Hypothetical extractions from a global perspective

ABSTRACT

The hypothetical extraction method (HEM) has been widely used to measure interindustry linkages and the importance of industries. HEM considers the hypothetical situation in which a certain industry is no longer operational. HEM was developed for national economies, using national input–output tables. When performing HEM, it is assumed (often implicitly) that the input requirements that were originally provided by the extracted industry are met by additional imports in the post-extraction situation. Applying HEM to global multiregional input–output tables then causes serious problems. It is no longer sufficient to assume that the required inputs are imported. Instead, it is necessary to indicate explicitly how much is imported from each origin to replace the original inputs. Our adaptation of HEM is the global extraction method (GEM). As an illustration, GEM is applied to the extraction of the motor vehicle industry in China, the US, and Germany, using the 2014 WIOD input–output table.

KEYWORDS:

• Hypothetical extraction method
• globalization effects
• industry linkage evaluation

1. Introduction

Globalization and the fragmentation of production processes have led to an enormous increase in the trade of intermediate products (see Baldwin, 2006, 2011, on the second great unbundling). A consequence is that an industry in one country requires inputs from another industry in a different country and interindustry linkages cross borders more and more often. Another consequence is that certain measures that are traditionally based on gross exports have become less meaningful. For example, Timmer et al. (2013) suggest to replace the standard measure for competitiveness by one that is based on the value added embodied in exports. That is, a measure that takes the global dimension of interindustry linkages into full account.

A method that has been widely used to measure interindustry linkages and the importance of industries is the hypothetical extraction method (HEM, developed by Paelinck et al., 1965, see Miller and Lahr, 2001, for an excellent overview). HEM considers the hypothetical situation in which a certain industry is no longer operational. Using the input–output framework, HEM calculates the outputs in the entire economy that are necessary for the original final demands. The difference between the original outputs and the HEM outputs (which are smaller than the original outputs) is a measure of the linkages of the deleted industry.

Deleting an industry or nullifying a sub-industry may adequately describe what happens to the production process in case of a disruption. To study the (in particular backward) impacts of disasters or disruptions, the inoperability input–output model has been widely used (see the introduction by Okuyama and Santos, 2014, to a recent special issue of Economic Systems Research on disaster impacts). Recently, however, Muldrow and Robinson (2014) and Dietzenbacher and Miller (2015) proposed HEM as an alternative input–output approach.

HEM was developed for national economies, using national input–output tables. When performing HEM, it is important that other things remain the same in order to single out the actual effect of the extraction. At the national level, this means that the remaining industries still receive the inputs they require. It is therefore assumed (often implicitly) that the input requirements that were originally provided by the extracted industry are met by additional imports in the post-extraction situation (Cai and Leung, 2004; Dietzenbacher and Lahr, 2013).

HEM was extended to the case of intercountry linkages in Dietzenbacher et al. (1993). Using multicountry input–output tables for part of the European Union (covering 5 countries for 1970 and 7 countries for 1980), one of the countries was hypothetically extracted (or isolated).¹ The same assumption that had been used in a national framework could be used here as well. For example, in the case of extracting Germany, the German agricultural inputs that are required by the French food processing industry are – in the post-extraction case – assumed to be imported from outside ‘the system’ (i.e. the EU5 or EU7, or the country when working in a national context).

Given the recent availability of a number of databases with world (global multiregional) input–output tables (see Tukker and Dietzenbacher, 2013, for an overview), it seems tempting to apply HEM also at the global level (e.g. Los et al., 2016). Unfortunately, however, this causes serious problems. Whereas HEM ‘has a clear economic intuition and can be easily taken to the data’ (Los et al., 2016, p. 1958) in the case of a national context, the intuition is far from clear in a global context. The assumption that has been used so far becomes problematic. If the German agriculture industry is extracted and can no longer export to, for example, the French food processing, the question arises where the French get the required agricultural inputs from? In a world input–output table, all countries are part of ‘the system’. The assumption to import the required inputs (that were originally provided by the – now – extracted industry) from outside ‘the system’ is no longer possible. It would assume importing from Mars. Simply nullifying the German agricultural exports would involve another heroic assumption. Namely that the French food processing industry is suddenly able to produce exactly the same output without any German agricultural input (and all other inputs, such as Spanish and Belgian agricultural inputs, remaining the same).

All this implies that the standard HEM, as developed for a national context, cannot be transferred straightforwardly to a global context and needs to be adapted. As a matter of fact, because there is nothing outside the system, the system has to replace the original (and now nullified) inputs itself. Moreover, we are forced to be explicit about how much of the inputs originate from which location. In this paper, we present our adaptation of HEM: the global extraction method (GEM). In the case of extracting German agriculture, it basically means that the French food processing industry replaces German agricultural inputs by Spanish agricultural inputs, and Belgian agricultural inputs, and so forth. The next section provides the details of the approach and Section 3 provides an empirical illustration of GEM for the motor vehicle industry.

