In recent years, the multiregional input–output (MRIO) models (Lenzen et al.
2004b; Wiedmann
2009; Moran and Wood
2014) that describe the input–output structure of international supply chains (Yamano and Ahmad
2006; Lenzen et al.
2012a; Tukker et al.
2013; Dietzenbacher et al.
2013; Wood et al.
2014) have also been used for environmental footprint calculations (e.g., Hertwich and Peters
2009; Feng et al.
2011; Lenzen et al.
2012b; Weinzettel et al.
2013; Wiedmann et al.
2015). The benefit of employing MRIOs for footprint analysis is that they clearly identify and represent the production technologies of individual nations, and that national system boundaries can be extended to include international supply chains (Weinzettel et al.
2014). In order to quantify carbon and material footprints for Japanese households, we clarified the expenditure on commodities by each household (million Japanese yen: M-JPY). The footprint per unit expenditure, or the footprint intensities, was calculated using a global link input–output model (GLIO) (Nansai et al.
2009,
2013a,
b). The GLIO is a MRIO composed of a Japanese input–output structure with 409 sectors of domestic commodities and 409 sectors of imported commodities, and overseas sectors covering 230 countries and regions.
Derivation of the carbon and material footprint intensities is elaborated in Nansai et al. (
2012) and Shigetomi et al. (
2015), respectively. However, to introduce the structure of the GLIO, the method used to calculate material footprint intensities is described briefly below. Vector
q, whose elements represent the material footprint intensities of commodities supplied to Japanese households, is calculated as shown in Eq.
1:
$${\mathbf{q}} = {\mathbf{d}}\left( {{\mathbf{I}} - {\mathbf{A}}} \right)^{ - 1}$$
(1)
Vector \({\mathbf{q}} = \left( {\begin{array}{*{20}c} {{\mathbf{q}}^{JD} } & {{\mathbf{q}}^{JI} } & {{\mathbf{q}}^{G} } \\ \end{array} } \right)^{\prime }\) consists of sub-vectors \({\mathbf{q}}^{JD} = \left( {q_{i}^{JD} } \right)\), \({\mathbf{q}}^{JI} = \left( {q_{i}^{JI} } \right),\) and \({\mathbf{q}}^{G} = \left( {q_{q}^{G} } \right)\), where elements \(q_{i}^{JD}\) and \(q_{i}^{JI}\) denote the material footprint intensities (t/M-JPY) of Japanese domestic commodity i = (1…n
JP
; n
JP
= 409) and of directly imported commodity i, respectively. As an aside, \(q_{q}^{G}\) represents the material footprint intensities (t/M-JPY) of overseas commodities q = (1…n
G
; n
G
= 230), but this is not used further in the present study. Row vector \({\mathbf{d}} = \left( {\begin{array}{*{20}c} {\mathbf{0}} & {\mathbf{0}} & {{\mathbf{i}}^{G} } \\ \end{array} } \right)\) has the same dimensions as vector q and includes the summation vector \({\mathbf{i}}^{G}\) in which all elements are unity. Matrix I is an identity matrix.
Matrix
A is a mixed-unit input coefficient matrix consisting of block matrices
\({\mathbf{A}}_{ 1 1}\),
\({\tilde{\mathbf{A}}}_{ 1 3}\),
\({\tilde{\mathbf{A}}}_{ 3 1}^{ (k )}\),
\({\tilde{\mathbf{A}}}_{ 3 2}^{ (k )},\) and
\({\tilde{\mathbf{A}}}_{ 3 3}^{ (k )}\), as is shown in Eq.
2:
$${\mathbf{A}} = \left( {\begin{array}{*{20}c} {{\mathbf{A}}_{ 1 1} } & {\mathbf{0}} & {{\tilde{\mathbf{A}}}_{ 1 3} } \\ {\mathbf{0}} & {\mathbf{0}} & {\mathbf{0}} \\ {\sum\limits_{k = 1}^{l} {{\tilde{\mathbf{A}}}_{ 3 1}^{ (k )} } } & {\sum\limits_{k = 1}^{l} {{\tilde{\mathbf{A}}}_{ 3 2}^{ (k )} } } & {\sum\limits_{k = 1}^{l} {{\tilde{\mathbf{A}}}_{ 3 3}^{ (k )} } } \\ \end{array} } \right),$$
(2)
where
\({\mathbf{A}}_{ 1 1}\) is the input coefficient matrix based on monetary units describing the input structure of domestic commodities
i with regard to Japanese domestic commodities
\(j = \left( {1 \ldots n^{JP} } \right)\), and
\({\tilde{\mathbf{A}}}_{ 1 3}\) is a matrix showing the import structure of domestic commodities
i in overseas sector
q.
\({\tilde{\mathbf{A}}}_{ 3 1}^{ (k )}\) is a matrix showing the input structure of critical metals contained in traded goods
k in overseas sector
\(p = \left( {1 \ldots n^{G} } \right)\) for Japanese domestic commodities
j, and
\({\tilde{\mathbf{A}}}_{ 3 2}^{ (k )}\) is a matrix showing the input structure of critical metals contained in traded goods
k of overseas sector
p for the input of commodities
j imported directly to Japanese final demand.
\({\tilde{\mathbf{A}}}_{ 3 3}^{ (k )}\) is a matrix showing the input structure for critical metals contained in traded goods
k of overseas sector
p for overseas sector
q. The superscript—denotes a matrix whose coefficients are based on mass units.
