Analyzing the effects of the choice of model in the context of marginal changes in final demand
Abstract
Literature on the choice of model for deriving an input–output table (IOT) from a pair of supply–use tables (SUTs) has focused on the consequences for the IOT and the Leontief inverse. Analyzing the technology and fixed sales structure transformation models and their applications involving impact analysis and multipliers of factor inputs or environmental extensions, we prove that the product technology and fixed sales structure assumption models are effectively identical and so are the industry technology and fixed product sales structure models. A dimensional analysis shows that the product technology and fixed sales structure assumption models maintain consistency in accounting units, while the industry technology and fixed product sales structure models do not. Comparison with selected topics in environmental life cycle assessment (LCA) shows that the commodity technology and fixed industry sales structure models yield results that are compatible with mainstream LCA. We conclude these models are “correct” in the context of impact analysis and multipliers of the satellite of a SUT/IOT system, despite the fact that they may result in “negatives.” We propose a new quantity, the intensity matrix, and highlight its benefits in terms of the consistency of dimension and ease of interpretation. We illustrate our findings with examples of a SUT/IOT for several EU countries. We finally discuss briefly the possibility of calculating contributions to multipliers, where it is shown that models that are equivalent in terms of observable results (multipliers) disagree on unobservable quantities (contributions to multipliers).
Keywords
Supply–use table (SUT) Input–output table (IOT) Choice of model Multiplier Life cycle assessment (LCA)1 Introduction
The main issue in the choice of model is the treatment of coproducts. BEA’s supply table of the USA in 2010 in 61 × 61 resolution shows that the average industry produces more than ten products, which demonstrates that coproduction is ubiquitous. Even though there may typically be a single primary product for each industry, the supply of many coproducts by many industries does create a multifunctionality problem. When an industry has coproducts, it is not clear which part of the labor inputs (and emissions) of that industry is related to the output of a particular (co)product. Quoting Sraffa (1960) “For in the case of jointproducts there is no obvious criterion for apportioning the labor among individual products” (p. 56). The different SUTtoIOT transformation model essentially creates singleoutput systems, solving the multifunctionality problem using different assumptions on the allocation of coproducts in a mechanistic manner. With the increase of the extent and detail of the SUTs, as witnessed by the advent of large databases, such as GTAP (http://www.gtap.org/), WIOD (http://www.wiod.org/), EORA (http://www.worldmrio.com/) and EXIOBASE (http://www.exiobase.eu/), this issue shows up more pervasively than before and the only practical solution is the application of the transformation model. The different transformation models yield different results, and these results are used for impact studies for policy on employment, innovation, climate change and so on. In addition, IOTs from consecutive years are used for econometric estimation, which again provide a basis for decisionmaking. Given the differences and the possible realworld implications, this paper revisits the choice of model problem, bringing in three new elements in the discussion.
In the first place, it argues that the data in an IOT are not empirically observable, but that only some of the results of calculations made with an IOT are empirically observable, for instance, the total input of labor or the total CO_{2} emission. Thus, disputes on the “true” IOT are not amenable to scientific discourse, while disputes on results obtained with them are. In this positivist emphasis on the empirically observable results, we to some extent follow RuedaCantuche and Ten Raa (2013). We also build on a paper by Suh et al. (2010), developed in the context of environmental IOA (EIOA), where it is recognized that adding a satellite matrix containing environmental emission and resource consumption data shifts the attention from the total output vector to the vector with environmental extensions. We extend their argument to the traditional valueadded vector, effectively bringing the discussion back to the economic domain via the detour of environmental accounting. Finally, we connect to Eurostat’s observation (2008, p. 310) that “it remains to be seen in empirical research which type of tables is the better option.”
Secondly, we argue that a change in accounting units should in the end not affect the final empirically observable result, although nonempirically observable intermediate results may be affected by such changes. Here, we build on Dietzenbacher et al. (2009) and Weisz and Duchin (2006) who study the difference between physical and monetary IOT, i.e., IOTs that are expressed in physical accounting units (such as kg) and IOTs that are expressed in monetary accounting units (such as USD).
A third line that enters our argument is that of environmental science, where scientists have developed their own analytical approaches [notably life cycle assessment (LCA); see Heijungs and Suh (2002)] and where a close comparison with IOA reveals interesting points of correspondence and divergence (Suh et al. 2010; MajeauBettez et al. 2014, 2016, 2018).
We demonstrate that some of the established models A–D, described by Eurostat (2008), that have been understood as distinctively different approaches to convert SUT into IOT are identical when it comes to calculating empirically observable results (argument 1). We also demonstrate that some of the models are consistent with the requirements from dimensional analysis, but some others are not (argument 2). Finally, we strengthen the conclusion by Suh et al. (2010) that the models in LCA are comparable to the models in IOA and EIOA.
The paper is structured as follows. Section 2 briefly summarizes the status quo of the SUT–IOT debate and introduces notation. Section 3 discusses the different types of use of IOA. Section 4 repeats the argument from Suh et al. (2010), but with the focus shifted to economic analysis instead of environmental analysis, and adding more types of IOT transformation models into the discussion. Section 5 analyzes the effects of changing the accounting units. Section 6 analyzes to what extent use of transformation models A–D can be circumvented, thus connecting to the practice in LCA, where a consistent calculation procedure has been developed which not necessarily involves the detour to coefficient matrices. Section 7 provides an empirical case study as an illustration of the mathematical theory, including an analysis of the way sectors or products contribute to a result. Section 8 addresses the relevance of the result and discusses the issue of negatives, one of the traditional key ingredients of the debate on the choice of models.
2 The transformation from SUT to IOT
This section recaps the mainstream literature on the derivation of IOTs from SUTs. It primarily builds on Eurostat’s manual (2008), but goes on the one hand further in preparing for the critical arguments of Sects. 4 until 6, and on the other hand skips many points (such as valuation layers and international trade) that are irrelevant given our focus.
Model  Format  Assumption 