2. The extraction methods

2.1. HEM at the national level

The original HEM was proposed to measure the importance of an industry (or its linkages) within a national economy. Suppose there are n industries. The typical element $z_{ij}$ of the $n\times n$ matrix $\mathbf{Z}$ gives the money value of intermediate deliveries from industry i to industry j, element $f_i$ of the n-element column vector $\mathbf{\text{f}}$ gives the deliveries from industry i to final users (i.e. for final demand purposes, including household consumption, investments, government expenditures, and exports), element $x_i$ of the n-element column vector $\mathbf{\text{x}}$ gives the output of industry i, element $v_j$ of the n-element column vector $\mathbf{\text{v}}$ gives the value added generated by industry j, and element $m_j$ of the n-element column vector $\mathbf{\text{m}}$ gives the imports by industry j.² Let the $n\times n$ matrix with input coefficients be given by $\mathbf{A}=\mathbf{Z}{\hat{\mathbf{x}}}^{-1}$, or $a_{ij}^{}=z_{ij}^{}/x_j^{}$ which gives the intermediate inputs from industry i to industry j, per unit of industry j’s output. In the same fashion, value added coefficients are given by ${\mathit{\pi }}^{\mathrm{\prime }}={\mathbf{\text{v}}}^{\mathrm{\prime }}{\widehat{\mathbf{\text{x}}}}^{-1}$, or ${\pi }_j=v_j/x_j$, and the import coefficients by ${\mathit{\mu }}^{\mathrm{\prime }}={\mathbf{\text{m}}}^{\mathrm{\prime }}{\widehat{\mathbf{\text{x}}}}^{-1}$, or ${\mu }_j=m_j/x_j$. The standard input–output equation is given by $\mathbf{\text{x}}=\mathbf{A}\mathbf{\text{x}}+\mathbf{\text{f}}$, or $\mathbf{\text{x}}=(\mathbf{I}-\mathbf{A}{)}^{-1}\mathbf{\text{f}}=\mathbf{L}\mathbf{f}$, with $\mathbf{L}$ the Leontief inverse. The total value added is obtained by $VA={\mathit{\pi }}^{\mathrm{\prime }}\mathbf{\text{x}}={\mathit{\pi }}^{\mathrm{\prime }}\mathbf{\text{Lf}}$, and the total imports of intermediate inputs by $IMPINT={\mathit{\mu }}^{\mathrm{\prime }}\mathbf{\text{x}}={\mathit{\mu }}^{\mathrm{\prime }}\mathbf{L}\mathbf{f}$.

Suppose now that industry k is hypothetically extracted from the domestic economy. The input coefficients in the kth row and column are then nullified, and so is the final demand for products from this industry. This yields a new input matrix $\overline{\mathbf{A}}$ and a new final demand vector $\overline{\mathbf{f}}$. That is (1a) ${\overline{a}}_{kj}={\overline{a}}_{ik}=0\mathrm{\forall }i,j,$(1a) (1b) ${\overline{a}}_{ij}=a_{ij}\mathrm{\forall }i,j\ne k,$(1b) (1c) ${\overline{f}}_k=0,$(1c) (1d) ${\overline{f}}_i=f_i\mathrm{\forall }i\ne k.$(1d)

When HEM is performed on national input–output tables, it is – often implicitly – assumed that industry j (≠k) now imports the intermediate inputs of product k instead of buying them at home. The underlying idea is that the demand for intermediate products is determined technologically and is fixed. This means that every unit of output of industry j requires a certain amount of product i as intermediate input, no matter whether produced domestically or imported. The same applies to satisfying the final demands of product k. Making this implicit assumption explicit, we would have for the import coefficients: ${\overline{\mu }}_j={\mu }_j+a_{kj},\mathrm{\forall }j\ne k$, and ${\overline{\mu }}_k=0$. The imports of final goods would increase by $f_k$.

Satisfying the same final demands in the hypothetical economy would imply the following levels of outputs, total value added, and total intermediate imports: (2) $\overline{\mathbf{x}}=(\mathbf{I}-\overline{\mathbf{A}}{)}^{-1}\overline{\mathbf{f}}=\overline{\mathbf{L}}\overline{\mathbf{f}},$(2) (3) $\overline{VA}={\mathit{\pi }}^{\mathrm{\prime }}\overline{\mathbf{\text{x}}}={\mathit{\pi }}^{\mathrm{\prime }}\overline{\mathbf{L}}\overline{\mathbf{f}},$(3) (4) $\overline{IMPINT}={\overline{\mathit{\mu }}}^{\mathrm{\prime }}\overline{\mathbf{x}}={\overline{\mathit{\mu }}}^{\mathrm{\prime }}\overline{\mathbf{L}}\overline{\mathbf{f}}.$(4)

The differences between the outputs under HEM and the original outputs have been proposed as an indicator for industry k’s importance for the national production. $\overline{\mathbf{x}}-\mathbf{x}$ (or the difference in total output $\mathrm{\Delta }\mathit{\text{OUTPUT}}={\mathbf{e}}_n^{\mathrm{\prime }}\left(\overline{\mathbf{x}}-\mathbf{x}\right)$, with ${\mathbf{e}}_n$ the n-element column summation vector consisting of ones) indicates the change in output levels if industry k ceases to exist. It is well known (and easy to prove) that, under the usual assumptions, the outputs will decrease under HEM, i.e. ${\overline{x}}_i<x_i,\mathrm{\forall }i$, and ${\overline{x}}_k=0$. The output loss in industries other than k is because they do not have to produce intermediate inputs for industry k any more. The change in total value added is $\mathrm{\Delta }VA=\overline{VA}-VA={\mathit{\pi }}^{\mathrm{\prime }}\left(\overline{\mathbf{x}}-\mathbf{x}\right)$, which is negative so that extracting any industry from the economy decreases the total value added. The total imports (of intermediate inputs and final goods) will also change under HEM, i.e. $\mathrm{\Delta }IMP=\mathrm{\Delta }IMPINT+\mathrm{\Delta }IMPFIN=\left(\overline{IMPINT}-IMPINT\right)+\left(\overline{IMPFIN}-IMPFIN\right)=\left({\overline{\mathit{\mu }}}^{\mathrm{\prime }}\overline{\mathbf{\text{x}}}-{\mathit{\mu }}^{\mathrm{\prime }}\mathbf{x}\right)+f_k$. The change in the imports turns out to be positive, which is proved next.

We have made the explicit assumption that the extracted goods and services are replaced by the same goods and services from abroad. We can therefore study the effects on the imports and (in combination with domestic GDP) on global GDP. Past research on HEM primarily restricted the focus to the domestic effects.

