An extended approach to value chain analysis

Knez, Klemen; Jaklič, Andreja; Stare, Metka Article An extended approach to value chain analysis Journal of Economic Structures Provided in Cooperation with: Pan-Pacific Association of Input-Output Studies (PAPAIOS) Suggested Citation: Knez, Klemen; Jaklič, Andreja; Stare, Metka (2021) : An extended approach to value chain analysis, Journal of Economic Structures, ISSN 2193-2409, Springer, Heidelberg, Vol. 10, pp. 1-37, https://doi.org/10.1186/s40008-021-00244-6 This Version is available at: https://hdl.handle.net/10419/261614 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. 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If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. https://creativecommons.org/licenses/by/4.0/ An extended approach tovalue chain analysis Klemen Knez* , Andreja Jaklič and Metka Stare 1 Introduction In recent decades, the growing complexity of the division of labour has been reflected in the fact that ever more production is occurring within value chains, both at home and abroad. Theoretical and empirical approaches to the analysis of value chains have advanced rapidly, yet are very eclectic and heterogeneous. The earliest definitions of commodity chains1 date back to the world-systems2 theory: “What we mean by such chains is the following: take an ultimate consumable item and trace back the set of inputs that culminated in this item— the prior transformations, the raw materials, the transportation mechanisms, the labour input into each of the material processes, the food Abstract In the article, we propose a comprehensive methodology of value chain analysis in the international input–output framework that introduces a new measure of value chain participation and an extended typology of value chains, with the novel inclusion of domestic value chain to address the extent of fragmentation of purely domestic production. This allows for the simultaneous analysis of both global and domestic production fragmentation, the complex patterns of their evolution and their impact on economic development. The main contribution of the proposed methodology is conceptual: it permits the measurement of all value chain paths that pass through each country-sector from production to final consumption, whether the path includes downstream linkages, upstream linkages or their combination. Empirical application of this methodology shows the importance of including domestic fragmentation in value chain analysis: The fragmentation of both global and domestic levels of production has a significant positive correlation with economic growth. This implies that the effects of global production fragmentation must be analysed together with the changing structure of the fragmentation of domestic production to obtain the whole picture, one that might provide important information for policymaking and industrial policy. Keywords: Value chain typology, Value chains, Global value chain, Domestic value chain, Participation rate, Input–output framework JEL Classification: F1, F4, F6 Open Access © The Author(s), 2021. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http:// creat iveco mmons. org/ licen ses/ by/4. 0/. RESEARCH Knezetal. Economic Structures (2021) 10:13 https://doi.org/10.1186/s40008-021-00244-6 *Correspondence: [email protected] Centre of International Relations, University of Ljubljana, Ljubljana, Slovenia 1 The term global commodity chain is a predecessor of global value chain. 2 Embracing a historical and macroeconomic approach to the analysis of the global division of labour, the world-systems approach examines the unequal patterns of exchange along global commodity chains as well as different structural patterns of the international integration of the core, periphery and semi-periphery (Arrighi and Drangel 1986). Page 2 of 37 Knezetal. Economic Structures (2021) 10:13 inputs into the labour. This linked set of processes we call a commodity chain (Hopkins and Wallerstein 1977)”. In the 1990s, the research programme of global commodity chains was first systematically outlined by Gereffi’s seminal contribution (Gereffi 1994) that defined three interlocking dimensions of the research: the input–output dimension, the spatial dimension, and the question of commodity chain governance3. This research period was characterised by moving away from a historical and macroeconomic perspective towards a special focus on industrial chains and the inter-firm cooperation perspective, with numerous case studies on value chains. The global value chain framework emerged early in the new century with the express aim of unifying the previous heterogeneous research (Gereffi 1999;Gereffi etal. 2001). On one hand, the global value chain approach increased the focus on the enterprise level and merged with the literature from international business and management4, while also drawing from the new institutional transaction cost approach5. On the other hand, the creation of international input–output tables6 led to a revival of the aggregated macroeconomic approach to global value chains, albeit with a different focus than the world-systems approach.7 In this article, we present a new methodology for measuring different value chain participation rates in the international input–output framework. Compared to the most widely used measurement of value chain participation introduced by Wang etal. (2017), we make two fundamental conceptual enhancements. First, our methodology creates a single and consistent measurement of value chain participation on the country-sector level, as opposed to the two (upstream and downstream) participation rates that feature in Wang’s methodology. The argumentation and logic used to derive a single value chain participation share on the country-sector level is very similar to the approach of Arto etal. (2019), which combines the sourceand sinkbased approaches to export decomposition. The idea is that decomposition based on final demand (sink-based decomposition) is independent of the decomposition of downstream value added (source-based) and thus both can be linearly combined to grasp both the information regarding the source of value added as well as the path to final demand simultaneously. Methodologies of export decomposition have recently seen significant improvements (Arto etal. 2019; Borin and Mancini 2019; Miroudot and Ye 2021). However, the value chain participation rate methodologies either still chiefly rely on the value-added export matrix to describe the value flows between any two country-sectors in the economy (Johnson and Noguera 2012) and result in separate upstream and downstream participation rate measures or combine a sinkand a source-based measure in 3 Governance was conceived as either consumer-driven (apparel sector) or producer-driven (automotive sector).This approach was further extended by Ponte and Sturgeon (2013). 4 Porter’s (1985) concept of the intra-firm value chain is often used to discuss the specialisation of enterprises, and core competencies and business literature on multinational enterprises overlap with the global value chain framework. 5 Which was used to extend the producer-driven and consumer-driven governance typology of commodity chain research to a more general typology of value chain linkages, from transactions in a completely free market to a strict hierarchy (Gereffi etal. 2005). 6 In international economics, use of the input–output methodology grew in importance as researchers of various international incentives integrated nationally based input–output tables into harmonised global input–output tables. The most prominent are the World Input–Output Database (Timmer etal. 2015), the OECD’s Trade in Value Added and the EORA (Lenzen etal. 2013). 