A (PTAp*p)  Productbyproduct  Product technology assumption 
B (ITAp*p)  Productbyproduct  Industry technology assumption 
C (ISAi*i)  Industrybyindustry  Fixed industry sales structure assumption 
D (PSAi*i)  Industrybyindustry  Fixed product sales structure assumption 
Basic transformation models to derive an IOT (\({\mathbf{S}}\) and \({\mathbf{E}}\) or \({\mathbf{B}}\) and \({\mathbf{h}}\)) from a SUT (\({\mathbf{V}}\) and \({\mathbf{U}}\)).
Source: Eurostat (2008, p. 349)
Model  Formulas for \({\mathbf{S}}\) or \({\mathbf{B}}\)  Formulas for \({\mathbf{E}}\)  Formulas for \({\mathbf{h}}\) 

A (PTAp*p)  \({\mathbf{S}}^{{\left( {\text{A}} \right)}} = {\mathbf{UV}}^{{  {\text{T}}}} {\hat{\mathbf{q}}}\)  \({\mathbf{E}}^{{\left( {\text{A}} \right)}} = {\mathbf{WV}}^{{  {\text{T}}}} {\hat{\mathbf{q}}}\)  – 
B (ITAp*p)  \({\mathbf{S}}^{{\left( {\text{B}} \right)}} = {\mathbf{U}}{\hat{\mathbf{g}}}^{  1} {\mathbf{V}}\)  \({\mathbf{E}}^{{\left( {\text{B}} \right)}} = {\mathbf{W}}{\hat{\mathbf{g}}}^{  1} {\mathbf{V}}\)  – 
C (ISAi*i)  \({\mathbf{B}}^{{\left( {\text{C}} \right)}} = {\hat{\mathbf{g}}}{\mathbf{V}}^{{  {\text{T}}}} {\mathbf{U}}\)  –  \({\mathbf{h}}^{{\left( {\text{C}} \right)}} = {\hat{\mathbf{g}}}{\mathbf{V}}^{{  {\text{T}}}} {\mathbf{d}}\) 
D (PSAi*i)  \({\mathbf{B}}^{{\left( {\text{D}} \right)}} = {\mathbf{V}}{\hat{\mathbf{q}}}^{  1} {\mathbf{U}}\)  –  \({\mathbf{h}}^{{\left( {\text{D}} \right)}} = {\mathbf{V}}{\hat{\mathbf{q}}}^{  1} {\mathbf{d}}\) 
To distinguish the coefficient tables made by the four models, we have added superscripts (A), (B), (C) and (D) to the different symbols in Table 2, where needed. Some of the formulas in Table 2 assume, with many other texts (like Eurostat 2008), that the matrices \({\mathbf{V}}\) and \({\mathbf{U}}\) are square, i.e., that the number of products is equal to the number of industries. In the discussion, we will return to the case of nonsquare (“rectangular”) matrices. It should be noted that origin and naming of these transformation models at least partly resides in the problem of treating industries that produce more than one product and products that are produced by more than one industry. Eurostat (2008, p. 327) writes that with “supply and use tables, it is no longer difficult to describe byproducts properly in the system… However, they still create a problem for symmetric input–output tables.” Preprocessing steps may be needed to fix this. Eurostat (2008, p. 325) recommends that “before applying the product technology, each product should be assigned to a primary producer.” As we will see in our final discussion, this step can be omitted. In fact, we believe that omitting it will avoid several complications.
Using the same initial data (\({\mathbf{U}}\), \({\mathbf{V}}\), \({\mathbf{q}}\), \({\mathbf{g}}\), \({\mathbf{d}}\) and \({\mathbf{W}}\)), the four basic models yield different IOTs (\({\mathbf{S}}\) or \({\mathbf{B}}\)) with different extra tables (\({\mathbf{d}}\) or \({\mathbf{h}}\) and \({\mathbf{E}}\) or \({\mathbf{W}}\)). Some of these forms (namely models A and C) may result in socalled negatives, which is deemed as problematic (Konijn 1994; Ten Raa and RuedaCantuche 2003; Eurostat 2008); see also the discussion.
Basic transformation models to derive IO coefficient matrices \({\mathbf{A}}\) and \({\mathbf{R}}\) from a SUT.
After Eurostat (2008, p. 349)
Model  Formulas for \({\mathbf{A}}\)  Formulas for \({\mathbf{R}}\) 