Consider the change in world GDP. The change in national GDP is given by the change in domestic value added, i.e. $\mathrm{\Delta }VA={\mathit{\pi }}^{\mathrm{\prime }}\left(\overline{\mathbf{x}}-\mathbf{x}\right)$. Imports are (in the current framework) foreign value added, so the change in foreign GDP is given by $\mathrm{\Delta }IMP=\left({\overline{\mathit{\mu }}}^{\mathrm{\prime }}\overline{\mathbf{x}}-{\mathit{\mu }}^{\mathrm{\prime }}\mathbf{\text{x}}\right)+f_k$. The change in world GDP (WGDP) is then given by $\mathrm{\Delta }WGDP=\mathrm{\Delta }VA+\mathrm{\Delta }IMP={\mathit{\pi }}^{\mathrm{\prime }}\left(\overline{\mathbf{x}}-\mathbf{x}\right)+\left({\overline{\mathit{\mu }}}^{\mathrm{\prime }}\overline{\mathbf{x}}-{\mathit{\mu }}^{\mathrm{\prime }}\mathbf{\text{x}}\right)+f_k$.

Note that ${\mathit{\pi }}^{\mathrm{\prime }}+{\mathit{\mu }}^{\mathrm{\prime }}+{\mathbf{e}}_n^{\mathrm{\prime }}\mathbf{A}={\mathbf{e}}_n^{\mathrm{\prime }}$ and ${\overline{\mathit{\pi }}}^{\mathrm{\prime }}+{\overline{\mathit{\mu }}}^{\mathrm{\prime }}+{\mathbf{e}}_n^{\mathrm{\prime }}\overline{\mathbf{A}}={\mathbf{e}}_n^{\mathrm{\prime }}$. This implies ${\mathit{\pi }}^{\mathrm{\prime }}+{\mathit{\mu }}^{\mathrm{\prime }}={\mathbf{e}}_n^{\mathrm{\prime }}\left(\mathbf{I}-\mathbf{A}\right)$ and post-multiplying with $\mathbf{L}=(\mathbf{I}-\mathbf{A}{)}^{-1}$ gives ${\mathit{\pi }}^{\mathrm{\prime }}\mathbf{L}+{\mathit{\mu }}^{\mathrm{\prime }}\mathbf{L}={\mathbf{e}}_n^{\mathrm{\prime }}$ or ${\mathit{\pi }}^{\mathrm{\prime }}\mathbf{L}+{\mathit{\mu }}^{\mathrm{\prime }}\mathbf{L}-{\mathbf{e}}_n^{\mathrm{\prime }}=0$. In the same way, we have ${\mathit{\pi }}^{\mathrm{\prime }}\overline{\mathbf{L}}+{\overline{\mathit{\mu }}}^{\mathrm{\prime }}\overline{\mathbf{L}}-{\mathbf{e}}_n^{\mathrm{\prime }}=0^{\mathrm{\prime }}$, as $\mathit{\pi }$ has not changed. We can now write $\mathrm{\Delta }WGDP=\left({\mathit{\pi }}^{\mathrm{\prime }}\overline{\mathbf{x}}+{\overline{\mathit{\mu }}}^{\mathrm{\prime }}\overline{\mathbf{x}}+f_k\right)-\left({\mathit{\pi }}^{\mathrm{\prime }}\mathbf{x}+{\overline{\mathit{\mu }}}^{\mathrm{\prime }}\mathbf{x}\right)$. Using $f_k={\mathbf{e}}_n^{\mathrm{\prime }}\mathbf{f}-{\mathbf{e}}_n^{\mathrm{\prime }}\overline{\mathbf{f}}$, $\overline{\mathbf{x}}=\overline{\mathbf{L}}\overline{\mathbf{f}}$ and $\mathbf{\text{x}}=\mathbf{L}\mathbf{f}$ gives $\mathrm{\Delta }WGDP=\left({\mathit{\pi }}^{\mathrm{\prime }}\overline{\mathbf{L}}+{\overline{\mathit{\mu }}}^{\mathrm{\prime }}\overline{\mathbf{L}}-{\mathbf{e}}_n^{\mathrm{\prime }}\right)\overline{\mathbf{f}}-({\mathit{\pi }}^{\mathrm{\prime }}\mathbf{L}+{\mathit{\mu }}^{\mathrm{\prime }}\mathbf{L}-{\mathbf{e}}_n^{\mathrm{\prime }})\mathbf{f}=0$. Hence, world GDP does not change if an industry is extracted in HEM. Value added is redistributed between countries. Domestic VA decreases and the total imports are increased by the same amount.

It should be stressed that deleting an entire industry is a heroic assumption. However, this case should be viewed as a benchmark case. Dietzenbacher and Lahr (2013, p. 349) analyzed also partial extractions, which are more realistic. They may follow from the nullification of one or more sub-industries or from the partial reduction of some of the input coefficients. In one of their applications, they looked at the relationship between the decrease in value added and the size of the reduction in selected input coefficients. They concluded that ‘although the relationship … is nonlinear, it is very nearly linear. It, thus, follows that in this application, the result for partial extraction can be well estimated from the result for full extraction.’