7 While all heterogeneous approaches to value chains focus on a development issue, the recent GVC approach has been adopted by international institutions to highlight the gains from liberalisation and industrial upgrading, while the worldsystems approach critically examines unequal rewards along the value chain and different structural integration patterns that may cause the perpetuation of unequal development (Gereffi 2018; Taglioni and Winkler 2016). Page 3 of 37 Knezetal. Economic Structures (2021) 10:13 merely one-sided, forward-looking measures. Our approach to value chain decomposition no longer uses the value-added export matrix and instead breaks down the asymmetric value chain stemming both downstream and upstream from each country-sector concerned simultaneously. Creating a single consistent variable on the country-sector level that measures the overall level of participation in value chains enables the empirical testing of many research theses that were previously either limited to the aggregate level or had to be articulated separately in terms of measuring the impacts of upstream and downstream value chain integration. Second, our methodology allows extensions of the value chain typology that are not possible with Wang’s approach to the decomposition of production activities or with export decompositions. We introduce a novel measure of the domestic value chain participation rate to measure the share of production which represents the extent of the fragmentation of domestic production. In place of a single and undifferentiated domestic component, we distinguish domestic production, which is fragmented (involving measurable cooperation among domestic firms), and domestic production, which is not fragmented (consisting of producing direct value for consumption without the cooperation of domestic firms). This makes our concept of the domestic value chain a completely new and different concept compared to Wang’s domestic component, which does not distinguish the two and combines both categories within a single undifferentiated concept. While Wang’s share of the domestic component is only a simple residual—a negation of the share of the fragmentation of global production and the global Ricardian trade share that does not provide information about the nature of the domestic economy, our novel methodology allows us to measure the extent of fragmentation of domestic production in addition to the usual study of the fragmentation of international production. We aim to use our approach to provide methodological tools that facilitate exploration of the complex interrelationship of global and domestic value chains and their evolution over time. We believe this will add to understanding of the diverse patterns of the structural integration of various countries/sectors and the different effects of such patterns on economic development. While this is primarily a methodological contribution, we shall use elementary empirical data to try to show the possible link between the level of fragmentation of global and domestic production and overall economic growth. The article is structured as follows: In Sect.2, we review the existing value chain indicators and address their shortcomings. In Sect.3, we present our methodology. In Sect.3.1, we present a new conceptualisation of value chain in the international I–O framework and define our object of disaggregation. A new value chain typology is presented in Sect.3.2 where we also derive participation shares. In Sect.4, we present an example of empirical application and some basic empirical results of the new methodology to show the insights into economic structures that can be gained by using the new value chain measures and which links exist between value chain integration patterns and overall economic growth. Finally, we discuss the contributions of the paper, its limitations and possibilities for further research. Page 4 of 37 Knezetal. Economic Structures (2021) 10:13 2 Background The most recent macroeconomic analyses of global value chains rely on the international input–output methodology. As international I–O data are essentially an integrated standard accounting data set harmonised on the sectoral level, information is lacking on the typology of value chain governance. This means the international I–O database cannot be the sole source for the study of production networks, which theoretically differ from purely open trade transactions by including at least some level of hierarchy, and which investigate the local embedding of production linkages (Buckley2009;Henderson etal. 2002; Hess and Coe 2006; Hortaçsu and Syverson 2009). However, the general framework of global value chains can function without such distinctions and this makes the international I–O data set one of its most important sources of information. The key benefit of applying the I–O methodology in global value chain analysis is that aggregated information about the structure of value chains can be obtained, as opposed to isolated firm-specific case studies that can provide a more detailed understanding of different aspects of a given value chain. Thus, of the three dimensions of commodity chain research noted by Gereffi (1994), both the I–O aspect and the spatial dimension, can be considered in the international I–O approach, while the governance aspect cannot. Various aggregated and sectoral global value chain indicators, indices and measures have been proposed, all derived from the international I–O framework. GVC indicators may be roughly divided into measures of length8 and participation rates, which we will discuss briefly. Early I–O measures of the GVC structure were simple upstream and downstream indicators that corresponded to the measure of distance to final demand (upstream) and the Leontief measure of backward linkage (downstream) and were often referred to as the length of a value chain (Ahmad etal. 2017). Fally (2011) and Antràs etal. (2012) defined the downstream indicator to “reflect how many plants (stages) are involved in production one after the other” up to the point observed and the upstream indicator to “measure how many plants this product will pass through (e.g. by assembly with other products) before it reaches final demand (Fally 2011, 10)”. Fally (2011) defined them as the number of vertical stages weighted by the value added of each stage, with the distance between each stage set to 1.9 Since then, the average vertical distance has been the basic measure of the length of the value chain in the international I–O framework. Miller and Temurshoev (2015) further specified the existing measures by presenting upstream and downstream indicators in a matrix formulation using Ghosh’s forward and Leontief’s backward coefficient matrices (Ghosh 1958; Leontief 1936). These upstream and downstream measures are simple measures of the upstream and downstream length of value chains measured by the average vertical distance. Within this framework, further improvements were introduced by Muradov (2016), who focused on separating the domestic from the global production component while calculating the length of value chains. The existing dominant conceptualisation of GVC participation measures is largely based on the work of Johnson and Noguera (2012), who produced a value-added export matrix that captures information on value flows in the economy between any two points 8 Relative position indices can easily be derived from length measures as simple ratios. 