A (PTAp*p)  \({\mathbf{A}}^{{\left( {\text{A}} \right)}} = {\mathbf{S}}^{{\left( {\text{A}} \right)}} {\hat{\mathbf{q}}}^{  1}\)  \({\mathbf{R}}^{{\left( {\text{A}} \right)}} = {\mathbf{E}}^{{\left( {\text{A}} \right)}} {\hat{\mathbf{q}}}^{  1}\) 
B (ITAp*p)  \({\mathbf{A}}^{{\left( {\text{B}} \right)}} = {\mathbf{S}}^{{\left( {\text{B}} \right)}} {\hat{\mathbf{q}}}^{  1}\)  \({\mathbf{R}}^{{\left( {\text{B}} \right)}} = {\mathbf{E}}^{{\left( {\text{B}} \right)}} {\hat{\mathbf{q}}}^{  1}\) 
C (ISAi*i)  \({\mathbf{A}}^{{\left( {\text{C}} \right)}} = {\mathbf{B}}^{{\left( {\text{C}} \right)}} {\hat{\mathbf{g}}}^{  1}\)  \({\mathbf{R}}^{{\left( {\text{C}} \right)}} = {\mathbf{W}}{\hat{\mathbf{g}}}^{  1}\) 
D (PSAi*i)  \({\mathbf{A}}^{{\left( {\text{D}} \right)}} = {\mathbf{B}}^{{\left( {\text{D}} \right)}} {\hat{\mathbf{g}}}^{  1}\)  \({\mathbf{R}}^{{\left( {\text{D}} \right)}} = {\mathbf{W}}{\hat{\mathbf{g}}}^{  1}\) 
3 The use of IOT for IOA

to calculate the consequences (GDP, value added, emissions, etc.) of an exogenous final demand scenario;

to calculate multipliers, which represent the consequences (GDP, value added, emissions, etc.) of a marginal change in the final demand;

to perform analytical calculations without a change, e.g., the contributions made by different final demand categories, structural decomposition analysis and structural path analysis.
In this paper, the emphasis will be on the second group of application: marginal changes. In the discussion, we will briefly consider the other two types of application. Note that we write superscripts (A/B), (A/B/C/D), etc. whenever we write one equation that holds for several models.
4 Use of IOT for impact analysis
In the SUT scheme and in the productbyproduct IOT (models A and B), a marginal change in final demand for products will be written as a change from \({\mathbf{d}}\) to \({\mathbf{d}} + \Delta {\mathbf{d}}\). For the industrybyindustry IOT (models C and D), things are more complicated. A marginal change in product demand \(\Delta {\mathbf{d}}\) transforms into a marginal change in industry output demand given by \(\Delta {\mathbf{h}}^{{\left( {\text{C}} \right)}} = {\hat{\mathbf{g}}\mathbf{V}}^{{  {\text{T}}}} \Delta {\mathbf{d}}\) or \(\Delta {\mathbf{h}}^{{\left( {\text{D}} \right)}} = {\mathbf{V}}{\hat{\mathbf{q}}}^{  1} \Delta {\mathbf{d}}\) according to Table 2.
Expressions for the satellite multiplier for models A–D
Model  Formula for impact analysis in terms of \({\mathbf{\Delta h}}\)  Formula for impact analysis in terms of \({\mathbf{\Delta d}}\) 