\(k = \left( {1 \ldots l} \right)\) represents the type of traded goods that contain target metals, with
l = 153 used for neodymium,
l = 160 for cobalt, and
l = 151 for platinum. These traded goods were selected from the Base pour l’Analyze du Commerce International (BACI) database, which is an improvement of the UN Comtrade database and defines traded goods based on the Harmonized Commodity (HS) code. See Nansai et al. (
2015) for a detailed explanation of the mixed-unit input coefficient matrix
A and Nansai et al. (
2014) for the selected traded goods.
2.1.2 Household expenditures by income level
To estimate the consumption trends of different household income levels, this study used the method of Shigetomi et al. (
2014). Briefly, the method provides domestic household consumption expenditure data for household attributes (e.g., household size) based on values in the Japan input–output table (JIOT) using household survey data (National Survey of Family Income and Expenditure: NSFIE). In addition to the study of Shigetomi et al. (
2014), which estimated consumption expenditures for six householder age groups, we also obtained household expenditures for each income quintile using the values from the JIOT (
2005) and NSFIE (
2004), as follows. Household income quintiles were calculated by dividing all of the households into five groups (quintiles) according to income (i.e., 20 % of all households in each group). These income groups were then ordered from the lowest to the highest, i.e., Quintile1 to Quintile5, abbreviated as Q1, Q2, Q3, Q4, and Q5, respectively.
First, we obtained
\(r_{ib}\), which represents the expenditure ratio of commodity
i per unit expenditure by each household income quintile (
b = 1…5) for Q1 to Q5 using Eq.
3.
$$r_{ib} = \frac{{P_{ib} }}{{\sum\limits_{i = 1}^{N} {P_{ib} } }}.$$
(3)
Here \(P_{ib}\) is expenditure per month (M-JPY/m) on commodity i by each household income quintile. N = 409 is the number of commodity sectors.
In Eq.
4,
\(s_{ib}\) denotes the market share of commodity
i among households (
b = 1…5).
M = 5 denotes the number of household attributes.
$$s_{ib} = \frac{{P_{ib} }}{{\sum\limits_{b = 1}^{M} {P_{ib} } }}.$$
(4)
A quadratic programming (QP) algorithm was used to determine the optimal solution for variables
\(\tilde{r}_{ib}\) and
\(\tilde{s}_{ib}\) with the objective function defined in Eq.
5 which minimizes the sum of the differences between
\(r_{ib}\) and
\(\tilde{r}_{ib}\) and between
\(s_{ib}\) and
\(\tilde{s}_{ib}\) under the constraints of Eqs. (
6) through (
9).
$$\mathop {{\text{Min}} .}\limits_{{\tilde{r}_{ib} ,\tilde{s}_{ib} }} \sum\limits_{b = 1}^{M} {\sum\limits_{i = 1}^{N} {\left( {\frac{{\tilde{r}_{ib} - r_{ib} }}{{r_{ib} }}} \right)} }^{2} + \sum\limits_{b = 1}^{M} {\sum\limits_{i = 1}^{N} {\left( {\frac{{\tilde{s}_{ib} - s_{ib} }}{{s_{ib} }}} \right)} }^{2}$$
(5)
s.t.
$$g_{i} = \sum\limits_{b = 1}^{M} {\tilde{r}_{ib} g_{b} }$$
(6)
$$\sum\limits_{i = 1}^{N} {\tilde{r}_{ib} = 1}$$
(7)
$$\tilde{r}_{ib} \ge 0$$
(8)
$$\tilde{s}_{ib} = \tilde{r}_{ib} g_{b} /g_{i}.$$
(9)
Here
\(g_{i}\) and
\(g_{b}\) represent the total consumption expenditure of commodity
i based on the JIOT and the total consumption expenditure by household income quintile, respectively. Equation (
6) shows that
\(g_{i}\) should be equal to the sum of consumption expenditure of commodity
i for each of the households. Since
\(\tilde{r}_{ib}\) is a ratio, and the total of each household is 1 (nonnegative), Eqs. (
7) and (
8) are satisfied. Equation (
9) expresses the relationship between
\(\tilde{r}_{ib}\) and
\(\tilde{s}_{ib}\).
We determined
\(g_{ib}\)(M-JPY/y), which is the consumption expenditure of commodity
i by household income quintile, by multiplying the optimal solutions of the above QP problem,
\(\hat{\tilde{r}}_{ib}\), and
\(g_{b}\). Since
\(g_{ib}\) is based on consumers’ prices and the carbon footprint intensity (t-CO
2eq/M-JPY) and material footprint intensity (t/M-JPY) are calculated on a producers’ price basis,
\(g_{ib}\) was converted to a producers’ price basis,
\(f_{ib}\) (see Shigetomi et al.
2014).
By multiplying the ratio of imported commodities
\(im_{i} \left( {0 \le im_{i} \le 1} \right)\), obtained from the JIOT, by
\(f_{{i_{} b}}\), the consumption expenditure for domestic commodities
\(f_{{i_{} b}}^{JD}\) (M-JPY/y) and the consumption expenditure for imported commodities
\(f_{{i_{} b}}^{JI}\) (M-JPY/y) were determined as follows:
$$f_{{i_{} b}}^{JD} = \left( {1 - im_{i} } \right)f_{ib}$$
(10)
$$f_{{i_{} b}}^{JI} = im_{i} f_{ib}.$$
(11)
Accordingly, the sum of consumption expenditure for each household income quintile estimated in this study is consistent with the total household expenditure in the JIOT.