9 Using a method similar to that used to calculate the average propagation length required for the analysis of the dynamic response to shocks, defined by Dietzenbacher and Romero (2016). Page 5 of 37 Knezetal. Economic Structures (2021) 10:13 (country-sectors) in the economy. This provides the basis for the disaggregation of value on the country-sector level, depending on whether the value was produced domestically for domestic consumption or involved cross-border transactions for either final or productive consumption (Koopman etal. 2014; Los etal. 2015;Wang etal. 2017). Since the valueadded export matrix tells us about the source and destination of value added and covers all possible paths between any two country-sectors in the economy, there are two indicators of the share of GVC participation—the upstream and downstream share. The conception of the upstream participation share of participation starts from the value added of individual industries (country-sectors), disaggregating all possible paths leading to the realisation of their value, while the conception of the downstream share of participation starts with final consumption, disaggregating all possible paths of the downstream production linkages. Within this framework, disaggregation is defined on the domestic part, the “Ricardian trade” in finished goods, the simple GVC and the complex GVC, which is currently the most widely used accounting framework for GVC participation and thus far has been used by the best-known research on GVC carried out jointly by the WTO, the WB group, the OECD, IDE-JETRO, RCGVC-UIBE and the China Development Research Foundation (GVC Development Reports). Further improvements and clarifications of the framework were made by Borin and Mancini (2019), who derive their own measure of GVC-related bilateral trade flows by decomposing export to that attributable to traditional trade and GVC trade. Their indicator is composed of source-based backward and sink-based forward parts of their export decomposition, which can be calculated in a bilateral, country and world setting. The development of I–O participation share measures of value chains, which are the primary interest of this article, evolved simultaneously with the development of methodologies of decomposing trade in value added (Johnson and Noguera 2012) as well as value added in trade (Arto etal. 2019; Borin and Mancini 2019; Miroudot and Ye 2021). However, despite similarities and some conceptual and formal mathematical overlapping, the fields of value chain participation share measures and value added in trade are driven by quite distinct research questions and research interests. On one hand, principal interest in decomposing exports is the correct evaluation of cross-border flows (properly removing double counting), assessing trade policy impacts and conducting overall impact analysis, either in a bilateral setting or with a focus on a specific country. On the other hand, value chain participation measures attempt to grasp the structure of an economy, sectoral and country interdependencies and the specific embeddedness of each production unit in different value chain structures, both at home and abroad. Value chain participation share measures usually correspond to a share of production, which statistically satisfies certain a priori criteria, such as “at least two cross-border transactions” or “at least one cross-border production sharing transaction”. The reviewed literature has contributed to better understanding of value chains and their I–O applied research, but still suffers two shortcomings that we try to address and improve with our approach. The first main shortcoming of all current value chain participation share indicators is the lack of a single uniform measure for different value chain participation rates on the country-sector level. First, the value chain decomposition of Wang etal. (2017) results in downstream and upstream value chain participation rates, which provide two different types of information at the country-sector level. This is relevant for some types of analysis Page 6 of 37 Knezetal. Economic Structures (2021) 10:13 that deal with the relationship between upstream and downstream participation in GVCs, but there is a variety of situations where a common measure of GVC participation, defined uniformly on the country-sector level, is required either as the main object of the analysis or as a supplementary or control variable.10 Second, GVC measures based on the decomposition of exports, even though they overcome the sinkand source-based decomposition in one unifying framework of export decomposition (Arto etal. 2019; Borin and Mancini 2019), are conceptually unable to offer a consistent solution to the question of a single country-sector value chain participation measure. That is because the criteria for export decomposition (separating domestic value added from foreign value added and the removal of double counting) do not correspond with the general criteria for different value chains on the country-sector level (the share of production with a certain number of crossborder transactions). Although export can be decomposed both with regard to the origin of the value added as well as the final demand, the very fact that the object of decomposition is export means it has a one-sided, forward orientation since export decomposition cannot address the fragmentation of production of a country-sector that has little or no exports (but can still form part of the fragmentation of a global value chain downstream). In this sense, the attempt by Borin and Mancini (2019) to provide a GVC measure of bilateral trade by decomposing exports cannot identify the share of production of a given country-sector which satisfies the criterion of a certain number of cross-border transactions, but only examines its forward part and is hence conceptually similar to Wang’s forward GVC measure. Our attempt to solve this issue demands the decomposition of the gross output (total output) of each country-sector to simultaneously account for both downstream and upstream value chain linkages. The second major shortcoming of existing value chain indicators is the lack of a measure of domestic value chain fragmentation. The decomposition put forward by Wang etal. (2017) includes a broadly defined “domestic component”, which covers all of the value that does not comply with the GVC and Ricardian trade criteria. One of the major contributions of this article is to conceptually further divide this broad domestic component into a first part which comprises domestic production fragmentation (involving production sharing between at least two domestic firms) and the second part which does not. This yields new information regarding the share of production not involved in the fragmentation of global production, but is part of the fragmentation of domestic production and enables research into the role of domestic production fragmentation, which was impossible with the existing conceptualisations. As a result of the present disaggregation of participation shares into the “domestic component” and the GVC participation rates (and the Ricardian trade share) consisting of a simple duality that in its construction sums to 1, the share of the domestic component is never used in regressions (due to collinearity) and never even examined as a theoretical concept. It is simply a residual, a share that does not interest researchers given that all the information they disaggregate is included in their GVC 10 It is also obvious that a simple solution, such as using the average of existing upstream and downstream indicators, cannot be justified in theory. If, for example, a given country-sector’s share in the upstream global value chain is high (close to 100%) and its share in the downstream global value chain is relatively low (close to 0%), then the average share in the value chain would be around 50%, which is misleading because the value chain as a whole is almost entirely global (using the criterion that the value crosses a border at least once). As far as value chain paths are concerned, despite the small share of downstream global value chain paths, a high share (close to 100%) of the same paths continues in the upstream global value chain such that production as a whole has a very high global share (close to 100%), while the use of the average of the upstream and downstream indicators does not correspond to the definition of the global value chain. Page 7 of 37 Knezetal. Economic Structures (2021) 10:13 participation rates. The existing approaches are used by researchers to focus exclusively on the international dimension of the fragmentation of production, neglecting the potential held by the international I–O methodology that allows analysis of domestic production fragmentation. Our approach is breaks ground in this area as it proposes a new concept of domestic fragmentation able to be measured on its own and according to its own definition and that is not collinear with the sum of the GVC participation rate. Our methodological approach starts with the formal criteria, which is common for most of the GVC literature where value chains are defined according to certain transaction criteria (number of cross-border production-sharing transactions or similar). It is important to note