Handbook on Extended Supply and Use Tables and Extended Input‑Output Tables

6. Extended input-output tables

Abstract

This chapter outlines the strategies and practical guidelines to construct extended input-output tables from extended supply and use tables. It begins with a discussion of the underlying theory of input-output tables. It then considers the choice of dimension for such tables: product-by-product or industry-by-industry. The following section examines construction methods for input-output tables and extended input-output tables. The final section provides concluding remarks and recommendations.

Introduction

This chapter outlines the strategies and practical guidelines to construct extended input-output tables (EIOTs) from extended supply and use tables (ESUTs) based on the literature on the construction of input-output tables (IOTs) from supply and use tables (SUTs). SUTs describe the supply and demand of products by industries, households and government. However, input-output analysis uses IOTs, which are mathematical constructs that describe the supply and demand of products by product-adjusted1 industries (in product-by-product IOTs) and the supply and demand of industry-adjusted2 products by industries (in industry-by-industry IOTs). As a result, the problem to derive IOTs is twofold. It consists of adjusting industries in SUTs (by columns)3 to construct product-by-product IOTs or adjusting products in SUTs (by rows)4 to construct industry-by-industry IOTs.

The first section of this chapter describes the underlying theory to construct IOTs from SUTs according to the literature. Choices must be made with respect to the dimension of a table, product-by-product or industry-by-industry, and with respect to the method of construction. The next two sections discuss the alternative dimensions of an IOT and an EIOT, respectively. The chapter then discusses the alternative methods of construction of IOTs and EIOTs. The final section provides concluding remarks and practical recommendations for compilers of EIOTs from ESUTs, who must make the two choices.

Underlying theory of input-output tables

This section briefly describes the underlying theory of the methods to construct IOTs from SUTs under a generalised framework based on Rueda-Cantuche (2017[1]) and reported in the United Nations (UN) Handbook on Supply, Use and Input-Output Tables (United Nations, 2018[2]). The UN generalised framework considers the amount of product i used by industry j to produce product k:

product i → industry j → product k

This suggests input-output coefficients a_ijk, i.e. with three indices. The industry technology assumption states that an industry uses the same amount of product i irrespective of the products it produces (i.e. for all k), reduces the number of indices to two5 and yields a product-by-product6 IOT. Alternatively, the product technology assumption states that all products have unique input structures, irrespective of the industry of fabrication (for all j).

The industry technology assumption yields non-negative IOTs and accommodates rectangular SUTs (different numbers of products and industries). The product technology assumption yields square IOTs (equal numbers of products and industries), with some cells possibly negative. The final use components of the use tables remain unchanged in product-by-product IOTs.

A similar framework can be set up in terms of the contribution of product i to the delivery from industry j to industry k:

industry j → product i → industry k

This suggests input-output coefficients a_jik, i.e. with three indices. The fixed industry sales structures assumption states that all industries have unique sales structures to other industries, irrespective of the products they produce (for all i), reduces the number of indices to two and yields an industry-by-industry IOT. Alternatively, the fixed product sales structures assumption states that all products have unique output structures, irrespective of the supplier industry (for all j).

The fixed product sales structures assumption yields non-negative IOTs and accommodates rectangular SUTs. Conversely, the SUTs need to be square if fixed industry sales structures are assumed and this assumption may yield new negatives in the resulting IOT. The fixed industry sales structures assumption yields square IOTs with some cells possibly negative. The gross value added components of the use tables remain unchanged in industry-by-industry IOTs.

For details about the different methods of construction of IOTs from SUTs, see Chapter 12 and annexes in United Nations (2018[2]), Beutel (2017[3]), and Eurostat (2008[4]), including the breakdown of domestic and import use matrices.