A (PTAp*p)  \(\Delta {\mathbf{e}}^{{\left( {\text{A}} \right)}} = {\mathbf{W}}\left( {{\mathbf{V}}^{\text{T}}  {\mathbf{U}}} \right)^{  1} \Delta {\mathbf{d}}\)  
B (ITAp*p)  \(\Delta {\mathbf{e}}^{{\left( {\text{B}} \right)}} = {\mathbf{W}}\left( {{\hat{\mathbf{q}}\mathbf{V}}^{  1} {\hat{\mathbf{g}}}  {\mathbf{U}}} \right)^{  1} \Delta {\mathbf{d}}\)  
C (ISAi*i)  \(\Delta {\mathbf{w}}^{{\left( {\text{C}} \right)}} = {\mathbf{W}}\left( {{\mathbf{V}}^{\text{T}}  {\mathbf{U}}} \right)^{  1} {\hat{\mathbf{g}}\mathbf{V}}^{{  {\text{T}}}} \Delta {\mathbf{h}}\)  \(\Delta {\mathbf{w}}^{{\left( {\text{C}} \right)}} = {\mathbf{W}}\left( {{\mathbf{V}}^{\text{T}}  {\mathbf{U}}} \right)^{  1} \Delta {\mathbf{d}}\) 
D (PSAi*i)  \(\Delta {\mathbf{w}}^{{\left( {\text{D}} \right)}} = {\mathbf{W}}\left( {{\hat{\mathbf{q}}\mathbf{V}}^{  1} {\hat{\mathbf{g}}}  {\mathbf{U}}} \right)^{  1} {\mathbf{V}}{\hat{\mathbf{q}}}^{  1} \Delta {\mathbf{h}}\)  \(\Delta {\mathbf{w}}^{{\left( {\text{D}} \right)}} = {\mathbf{W}}\left( {{\hat{\mathbf{q}}\mathbf{V}}^{  1} {\hat{\mathbf{g}}}  {\mathbf{U}}} \right)^{  1} \Delta {\mathbf{d}}\) 
Table 4 shows that \(\Delta {\mathbf{e}}^{{\left( {\text{A}} \right)}} = \Delta {\mathbf{w}}^{{\left( {\text{C}} \right)}}\) and that \(\Delta {\mathbf{e}}^{{\left( {\text{B}} \right)}} = \Delta {\mathbf{w}}^{{\left( {\text{D}} \right)}}\). So, while there are four models, there are just two different answers, at least when it comes to observable change in value added. In the remaining text, we will no longer separate \(\Delta {\mathbf{e}}\) and \(\Delta {\mathbf{w}}\), but always write \(\Delta {\mathbf{w}}\), also for models A and C.
The intermediate (at least considered by the authors to be intermediate) results of model A/C and B/D are the coefficient matrices \({\mathbf{A}}\) and \({\mathbf{R}}\), the Leontief inverse \({\mathbf{L}} = \left( {{\mathbf{I}}  {\mathbf{A}}} \right)^{  1}\) and the output vectors \(\Delta {\mathbf{g}}\) and \(\Delta {\mathbf{q}}\). These are different in all four models. Most of the literature has focused on these differences [see, e.g., Kop Jansen and Ten Raa (1990), Ten Raa and RuedaCantuche (2003), RuedaCantuche and Ten Raa (2009)]. In addition, substantive arguments have been introduced on the principles behind these methods, for instance, relating to their intended use (RuedaCantuche and Ten Raa 2009; Oosterhaven 2012). If, by contrast, we study the observable end result of the models, namely \(\Delta {\mathbf{w}}\), we see only two classes of results: models A and C on the one hand and models B and D on the other hand.
As far as we know, the pairwise equivalence of models A and C and of models B and D in the context of impact analysis and multipliers has never been discussed in the literature. As a matter of fact, Suh et al. (2010) do discuss the equivalence between models A and another model (the socalled byproduct technology model). They write that the choice between these models “does not have significant practical meaning when it comes to actual application” and “acknowledge that these two methods have different underlying economic implications, which nevertheless does not have any effect in the results of [a] study” (Suh et al. 2010, p. 341). However, they do not fully address the equivalence of models A and C and certainly not of B and D. In a broader historical context, the pairwise similarity has been stressed before, for instance, by not distinguishing four models (A–D) but only two [UN (1999), p. 86): “There are basically two methods to combine the use and supply matrices mathematically to generate the traditional symmetric input–output matrix. These methods are based on either the industry technology assumption or the commodity technology assumption.],” but not so much in terms of an equivalence of multipliers.
5 Change in accounting units in impact analysis
The consideration of accounting units adds another insight. Products can be expressed in different accounting units: monetary (dollar, yen, etc.) or physical (kg, MJ, etc.). Although there is a “natural” accounting unit for some products, many products can be and are expressed in several alternative accounting units. For instance, gasoline may be expressed in liter, in kg, in MJ, in dollar, etc.
In this section, we will explore to what extent conclusions drawn from IO impact analysis depend on a change in accounting units. The general idea is that numbers will change when accounting units are changed, but only in a very specific way. For instance, if an extra demand of 1 L of gasoline leads to an extra value added of 0.1 dollar, an extra demand of 1 gallon of gasoline should lead to an extra value added of approximately 0.4 dollar, because 1 gallon corresponds to approximately 4 L, and because the multiplier model of IOA is a linear model. We implement changes in accounting unit in the product part (the rows of \({\mathbf{U}}\), \({\mathbf{V}}^{\text{T}}\), \({\mathbf{d}}\) and \({\mathbf{q}}\)) and in the satellite part (the rows of \({\mathbf{W}}\) and \({\mathbf{w}}\)) and will develop formulas for the effects of such changes in models A–D.
In general, the topic of accounting units is underemphasized in the field of economics. While textbooks in physics and chemistry mention the importance of units and dimensional analysis (Bridgman 1922), textbooks in economics hardly even mention the existence of units, let alone the way to properly do the algebra of units (Barnett 2004). It is no wonder that critical remarks on the failure to address units in a proper way have been made mostly within the more physical areas of economics, such as ecological economics (Mayumi and Giampietro 2010) and physical input–output analysis (Weisz and Duchin 2006). Few treatments of this topic within economics are available but De Jong (1967) is a good source. A further complication is that the term unit is in the SUT and IOTliterature typically used for something else, namely the statistical unit, which is the “entity for which the required statistics are compiled” (Eurostat 2017a) which may be an enterprise, a kindofactivity unit, etc. Finally, the literature is not always sharp on the distinction between units and dimensions. Dimensions represent generic classes of measurement quantities, such as length and time. A unit is a specific choice for operationalizing a dimension, such as meter or mile within the dimension length or second and hour within the dimension time. We will primarily study changes in unit within the same dimension. But as one of the principles of SUT and IOT is that products within one row are homogeneous, we might also make a change from mass to price or from price to volume. As a final remark, RuedaCantuche and Ten Raa (2009) study the degree of “scale invariance” of the models C and D and conclude that model C is “superior from an axiomatic point of view.” Basically, we reinterpret their analysis in terms of unit invariance and also broaden their argument by including models A and B.
Results of a change in accounting units in the expressions for the satellite vector for models A–D
Model  Unitchanged formula for impact analysis  Invariant 