that any such criteria are arbitrary and potential multiplicity of such criteria and hence value chain typologies can coexist and offer researchers some leeway in their empirical applications.11 With a view to creating a uniform value chain measure on the country-sector level, we use the total output of each country-sector as the starting point of our disaggregation. Decomposing total output (as opposed to export or total value added) enables us to simultaneously grasp both the downstream and upstream value chain paths as well as the structure of the economy that is entirely domestic. Our decomposition begins with a set of the presented value chain tree matrices ( τi ) which describe all of the value chain paths, from any country-sector of primary origin to any countrysector of production for final consumption that passes through (include a production stage of) a single particular country-sector. The logic of our approach is very similar to that of Arto etal. (2019) for combining the sinkand source-based decomposition of exports: because the decomposition of paths to final demand is independent of the decomposition of downstream value added, these decompositions can be linearly combined to capture both types of information in a single decomposition along two different dimensions at the same time. The big distinction with this approach is that object of decomposition is different—in our case, it is the total output (gross output) of each country-sector. Our choice of the object of decomposition is a prerequisite for properly capturing downstream linkages and, more importantly, properly accounting for the domestic structure of the economy. This formulation is the first attempt to capture information concerning the asymmetric value chain tree, which is a specific feature of each individual country-sector (Fig.1). The proposed value chain tree matrices are unique in that they allow us to simultaneously capture the structure of the downstream and upstream value chain paths and to define value chain participation rates as a single measure for each country-sector. The crucial point of the proposed methodology is to enable the disaggregation of value chains based solely on the structure of value chain paths—taking into account whether these paths include only domestic production fragmentation, international production fragmentation or no production fragmentation at all. This allows us to introduce the concept of domestic value chain fragmentation that simply cannot be created within the existing framework of 2 separate participation indices. This multiplies the research opportunities offered by the value chain methodology based on the international input–output structure by permitting general analysis of the fragmentation of both domestic and global production and their interdependence along with any mutual effects of their development. 11 For example, in a forthcoming article we explore the decomposition of value chains based on the criterion of the number of domestic transactions subject to meeting the usual global value chain criterion of having at least one production-sharing cross-border transaction. In this setting, we decompose the global value chain share into a GVC with no domestic cooperation, a GVC with simple domestic cooperation, and a GVC with complex domestic cooperation, offering information on the specific pattern of the EU periphery’s integration. Page 8 of 37 Knezetal. Economic Structures (2021) 10:13 Applying this methodology, we show that increasing fragmentation of global production in recent decades has been a general trend for most countries (with a backlash in later years), but different institutional arrangements and structural economic positions led to various types of global economic integration, bringing diverse effects for domestic fragmentation. With our methodology, we shall empirically demonstrate that in many countries with high growth and ever stronger global integration domestic fragmentation also increased. However, one can find cases where domestic fragmentation stagnated or even declined whereas fragmentation of the global value chain increased. The different types of integration in global value chains are the outcome of several structural and institutional developments.12 On one hand, the simultaneous increase in domestic and global fragmentation might only be a consequence of the growing complexity and division of labour. Yet, on the other hand, the simultaneous rise in global fragmentation and drastic decline in domestic integration might be due to the fracturing of domestic vertically integrated companies, parts of which are integrated into global value chains as subsidiaries, or due VA FC VA FC Valueadded (VA) Finalconsumption (FC) The unitin focus (Thesiphon) Pathsofvalue realisation (The leaves) (The roots) Pathsofvalue creation VA FC VA FC Cross-border Domestic Cross-border Domestic Cross-border Domestic Cross-border Domestic Cross-border Domestic Cross-border Domestic F. D. F. D. F. D. F. D. F. D. F. D. F. D. F. D. Fig. 1 Value chain tree. Source: own conceptualisation and design. Arrows represent production-sharing transactions—buying and selling of intermediate products for production. Orange colour denotes production that does not involve any production sharing, while any combination of red or orange paths denotes domestic production fragmentation. Any value chain path which includes a cross-country production-sharing transaction (a black arrow) is part of a global value chain from the perspective of the particular unit in focus. The paths of value creating and value realisation in a general case continue to branch ad infinitum (three levels are chosen only for demonstration purposes) 12 For example, the concept of integrated periphery was introduced to describe a specific type of integration in the case of the Slovak and Czech car industries, characterised by their proximity to consumer markets, cheaper labour force, the absence of positive spillover effects and lack of domestic linkages (Oldřich and Vladan 2019; Pavlínek 2018). Page 15 of 37 Knezetal. Economic Structures (2021) 10:13 decomposition gives information about how the product of the ith country-sector is consumed either directly or as part of the final product of other country-sectors. Two separate vectors which disaggregate the value chain paths of the downstream (ith column of Z) and upstream value chain (ith row of W) thus span an entire matrix of total output shares that capture the value chain tree structure of the ith country-sector. We combine them with the direct product that defines the matrix of the value chain tree for each country-sector (i) by multiplying each element of Zei (the ith column of Z) by each element of  ei TW (the ith row of W). Definition 3 Value chain tree matrix τi=Z�ei⊗�ei TW ; τi∈Rn×n , where �ei∈ R n represents the standard orthonormal basis of Rn. This defines each element of the value chain tree matrix tijk ∈τi as tijk =wijzki . Each element of the value chain tree matrix τi thus represents a share of the total output of country-sector i, which is primarily produced in country-sector k and consumed as an end product of country-sector j, along any upstream and downstream value chain path. The main point of our derivation is not the expressed final value distribution of the total output of each country-sector along any of its upstream and downstream value chain paths, but the expression of the total output distribution (of the respective country-sector) along any value chain path, be it a downstream value chain path, an upstream value chain path or any combination of both paths at the same time. The structure of the value chain tree matrices allows us to focus our disaggregation on the composition of the value chain paths covered by the two global Leontief inverses in the equation, the first representing all downstream parts of the value chain paths and the second representing all upstream parts of the value chain paths. A single value chain path is determined by a series of concrete transactions between companies: It is a unique path from primary value creation (value created in production, not transferred from intermediate products) to