiplier M1, which plays an important role in the new approach, all kinds of calculations and analysis can be carried out. Then, the core problem discussed by the previous papers in this debate, i.e., the imputation of land use to extended domestic final demand and extended exports can be easily given by ĉM1d̂ and ĉM1ê, respectively. The waste generation required directly and indirectly to produce domestic final demand and exports can also be easily given by q̂M1d̂ and q̂M1ê, respectively. Moreover, the new approach can also calculate appropriated land that can be attributed to generating the waste that is necessary for a certain final demand component. The land use multipliers of nonwaste part and waste part in the new approach are ĉM and ĉMq̂M1 respectively, which can provide further details on land use. For example, the non-waste part and waste part of land use imputed to domestic final demands can then be determined by ĉMd̂ and ĉMq̂M1d̂ respectively. Similarly, the non-waste part and waste part of land use imputed to exports can be determined by ĉMê and ĉMq̂M1ê. This is also an advantage acclaimed by Dietzenbacher of his approach. 3. Case study Three-sector PIOT for Germany 1990 (shown in Table 3), aggregated by Hubacek and Giljum (2003) and used by the previous papers, is taken to examine the new approach. Using the new approach, calculations are carried out. The results of imputation of land use and distribution of waste are listed in Tables 4 and 5 respectively. Compared with the results of Suh's approach and Dietzenbacher's approach, the results of the new approach are exactly the same. The land use multipliers are listed in Table 6, which are also the same as that of Dietzenbacher's approach. This is not simply a coincidence. The equivalence of the new approach, Dietzenbacher's approach and Suh's approach, and the equivalence of the multiplier of the new approach and Dietzenbacher's approach are proved in Appendix A. Let us compare these three approaches briefly. In Suh's and Dietzenbacher's approaches, two patterns of PIOT (both PIOT pattern
Table 2 PIOT pattern 2: measured in single mass unit, with waste as negative “input”. Supply
−1
q V= w Vx̂
:
ð4Þ
According to the mass balance, we have, Ax + d + e + w = x.
1 A diagonal matrix with the elements of vector x on its main diagonal and all other entries equal to zero is indicated by a circumflex (e.g. x̂). Vectors are columns by definition, so row vectors are obtained by transposition, indicated by a prime (e.g. x′).
ð5Þ
Sectors (1,…, n) Primary material inputs (domestic extraction and imports) Disposal to nature Total input Land appropriation
Use Sectors (1,…, n)
Final demand Domestic
Exports
Total output
Z r′
d
e
x2
− w′ x2′ s′
Y. Xu, T. Zhang / Ecological Economics 68 (2009) 2475–2478
2477
Table 3 Three-sector PIOT for Germany 1990 (million tons). Supply
Use Agriculture
Agriculture Manufacturing Services Primary material inputs (domestic extraction and imports) Total input Land appropriation (in hectares)
Manufacturing
Services
2247.7 27.4 5.1 4234.2
1442.2 1045.4 68.5 277.9
336.2 206.2 50.9 567.9
6514.4 21,019,662
2834.0 1,912,694
1161.2 1,551,786
Domestic
Final demand Exports
Disposal to nature
46.8 552.5 16.3
36.7 155.9 20.0
2404.8 846.6 1000.4
Total output 6514.4 2834.0 1161.2
24,484,142
Table 4 Imputation of land use in the new approach. Imputation of land use (in hectares) to Extended domestic final demand
Agriculture Manufacturing Services Total
Extended exports
A
M
S
Total
A
M
(1)
(2)
(3)
(4)
(5)
555,387 2297 3064 560,748
13,313,176 1,354,125 730,369 15,397,670
1,328,682 77,402 273,849 1,679,934
15,197,245 1,433,824 1,007,282 17,638,351
435,528 1801 2403 439,732
Total S
Total
Sum
(6)
(7)
(8)
(9)
3,756,605 382,096 206,090 4,344,790
1,630,285 94,973 336,011 2,061,268
5,822,417 478,870 544,504 6,845,791
21,019,662 1,912,694 1,551,786 24,484,142
Sum
(4) = (1) + (2) + (3), (8) = (5) + (6) + (7), (9) = (4) + (8).