2.2. The global extraction method

To explain the GEM, suppose there are N countries with n industries. The $Nn\times Nn$ matrix $\mathbf{Z}$ of intermediate deliveries, the $Nn\times N$ matrix $\mathbf{F}$ of final demands, the Nn-element output vector $\mathbf{x}$, and the Nn-element value added vector $\mathbf{v}$ are (in partitioned form) given by $\begin{array}{rl}\mathbf{Z}& =\left[\begin{array}{ccc}\begin{array}{cc}{\mathbf{Z}}^{11}& \cdots \\ ⋮& \ddots \end{array}& \begin{array}{c}{\mathbf{Z}}^{1R}\\ ⋮\end{array}& \begin{array}{cc}\cdots & {\mathbf{Z}}^{1N}\\ \text{.}\text{.}\text{.}& ⋮\end{array}\\ \begin{array}{cc}{\mathbf{Z}}^{R1}& \cdots \end{array}& {\mathbf{Z}}^{RR}& \begin{array}{cc}\cdots & {\mathbf{Z}}^{RN}\end{array}\\ \begin{array}{cc}⋮& \text{.}\text{.}\text{.}\\ {\mathbf{Z}}^{N1}& \cdots \end{array}& \begin{array}{c}⋮\\ {\mathbf{Z}}^{NR}\end{array}& \begin{array}{cc}\ddots & ⋮\\ \cdots & {\mathbf{Z}}^{NN}\end{array}\end{array}\right],\mathbf{F}=\left[\begin{array}{ccc}\begin{array}{cc}{\mathbf{f}}^{11}& \cdots \\ ⋮& \ddots \end{array}& \begin{array}{c}{\mathbf{f}}^{1R}\\ ⋮\end{array}& \begin{array}{cc}\cdots & {\mathbf{f}}^{1N}\\ \text{.}\text{.}\text{.}& ⋮\end{array}\\ \begin{array}{cc}{\mathbf{f}}^{R1}& \cdots \end{array}& {\mathbf{f}}^{RR}& \begin{array}{cc}\cdots & {\mathbf{f}}^{RN}\end{array}\\ \begin{array}{cc}⋮& \text{.}\text{.}\text{.}\\ {\mathbf{f}}^{N1}& \cdots \end{array}& \begin{array}{c}⋮\\ {\mathbf{f}}^{NR}\end{array}& \begin{array}{cc}\ddots & ⋮\\ \cdots & {\mathbf{f}}^{NN}\end{array}\end{array}\right],\\ \mathbf{x}& =\left(\begin{array}{c}{\mathbf{x}}^1\\ ⋮\\ {\mathbf{x}}^R\\ ⋮\\ {\mathbf{x}}^N\end{array}\right);\mathbf{v}=\left(\begin{array}{c}{\mathbf{v}}^1\\ ⋮\\ {\mathbf{v}}^R\\ ⋮\\ {\mathbf{v}}^N\end{array}\right)\end{array}$ Element $z_{ij}^{RS}$ of the $n\times n$ matrix ${\mathbf{Z}}^{RS}$ gives the money value of intermediate deliveries from industry i in country R to industry j in country S, element $f_i^{RS}$ of the n-element vector ${\mathbf{f}}^{RS}$ gives the deliveries from industry i in country R for final demands in country S, element $x_i^R$ of the n-element vector ${\mathbf{x}}^R$ gives the output of industry i in country R, and element $v_j^R$ of the n-element vector ${\mathbf{v}}^R$ gives the value added generated by industry i in country R. The $Nn\times Nn$ matrix with input coefficients is given by $\mathbf{A}=\mathbf{Z}{\hat{\mathbf{x}}}^{-1}$, implying ${\mathbf{A}}^{RS}={\mathbf{Z}}^{RS}({\hat{\mathbf{x}}}^S{)}^{-1}$ or $a_{ij}^{RS}=z_{ij}^{RS}/x_j^S$ which gives the intermediate inputs per unit of the receiving industry’s output. The Nn-element final demand vector is given by $\mathbf{f}=\mathbf{F}{\mathbf{e}}_N$ (with ${\mathbf{f}}^R=\sum _S{\mathbf{f}}^{RS}$).

Suppose that industry k in country H is hypothetically extracted. That is, (5a) ${\overline{a}}_{kj}^{HS}={\overline{a}}_{ik}^{TH}=0\mathrm{\forall }i,j,\mathrm{\forall }S,T,$(5a) (5b) ${\overline{f}}_k^{HS}=0\mathrm{\forall }S.$(5b) Equations 5a and 5b express that industry k in country H (hereafter denoted as k-H) does not buy any inputs (${\overline{a}}_{ik}^{TH}=0$) and does not sell any outputs, neither to an industry (${\overline{a}}_{kj}^{HS}=0$) nor to a final user (${\overline{f}}_k^{HS}=0$).

Any industry other than k-H that used to require inputs from k-H now needs to buy input k from another source. We distinguish between industries (other than k) in country H and industries in other countries. As an example, suppose the Belgian agricultural industry is extracted. The Belgian food processing requires agricultural inputs and suppose that 40% of them originate in Belgium itself, 30% come from France, 20% from Germany, and the remaining 10% from the Netherlands. Because the Belgian food processing can no longer buy inputs from the Belgian agricultural industry, we assume that the imported agricultural inputs are all increased by the same percentage (in this case 66.7%), so that they add up to 100% again. This means that the Belgian food processing now imports 50% of its required agricultural inputs from France, 33.3% from Germany, and 16.7% from the Netherlands. Note that it is assumed that the matrix of technological coefficients (which, for country S, is obtained as $\sum _R{\mathbf{A}}^{RS}$) remains fixed. From a technological perspective, one unit of output in industry j-H requires good k as an intermediate input to the amount of $\sum _R{a}_{kj}^{RH}$. So, whenever inputs of good k can no longer come from country H, they must come from one of the other countries.

For any foreign industry, the situation is slightly different. Suppose again that the Belgian agricultural industry is extracted. Suppose that originally the German food processing requires agricultural inputs from the own country (60%), from Belgium (12%), from France (20%), and from the Netherlands (8%). After extraction, the German food processing can no longer buy agricultural inputs from Belgium. We assume that the German food processing now imports the Belgian agricultural inputs from the other original exporters in the same proportion. German agricultural imports from France and the Netherlands are increased by 12/28 (which is 42.9%) so that they add up to 40% again. This means that the German food processing still buys 60% of its required agricultural inputs at home in Germany, 28.6% from France, and 11.4% from the Netherlands. Observe that the use of German agricultural inputs does not change. Our reasoning is that the German food processing buys Belgian agricultural inputs (in the original situation) for some reason (e.g. because the products are not available in Germany, or are too expensive). We assume therefore that Belgian agricultural inputs are not replaced by German agricultural products, but only by other non-German agricultural products.

The same assumptions that have been made for intermediate inputs are also made for replacing the deliveries by industry k-H to final users. Mathematically, the assumptions are as follows. (5c) ${\overline{a}}_{kj}^{TS}=a_{kj}^{TS}+a_{kj}^{HS}\frac{a_{kj}^{TS}}{{\sum }_{R\ne H,S}{a}_{kj}^{RS}}\mathrm{\forall }j,\mathrm{\forall }S,\mathrm{\forall }T\ne H,S,$(5c) (5d) ${\overline{f}}_k^{TS}=f_k^{TS}+f_k^{HS}\frac{f_k^{TS}}{{\sum }_{R\ne H,S}{f}_k^{RS}}\mathrm{\forall }S,\mathrm{\forall }T\ne H,S.$(5d) For any other elements that are not covered by the cases in Equations 5a–5d, we have ${\overline{a}}_{ij}^{RS}=a_{ij}^{RS}$ and ${\overline{f}}_i^{RS}=f_i^{RS}$.