value realisation (final consumption, not productive consumption of intermediate products), which passes through the production stage of the ith country-sector. The total output of i is not only disaggregated along all possible paths leading from any country-sector of origin via country-sector i to any countrysector of final stage production (as determined by τi ), but is also disaggregated in much finer detail, along all the unique value chain paths that pass through i. That a concrete value chain path only forms part of the value chain tree matrix can easily be recognised if both inverses in τi are replaced by an infinite series ( (I − A)−1 = I + A + A2+··· ). Such disaggregation then results in an infinite number of value chain paths, and the total output of the ith country-sector is distributed over all of these paths. A certain value chain path share of the total output of i is determined by the Leontief technical coefficients aij ∈A . For example, take a value chain path consisting of value (3.11) τi =ˆ vC(I − A)−1 � ei ⊗� eiT ˆ x−1(I − A)−1ˆ f Page 16 of 37 Knezetal. Economic Structures (2021) 10:13 primarily produced in country-sector CS1 20, then used as an intermediate in CS2 , which in turn is used as an intermediate in i (the country-sector whose value chain is broken down), and then sent as an intermediate to CS3 , which is then sent as an intermediate to CS4 , where it is finished and sold for consumption. This value chain path has an origin ( CS1 ), a midpoint (i) and a final destination of production ( CS4 ), as well as a concrete path with a length of 5 (5 country-sectors contribute to production from origin to final consumption). The share of the total output of the ith country-sector that may be attributed to this specific path is: A specific unique value chain path of the ith country-sector’s value chain tree, that has its origin in k and final stage in j, can be written as: Such a path has a downstream length of d and an upstream length of u−1−d and the path is determined by a unique set of production-sharing transactions from the origin to the final stage (from origin j=CS0 , to CS1 , to CS2 , ..., to i=CSd , and further to CSd+1 , CSd+2 , ..., to k=CSu ). Leontief technical coefficients aCSp−1CSp determine each production-sharing transaction. The summation along the total output shares of i attributed to all such unique value chain paths, taking into account all permutations of possible transaction sequences and also all possible lengths (all possible length combinations of downstream and upstream lengths) as well as all possible origins and final stage destinations, results in a unit: Our conceptualisation allows us to define decomposition criteria applicable to each value chain path of the value chain tree of the ith country-sector. Based on this property, we will decompose the value chain structure of each country-sector separately in the following section. 3.2 The value chain typology 3.2.1 Definitions The framework of the international I–O analysis allows the separate analysis of final transactions to consumers and transactions between companies. Based on this characteristic, we propose a typology of value chains based solely on the structure of linkages between enterprises, while adding a further decomposition with regard to different possible (3.12) vCS1 a CS1CS2 a CS2i x −1 i a iCS3 a CS3CS4 f CS4. (3.13) d p=1vCS0aCSp−1CSpx −1 i  u−1 p = d aCSpCSp+1fCSu . (3.14) 1 Tτ i 1=1Tˆ v C (I−A) − 1�e i ⊗�e i Tˆ x − 1(I−A) − 1 ˆ f1T= 1. 20 CSk represents an index for different country-sectors. aCS1CS2 thus represents a single Leontief technical coefficient indicating that the value produced by CS2 requires a aCS1CS2 share of CS1 input. Page 17 of 37 Knezetal. Economic Structures (2021) 10:13 transactions to reach the final consumption post festum.21 Each matrix τi expressed by equation3.13 represents the desegmentation of the total product of country-sector i along different downstream and upstream paths. When we refer to a value chain, we refer to the specific share of value (share of output) that corresponds to a particular value chain path. Path22 of each value share generally includes any combination of domestic and crossborder production-sharing transactions, which can take place both downstream and upstream relative to the respective country-sector. Our criteria for the value chain typology thus refer to each specific value share corresponding to a single path within a value chain tree specific to each country-sector. Definition 4 Domestic value chain Domestic value chain (DVC) is a value that involves at least 1 domestic production-sharing transaction and involves only domestic production-sharing transactions along its path. Definition 5 Global value chain Global value chain (GVC) is a value that involves at least 1 cross-border production-sharing transaction along its path. We further distinguish two types of global value chains: simple and complex. Definition 5.1 Simple global value chain Simple global value chain (SGVC) is a value that involves exactly 1 cross-border production-sharing transaction anywhere along its path. Definition 5.2 Complex global value chain Complex global value chain (CGVC) is a value that involves more than 1 cross-border production-sharing transaction along its path. Definition 6 No value chain No value chain (NVC) is a value that does not involve any production-sharing transactions and has no value chain path within production. 21 For example, Wang’s disaggregation into simple and complex GVCs uses the number of cross-border transactions, regardless of whether the value crossed a border for production or whether it is only an export to end users. Such a criterion mixes two conceptually different transactions, leading to unnecessary calculation complexity and the impossibility of further conceptual disaggregation. Existing definitions of the typology of value chains, like all such definitions, are constructed in a relatively arbitrary way. More important than strict adherence to the prevailing definitions is the clarity of the proposed revision and the presentation of the conceptual relationship of the new concepts with the old ones. Our proposal facilitates a more detailed decomposition that will allow researchers to construct an indicator better suited to their research questions. Since the revised typology is based on a more detailed decomposition compared to the currently prevailing typology, researchers can (by simply adding components of the revised decomposition) also replicate objects that correspond to existing studies. 22 Here we examine the path of production fragmentation, while the path to final consumption, which represents an additional transaction, is analysed in Sect.3.2.5. Page 18 of 37 Knezetal. Economic Structures (2021) 10:13 A few brief comments are appropriate on our definitions and their interpretation. No material product or service belongs to a single classification of value chain, and no enterprise can be considered part of a single type of value chain. The output of each enterprise belongs to a variety of value chain paths. In general, one part of the output comprises many cross-border transactions, another part only domestic transactions, and yet another part their relatively complex interrelationship. Each product (or country-sector in our case) can be assigned different shares of the value chain paths. These shares are objects that provide information about the structure of the economy. For example, virtually no enterprise could be classified exclusively as part of a no value chain, but some enterprises that provide services (e.g. domestic services) have a relatively high share of output that has no value chain path, especially in services, where salaries account for almost all of the enterprise’s expenditure and where their product directly satisfies final demand. On one hand, enterprises that specialise in intermediate goods are always part of a value chain, whether domestic or global. On the other hand, even modern industries such as food-processing and pharmaceuticals, also have a certain (usually small) share of value added that is not part of any value chain (no value chain share), corresponding to the share of domestic value added in these industries that is also directly consumed (part of output that has no value chain path). The value chain shares and their changes are the object that provide information about the structure of the economy, whether on the sector or country level. As the economy develops, the division of labour also increases, which corresponds to the growing fragmentation of production, in particular international production fragmentation, and a decrease in shares where there is limited or no value chain fragmentation. Compared to the existing typology of value chains, this revised typology allows for analysis of the relationship between global and domestic fragmentation, which might prove especially relevant for the policies of developing countries. 