Table 5 The distribution of waste in the new approach. Waste (in mt) imputed to Domestic final demand
Agriculture Manufacturing Services Total
Exports
A
M
S
Total
(1)
(2)
(3)
(4)
63.54 1.02 1.98 66.53
1523.12 599.37 470.85 2593.34
152.01 34.26 176.54 362.81
1738.67 634.64 649.37 3022.69
A
Total M
S
Total
(5)
(6)
(7)
(8)
(9)
49.83 0.80 1.55 52.17
429.78 169.12 132.86 731.77
186.52 42.04 216.62 445.17
666.13 211.96 351.03 1229.11
2404.80 846.60 1000.40 4251.80
(4) = (1) + (2) + (3), (8) = (5) + (6) + (7), (9) = (4) + (8).
1 and pattern2), the variables of A, M, x, c, A2, M2, x2, c2, and q2 are used (the definitions of A2, M2, c2, and q2 are listed in Appendix A.1). While in the new approach, only one pattern of PIOT (PIOT pattern 1), fewer variables of A, M, x, c, M1 and q are used to yield the same results. Therefore, the new approach is simpler than Suh's and Dietzenbacher's approaches.
Table 6 Land use multiplier of the new approach. Land use (in hectares) imputed to 1 mt of final demand Non-waste part Agriculture Manufacturing Services Total
Waste part
A
M
S
A
M
S
4955.36 7.20 2.04 4964.60
4082.15 1082.97 55.59 5220.71
2258.61 203.30 1408.57 3870.48
6911.88 41.88 63.43 7017.19
20,014.10 1367.93 1266.34 22,648.37
79,255.62 4545.34 15,391.99 99,192.94
4. Conclusions PIOT pattern 1, in a single mass unit, is a good way for bookkeeping of material flows across economic systems. It clearly shows different kinds of inputs and outputs of economic systems. It is also appropriate for input–output analysis, which is proved by the new approach to modeling waste in physical input–output analysis presented in this paper. Without changing the structure of PIOT and the sectoral input, i.e. totally based on PIOT pattern 1, the new approach yields the same results as Suh's approach and Dietzenbacher's approach. It not only is simpler but also can well reflect the physical reality of economic systems. Besides, the newly defined multiplier M1, which plays an important role in the new approach, builds a bridge between the total input and the final demand in PIOT, just like the traditional Leontief inverse in MIOT. The equivalence of the three approaches and the equivalence of non-waste part and waste part multiplier of the new approach and Dietzenbacher's approach are also proved. As Weisz and Duchin (2006) and Suh (2004) pointed out in their papers, it may be better to measure the sectors' products, or the usable
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output in their most appropriate units. Therefore, the development of PIOT measured in mixed units instead of single mass unit, and subsequently the research of input–output analysis based on PIOT measured in mixed units could be the future work. Acknowledgements The authors acknowledge funding by the Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP), project Number 20050003039. The authors also gratefully acknowledge the anonymous reviewers for their suggestions. Appendix A A.1. Proof of equivalence of the new approach, Dietzenbacher's approach and Suh's approach In order to carry out the physical input–output analysis correctly, Suh and Dietzenbacher used PIOT pattern 2 also. The matrix of input coefficients A2, Leontief inverse M2 and the vector of land appropriation coefficients c2 and waste coefficients q2 are redefined by Eqs. (11)–(14) respectively. −1
ð11Þ
A2 = Z x̂ 2
−1
M2 = ðI −A2 Þ
ð12Þ
−1
c2V = sVx̂2
−1
q2V = wVx̂2 :
−1 −1 −1 −1 M1 = x̂ x̂ − zx̂ −ŵ x̂ h i −1 −1 −1 = ðx̂ − z −ŵ Þx̂ = x̂ ðx̂ −z −ŵ Þ : Note that x = x2 + w, so x̂ = x̂2 + ŵ. Therefore, M1 = x̂ ðx̂
2 −zÞ
−1
= x̂ ðx̂
2 − A2 x̂ 2 Þ
−1
− 1 −1 −1 −1 = x̂ ðI − A2 Þx̂2 = x̂ x̂ 2 ðI −A2 Þ = x̂ x̂ 2 M2
:
ĉ M1 = ŝ x̂
−1
−1
x̂ x̂ 2
−1
M2 = ŝ x̂ 2
M2 = ĉ 2 M2 :