A/C (PTAp*p/ISAi*i)  \(\Delta {\tilde{\mathbf{w}}}^{{\left( {{\text{A}}/{\text{C}}} \right)}} = {\hat{\boldsymbol{\beta }}}\Delta {\mathbf{w}}^{{\left( {{\text{A}}/{\text{C}}} \right)}}\)  Yes 
B/D (ITAp*p/PSAi*i)  \(\Delta {\tilde{\mathbf{w}}}^{{\left( {{\text{B}}/{\text{D}}} \right)}} = \hat{{\boldsymbol{\beta }}}{\bf W}\left( {{\hat{\mathbf{q}}\mathbf{V}}^{  1} {\hat{\mathbf{g}}}  {\mathbf{U}}} \right)\Delta {\mathbf{d}} \ne \hat{\boldsymbol{\beta }}\Delta {\mathbf{w}}^{{\left( {{\text{B}}/{\text{D}}} \right)}}\)  No 
Thus, it is demonstrated that models A and C are unit invariant, while models B and D are not. In particular, models B and D are invariant for changes in the accounting units of the extensions (\({\mathbf{W}}\)) but not for changes in the accounting units of the products (\({\mathbf{V}}^{\text{T}}\) and \({\mathbf{U}}\)). In other words, two different choices of the accounting unit of products in the SUT will lead to two different satellite vectors when model B or D is applied, but to identical results when model A or C is applied. This is not only true for a dimensionaffecting change in units (e.g., from dollar to kg), but also for a dimensionpreserving more trivial change in units, e.g., from kilogram to gram or pounds or from dollar to millions of dollar or yen.
In hindsight, there is some logic in this. The problem is in the vector \({\mathbf{g}}\) in Fig. 2, which is constructed by adding rows, so by adding products. But if the rows denote products measured in different units, this sum is not defined, mathematically. One cannot add kg and MJ, and neither can one add kg and tonne. And if one would neglect the units and just add the numbers, there is no way of converting the result of kg and MJ into that of kg and kWh. In fact, applying the equation of model B (Table 2) to a mixedunit SUT leads to a representation that actually violates the industry technology assumption, i.e., the assumption that all coproducts of an industry obey the same production function. Model A does not contain a term with \({\mathbf{g}}\), while models B, C and D do. In model C, the terms with \({\hat{\mathbf{g}}}\) and \({\hat{\mathbf{g}}}^{  1}\) cancel in the formula for \(\Delta {\mathbf{w}}\) in Appendix A. As a consequence, model C is, despite the presence of the illdefined \({\mathbf{g}}\), dimensionally consistent, at least, at the level of the observable result \(\Delta {\mathbf{w}}\). The issue of mixedunit systems has been addressed before (see, e.g., Pauliuk et al. (2015), MajeauBettez et al. (2016)), but the real issue is not so much storing and balancing data, but using incommensurable data in a mathematical framework. Formulas like \({\hat{\mathbf{q}}\mathbf{V}}^{  1} {\hat{\mathbf{g}}}  {\mathbf{U}}\) do not make sense when \({\mathbf{g}}\) is invalid. While we recognize this theoretical limitation, we nevertheless proceed as if we are ignorant about it. This can be justified on the basis of mainstream documents, such Eurostat (2008), which do not explicitly warn us that the accounting units of the products must be equal. If SUTs are primarily used as accounting tables, mixed units should be used, materials in kg, electricity in kWh, services in dollar. That they may also be used for “analytical” purposes, IOT and IOA, is then creating problems, at least with models B and D.
Within the IOliterature, some authors (Kop Jansen and Ten Raa 1990; RuedaCantuche and Ten Raa 2009) have introduced changes in scale in a comparable axiomatic framework to sort out “correct” and “incorrect” models. Here, we offer a reinterpretation in terms of a change in the accounting unit of the products. In Appendix C, we show that the columnwise scale invariance axiom gives in the end the same result as the rowwise unit variance axiom: Models A and C satisfy both axioms, while models B and D violate it. In a different context, Dietzenbacher and Stage (2006) observe consistency problems in carrying out a structural decomposition analysis in IOTs with mixed units.
6 Impact analysis without IOT and without coefficient matrices
As noted above, there are different models for moving from a SUT to an IOT, and to construct a coefficient matrix from transaction matrices. Several authors have argued that it is well possible to do impact analysis without constructing an IOT and without calculating coefficient matrices (Rosenbluth 1968; Heijungs 2001; Suh et al. 2010; Lenzen and RuedaCantuche 2012). For instance, Rosenbluth (1968, p. 255) argued that “there is nothing [IOA] can do that cannot be done equally well by [SUT] analysis, and a good many things that the latter can do better,” and Suh et al. (2010, p. 341) state that the IOliterature “has overlooked the fact that coefficient matrices… are rarely, if ever, used alone… [they] fulfill an intermediate function.” On the other hand, RuedaCantuche (2011b, p. 36) observed that “there has been very little research on the application of supply and use tables to impact analysis.” In a followup paper, however, Lenzen and RuedaCantuche (2012, p. 151) showed that “the use of supply–use tables in a common framework concerning product and industryrelated assumptions may overcome the undesirable limitations of symmetric input–output tables.” In this section, we will discuss in more detail the connection with the LCA literature, where working without a coefficient form was already discussed much earlier.