3.2.2 The decomposition ofpaths Our value chain typology is established according to criteria along the entire value chain. For this reason, we disaggregate the value chain tree matrices τi in terms of criteria for different types of value chain paths. Our decomposition consists of the decomposition of two Leontief inverses, which may be interpreted as the decomposition of the downstream part and upstream part of each value chain path, as defined by equation3.11: τi =ˆ vC(I − A)− 1� ei ⊗� eiT ˆ x− 1 (I − A)− 1 ˆ f . The decomposition is constructed based on of the criteria of the number of cross-border and domestic production-sharing transactions that are consistent with the revised value chain typology. First, we investigate the decomposition of only a single Leontief inverse (interpreted symmetrically with respect to our criteria in the upstream and downstream value chain) and only then do we analyse the decomposition of all value chain paths characterised by the two Leontief inverses. The international I–O data have a specific block matrix structure in which the block diagonal elements represent domestic production-sharing transactions and the block off-diagonal elements represent international production-sharing transactions ( AD denotes domestic—block diagonal—and ACB cross-border—block-off diagonal—part of A), which allows us to decompose the Leontief inverse in the following way: Page 19 of 37 Knezetal. Economic Structures (2021) 10:13 (1) I obviously represents that part of the output which contains no production-sharing transactions—no value chain linkages. In the upstream part, it represents the share of total output that directly satisfies final demand (i.e. no upstream value chain), while in the downstream part it represents the direct value added of the country-sector whose production is being decomposed (i.e. no downstream value chain). (2) A D (I−A D ) −1 =A D +A 2 D +A 3 D + ... represents that part of output which contains at least 1 domestic production-sharing transaction and contains only domestic production-sharing transactions. (3) (I − AD)−1ACB(I − AD)−1 represents that part of the output which contains at least 1 production-sharing transaction and contains exactly one cross-border productionsharing transaction somewhere along its value chain path. This can be demonstrated by paraphrasing the part as all possible combinations of a single cross-border transaction among any possible set of domestic production-sharing transactions that occur before or after the single cross-border production-sharing transaction: (4) (I − A)−1 − (I − AD)−1 − (I − AD)−1ACB(I − AD)−1 represents that part of the output which contains at least two or more production-sharing transactions, of which at least two are cross-border production-sharing transactions. This logically follows from the fact that parts (1), (2) and (3) cover the total output that contains less than two cross-border transactions, and that the full Leontief inverse covers the total output. 3.2.3 Value chain tree matrix decomposition We proceed by disaggregating all of the value chain paths as they are structured in the value chain tree matrices. Using the decomposition of the Leontief inverse that we disaggregated in the previous subsection and inserting it into Eq.3.11, we obtain 16 components (3.15) ( I−A)− 1 =(I−AD)− 1 +(I−A)− 1 −(I−AD)− 1 =I+AD+A2 D+A3 D+···+(I−A)−1−(I−AD)−1 =I+AD(I+AD+A2 D+...)+(I−A)−1−(I−AD)−1 =I+AD(I−AD)−1+(I−A)−1−(I−AD)−1 =I  1.) +AD(I−AD)−1   2.) +(I−AD)−1ACB(I−AD)−1    3.) +(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1    4.) ( I−AD) −1 ACB(I−AD) −1 =ACB +ACBAD+ACBA 2 D··· +ADACB +ADACBAD+ADACBA2 D··· +A2 DACB +A2 DACBAD+A2 DACBA2 D +···+ . . . Page 20 of 37 Knezetal. Economic Structures (2021) 10:13 ( 4×4 product) for each matrix τi .23 This disaggregation along both the upstream and downstream paths is the basis for deriving value chain shares that correspond to our typology. We decompose each τi matrix describing all possible value chain paths of the output of the ith country-sector into a matrix consisting of domestic value chain paths only, a matrix containing all possible global value chain paths (as well as simple and complex global value chain paths separately), and a matrix consisting only of the value that has no value chain path. Definition 7 Domestic value chain tree τDVC i The domestic value chain tree represents all value chain paths of the output of each country-sector which, according to Definition 4, are part of the domestic value chains. In Fig.1, the domestic value chain paths are marked in red. Domestic value chain paths are defined as all paths that contain at least one red-coloured linkage (representing transactions between domestic enterprises) and include only red-coloured linkages and orange paths (representing the value creation or realisation in the respective country-sector in focus). The first part ( ˆ vCAD(I − AD)− 1� ei ⊗� eiT ˆ x− 1 ˆ f ) covers the downstream domestic value added (downstream domestic path), which ends as the ith country-sector final stage (no upstream path), the second part ( ˆ vC � ei ⊗� ei Tˆ x −1 AD(I − AD) −1 ˆ f ) covers the value added of the ith country-sector (no downstream path) that is transferred via the upstream domestic value chain (upstream domestic path), and the third part ( ˆ vCAD ( I − AD ) − 1� ei ⊗� eiT ˆ x− 1 AD ( I − AD ) − 1 ˆ f ) comprises the downstream domestic value added that is used as an intermediate product in the production of i and then used as an intermediary further in the upstream domestic value chain until it reaches final demand (both downstream and upstream domestic paths). All three cases meet the definition of a domestic value chain. Definition 8 Global value chain tree τGVC i The global value chain tree represents all paths of the output of the individual countrysector, which form part of global value chains according to Definition 5. In Fig.1, the global value chain paths are represented by all paths containing at least one black-coloured linkage (representing cross-border transactions between enterprises). Global value chain paths can contain any number of red (domestic) and orange (no value chain) linkages provided there is at least one black (cross-border) linkage along their path. The first element ( ˆ vC (I−A D )−1�e i ⊗�e i Tˆ x−1  (I−A)−1−(I−A D )−1 ˆ f ) covers the downstream τ DVC i=ˆ vCAD(I−AD) − 1�ei⊗�eiTˆ x − 1 ˆ f+ˆ vC�ei⊗�eiTˆ x − 1AD(I−AD) − 1 ˆ f +ˆ vCAD ( I − AD )−1� ei ⊗� ei Tˆ x −1 AD ( I − AD )−1ˆ f . τGVC i=ˆ vC(I−AD) − 1�ei⊗�ei T ˆ x − 1(I−A) − 1−(I−AD) − 1 ˆ f +ˆ vC(I−A)−1−(I−AD)−1�ei⊗�eiTˆ x−1(I−AD)−1ˆ f +ˆ v C (I−A)−1−(I−A D )−1)�e i ⊗�e i Tˆ x−1  (I−A)−1−(I−A D )−1  ˆ f . 