Product-by-product versus industry-by-industry input-output tables

This section discusses the choice of dimension of an IOT, either product-by-product or industry-by-industry. Rueda-Cantuche (2017[1]) reviewed the literature on the choice of dimension of an IOT, specifically comparing product-by-product and industry-by-industry IOTs. Prior to the 2008 System of National Accounts (European Commission; IMF; OECD; UN; World Bank, 2009[5]), industry-by-industry IOTs were not considered as useful as product-by-product IOTs. This was mainly because final use components are rarely expressed in terms of industries and that an industry group might be less realistic due to the estimation methods used to obtain an overall total industry output of the domestic economy.

The 2008 System of National Accounts was the first to acknowledge that IOTs of different dimensions serve different analytical purposes and recognised a change of emphasis from product-by-product IOTs to industry-by-industry IOTs (para. A4.21). On the European side, neither the 1995 European System of Accounts (Eurostat, 1995[6]) nor the 2010 European System of Accounts (Eurostat, 2013[7]) explicitly mentioned the issue of the choice of the dimension of an IOT. However, Eurostat Manual of Supply, Use and Input-output Tables (Eurostat, 2008, p. 301[4]) recognised the pros and cons of the use of product-by-product IOTs and industry-by-industry IOTs. In this sense:

• Product-by-product IOTs are theoretically more homogenous than industry-by-industry ones since a single element of the industry-by-industry IOTs can refer to products that are primary for other industries as well.

• Product-by-product IOTs have clearer input structures in terms of products and factor inputs used. Industry-by-industry IOTs report mixed bundles of goods and services produced within each industry.

• Industry-by-industry IOTs are more transparent in the sense that their construction is relatively simple and often easily traceable back to the original (rectangular) SUTs when using the fixed product7 sales assumption. This is not the case for the product8 technology assumption for product-by-product IOTs, as it requires a procedure after removing the negative coefficients that may arise.

• Industry-by-industry IOTs are close to national accounts data and their statistical sources while product-by-product IOTs have been compiled using analytical methods (e.g. to remove negatives) that result in less comparability with original data sources and, sometimes, across countries.

• Product-by-product IOTs are well suited for specific analytical purposes involving homogenous products (e.g. environmental and energy policies, productivity-related analysis, or the analysis of new product technologies in the economy). However, industry-by-industry IOTs are well suited for other types of analyses such as tax reforms or employment impacts of industrial, fiscal and monetary policies. While this distinction and list of potential applications might be useful, there is currently no clear guidance on the choice of IOT dimension in the literature.

Another important aspect highlighted by Eurostat (2008[4]) is the relevance of secondary production in the choice of the dimension of an IOT. Relatively low levels of secondary production reported in the supply tables would imply that the distinction between products and industries would become superfluous, thus making the dimension of the IOT a minor issue. However, the level of secondary activities can vary considerably across industries and most times the choice of type of IOT is relevant.

Rueda-Cantuche (2011[8]) and Lenzen and Rueda-Cantuche (2012[9]) advocate the use of SUTs instead of product-by-product IOTs and industry-by-industry IOTs for impact analysis. This alternative approach has the potential to circumvent product-by-product and industry-by-industry IOT issues, which are due to the symmetry of their dimensions, unlike the SUTs, which have a product-by-industry dimension. For instance, environmental policies can alter households’ consumption patterns of products, which may affect the employment of industries (not products). Another example is the price effects on products (not industries) due to increased costs in the labour inputs used by industries.

As also noted by United Nations (2018[2]), users of IOTs will often be required to choose the dimension of an IOT – product-by-product or industry-by-industry – rather than the type of technology or sales assumption that has been applied (as explained in the next section). This is because users often need to combine the IOTs with other types of data to conduct their analysis. For many types of analysis, the IOTs must be combined with structural or time series data, which are based on industry-based classifications, such as energy and productivity analysis. For other types of analysis, such as those related to prices, the matching data are usually available and based on products.

However, as shown by Rueda-Cantuche (2011[8]) and Lenzen and Rueda-Cantuche (2012[9]), the chosen dimension of an IOT should not exclude a priori any type of analysis. This is because the information contained in the supply table can be used to transform product-classified data into the industry classification, and vice versa9 (United Nations, 2018[2]).