The assumption that Belgian agricultural products are not replaced by German agricultural products but only by other non-German agricultural products, can be motivated as follows. First, there is the ‘love of variety’. In line with new trade theory, it has been argued that love of variety leads to differentiation between firms operating in the same industry (Bernard et al., 2007). The demand for variety in receiving country $S$ implies that it is unlikely that the import of good i is replaced by the domestically produced good i. Second, industries in input–output tables are generally very broad. For our empirical application in the next section, we have used the WIOD tables (Dietzenbacher et al., 2013, Timmer et al., 2015), which distinguish 56 industries. Each of these industries has a considerable number of sub-industries. It may thus be the case that country $S\left(\ne H\right)$ imports from a specific sub-industry of industry k in H because this sub-industry is not present (or not well developed, or very small) in S. The extraction of industry k-H will not change the situation in country S, in the sense that this specific sub-industry will still be absent in S. Imports by S from H will therefore not be replaced by domestically produced inputs.

It should be admitted that the adaptations in Equations 5a–5d are somewhat mechanical. At the same time, however, it should be stressed that the extraction method is sufficiently flexible to cover other assumptions based on additional, more detailed information (see Dietzenbacher and Lahr, 2013).

The calculations for GEM are very similar to those for the standard HEM in the previous subsection. The differences between the outputs in the new situation and the original outputs indicate the importance of industry k-H for production. That is, ${\overline{\mathbf{x}}}^T-{\mathbf{x}}^T$ gives the effect on the output levels in country T (or the difference in total output $\mathrm{\Delta }OUTPU{T}^T={\mathbf{e}}_n^{\mathrm{\prime }}\left({\overline{\mathbf{x}}}^T-{\mathbf{x}}^T\right)$). The change in total value added is $\mathrm{\Delta }V{A}^T={\overline{VA}}^T-V{A}^T=\left({\mathit{\pi }}^T{)}^{\mathrm{\prime }}\right({\overline{\mathbf{x}}}^T-{\mathbf{x}}^T)$. It should be noted that the sum of values added over the countries (i.e. world GDP) does not change. This was also the case with HEM.

To prove that world GDP does not change, we use the following theorem that has wider application than GEM or HEM.

Theorem 1:

Assume the input–output table changes but remains consistent (i.e. the sum of the intermediate input coefficients and the value added coefficient equals one in each industry). World GDP remains constant if (and only if) the global sum of final demands remains constant.

Proof.

We have to show that $\sum _T\mathrm{\Delta }V{A}^T=0$. Consistency of the input–output tables implies ${\mathit{\pi }}^{\mathrm{\prime }}={\mathbf{\text{e}}}_{Nn}^{\mathrm{\prime }}\left(\mathbf{I}-\mathbf{A}\right)$ before the change and ${\overline{\mathit{\pi }}}^{\mathrm{\prime }}={\mathbf{\text{e}}}_{Nn}^{\mathrm{\prime }}\left(\mathbf{I}-\overline{\mathbf{A}}\right)$ after the change. World GDP is given by ${\mathit{\pi }}^{\mathrm{\prime }}\mathbf{x}={\mathbf{\text{e}}}_{Nn}^{\mathrm{\prime }}\left(\mathbf{I}-\mathbf{A}\right)(\mathbf{I}-\mathbf{A}{)}^{-1}\mathbf{f}={\mathbf{\text{e}}}_{Nn}^{\mathrm{\prime }}\mathbf{f}$ in the original situation and by ${\overline{\mathit{\pi }}}^{\mathrm{\prime }}\overline{\mathbf{x}}={\mathbf{\text{e}}}_{Nn}^{\mathrm{\prime }}\overline{\mathbf{f}}$ after the change. Because ${\mathbf{\text{e}}}_{Nn}^{\mathrm{\prime }}\overline{\mathbf{f}}={\mathbf{\text{e}}}_{Nn}^{\mathrm{\prime }}\mathbf{f}$, we have ${\overline{\mathit{\pi }}}^{\mathrm{\prime }}\overline{\mathbf{x}}={\mathit{\pi }}^{\mathrm{\prime }}\mathbf{x}$. This implies that world GDP is not affected by the changes, just redistributed.

Note that GEM is a special case in the sense that the value added coefficient in any (but the extracted) industry is the same before and after extraction. In other words, the intermediate inputs are replaced by the same intermediate inputs from another country. Theorem 1 also allows other changes. For example, substituting a certain intermediate input for labor or making the production process more productive. Also HEM can be viewed as a special case of Theorem 1 (see Supplemental material).

The intuition for the world GDP to remain constant is as follows. Any final product that is consumed (i.e. used as final demand) must be produced. Ultimately (using the round-by-round approach), it exists entirely of values added, nothing else. These are values added generated in all industries, domestic and foreign. Consequently, the sum of all final demands must equal the sum of all values added. If the one does not change, also the other must remain constant.

3. An illustration for the motor vehicle industry

To demonstrate its merit, we have applied GEM to the 2016 Release of the World Input–Output Database (WIOD), which involves 56 industries and 44 countries including the rest-of-the-world (RoW) region (Timmer et al., 2015). Of the time series of the WIOD world input–output tables, we have used the table for 2014, which is the latest year available. We have focused on the industry ‘Manufacture of motor vehicles, trailers and semi-trailers’ (hereafter, the motor vehicle industry). We studied the cases in which the motor vehicle industry in China, in Germany and in the United States were – each separately – extracted. The motor vehicle industries in these three countries have the largest shares in the world total of VA in motor vehicle industries (China 21.6%, Germany 13.9%, and the US 13.6%).