23 Details of the disaggregation are given in Appendix B. Page 21 of 37 Knezetal. Economic Structures (2021) 10:13 domestic and no value chain paths, which have global upstream linkages (simple or complex), the second element ( ˆ vC (I−A) − 1−(I−A D ) − 1  �e i ⊗�e iT ˆ x − 1(I−A D ) − 1 ˆ f ) covers downstream global linkages (simple or complex), which have a upstream domestic or no value chain path and the third element ( ˆ vC (I−A)−1−(I−A D )−1  �e i ⊗�e i Tˆ x−1  (I−A)−1−(I−A D )−1 ˆ f ) covers the value that has global paths both upstream and downstream. All of these cases correspond to our definition of a global value chain. Definition 8.1 Simple global value chain tree τSGVC i The simple global value chain tree represents all paths of the output of each countrysector that are part of simple global value chains as defined by 5.1 The first element ( ˆ vC(I − AD) −1� ei ⊗� ei Tˆ x −1 (I − AD) −1 ACB(I − AD) −1 ˆ f ) covers a downstream domestic and no value chain path that has simple global upstream linkages and the second element ( ˆ vC ( I − AD )−1 ACB ( I − AD )−1� ei ⊗� ei Tˆ x −1( I − AD )−1 ˆ f ) covers downstream simple global linkages that have an upstream domestic or no value chain path. These are the only cases that fit our definition of a simple global value chain. A value chain path covering both downstream and upstream simple global linkages already has more than 1 cross-border transaction and is hence part of a complex global value chain. Definition 8.2 Complex global value chain tree τCGVC i The complex global value chain tree represents all paths of the output of individual country-sectors that form part of complex global value chains as defined in 5.2 The first element ( ˆ vC (I−A D ) − 1�e i ⊗�e iT ˆ x − 1  (I−A) − 1−(I−A D ) − 1−(I−A D ) − 1A CB (I−A D ) − 1 ˆ f ) covers the downstream domestic and no value chain path, having complex global upstream linkages, the second element ( ˆ v C  (I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1  �ei⊗�eiTˆ x−1(I−AD)−1 ˆ f ) comprises downstream complex global linkages, which have an upstream domestic or no value chain path, and the third element ( ˆ vC (I−A) − 1−(I−A D ) − 1  �e i ⊗�e iT ˆ x − 1  (I−A) − 1−(I−A D ) − 1 ˆ f ) represents combinations of global downstream and upstream paths (simple-simple, simple-complex, complex-simple, complex-complex). All of these elements meet our definition of a complex global value chain because the value in all cases crosses borders for production at least twice. Definition 9 No value chain tree τNVC i τ SGVC i=ˆ vC(I−AD) − 1�ei⊗�ei T ˆ x − 1(I−AD) − 1ACB(I−AD) − 1 ˆ f +ˆ vC ( I − AD )−1 ACB ( I − AD )−1� ei ⊗� ei Tˆ x −1( I − AD )−1ˆ f. τCGVC i=ˆ vC(I−AD) −1 �ei⊗�ei T ˆ x −1 (I−A) −1 −(I−AD) −1 −(I−AD) −1 ACB(I−AD) −1  ˆ f +ˆ vC(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ x−1(I−AD)−1ˆ f +ˆ v C (I−A)−1−(I−A D )−1  �e i ⊗�e i Tˆ x−1  (I−A)−1−(I−A D )−1  ˆ f. τNVC i =ˆ v C �e i ⊗�e iT ˆ x −1ˆ f . Page 22 of 37 Knezetal. Economic Structures (2021) 10:13 A no value chain tree represents that part of the output of each country-sector which is not part of a value chain according to Definition 6. In Fig.1, a no value chain path is represented by the orange colour only (any other linkage represents a value chain path). Solely the share of value added produced in the respective country-sector in focus (no downstream stages) and also completed for final consumption (no upstream stages) in the same production phase satisfies this criterion. Since the I–O method distinguishes between a product used as an intermediate product within the same sector24 and the product manufactured for final consumption, the use of this definition as no value chain does not depend on the level of detail of I–O data disaggregation. The cyclical effect of the production of intermediate goods within the same country-sector is already included in the domestic value chain tree and, after taking into account all of the defined value chain paths (domestic, simple and complex global value chain paths), a value share remains without a value chain path and with a simple representation as the value added of the country-sector which is also directly consumed. This represents a value that has no path in terms of transactions that represent the fragmentation of production. This concludes the value chain tree decomposition, which can be written as: 3.2.4 The value chain participation rates In Sect.3.1, we showed that a set of value chain tree matrices τi represents all possible value chain paths of the output of each country-sector and that the summation along all shares of total output assigned to all such unique value chain paths yields a unity for each value chain tree (Eq.3.14). Namely, we presented a unique disaggregation of the output of each country-sector along all of its value chain paths. In the same way, the summation along the two disaggregating dimensions of our decomposed set of matrices (global, domestic and no value chain tree matrices) captures the overall share of the total output of each countrysector i that meets the criteria by which the value chain paths were decomposed by including either only domestic value chain paths, only global value chain paths, or only values that have no value chain paths at all. In other words, the summation of the disaggregated value chain matrices along any origin and end stage represents the share of output of each country-sector that has either a domestic, a global or a no value chain. Definition 10 Domestic value chain share DVCs DVCs ∈IR n ; DVCsi= n j = 1n k = 1 t DVC ijk ; DVCs =      1 T τ DVC 11 1TτDVC 21 . . . 1TτDVC n 1      . (3.16) τi =τ DVC i +τ GVC i +τ NVC i, (3.17) τGVC i =τ SGVC i +τ CGVC i. 24 This is determined by the pure diagonal elements of the Leontief technical matrix A. Each aii represents the portion of the total product of the ith country-sector that requires the use of the intermediate product of the same countrysector in the production process, thereby covering cyclical transactions within a sector. These cyclical transactions are of course included in the decomposition of the domestic value chain and not the no value chain since cyclical transactions represent the fragmentation of a domestic value chain. Page 23 of 37 Knezetal. Economic Structures (2021) 10:13 Domestic value chain share represents the share of each country-sector’s output that has a domestic value chain path. Definition 11 Global value chain share GVCs GVCs ∈IR n ; GVCs i= n j=1 n k=1t GVC ijk ; GVCs =      1 T τ GVC 11 1TτGVC 21 . . . 1TτGVC n 1      . Global value chain share represents the share of each country-sector’s output that has a global value chain path. Definition 11.1 Simple global value chain share SGVCs SGVCs ∈IR n ; SGVCs i= n j=1 n k=1t SGVC ijk ; SGVCs =      1 T τ SGVC 11 1TτSGVC 21 . . . 1TτSGVC n 1      . Simple global value chain share represents the share of each country-sector’s output that has a simple global value chain path. Definition 11.2 Complex global value chain share CGVCs CGVCs ∈IR n ; CGVCs i= n j =1 n k =1t CGVC ijk ; CGVCs =      1 T τ CGVC 11 1TτCGVC 21 . . . 1T τ CGVC n 1      . Complex global value chain share represents the share of each country-sector’s output that has a complex global value chain path. Definition 12 No value chain share NVCs NVCs ∈IR n ; NVCsi= n j =1 n k =1t NVC ijk ; NVCs =      1 T τ NVC 11 1TτNVC 21 . . . 1TτNVC n 1      . A no value chain share represents the share of each country-sector’s output that has a no value chain path. (3.18) DVCs i+GVCsi+NVCsi= n  j = 1 n  k = 1 tDVC ijk +tGVC ijk +tNVC ijk  = n  j = 1 n  k = 1 tijk = 1, (3.19) SGVCs i+CGVCsi= n  j =1 n  k = 1 tSGVC ijk +tCGVC ijk  = n  j =1 n  k = 1 tGVC ijk =GVCsi . Page 24 of 37 Knezetal. Economic Structures (2021) 10:13 With this, we conclude our disaggregation of each country-sector’s total output with respect to its specific value chain integration based on production-sharing linkages. We can summarise our decomposition in the simple vector form: 3.2.5 Decomposition ofthetransaction tothefinal consumer Since all value chain paths within production are covered and decomposed, we still have one last transaction to the consumer to complete the value chain path from production to consumption. We can decompose the final transaction to the consumer upon the criterion of whether it is a transaction to domestic consumers or a cross-border transaction (export of the final product for consumption). Domestic consumption here refers to the countrysector in which the last stage of production took place and not the country-sector whose value chain we are analysing. Each country-sector has a unique value chain and a specific structure of value chain paths. The completion of each value chain path by a transaction to the consumer can be achieved by an additional cross-border transaction of exporting the final product or consumption