Now, we reach the conclusion that ĉM1 = ĉ2M2, which proves the equivalence of the new approach and Suh's approach. The equivalence of Dietzenbacher's approach and Suh's approach has been proved by Dietzenbacher, so the equivalence of the three approaches is proved. A.2. Proof of equivalence of the multiplier of the new approach and Dietzenbacher's approach The multipliers of non-waste part and waste part in Dietzenbacher's approach are ĉM and ĉMq̂2M2, respectively. The multipliers of non-waste part and waste part in the new approach are ĉM and ĉMq̂M1, respectively. Therefore, if the equivalence of q̂2M2 and q̂M1 is proved, the equivalence of the multiplier of the new approach and Dietzenbacher's approach is proved. In Appendix A.1, ĉM1 = ĉ2M2 is proved. Using the definitions of c and c2, we have −1
−1
ŝ x̂ 2 M2 = ŝ x̂
M1 :
Replacing s with w, we have −1
−1
ŵ x̂ 2 M2 = ŵ x̂
M1 :
ð13Þ
̂ 1M2 and q̂M1 = ŵx− ̂ 1M1, we can reach the Note that q̂2M2 = ŵx− 2 conclusion that q̂2M2 = q̂M1.
ð14Þ
References
Take the new approach as a starting point and consider M1 =(I −A− q̂) . Using the definitions of q and A, we have −1
Using the definition of c, we have
CEC/IMF/OECD/UN/WB, 1993. System of National Accounts 1993. CEC/IMF/OECD/ UN/WB, Brussels, New York, Paris, Washington, DC. Dietzenbacher, E., 2005. Waste treatment in physical input–output analysis. Ecological Economics 55 (1), 11–23. Giljum, S., Hubacek, K., 2004. Alternative approaches of physical input–output analysis to estimate primary material inputs of production and consumption activities. Economic Systems Research 16 (3), 301–310. Hubacek, K., Giljum, S., 2003. Applying physical input–output analysis to estimate land appropriation (ecological footprints) of international trade activities. Ecological Economics 44 (1), 137–151. Suh, S., 2004. A note on the calculus for physical input–output analysis and its application to land appropriation of international trade activities. Ecological Economics 48 (1), 9–17. Weisz, H., Duchin, F., 2006. Physical and monetary input–output analysis: what makes the difference? Ecological Economics 57 (3), 534–541.
Contents lists available at ScienceDirect
Ecological Economics j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / e c o l e c o n
Commentary
A new approach to modeling waste in physical input–output analysis Yijian Xu ⁎, Tianzhu Zhang Department of Environmental Science and Engineering, Tsinghua University, Beijing 100084, China
a r t i c l e
i n f o
Article history: Received 11 January 2006 Received in revised form 16 December 2008 Accepted 11 April 2009 Available online 23 May 2009 Keywords: Physical input–output tables Input–output analysis Waste Land appropriation
a b s t r a c t It is important to treat waste properly in physical input–output analysis and a series of publications discussed this topic in this journal recently. In this paper, we propose a new approach to deal with physical input– output table (PIOT) measured in a single mass unit, by which the structure of PIOT need not be changed. The new approach yields consistent and reasonable results. It not only is simpler than the existing approaches but also can reflect the physical reality of economic systems. We first discuss and clarify the concept of different kinds of inputs and outputs of economic systems. We then present the details of the new approach. During the process we define a new multiplier, which builds a bridge between the total input and the final demand in PIOT, just like the traditional Leontief inverse in MIOT. We select the three-sector PIOT for Germany 1990 as a case study to show the validity of the new approach. Finally, we prove the equivalence between the new approach, Suh's approach and Dietzenbacher's approach and the equivalence of non-waste part and waste part multiplier of the new approach and Dietzenbacher's approach. © 2009 Elsevier B.V. All rights reserved.