Extraction of the Chinese motor vehicle industry by GEM decreases Chinese GDP by 608.7 billion USD (which is 5.9% of GDP). This decrease in Chinese GDP can be divided into two parts. One is a decrease of 230.0 billion USD in the value added in the extracted sector itself (i.e. the motor vehicle industry), which we call the internal effect. The other part is a decrease of 378.7 billion USD in the value added in the other industries in China, which we call the external effect. Figure 1 shows the changes in the GDP of selected countries, the results for all countries are given in Table 1. The internal effect measures the change in the VA in the motor vehicle industry, the external effect the total change in the VA of the other industries. Germany, Japan, the United States, the United Kingdom, and Korea are greatly affected by the extraction of Chinese motor vehicle industry, in the sense that their GDP is substantially increased. It should be noted that they are all top 10 motor vehicle-producing countries in terms of VA generated in the motor vehicle industry. It appears that Chinese motor vehicles are replaced by motor vehicles produced in these countries. In contrast, and perhaps surprisingly, Australia and Taiwan are negatively affected. GDP is decreased by 1.3 billion USD in Australia (with a negative external effect of 1.7 billion USD). In Taiwan, GDP is slightly increased (0.4 billion USD) but also has a negative external effect, of 0.6 billion USD (see Table 1). This can be attributed to the fact that Australia and Taiwan have significant contributions of upstream industries in the supply chain of Chinese motor vehicles. This holds, in particular, for the ‘Mining and quarrying’ industry in Australia and the ‘Manufacture of computer, electronic and optical products’ industry in Taiwan. (Among the elements in the column of $\hat{\mathit{\pi }}\mathbf{L}=\hat{\mathit{\pi }}(\mathbf{I}-\mathbf{A}{)}^{-1}$ for the Chinese motor vehicle industry, the elements for the ‘Mining and quarrying’ sector in Australia and the ‘Manufacture of computer, electronic and optical products’ sector in Taiwan are the second and fourth largest, respectively, except for the elements referring to Chinese and RoW industries.)

FIGURE 1. Change in GDP and internal and external effects in selected countries when the Chinese motor vehicle industry is extracted.

Table 1. Change in GDP after extraction of the motor vehicle industry and the split into internal and external effect (results by country, in million USD).

	China			Germany			the US
Country	Internal	External	ΔGDP	Internal	External	ΔGDP	Internal	External	ΔGDP
AUS	349	−1673	−1323	135	1242	1376	155	1626	1780
AUT	3468	5399	8867	1852	−1048	803	1578	1926	3504
BEL	1851	3883	5734	3327	807	4133	492	1042	1534
BGR	50	315	365	107	120	227	20	122	143
BRA	897	1016	1914	925	1631	2556	1222	1342	2564
CAN	1730	4231	5961	2584	3143	5728	16604	4342	20946
CHE	242	2784	3025	231	−951	−720	197	823	1020
CHN	−229991	−378738	−608728	5587	14140	19727	8471	16587	25058
CYP	0	45	46	2	12	14	0	18	18
CZE	2627	4217	6844	4769	1878	6647	1034	1390	2423
DEU	101629	88392	190021	−147494	−98844	−246338	34192	29091	63283
DNK	95	1165	1260	207	−7	200	48	383	432
ESP	1866	5196	7063	7163	8553	15715	1604	2846	4450
EST	22	94	116	81	34	115	13	33	46
FIN	218	939	1157	427	200	627	90	265	356
FRA	3442	11949	15392	6725	9412	16138	1397	4006	5403
GBR	16844	26068	42912	5256	6347	11603	3668	5356	9023
GRC	1	307	308	1	75	77	1	122	123
HRV	15	225	240	39	5	44	8	84	92
HUN	3166	2422	5588	3103	577	3679	1393	885	2277
IDN	1115	964	2079	783	908	1691	731	1095	1826
IND	1008	2888	3896	1239	2631	3870	811	1586	2396
IRL	79	801	880	171	145	316	36	27	63
ITA	4809	16166	20975	4806	8034	12839	2773	6938	9712
JPN	50274	43820	94094	19063	18155	37218	42837	37442	80279
KOR	15964	15312	31277	7916	9666	17581	14166	15215	29381
LTU	20	185	204	50	7	57	9	67	76
LUX	9	417	426	33	−130	−97	5	157	162
LVA	9	88	98	33	14	47	7	33	39
MEX	6161	6134	12295	5132	4481	9614	27903	14649	42553
MLT	3	34	37	4	10	14	1	12	13
NLD	706	6349	7055	1541	−2326	−785	265	1836	2101
NOR	200	2070	2270	222	−274	−52	51	662	713
POL	1778	7175	8953	2724	965	3689	761	2545	3306
PRT	1632	1369	3001	1229	779	2008	211	303	515
ROU	590	1877	2467	719	765	1484	234	687	921
RUS	565	4594	5159	273	270	544	115	2189	2304
SVK	4652	3948	8600	2086	1136	3222	719	800	1519
SVN	181	568	749	558	77	635	86	202	288
SWE	2870	4302	7172	3563	2841	6404	1132	1427	2559
TUR	881	3383	4264	3010	3164	6174	547	1204	1751
TWN	1005	−559	447	581	1079	1660	1396	1121	2517
USA	25925	49084	75009	8499	16168	24668	−145060	−206630	−351689
ROW	13408	8428	21836	7471	17376	24847	6980	15244	22225

Notes: The column labeled ‘Internal’ shows the internal effect in each country, which refers to the change in value added in the motor vehicle industry in that country. The column labeled ‘External’ refers to the change in value added in the other industries. The column labeled ‘ΔGDP’ refers to the change in GDP of a country.