Product-by-product versus industry-by-industry extended input-output tables

This section elaborates on the choice of the dimension of an EIOT, taking into account the product and industry dimensions of the ESUTs. EIOTs are preferably derived from ESUTs. The additional information by size, ownership and/or export orientation, by either products or industries, provides a better statistical ground to compile EIOTs and even subsequent regular IOTs, by aggregation. The use of enterprise heterogeneity data provides new insights into the input structures of industries and product deliveries that could not be captured otherwise. Alternatively, although EIOTs could also be constructed based on regular IOTs, the information required to break down product-adjusted industries into product-by-product IOTs (i.e. columns) is not readily available or directly observable in enterprise surveys. The same applies to the required information to split industry-adjusted products (i.e. rows) into industry-by-industry IOTs.

United Nations (2018, p. 397[2]) provides a compelling illustration of how industry-by-industry IOTs generated from rectangular SUTs (which have more products than industries) can produce different outcomes compared to those derived from aggregated SUTs (where the number of products and industries are equal), even when both are built on the premise of fixed product sales structures.

When building EIOTs from ESUTs, similar to standard IOTs, the initial step is to choose the dimension of the IOT – product-by-product or industry-by-industry. Next an assumption for its construction must be selected, which involves deciding how to handle secondary products or industries.

Breakdown only by industries

Assuming square10 or rectangular ESUTs with more industries than products, due to the breakdown of certain industries by size, ownership and/or export status, the construction of product-by-product EIOTs would imply the same input structures11 across various sizes of enterprises (multinational enterprises [MNEs], small and medium-sized enterprises [SMEs], etc.) and/or different types of enterprises (by ownership or export status). Indeed, this is precisely the limitation that ESUTs seek to address, thus making little sense in this case12 to construct product-by-product EIOTs (i.e. product-by-industry adjusted products). Therefore, when breaking down industries only, industry-by-industry EIOTs have the recommended dimension, irrespective of the dimensions of the regular SUTs.

Breakdown only by products

Analogously, assuming square13 or rectangular ESUTs with more products than industries, due to the breakdown of certain products by the size, ownership and/or export status of the industries that produce them, the construction of industry-by-industry EIOTs would imply the same output structures14 across various extended types of products within industries. Similarly, this is particularly the caveat that ESUTs seek to address, thus making little sense to compile industry-by-industry EIOTs (i.e. product-adjusted industries by industry). Therefore, when breaking down products only, product-by-product EIOTs have the recommended dimension, irrespective of the dimensions of the regular SUTs.

However, due to the lack of observable data on product breakdowns and the existing experience of national statistical offices, real examples of breakdowns by product only are seldom found.

Breakdown by industries and products

When both industries and products are broken down simultaneously, there is no specific criterion to determine the most suitable dimension of the EIOT, unless either industries (or products) are split into more types of categories than products (or industries) are. If industries are categorised by size (small, medium and large) and ownership (foreign and domestically owned) into six categories, it would be best to create industry-by-industry EIOTs to maintain the detailed information captured by the extended ESUTs. This is especially important because products may only be categorised by size due to limited survey-based data.15

The most common breakdown of regular SUTs leads to the same numbers of extension categories by industries and by products and hence yields square EIOTs. In this case, the choice of the dimension of the EIOT can be driven by the purpose of the analysis, as described for regular IOTs.

Construction methods for input-output tables

This section addresses the choice of assumption in the construction of product-by-product and of industry-by-industry IOTs.

From national statistical offices’ current experiences, the fixed product sales structures assumption is the preferred option to compile industry-by-industry IOTs. It seems logical to assume that the market sales structure of products remains unchanged independent of who the supplier industries are. In other words, secondary outputs typically have different destinations than the primary outputs (Thage and Ten Raa, 2006[10]; Yamano and Ahmad, 2006[11]).