1. Introduction Recently, a debate over how to treat waste properly in physical input– output analysis was initiated in this journal. The debate was triggered by a paper by Hubacek and Giljum (2003), in which the embodied land appropriation by international trade activities of the EU-15 is analyzed using a physical input–output table (PIOT). But their approach causes a problem of inconsistency, i.e., none of the calculated sectoral land use is equal to the actual sectoral land use; even the calculated total land use is not equal to the actual total land use. The cause is that they violate the mass balance during the allocation of waste. In a reply to this paper, Suh (2004) pointed out this inconsistency and afforded three approaches, which yields two different results (the results of Suh's approach 1 and approach 3 are the same). All the three approaches are consistent, because they are all based on the overall balance. However, Suh's approach 2 is somewhat unreasonable, because it assumes that the waste and the usable output of an industry are equally responsible for the factor inputs to the industry, in proportion to their mass. As a reply, Giljum and Hubacek (2004) raised another approach, which has corrected the inconsistency and yields the same result as Suh's approach 2. However, it is also somewhat unreasonable, since the assumption is the same as Suh's approach 2. Then, Dietzenbacher (2005) reviewed all the existing methods and introduced another approach to reconcile Suh's approach 1 and GH approach. Dietzenbacher's approach yields the same result as Suh's approach 1 (from now on, if not specified, Suh's approach refers to Suh's approach 1). ⁎ Corresponding author. Tel.: +86 10 62794144; fax: +86 10 62796956. E-mail address: [email protected] (Y. Xu). 0921-8009/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.ecolecon.2009.04.010
Both Suh's approach and Dietzenbacher's approach yield consistent and reasonable results. Because they are all based on the overall balances and assume that the usable output of an industry is responsible for the whole factor inputs to the industry regardless of whether they are actually used to produce usable product or to generate wastes, which means that they do not rely on the volume of final demand, where the wastes are included, but the indirect supplychain effect through input–output relations. However, they regard the waste as negative input so that the structure of PIOT must be changed. Due to the structural change, the sectoral input in the model is smaller than the real sectoral input, which does not reflect the physical reality of an economic system. A new approach is presented in this paper, which solves this problem. Without changing the structure of PIOT, the new approach yields consistent and reasonable results. It not only is simpler than the existing approaches but also can reflect the physical reality of economic systems. 2. Method 2.1. Inputs and outputs of economic systems Before presenting the new approach, we would like to discuss and clarify different types of inputs and outputs of economic systems first. The PIOT given by Hubacek and Giljum (2003) can be summarized as Table 1, which is called PIOT pattern 1 in this paper. The n × n matrix Z denotes the intermediate deliveries of secondary inputs, d the vector of domestic final demand, e the vector of foreign final demand, w the vector of waste, and x the vector of total input/output.
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Y. Xu, T. Zhang / Ecological Economics 68 (2009) 2475–2478
From Eq. (4), we have, w = q̂x, so
Table 1 PIOT pattern 1: measured in single mass unit. Supply
ðI − A − q̂ Þx = d + e:
Use Sectors (1,…, n)
Sectors (1,…, n) Primary material inputs (domestic extraction and imports) Total input Land appropriation
Z r′
Domestic
Final demand Exports
Disposal to nature
Total output
d
e
w
x
M1 = ðI −A −q̂ Þ
x′ s′
Dietzenbacher (2005) argued that Table 1 is entirely correct from a bookkeeping perspective, but waste is not an output but merely an outflow of the production process, thus the structure of PIOT should be changed into another pattern to better reflect the production principle (see CEC/IMF/OECD/UN/WB, 1993), which is shown in Table 2 and is called PIOT pattern 2 in this paper. In this way, the waste is treated as negative input, thus total input and output are changed into x2 accordingly. Both Suh's approach and Dietzenbacher's approach rely more on PIOT pattern 2 than PIOT pattern 1. However, PIOT is related to material flows, which is quite different from monetary input–output table (MIOT). Thus, a problem in PIOT pattern 2 would then be found, if we carefully examine the concept of inputs and outputs of economic systems combined with the view of MFA (Material Flow Analysis). Total input consists of primary input and intermediate input. Total output consists of usable output and waste, and usable output consists of intermediate use and final use (or called final demand). Here, waste refers to the waste that finally disposed to nature, in which recycled waste is not included. In the whole economic system, total input should be equal to total output. In each sector, total sectoral input should be equal to total sectoral output. This is the mass balance principle of physical input–output analysis. Total input equals usable output plus waste numerically, which means that total input is larger than usable output. In PIOT pattern 2, the so-called total output is actually usable output. The so-called total input is neither total input nor primary input, nor intermediate input. It is only equal to usable output numerically, while it is smaller than the real total input. Therefore, PIOT pattern 2 does not reflect the physical reality well, while PIOT pattern 1 do, which calls for a new approach totally based on PIOT pattern 1. 2.2. New approach to modeling waste The new approach presented here is totally based on PIOT pattern 1. The matrix of input coefficients A, the Leontief inverse M and the vector of land appropriation coefficients c are defined by Eqs. (1)–(3) respectively1. A = Z x̂
−1
ð1Þ −1
ð2Þ
:
ð3Þ
M = ðI − AÞ −1
c V= sVx̂
Now, define a new multiplier M1 by Eq. (5).