One caveat applies to the empirical results in our paper. That is, for our calculations of the change in GDP, we have taken – for each industry – the output at basic prices minus total intermediate consumption. We then have summed over the industries, which is also known as the sum of industries’ primary inputs. It equals GDP at basic prices plus the international transport margins paid by industries and taxes less subsidies on products paid by industries. In the previous section, we proved – both for GEM and HEM – that global GDP as measured by the sum of industries’ primary inputs remains the same.³ In this empirical case, however, the sum of industries’ primary inputs differs from the global GDP at basic prices. In 2014, the sum of industries’ primary inputs (75,447 billion USD) consists of global GDP at basic prices (73,807 billion USD, 97.8% of the sum of industries’ primary inputs), taxes less subsidies on products paid by industries (987 billion USD, 1.3%), and international transport margins paid by industries (653 billion USD, 0.9%). The results in our tables and figures are based on changes in the sum of industries’ primary inputs, because at the global level, it is not affected by the extraction and because the difference with global GDP at basic prices is small. Looking at what happens to GDP at basic prices (instead of the sum of industries’ primary inputs), we find that extraction of the Chinese motor vehicle industry decreases China’s GDP by 600.2 billion USD (instead of 608.7 billion USD dollars when the sum of industries’ primary inputs is used). The increase in foreign (i.e. non-Chinese) GDP amounts to 580.9 billion USD. Global GDP at basic prices therefore does change a little, it decreases by 19.3 billion USD (0.03% of global GDP at basic prices in 2014). The sum of industries’ primary inputs remains the same, of course.

When the German motor vehicle industry is extracted by GEM, German GDP is decreased by 246.3 billion USD (which is 6.8% of GDP), of which 59.9% are internal and 40.1% are external effects. As shown in Figure 2, Japan, the United States, China, Korea, France, Spain, Italy, and the United Kingdom are substantially and positively affected by the extraction of the German motor vehicle industry. Of these eight countries, France, Spain, and Italy are not major competitors. That is, their motor vehicle industries are not in the top 10 but only in the top 20 of VA generated by motor vehicle industries.

FIGURE 2. Change in GDP and internal and external effects in selected countries when the German motor vehicle industry is extracted.

The replacement of German motor vehicles leads to an increase in the GDP of countries that produce motor vehicles. However, because of close relationships among European countries, several of them are negatively affected by the extraction of the German motor vehicle industry. Austria has a negative external effect of 1.0 billion USD, while its GDP is increased by 0.8 billion USD. This means that the replacement of German by Austrian motor vehicles yields a larger positive effect on VA than the negative effect on VA in Austrian upstream industries in the supply chain of German motor vehicles. The GDPs of the Netherlands, Switzerland, Luxembourg, and Norway, in which the motor vehicle industry is not sizable, are decreased by 1.7 billion USD in total. The industries in which the value added is decreased more than 0.1 billion USD include ‘Mining and quarrying’ in the Netherlands and Norway; ‘Manufacture of basic metals’ in Austria and the Netherlands; and ‘Manufacture of fabricated metal products, except machinery and equipment’ in Austria and Switzerland.

When the US motor vehicle industry is extracted, the GDP of the United States is decreased by 351.7 billion USD (which is 2.0% of GDP). 41.2% of this decrease is internal effects and 58.8% is external effects. As shown in Figure 3, Mexico and Canada, which are not significantly affected in the cases of China and Germany, are greatly affected in the US case. Moreover, the extraction of the US motor vehicle industry does not induce negative external effects to any country. This indicates that the industrial structure of the US economy is more self-supporting than the structure of China and Germany. In other words, a substantial portion of the deliveries by upstream industries in the supply chain of the US motor vehicle industries are from industries located in the United States.

FIGURE 3. Change in GDP and internal and external effects in selected countries when the US motor vehicle industry is extracted.

Extracting the Chinese motor vehicle industry by HEM leads to a decrease in the Chinese GDP of 623.6 billion USD. Compared with the result with GEM (a decrease of 608.7 billion USD), HEM decreases China’s GDP 14.9 billion USD more than GEM does, which is 2.4% more (Table 2). When the German motor vehicle industry is extracted, HEM shows a decrease of 256.8 billion USD in German GDP, which is 10.5 billion USD or 4.3% more than GEM. The extraction of the US motor vehicle industry yields a decrease with HEM that is 16.3 billion USD or 4.6% larger than with GEM. In these three cases, HEM provides a larger decrease in GDP of the country, from which its motor vehicle industry is extracted, than GEM. It should be noted that the internal effect (i.e. the loss of the original VA of the motor vehicle industry) is the same for both methods. This means that the difference between the two methods is in the external effects. They are more negative for HEM than for GEM because the replacement of e.g. Chinese motor vehicles by foreign motor vehicles requires extra inputs from China when a global multiregional input–output framework is used (as is the case for GEM), but not when a national input–output framework is used (as is the case with HEM). This means that, when the extraction method is applied to a country in which raw materials are produced and exported, the difference between GEM and HEM is likely to be substantial.

Table 2. Comparison between GEM and HEM for the extraction of the motor vehicle industry.

	GEM	HEM	$\mathrm{\Delta }$	%$\mathrm{\Delta }$-all	%$\mathrm{\Delta }$-ext
Extraction in	(1)	(2)	(3)	(4)	(5)
China	−608.7	−623.6	14.9	−2.4	−3.9
Germany	−246.3	−256.8	10.5	−4.3	−10.6
USA	−351.7	−368.0	16.3	−4.6	−7.9

Notes: Columns (1), (2) and (3) are in billion USD, columns (4) and (5) are percentages. ‘$\mathrm{\Delta }$’ gives the difference between GEM and HEM, (3) = (1) – (2). ‘%$\mathrm{\Delta }$-all’ gives the difference as percentage of all effects in GEM, (4) = 100 × (3)/(1). ‘%$\mathrm{\Delta }$-ext’ gives the difference as a percentage of the external effects reported in Table 1. Each figure in this table refers to the change in the GDP of the country from which its motor vehicle industry is extracted. For GEM has the sum of the changes in the GDP in other countries the same size but the opposite sign. For HEM, this holds for the sum of all imports.

In addition, the differences as reported in column (4) of Table 2 underreport the percentage errors. Because the difference between GEM and HEM is in the external effects only, the last column in Table 2 provides the difference as a percentage of the part that varies across the methods (i.e. the external effects with GEM). Obviously, the percentages are larger and the error even becomes sizeable for Germany (10.6%).