Likewise, the product technology assumption is generally the preferred option to compile product-by-product IOTs, although limited by its requirement to have the same number of industries as products and the potential negatives that may arise. It seems logical to assume that one product can be produced with a unique input structure irrespective of the industry that actually produces it. Indeed, at a microdata level, the product technology assumption seemed to work best in the tests provided by Mattey and Ten Raa (1997[12]) and Ten Raa and Rueda-Cantuche (2013[13]). Moreover, Steenge (1990[14]) and Konijn (1994[15]) showed that it is theoretically possible to find a non-negative IOT using the product technology assumption.

However, the theoretical properties of input-output coefficients determine the superior model/assumption based on the fulfilment of the so-called material balance, financial balance, price invariance and scale invariance axioms16 (Rueda-Cantuche, 2017[1]; Kop Jansen and Ten Raa, 1990[16]; Ten Raa and Rueda-Cantuche, 2003[17]; Rueda-Cantuche and Ten Raa, 2009[18]). In the case of product-by-product IOTs, the product technology assumption satisfies all four axioms, whereas for industry-by-industry IOTs, it is the fixed industry sales structures assumption that satisfies all of them.

Hybrid technology assumptions have gained importance in national statistical offices’ compilation practice. For product-by-product IOTs, the common practice is to use the product technology assumption and then treat the biggest negatives17 as potential errors of measurement and/or aggregation. The industry technology assumption is used as a last resort to remove negative coefficients or when the bundle of products under one single industry is highly heterogeneous. This practice has been confirmed by a survey conducted by Eurostat among the EU member states (Eurostat, 2015[19]). For industry-by-industry IOTs, the practice is more straightforward since the fixed product sales structures assumption does not yield negatives by definition.

Rueda-Cantuche and Ten Raa (2013[20]) proposed an econometric framework using surveyed enterprises’ data to help IOT compilers to construct product-by-product IOTs, identifying the assumption that fits best with the empirical evidence, either the product technology assumption or the industry technology assumption. These tests lead to statistically significant conclusions that complement experts’ judgement in the compilation process. The Statistical Institute of Catalonia (IDESCAT, 2015[21]) successfully used this approach for the compilation of the 2011 IOTs of Catalonia. The power of these tests can be affected by the heterogeneity in the product classification, the lack of granularity in the surveyed data and/or potential errors of measurement at the establishment level. Conversely, industry-by-industry IOTs do not admit similar econometric tests.

Construction methods for extended input-output tables

This section provides further insights into the choice of assumption for EIOTs, for both product-by-product and industry-by-industry tables, based on the actual dimensions of the ESUTs.

The choice of assumption for the construction of EIOTs highly relies on their own definitions. For product-by-product EIOTs, the product technology assumption is the only one that allows one differentiated product technology for each category of enterprises (e.g. SMEs, MNEs, etc.) independent of the industry that actually produces them. Otherwise, differentiated products are assumed to be produced with the same industry technology.

For industry-by-industry EIOTs, the fixed product sales structures assumption is the only one that allows differentiated sales structures for each of the different products produced by SMEs, MNEs, etc., independent of the industry that actually produces them. Otherwise, the fixed industry sales structures assumption would imply that, for instance, differentiated products produced typically by SMEs and MNEs would have the same sales structure as other industries that can actually produce them, as secondary output.

Table 6.1 summarises the guidance and recommendations for the choice of assumption in the construction of EIOTs and provides information about the assumptions that might yield negative values. In essence, the recommended assumptions are those allowing capturing the heterogeneity in categories by size, ownership, etc. of ESUTs, i.e. the product technology assumption for product-by-product EIOTs and the fixed product sales structures assumption18 for industry-by-industry EIOTs. This also reflects the standard practice of national statistical offices.

Table 6.1. The choice of assumption for extended input-output tables: Extensions by size, as example

	Product technology assumption	Industry technology assumption	Fixed industry sales structures	Fixed product sales structures
Assumption	MNEs and SMEs have DIFFERENT technology for the production of their respective differentiated products	MNEs and SMEs have the SAME technology for the production of their respective differentiated products	MNEs and SMEs have the SAME fixed sales structures as the industry that actually produces their respective differentiated products	MNEs and SMEs have DIFFERENT fixed sales structures for their respective differentiated products
Dimension	Product	Product	Industry	Industry
Negatives	Yes	No	Yes	No

Note: MNE: multinational enterprise; SME: small and medium-sized enterprise.