In order to treat waste properly, the vector of waste coefficient is defined by Eq. (4).
−1
:
The multiplier M1, derived from Eq. (5), has its analytical significance. The typical element (i, j) of the matrix M1 gives the extra output in sector i that is (directly and indirectly) required to generate 1 mt of final demand in sector j. The multiplier M1 builds a bridge between the total input and the final demand in PIOT, just like the traditional Leontief inverse in MIOT. On the basis of the newly defined multiplier M1, which plays an important role in the new approach, all kinds of calculations and analysis can be carried out. Then, the core problem discussed by the previous papers in this debate, i.e., the imputation of land use to extended domestic final demand and extended exports can be easily given by ĉM1d̂ and ĉM1ê, respectively. The waste generation required directly and indirectly to produce domestic final demand and exports can also be easily given by q̂M1d̂ and q̂M1ê, respectively. Moreover, the new approach can also calculate appropriated land that can be attributed to generating the waste that is necessary for a certain final demand component. The land use multipliers of nonwaste part and waste part in the new approach are ĉM and ĉMq̂M1 respectively, which can provide further details on land use. For example, the non-waste part and waste part of land use imputed to domestic final demands can then be determined by ĉMd̂ and ĉMq̂M1d̂ respectively. Similarly, the non-waste part and waste part of land use imputed to exports can be determined by ĉMê and ĉMq̂M1ê. This is also an advantage acclaimed by Dietzenbacher of his approach. 3. Case study Three-sector PIOT for Germany 1990 (shown in Table 3), aggregated by Hubacek and Giljum (2003) and used by the previous papers, is taken to examine the new approach. Using the new approach, calculations are carried out. The results of imputation of land use and distribution of waste are listed in Tables 4 and 5 respectively. Compared with the results of Suh's approach and Dietzenbacher's approach, the results of the new approach are exactly the same. The land use multipliers are listed in Table 6, which are also the same as that of Dietzenbacher's approach. This is not simply a coincidence. The equivalence of the new approach, Dietzenbacher's approach and Suh's approach, and the equivalence of the multiplier of the new approach and Dietzenbacher's approach are proved in Appendix A. Let us compare these three approaches briefly. In Suh's and Dietzenbacher's approaches, two patterns of PIOT (both PIOT pattern
Table 2 PIOT pattern 2: measured in single mass unit, with waste as negative “input”. Supply
−1
q V= w Vx̂
:
ð4Þ
According to the mass balance, we have, Ax + d + e + w = x.
1 A diagonal matrix with the elements of vector x on its main diagonal and all other entries equal to zero is indicated by a circumflex (e.g. x̂). Vectors are columns by definition, so row vectors are obtained by transposition, indicated by a prime (e.g. x′).