4. Summary and conclusions

The HEM was originally developed – and has been widely used – to measure interindustry linkages at the national level. Recently, however, Muldrow and Robinson (2014) and Dietzenbacher and Miller (2015) proposed HEM for describing what happens in the short-run to production in case of a disaster or disruption. One of the industries in a country is hypothetically deleted (or nullified). The loss in, for example, GDP then indicates how interwoven this industry is with other industries in the country, which reflects this industry’s importance for the country. The silent assumption is that this industry’s product is replaced by an imported product whenever it is used as an input in other domestic industries. The imports increase, which equals the increase in foreign GDP.

Given the recent availability of databases with world (global multiregional) input–output tables, it seems an obvious step to apply HEM also at the global level. However, this is not possible. The silent assumption that was used for HEM can no longer be used. That is, at the global level, we must specify explicitly how the deleted inputs are replaced. In this paper, we have proposed the GEM and we have provided a very mechanistic way of replacing the deleted (or nullified) inputs. In practical real-world applications, researchers will probably have additional information. As Dietzenbacher and Lahr (2013) pointed out, the extraction method is extremely flexible and more realistic scenarios for replacing deleted inputs can easily be implemented.

To test the working of this extraction method, we have applied GEM and HEM to the extraction of the motor vehicle industry in China, in the US, and in Germany, using the 2014 WIOD input–output table. Summarizing the differences between GEM and HEM, we find the following. (1) GEM requires global multiregional input–output tables and is thus more demanding in terms of data than HEM, which only uses national input–output tables. (2) GEM calculates the effects in other countries. Most of the effects are positive in other countries. However, for the extraction of the German motor vehicle industry, we found small declines in GDP in the Netherlands, Switzerland, and Luxemburg. This is because the extraction of the motor vehicle industry in Germany reduces outputs in other German industries, including those that depend a lot on Swiss inputs. The losses in these Swiss industries are larger than the (small) gains in the Swiss motor vehicle industry. (3) GEM requires assumptions about how the outputs of the extracted industry (that are used either as inputs or as final products) are replaced. (4) For both methods, it is the case that world GDP remains the same when an industry is extracted implying that a redistribution of value added takes place.

In evaluating the extraction methods, let us distinguish between the case where one is only interested in the domestic effects of extraction and the case where one is also interested in the foreign effects.

In the first case, two related questions pop up. Why would we use the global input–output model and which method is preferred, GEM or HEM? The global model is theoretically superior to the national model, even if one is only interested in the national effects of extracting an industry. This is because the global model includes intercountry feedback effects. As an example, extraction of the Brazilian motor vehicle industry would reduce imports of US tires, which (supposedly) use Brazilian rubber as input. The output reduction in the Brazilian rubber industry is a feedback effect that is included in the global input–output model but not in a national model. The importance of these feedback effects has increased over time due to globalization and fragmentation of the production processes. However, although the global model is superior to the national model from a theoretical viewpoint, it remains to be seen whether this also holds for empirical applications. It may be the case that the theoretical ‘gains’ are smaller than the empirical ‘losses’.

Data quality and the amount of detail that is required are crucial in this respect. A major part of the deliveries in global input–output tables is estimated and global tables are only available at a high level of aggregation (whereas national tables are often available for a much more detailed industry classification). For the choice between GEM and HEM, it also matters whether one is interested in, for example, the decrease in total GDP or whether one would like to know the decrease in each industry’s value added. In addition, it should be mentioned that there is only one answer for HEM, whilst the outcome for GEM depends on the scenario for replacing the extracted inputs. So, even if GEM would be superior to HEM, there is not a single GEM. If one needs a benchmark, HEM might therefore be preferred.

If one is also interested in the foreign effects of extraction, GEM is to be preferred. It should be emphasized that point (2) above can be remedied for HEM if a full imports matrix is available for each source country. But, of course, this comes at a cost. Data requirements increase dramatically and assumptions about replacements are necessary. These were exactly the ‘advantages’ of HEM mentioned in points (1) and (3). In some cases also, a global perspective is simply necessary. Many environmental input–output studies deal with greenhouse gases (GHGs) and trade therein. Because GHGs are global pollutants, it does not suffice to consider only the national effects and are the effects in other countries equally important.

In conclusion, GEM is an interesting alternative to HEM that is richer from a theoretical perspective. The choice between GEM and HEM in empirical applications, however, should depend on the research question (and the amount of detail that is required for its answer) in combination with data quality and availability.

Acknowledgements

We thank Bert Steenge and the three referees for their comments, which – we think – have improved the exposition. All remaining deficiencies and errors are ours.

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The research of Kondo was partly supported by the Environment Research and Technology Development Fund (3-1704) of the Environmental Restoration and Conservation Agency of Japan.

Notes

1 In a similar fashion, Dietzenbacher and van der Linden (1997) extracted single industries from the EU5 and EU7 tables.

2 Matrices are indicated by bold, upright capital letters; vectors by bold, upright lower case letters; and scalars by italicized lower case letters. Vectors are columns by definition, so that row vectors are obtained by transposition, indicated by a prime. A diagonal matrix with the elements of any vector on its main diagonal and all other entries equal to zero is indicated by a circumflex.

3 In the previous section, we used ${\mathit{\pi }}^{\mathrm{\prime }}={\mathbf{e}}_{\mathit{N}\mathit{n}}^{\mathrm{\prime }}-{\mathbf{e}}_{Nn}^{\mathrm{\prime }}\mathbf{A}$. Post-multiplying both sides with $\hat{\mathbf{x}}$ gives ${\mathit{\pi }}^{\mathrm{\prime }}\hat{\mathbf{x}}={\mathbf{x}}^{\mathrm{\prime }}-{\mathbf{e}}_{Nn}^{\mathrm{\prime }}\mathbf{A}\hat{\mathbf{x}}$ or ${\mathbf{v}}^{\mathrm{\prime }}={\mathbf{x}}^{\mathrm{\prime }}-{\mathbf{e}}_{Nn}^{\mathrm{\prime }}\mathbf{Z}$, which defines value added in each industry as the output minus all (domestically produced and imported) intermediate inputs.