from productbyindustry (\({\mathbf{U}}\) and \({\mathbf{V}}\)) to productbyproduct (\({\mathbf{S}}\)) or industrytoindustry (\({\mathbf{B}}\)) format;

from transaction (\({\mathbf{S}}\) or \({\mathbf{B}}\)) to coefficient (\({\mathbf{A}}\)) format.

assignment to the correct column: is a cow breeding company that produces dairy and meat a dairy producer or a meat producer?

division by the correct total: should we specify its inputs per unit of dairy, per unit of meat or per unit of undifferentiated output?
As noted by Eurostat (2008, p. 327): “the issue has been debated a lot in literature, but a truly satisfactory solution has not yet been found.” The interesting aspect of adding the LCA model by this paper [and by Suh et al. (2010)] is that it works without making the step to a symmetric table and without making the step to a coefficient table. In doing so, it avoids to face the choice of model.
The basic idea underlying the use of \({\mathbf{N}} = {\mathbf{V}}^{\text{T}}  {\mathbf{U}}\) partly coincides with the byproduct technology assumption, also referred to as Stone’s method (Eurostat 2008), the byproduct technology model (Suh et al. 2010) or the byproduct technology construct (MajeauBettez et al. 2014), which assumes that coproducts within one industry are produced in fixed ratios. The LCA model, like the byproduct technology assumption, considers such extra outputs as negative inputs. As a consequence, its use in impact analysis may yield negatives, which is natural because this model coincides with models A and C, which were already known to potentially yield negatives. So, one might wonder, is the LCA model not the same as models A and/or C? The answer is negative: While model A uses \({\mathbf{A}} = {\mathbf{UV}}^{{  {\text{T}}}}\) and \({\mathbf{R}} = {\mathbf{WV}}^{{  {\text{T}}}}\) and model C uses \({\mathbf{A}} = {\hat{\mathbf{g}}\mathbf{V}}^{{  {\text{T}}}} {\mathbf{U}}{\hat{\mathbf{q}}}^{  1}\) and \({\mathbf{R}} = {\mathbf{W}}{\hat{\mathbf{g}}}^{  1}\), the LCA model refrains from constructing \({\mathbf{A}}\) and \({\mathbf{R}}\) altogether and is only interested in their implicit combination through \({\mathbf{W}}\left( {{\mathbf{V}}^{\text{T}}  {\mathbf{U}}} \right)^{  1}\). This avoidance of making \({\mathbf{A}}\) and \({\mathbf{R}}\) is precisely which makes the difference with the byproduct technology model, which still produces \({\mathbf{A}} = \left( {{\mathbf{U}}  {\mathbf{V}}_{\text{od}}^{\text{T}} } \right){\mathbf{V}}_{\text{d}}^{  1}\) and \({\mathbf{R}} = {\mathbf{WV}}_{\text{d}}^{  1}\), where the subscripts \({\text{d}}\) and \({\text{od}}\) code for diagonal and offdiagonal entries [Suh et al. (2010, p. 340)]. Indeed, in the byproduct technology model one still needs to decide on what is the main product and what are the coproducts of an industry: The model “assumes that production of coproducts is fully dependent on the production of the primary product of a process” (Suh et al. 2010, p. 339). Without that choice, we cannot figure out which numbers are on the diagonal and which are offdiagonal.

It is standard practice in LCA to remodel coproducing industries into industries with one output. For instance, the ISOstandard on LCA (ISO 2006, p. 14) prescribes that “the inputs and outputs shall be allocated to the different products.”