ð5Þ
Sectors (1,…, n) Primary material inputs (domestic extraction and imports) Disposal to nature Total input Land appropriation
Use Sectors (1,…, n)
Final demand Domestic
Exports
Total output
Z r′
d
e
x2
− w′ x2′ s′
Y. Xu, T. Zhang / Ecological Economics 68 (2009) 2475–2478
2477
Table 3 Three-sector PIOT for Germany 1990 (million tons). Supply
Use Agriculture
Agriculture Manufacturing Services Primary material inputs (domestic extraction and imports) Total input Land appropriation (in hectares)
Manufacturing
Services
2247.7 27.4 5.1 4234.2
1442.2 1045.4 68.5 277.9
336.2 206.2 50.9 567.9
6514.4 21,019,662
2834.0 1,912,694
1161.2 1,551,786
Domestic
Final demand Exports
Disposal to nature
46.8 552.5 16.3
36.7 155.9 20.0
2404.8 846.6 1000.4
Total output 6514.4 2834.0 1161.2
24,484,142
Table 4 Imputation of land use in the new approach. Imputation of land use (in hectares) to Extended domestic final demand
Agriculture Manufacturing Services Total
Extended exports
A
M
S
Total
A
M
(1)
(2)
(3)
(4)
(5)
555,387 2297 3064 560,748
13,313,176 1,354,125 730,369 15,397,670
1,328,682 77,402 273,849 1,679,934
15,197,245 1,433,824 1,007,282 17,638,351
435,528 1801 2403 439,732
Total S
Total
Sum
(6)
(7)
(8)
(9)
3,756,605 382,096 206,090 4,344,790
1,630,285 94,973 336,011 2,061,268
5,822,417 478,870 544,504 6,845,791
21,019,662 1,912,694 1,551,786 24,484,142
Sum
(4) = (1) + (2) + (3), (8) = (5) + (6) + (7), (9) = (4) + (8).
Table 5 The distribution of waste in the new approach. Waste (in mt) imputed to Domestic final demand
Agriculture Manufacturing Services Total
Exports
A
M
S
Total
(1)
(2)
(3)
(4)
63.54 1.02 1.98 66.53
1523.12 599.37 470.85 2593.34
152.01 34.26 176.54 362.81
1738.67 634.64 649.37 3022.69
A
Total M
S
Total
(5)
(6)
(7)
(8)
(9)
49.83 0.80 1.55 52.17
429.78 169.12 132.86 731.77
186.52 42.04 216.62 445.17
666.13 211.96 351.03 1229.11
2404.80 846.60 1000.40 4251.80
(4) = (1) + (2) + (3), (8) = (5) + (6) + (7), (9) = (4) + (8).
1 and pattern2), the variables of A, M, x, c, A2, M2, x2, c2, and q2 are used (the definitions of A2, M2, c2, and q2 are listed in Appendix A.1). While in the new approach, only one pattern of PIOT (PIOT pattern 1), fewer variables of A, M, x, c, M1 and q are used to yield the same results. Therefore, the new approach is simpler than Suh's and Dietzenbacher's approaches.
Table 6 Land use multiplier of the new approach. Land use (in hectares) imputed to 1 mt of final demand Non-waste part Agriculture Manufacturing Services Total
Waste part
A
M
S
A
M
S
4955.36 7.20 2.04 4964.60
4082.15 1082.97 55.59 5220.71
2258.61 203.30 1408.57 3870.48
6911.88 41.88 63.43 7017.19
20,014.10 1367.93 1266.34 22,648.37
79,255.62 4545.34 15,391.99 99,192.94
4. Conclusions PIOT pattern 1, in a single mass unit, is a good way for bookkeeping of material flows across economic systems. It clearly shows different kinds of inputs and outputs of economic systems. It is also appropriate for input–output analysis, which is proved by the new approach to modeling waste in physical input–output analysis presented in this paper. Without changing the structure of PIOT and the sectoral input, i.e. totally based on PIOT pattern 1, the new approach yields the same results as Suh's approach and Dietzenbacher's approach. It not only is simpler but also can well reflect the physical reality of economic systems. Besides, the newly defined multiplier M1, which plays an important role in the new approach, builds a bridge between the total input and the final demand in PIOT, just like the traditional Leontief inverse in MIOT. The equivalence of the three approaches and the equivalence of non-waste part and waste part multiplier of the new approach and Dietzenbacher's approach are also proved. As Weisz and Duchin (2006) and Suh (2004) pointed out in their papers, it may be better to measure the sectors' products, or the usable