Concluding remarks and recommendations

This section provides concluding remarks and practical recommendations for compilers of EIOTs from ESUTs regarding the choice of type of EIOT and the choice of assumption to construct product-by-product and industry-by-industry EIOTs.

When constructing EIOTs from ESUTs, as for regular IOTs, one must first decide on the type of IOT19 – product-by-product vs. industry-by-industry – then choose an assumption for its construction – a way to treat secondary products or industries (Rueda-Cantuche, 2017[1]).

Regarding the choice of the dimension of an EIOT, industry-by-industry EIOTs are preferable when the regular SUTs are broken down by industries only; analogously, product-by-product EIOTs are the preferred option when breaking down products only, although this is much less observable from national statistical offices’ current experience.

If products and industries are broken down simultaneously into a different number of categories/dimensions,20 it would be best to create industry-by-industry EIOTs to maintain the detailed information captured by the ESUTs, particularly when the number of industry categories is greater than the number of product categories. Otherwise, product-by-product EIOTs would be advisable but less likely given the lack of available survey-based data to break down products into much more detail than industries.

The typical scenario involves breaking down square regular SUTs into the same number of industry and product categories, resulting in the construction of square EIOTs. In this situation, the selection of the dimension of an EIOT would be solely determined by the specific purpose of the analysis, similar to the approach used for regular IOTs.

Regarding the choice of assumption for the construction of product-by-product EIOTs, the preferred option is the product technology assumption, for which, for instance, MNEs and SMEs have different technologies for the production of their respective differentiated products, independent of the industry that actually produces them. As usual, this assumption requires square ESUTs and a special treatment of the resulting negatives. This is in line with the theoretical approach of the existing literature and national statistical offices’ current practice for regular IOTs.

For industry-by-industry EIOTs, the preferred option is the fixed product sales structures assumption, for which, for instance, MNEs and SMEs have different sales structures for their respective differentiated products, independent of the industry that actually produces them. Opposite to product-by-product EIOTs, this assumption requires neither square ESUTs nor any special treatment of negatives.

Another important finding for EIOT compilers is that ESUTs, when broken down by industries only (rectangular), can be used to generate fully fledged industry-by-industry EIOTs, assuming a fixed product sales structures across different industry categories. This becomes especially important when resources are limited and only industry breakdowns are feasible. Similar reasoning applies to ESUTs broken down by products only, although it is less common due to the lack of observable data for product breakdowns. Additionally, the fact that the preferred option (i.e. product technology assumption) requires square ESUTs makes it more challenging, unless the less preferred industry technology assumption is used.

References

[3] Beutel, J. (2017), “The supply and use framework of national accounts”, in Ten Raa, T. (ed.), Handbook of Input-Output Analysis, pp. 41-132, Edward Elgar Publishing, Cheltenham, United Kingdom.

[5] European Commission; IMF; OECD; UN; World Bank (2009), System of National Accounts 2008, https://unstats.un.org/unsd/nationalaccount/docs/SNA2008.pdf.

[19] Eurostat (2015), Compilation of National Accounts Supply, Use and Input-Output Tables Questionnaire: Results, National Accounts Working Group Meeting, Doc. 860, Luxembourg, 21-22 May.

[7] Eurostat (2013), European System of Accounts – ESA 2010, Office for Official Publications of the European Communities, Luxembourg, https://doi.org/10.2785/16644.

[4] Eurostat (2008), Eurostat Manual of Supply, Use and Input-output Tables, Office for Official Publications of the European Communities, Luxembourg, https://ec.europa.eu/eurostat/web/products-manuals-and-guidelines/-/ks-ra-07-013.