Most of the LCA literature does not recognize the advantage of not needing a coefficient form and still insists on making this conversion step. For instance, ISO (2006 p. 13) states that “an appropriate flow shall be determined for each unit process. The quantitative input and output data of the unit process shall be calculated in relation to this flow.”
Clearly, both practices refer to unnecessary actions. The expression \({\mathbf{WN}}^{  1}\) just works whenever \({\mathbf{N}}\) is square and invertible, even when some of the offdiagonal elements are positive. And as long as \({\mathbf{N}} = {\mathbf{V}}^{\text{T}}  {\mathbf{U}}\) and \({\mathbf{W}}\) are standardized by the same (nonzero) vector (say, \({\mathbf{c}}\)), we have \(({\mathbf{W}}{\hat{\mathbf{{c}}}^{  1} })\left( {\left( {{\mathbf{V}}^{\text{T}}  {\mathbf{U}}} \right){\hat{\mathbf{c}}}^{  1} } \right)^{  1} = {\mathbf{W}}\left( {{\mathbf{V}}^{{  {\mathbf{T}}}}  {\mathbf{U}}} \right)\), so the choice of \({\mathbf{c}}\) does not matter. Tricks are only needed when \({\mathbf{N}}\) or \(\left( {{\mathbf{V}}^{\text{T}} ,{\mathbf{U}}} \right)\) is not square. But that is no different for models A and C, as these models work with \({\mathbf{V}}^{{  {\text{T}}}}\) and therefore are restricted to square SUTs as well. Finally, we mention the fact that ISO’s (2006) coproduct allocation has spawned a large literature which bears a lot of similarities with that of IOA. This literature features terms such as “partitioning,” “substitution,” “avoided impacts” and “system expansion.” Suh et al. (2010) and MajeauBettez et al. (2014, 2018) contain an extensive treatment, which we will not repeat here.
Observe that we have added a superscript (LCA) which will allow us to make easy comparisons with the earlier frameworks based on models A–D. Also observe that matrix \({\mathbf{N}}\) is often referred to by the symbol \({\mathbf{A}}\) (Heijungs and Suh 2002); here we choose for another letter to avoid confusion with the \({\mathbf{A}}\) in the Leontief inverse \(\left( {{\mathbf{I}}  {\mathbf{A}}} \right)^{  1}\).
Suh et al. (2010) demonstrate that the form in (13) is equivalent to model A. They also add a section on the historic origins of this observation (p. 348). They in fact show moreover that it is also equivalent to the byproduct model, another variant besides Eurostat’s A–D. They, however, do not discuss the equivalence with the productbyproduct version C.
The point is, however, that the LCAformat with \({\mathbf{N}} = {\mathbf{V}}^{\text{T}}  {\mathbf{U}}\) appears naturally when we look at the net production of a sector: When sector \(j\) produces an amount \(v_{ji}\) of product \(i\) and to do so uses an amount \(u_{ij}\) of the same product \(i\), its net production is simply \(n_{ij} = v_{ji}  u_{ij}\). There is no such a natural interpretation of \(m_{ij} = q_{j} v_{ji} g_{i}  u_{ij}\). Indeed, we are not aware of fields of science where a quantity like \({\mathbf{M}}\) has been proposed to measure the net effect of production. It has only been constructed with the aim of constructing coefficient matrices, which are—as argued here—an intermediate step at most.
7 Illustrative case study
As predicted by the theoretical analysis, the total output calculated for models A and C and that for models B and D are the same, while for models A and B and for models C and D there are sometimes substantial differences, up to 50%. An analysis of the size of these differences is beyond the scope of the present paper. For an example based on supply–use tables of Andalusia, see RuedaCantuche and Ten Raa (2013). Similar calculations have been done for 16 other European countries, all giving a similar outcome. These additional results are available in SI.
The graphs confirm the theoretical prediction that the satellite multipliers are equal for models A and C and that they are equal for models B and D. They also show that the differences between models A/C and B/D can be substantial in a concrete case.
8 Discussion