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Y. Xu, T. Zhang / Ecological Economics 68 (2009) 2475–2478
output in their most appropriate units. Therefore, the development of PIOT measured in mixed units instead of single mass unit, and subsequently the research of input–output analysis based on PIOT measured in mixed units could be the future work. Acknowledgements The authors acknowledge funding by the Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP), project Number 20050003039. The authors also gratefully acknowledge the anonymous reviewers for their suggestions. Appendix A A.1. Proof of equivalence of the new approach, Dietzenbacher's approach and Suh's approach In order to carry out the physical input–output analysis correctly, Suh and Dietzenbacher used PIOT pattern 2 also. The matrix of input coefficients A2, Leontief inverse M2 and the vector of land appropriation coefficients c2 and waste coefficients q2 are redefined by Eqs. (11)–(14) respectively. −1
ð11Þ
A2 = Z x̂ 2
−1
M2 = ðI −A2 Þ
ð12Þ
−1
c2V = sVx̂2
−1
q2V = wVx̂2 :
−1 −1 −1 −1 M1 = x̂ x̂ − zx̂ −ŵ x̂ h i −1 −1 −1 = ðx̂ − z −ŵ Þx̂ = x̂ ðx̂ −z −ŵ Þ : Note that x = x2 + w, so x̂ = x̂2 + ŵ. Therefore, M1 = x̂ ðx̂
2 −zÞ
−1
= x̂ ðx̂
2 − A2 x̂ 2 Þ
−1
− 1 −1 −1 −1 = x̂ ðI − A2 Þx̂2 = x̂ x̂ 2 ðI −A2 Þ = x̂ x̂ 2 M2
:
ĉ M1 = ŝ x̂
−1
−1
x̂ x̂ 2
−1
M2 = ŝ x̂ 2
M2 = ĉ 2 M2 :
Now, we reach the conclusion that ĉM1 = ĉ2M2, which proves the equivalence of the new approach and Suh's approach. The equivalence of Dietzenbacher's approach and Suh's approach has been proved by Dietzenbacher, so the equivalence of the three approaches is proved. A.2. Proof of equivalence of the multiplier of the new approach and Dietzenbacher's approach The multipliers of non-waste part and waste part in Dietzenbacher's approach are ĉM and ĉMq̂2M2, respectively. The multipliers of non-waste part and waste part in the new approach are ĉM and ĉMq̂M1, respectively. Therefore, if the equivalence of q̂2M2 and q̂M1 is proved, the equivalence of the multiplier of the new approach and Dietzenbacher's approach is proved. In Appendix A.1, ĉM1 = ĉ2M2 is proved. Using the definitions of c and c2, we have −1
−1
ŝ x̂ 2 M2 = ŝ x̂
M1 :
Replacing s with w, we have −1
−1
ŵ x̂ 2 M2 = ŵ x̂
M1 :
ð13Þ
̂ 1M2 and q̂M1 = ŵx− ̂ 1M1, we can reach the Note that q̂2M2 = ŵx− 2 conclusion that q̂2M2 = q̂M1.
ð14Þ
References
Take the new approach as a starting point and consider M1 =(I −A− q̂) . Using the definitions of q and A, we have −1
Using the definition of c, we have
CEC/IMF/OECD/UN/WB, 1993. System of National Accounts 1993. CEC/IMF/OECD/ UN/WB, Brussels, New York, Paris, Washington, DC. Dietzenbacher, E., 2005. Waste treatment in physical input–output analysis. Ecological Economics 55 (1), 11–23. Giljum, S., Hubacek, K., 2004. Alternative approaches of physical input–output analysis to estimate primary material inputs of production and consumption activities. Economic Systems Research 16 (3), 301–310. Hubacek, K., Giljum, S., 2003. Applying physical input–output analysis to estimate land appropriation (ecological footprints) of international trade activities. Ecological Economics 44 (1), 137–151. Suh, S., 2004. A note on the calculus for physical input–output analysis and its application to land appropriation of international trade activities. Ecological Economics 48 (1), 9–17. Weisz, H., Duchin, F., 2006. Physical and monetary input–output analysis: what makes the difference? Ecological Economics 57 (3), 534–541.