[6] Eurostat (1995), European System of Accounts – ESA 1995, Office for Official Publications of the European Communities, Luxembourg, https://ec.europa.eu/eurostat/web/products-manuals-and-guidelines/-/ca-15-96-001.

[21] IDESCAT (2015), Input-Output Framework of Catalonia, 2011 (in Spanish), Statistical Institute of Catalonia, Barcelona, Spain, http://www.idescat.cat/dades/mioc/2011/metod/4.html?lang=en.

[15] Konijn, P. (1994), The Make and Use of Products by Industries, PhD Thesis, University of Twente, Enschede.

[16] Kop Jansen, P. and T. Ten Raa (1990), “The choice of model in the construction of input-output coefficients matrices”, International Economic Review, Vol. 31/1, pp. 213-227, https://doi.org/10.2307/2526639.

[9] Lenzen, M. and J. Rueda-Cantuche (2012), “A note on the use of supply-use tables in impact analyses”, Statistics and Operations Research Transactions, Vol. 36/2, pp. 139-152.

[12] Mattey, J. and T. Ten Raa (1997), “Primary versus secondary production techniques in US manufacturing”, Review of Income and Wealth, Vol. 43/4, pp. 449-464, https://www.roiw.org/1997/449.pdf.

[1] Rueda-Cantuche, J. (2017), “The construction of input-output coefficients”, in Ten Raa, T. (ed.), Handbook of Input-Output Analysis, pp. 133-174, Edward Elgar Publishing, Cheltenham, United Kingdom.

[8] Rueda-Cantuche, J. (2011), “The choice of type of input-output table revisited: Moving towards the use of supply-use tables in impact analysis”, Statistics and Operations Research Transactions, Vol. 35/1, pp. 21-38, https://raco.cat/index.php/SORT/article/view/242561/325182.

[20] Rueda-Cantuche, J. and T. Ten Raa (2013), “Testing the assumptions made in the construction of input-output tables”, Economic Systems Research, Vol. 25/2, pp. 170-189, https://doi.org/10.1080/09535314.2013.774265.

[18] Rueda-Cantuche, J. and T. Ten Raa (2009), “The choice of model in the construction of industry input-output coefficients matrices”, Economic Systems Research, Vol. 21/4, pp. 363-376.

[22] Sallusti, F., E. Pallott and S. Cuicchio (2024), eIOT Framework for the Italian Economy: How to Compile it and its Analytical Potential, 6th Meeting of Eurostat’s Expert Group on Integrated Global Accounts, https://unece.org/sites/default/files/2024-04/1B_5_eIOT_Italy_gena.pdf.

[14] Steenge, A. (1990), “The product technology revisited: Theoretical basis and an application to error location in the make-use framework”, Economic Modelling, Vol. 7/4, pp. 376-387, https://doi.org/10.1016/0264-9993(90)90002-L.

[13] Ten Raa, T. and J. Rueda-Cantuche (2013), “The problem of negatives generated by the commodity technology model in input-output analysis: A review of the solutions”, Journal of Economic Structures, Vol. 2/5, https://doi.org/10.1186/2193-2409-2-5.

[17] Ten Raa, T. and J. Rueda-Cantuche (2003), “The construction of input-output coefficients matrices in an axiomatic context: Some further considerations”, Economic Systems Research, Vol. 15/4, pp. 439-455, https://doi.org/10.1080/0953531032000152317.

[23] Ten Raa, T. and R. Stahlie (2024), “Footprint analysis and the incidence of emission taxes”, Ecological Economics, Vol. 224, https://doi.org/10.1016/j.ecolecon.2024.108214.

[10] Thage, B. and T. Ten Raa (2006), Streamlining the SNA 1993 Chapter on Supply and Use Tables and Input-output, Paper presented at the 29th IARIW conference in Joensuu, http://old.iariw.org/papers/2006/thage.pdf.

[2] United Nations (2018), Handbook on Supply, Use and Input-output Tables with Extensions and Applications, United Nations, New York, NY, https://doi.org/10.18356/9789213582794.