that models A and C give identical results, and so do models B and D,

that models A and C are unit invariant while models B and D are not,

that it is possible to do impact analysis without an IOT and without coefficient matrices (referred to as the “LCA model”) and that the result of the latter analysis is identical to the results of models A and C.
From an industrial ecologist’s or ecological economist’s perspective where a proper physical representation is important, models A, C and LCA are preferred over models B and D because they are unit invariant. Of course, being a model, they are not correct in absolute terms. Any model is as weak as the validity of its assumptions, and the models presuppose, among others, a square SUT, besides the usual IOassumptions of linear technology and full market clearing. But models A, C and LCA survive at least a number of critical tests.
However, models A and C are not identical, so how can both be correct? As a matter of fact, at the level of a breakdown of a result into the contributing products or sectors, model A deviates from model C and LCA, so there are real differences. A closer consideration of the problem reveals that a productbyproduct structure (model A) and an industrybyindustry structure (model C) are in fact both difficult to interpret. The issue is that households and governments exert a demand for products, while the value added (and the emissions) is coming from the industries. A productbyproduct structure reallocates the value added (and the emissions) to products, while an industrybyindustry structure reallocates the final demand to industries. Both are artificial (RuedaCantuche 2011a). The LCA model does not do this: The demand is for products and the value added (and emissions) belongs to industries. The LCA model is thus closer to economic accounting structure. Recall that this was precisely the reason for switching from IOT to SUT by Stone: “the SUTs framework provides the natural statistical framework” (Eurostat 2008, p. 51). Our analysis shows that we can maintain this natural framework not only in accounting, but also in analysis, at least for the case of marginal changes, so for satellite multipliers.
The debate on models A/B and C/D has in part been fueled by the fact that models A and C may yield negatives. That is, it may happen that the input coefficient matrix (\({\mathbf{A}}^{{\left( {\text{A/C}} \right)}}\)) contains negative entries. As a consequence, the Leontief inverse (\({\mathbf{L}}\)) the product output (\({\mathbf{q}}\)), the industry output (\({\mathbf{g}}\)), or the value added (or emissions) (\({\mathbf{w}}\)) may contain negative elements. Such elements point to a negative amount of product, negative industrial activity, negative value added or negative emissions. Obviously, these negatives do not make sense; hence, the quest for transformations without negatives seems to be well grounded (Konijn 1994; Ten Raa and RuedaCantuche 2003). However, when we restrict the scope to small changes as in the case of multipliers, the argument gets much weaker (Suh et al. 2010). A marginal increase in final demand for one product may well decrease the marginal activity of some industries and thereby decrease the value added or the emissions. In a context of multipliers, indicating the effect of marginal changes, the issue of negatives is therefore not a valid argument in the choice of model.
The argument against negatives is much more valid in the context of larger changes or scenario analysis. Use of \({\mathbf{w}} = {\mathbf{R}}\left( {{\mathbf{I}}  {\mathbf{A}}} \right)^{  1} {\mathbf{d}}\) for predicting the values added (or emissions) of an economywide scenario \({\mathbf{d}}\) is an entirely different case. It is anyhow doubtful to what extent the underlying assumption of a linear technology with fixed coefficients is valid for such types of study. Whenever the result of a scenario study returns negative production, negative value added or negative emissions, we can be sure that the model was not appropriate. A choice for model B/D will only mask the intrinsic defects of the model by “saving the phenomena,” an expression introduced by Duhem (1969) to characterize the complex additions to save Ptolemy’s astronomical system, while it should have in fact been replaced by an entirely different system.
In the context of SUT, Konijn (1994, p. 7) has observed that “we cannot perform input–output analysis in the way we used to, without (re)constructing an input–output table from the system of [supply] and use matrices.” As we have demonstrated, this is true, but we should reflect on the question if we need to perform input–output analysis at all, since the LCA model is simpler, devoid of fundamental questions and equally effective for (environmental) impact analysis, at least when we restrict the discussion to square matrices. While the calculation of an IOT, a coefficient matrix, a Leontief inverse or an industry output is useful for certain types of analysis (such as structural path analysis), it is not necessary for the calculation of satellite multipliers, including environmental impact analysis, at least in the case of marginal changes. This paper argues that the intensity matrix \({\varvec{\Lambda}}^{{\left( {\text{LCA}} \right)}} = {\mathbf{W}}\left( {{\mathbf{V}}^{\text{T}}  {\mathbf{U}}} \right)^{  1}\) is crucial, for such analyses. It is uniquely available, obviating the choice between models A, B, C and D, or any of the other models [including those mentioned by Kop Jansen and Ten Raa (1990) and Eurostat (2008)]. So even when we have shown that models A and C are from a satellite point of view identical, we can still do without these models, and in fact we propose to abandon them entirely and use the less problematic LCA model, because they are different in their intermediate results and because they need assumptions that are problematic.
A final remaining issue with the intensity matrix \({\varvec{\Lambda}}^{{\left( {\text{LCA}} \right)}}\) is that it involves an inverse, so that the SUT must be square. This can be a problem (Konijn 1994; Duchin and Levine 2011), but most alternative approaches also start by assuming a square SUT [see, e.g., Kop Jansen and Ten Raa (1990, p. 213), RuedaCantuche and Ten Raa (2009, p. 364), RuedaCantuche et al. (2009, p. 63), Suh et al. (2010, p. 339)]. Models B and D have two seemingly attractive sides: They do not yield negatives (discussed above) and they rely on formulas that do not require a square SUT [see also Lenzen and RuedaCantuche (2012)]. However, they have been shown from a dimensional perspective to suffer from theoretical shortcomings, so the mere fact they can be employed for rectangular cases by no means justifies their use. Probably, the topic of the construction of an IOT from a rectangular SUT can benefit from a renewed analysis similar to the one undertaken in this paper for the square case. Ingredients are again an emphasis on how the IOT is used to calculate empirically observable results, instead of which intermediate results are traditionally constructed, and on the formulation and application of consistency requirements, such as that of dimensional invariance.
Notes
Authors’ contributions
RH conceived the argument and provided the mathematical proofs. AdK provided the data and code for the graphs. RH and AdK wrote the manuscript together.
Acknowledgements
Two reviewers gave very detailed comments, which helped us to improve and clarify our message. Richard Wood stretched his editorial role by providing several valuable suggestions.
Competing interests
The authors declare that they have no competing of interests.
Availability of data and materials
The online version contains a downloadable zip file (Additional file 1) with the input data, routines and outputs used in the illustrative example.
Funding
Not applicable.
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Supplementary material
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