[11] Yamano, N. and N. Ahmad (2006), “The OECD Input-Output Database: 2006 Edition”, OECD Science, Technology and Industry Working Papers, No. 2006/8, OECD Publishing, Paris, https://doi.org/10.1787/308077407044.

Notes

← 1.  As described in United Nations (2018, pp. 373-374[2]).

← 2.  As described in United Nations (2018, pp. 373-374[2]).

← 3.  This also includes adjustments to the components of the gross value added and taxes less subsidies on products by industries. However, the final use components remain unchanged.

← 4.  This also includes adjustments to the final use components by product. However, gross value added and taxes less subsidies on products remain unchanged.

← 5.  Other reductions are not considered here, e.g. using base year prices or different units of measurement.

← 6.  Strictly speaking, the two dimensions are “product-by-product adjusted industries” but for simplicity, they are denoted as product-by-product IOTs. The same applies for industry-by-industry IOTs, i.e. “industry-adjusted product by industry”, as stated in United Nations (2018, pp. 373-374[2]).

← 7.  This assumption is the preferred option for most of the national statistical institutes (Eurostat, 2008[4]) to compile industry-by-industry IOTs.

← 8.  This assumption is the preferred option for most of the national statistical institutes (Eurostat, 2008[4]) to compile product-by-product IOTs.

← 9.  A recent example is Ten Raa and Stahlie’s (2024[23]) footprint analysis.

← 10.  Square ESUTs here refer to rectangular SUTs with more products than industries, which become square ESUTs because of the new extensions by industries.

← 11.  This is independent of how the EIOT is constructed, with either the product technology assumption for square ESUTs or the industry technology assumption for rectangular ESUTs.

← 12.  This might seem contradictory since many observations to estimate the proportions between inputs and outputs should better inform the aggregated estimated technical coefficients. However, if the primary objective is to develop an extended industry-by-industry IOT that explicitly includes new industry extensions (for example, by size), it stands to reason that generating a product-by-product extended IOT would hide the industry heterogeneity (such as size distinctions), thereby rendering the extension of the SUTs ineffective.

← 13.  This applies to rectangular SUTs with more industries than products that become square ESUTs because of the new extensions by products.

← 14.  This is independent of how the EIOT is constructed, with either the fixed industry sales structures assumption for square ESUTs or the fixed product sales structures assumption for rectangular ones.

← 15.  The reasoning is similar when the number of product categories is greater than the number of industry categories, but less frequent, if not inexistent.

← 16.  For the material balance axiom, the theoretical intermediate input requirements by product must be equal to the observed values in the use table. For the financial balance axiom, the theoretical intermediate costs of each industry must be equal to the observed costs in the use table. For the price invariance axiom, input-output coefficients must remain unchanged with respect to units of measurement. For the scale invariance axiom, input-output coefficients must not change when industry inputs and outputs vary proportionally. See Rueda-Cantuche (2017[1]) for a summary of the theoretical framework, theorems and properties.

← 17.  See United Nations (2018[2]) and Rueda‑Cantuche (2017[1]) for a comprehensive list of the methods and solutions used for the treatment of negatives.

← 18.  Sallusti, Pallott and Cuicchio (2024[22]) already follow this approach for the compilation of the Italian ESUTs.

← 19.  There could be a previous step that is to decide whether to use ESUTs directly for impact analysis instead of EIOTs (see, for example, Rueda-Cantuche (2011[8]); Lenzen and Rueda-Cantuche (2012[9]); Ten Raa and Stahlie (2024[23])). Following this argument, the choice of dimensions (product-by-product, industry-by-industry or product-by-industry) should better be problem-driven, e.g. footprints are emissions (by industry) related to consumptions (by product) that are better suited for SUTs than for IOTs. Notwithstanding the importance of this issue, this chapter only focuses on providing guidelines for the use of EIOTs in impact analysis.

← 20.  Typically, more industry categories than product categories due to the lack of observable data.