in the country where the product was finalised. Such a further decomposition of the value chain paths allows a more detailed analysis of the value chains. The I–O data include information on the transaction to final consumers within matrix F, which can be decomposed into its cross-border and domestic flows to final consumers ( F=FCB +FD ) due to its block vector structure. We construct a matrix of all cross-border final consumption flows and a matrix of all domestic consumption flows: Every value chain path within production can thus be further decomposed with an additional criterion of a transaction to final consumers. Each set of disaggregated value chain matrices, defined by Eqs.3.16 and 3.17, can be separated on two matrices, one covering all of the production paths that end in domestic final consumption (no export - τNE i ) and the other all of the production value chain paths that end with exporting for final consumption ( τE i ). Due to their simple additive properties of operation, all of the decomposed value chain tree matrices are similarly decomposed to ones with exporting or with no exporting as the final transaction. (3.20) DVCs +GVCs +NVCs =1, (3.21) GVCs =SGVCs +CGVCs. (3.22) ˆ f = ˆ fD + ˆ fCB. (3.23) τi =ˆ vC(I − A)−1 � ei ⊗� eiT ˆ x−1(I − A)−1ˆ fD +ˆ vC(I − A)−1 � ei ⊗� eiT ˆ x−1(I − A)−1ˆ fCB = τ NE i + τ E i (3.24) τi =τ NE i +τ E i =τ GVC−NE i +τ DVC−NE i +τ NVC−NE i +τ GVC−E i +τ DVC−E i +τ NVC−E i (3.25) τGVC i =τ GVC−NE i +τ GVC−E i =τ SGVC−NE i +τ CGVC−NE i +τ SGVC−E i +τ CGVC−E i Page 31 of 37 Knezetal. Economic Structures (2021) 10:13 of production, that is chiefly correlated with growth, irrespective of its global or domestic nature. Accordingly, the proposed measure and the new typology of value chains, in particular the novel conceptualisation of domestic value chain fragmentation, could bring Fig. 7 USA manufacturing participation rates. creditSource: WIOD 2016; own calculations Table 1 Regression results Source: WIOD, 2016; WB; own calculations * p < 0.05 , ** p < 0.01 , *** p < 0.001 (1) (2) (3) Yearly growth Yearly growth Yearly growth logGDP − 0.013*** − 0.013*** − 0.013*** (0.00) (0.00) (0.00) DVC share 0.169*** 0.181*** 0.183*** (0.03) (0.04) (0.04) GVC share 0.163*** 0.160*** 0.162*** (0.03) (0.03) (0.03) logPOP − 0.001 − 0.001 (0.00) (0.00) EU − 0.003 (0.00) Constant 0.036 0.049 0.055 (0.03) (0.03) (0.04) R2 0.819 0.821 0.824 F57.126 42.314 33.806 Page 32 of 37 Knezetal. Economic Structures (2021) 10:13 to light important information that has been concealed in the existing typology, which conceptualises the domestic component only as a negation of the global value chain and thus did not allow research with explicit questions concerning domestic integration. The complex development of globalisation in recent decades and the shifts of late towards the localisation and regionalisation of economic integration caused by political, economic and external factors make this new approach increasingly relevant. The proposed measure, particularly in conjunction with data from other sources, could further deepen the theoretical discussion and empirical investigations. In conclusion, we believe that our new methodological approach and the new extended typology of value chains associated with it provide fertile grounds for obtaining deeper insights into different types of value chains as well as a broader set of tools of use for various extensions of research. Appendix A: Notations nS∈IN Number of sectors. nC∈IN Number of countries. n∈IN ; n=nS∗nC Number of country-sectors. 1∈IR n Vector of ones. � 1∈IR n C vector of ones. � ei ∈ IR n ; eij=δij Standard orthonormal basis of IR n . I∈IR n×n Identity matrix. x∈IR n Total output vector. ˆ x∈IR n×n ; ˆ x=diag(x) Total output matrix. C∈IR n×n Intermediate consumption matrix. F∈IR n×nC Final consumption matrix on the country level.27 f∈IR n ; f = F� 1 Total final consumption vector. ˆ f ∈IR n× n ; ˆ f = diag ( f) Total final consumption matrix. A∈IR n×n ; A=Cˆ x−1 Leontief technical coefficient matrix. G∈IR n×n ; G=ˆ x−1C Ghosh technical coefficient matrix. v∈IR n ; vT=xT− 1 TC= 1 ( ˆ x−A ˆ x)= 1 T(I−A)ˆ x Vector of total value added. ˆ v∈IR n×n ; ˆ v=diag(v) Total value-added matrix. vC∈IR n ; vT C =v T ˆ x −1 =1 T (I−A ) Vector of value-added coefficients – value-added share in total output. ˆ vC∈IR n×n ; ˆ vC=diag ( vC) Value-added coefficients matrix. C, A and G have a block-matrix structure IR (n S ×n S )×(n C ×n C ) , while F has a block vector structure IR n S ×(n C ×n C ) . Diagonal block elements with respect to countries represent domestic intermediate transfers and domestic consumption and off diagonal block elements represent transactions that cross a border either for intermediate use or final consumption. 27 In the international I–O framework, F is usually disaggregated on the country level as well as in an additional dimension of final consumption (household, government and non-profit consumption, fixed capital formation and changes in inventories), which in our derivation is irrelevant and left out. Disaggregation by countries is relevant for enabling the separation of domestic final consumption and export. Page 33 of 37 Knezetal. Economic Structures (2021) 10:13 C=CCB +CD A=ACB +AD G=GCB +GD F=FCB +FD fCB ∈IR n ; fCB = FCB� 1 Total final consumption by exporting. fD∈ IR n ; fD = FD� 1 Total final consumption by domestic transactions. ˆ fCB ∈IR n×n ; ˆ fCB = diag(fCB) Total final consumption by exporting matrix. ˆ fD ∈ IR n ×n ; ˆ fD = diag(fD) Total final consumption by domestic transactions matrix. Appendix B: τi decomposition τ i=ˆ vC(I−A) −1 �ei⊗�ei T ˆ x −1 (I−A) −1ˆ f =ˆ vCI+AD(I−AD)−1+(I−AD)−1ACB(I−AD)−1 +(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1�ei ⊗�eiTˆ x−1I+AD(I−AD)−1+(I−AD)−1ACB(I−AD)−1 +(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1ˆ f =ˆ vC�ei⊗�eiTˆ x−1ˆ f+ˆ vCAD(I−AD)−1�ei⊗�eiTˆ x−1ˆ f+ˆ vC�ei⊗�eiTˆ x−1AD(I−AD)−1ˆ f +ˆ vCAD(I−AD)−1�ei⊗�eiTˆ x−1AD(I−AD)−1ˆ f+ˆ vC�ei⊗�eiTˆ x−1(I−AD)−1ACB(I−AD)−1ˆ f +ˆ vCAD(I−AD)−1�ei⊗�eiTˆ x−1(I−AD)−1ACB(I−AD)−1ˆ f+ˆ vC(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ x−1ˆ f +ˆ vC(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ x−1AD(I−AD)−1ˆ f +ˆ vC(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ x−1(I−AD)−1ACB(I−AD)−1ˆ f +ˆ vC�ei⊗�eiTˆ x−1(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1ˆ f +ˆ vCAD(I−AD)−1�ei⊗�eiTˆ x−1(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1ˆ f +ˆ vC(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ x−1(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1ˆ f +ˆ vC(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ x−1ˆ f +ˆ vC(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ x−1AD(I−AD)−1ˆ f +ˆ vC(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ x−1(I−AD)−1ACB(I−AD)−1ˆ f +ˆ vC(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1�ei ⊗�eiTˆ x−1(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1ˆ f =ˆ vC�ei⊗�eiTˆ x−1ˆ f+ˆ vCAD(I−AD)−1�ei⊗�eiTˆ x−1ˆ f+ˆ vC�ei⊗�eiTˆ x−1AD(I−AD)−1ˆ f +ˆ vCAD(I−AD)−1�ei⊗�eiTˆ x−1AD(I−AD)−1ˆ f+ˆ vC(I−AD)−1�ei⊗�eiTˆ x−1(I−A)−1−(I−AD)−1ˆ f +ˆ vC(I−A)−1−(I−AD)−1�ei⊗�eiTˆ x−1(I−AD)−1ˆ f +ˆ vC(I−A)−1−(I−AD)−1�ei⊗�eiTˆ x−1(I−A)−1−(I−AD)−1ˆ f =τNVC i +τDVC i +τGVC i =τ i . Page 34 of 37 Knezetal. Economic Structures (2021) 10:13 Appendix C We make a demonstration of the methodology on a simple 2 sector 2 countries numerical example.28 This simple case of international economy has following intermediate consumption matrix and final demand: Total output is the sum of all the intermediate and final demand: Calculation of value added coefficients and Leontief technical coefficients: We continue with separate upstream and downstream decompositions, W and Z: τ GVC i=ˆ vC(I−AD) − 1�ei⊗�eiTˆ x − 1(I−A) − 1−(I−AD) − 1 ˆ f +ˆ vC(I−A)−1−(I−AD)−1�ei⊗�eiTˆ x−1(I−AD)−1ˆ f +ˆ vC(I−A)−1−(I−AD)−1�ei⊗�eiTˆ x−1(I−A)−1−(I−AD)−1ˆ f =ˆ vC(I−AD)−1�ei⊗�eiTˆ x−1(I−AD)−1ACB(I−AD)−1ˆ f +ˆ vC(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ x−1(I−AD)−1ˆ f +ˆ vC(I−AD)−1�ei⊗�eiTˆ x−1(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1ˆ f +ˆ vC(I−A)−1−(I−AD)−1−(I−AD)−1ACB(I−AD)−1�ei⊗�eiTˆ x−1(I−AD)−1ˆ f +ˆ vC(I−A)−1−(I−AD)−1�ei⊗�eiTˆ x−1(I−A)−1−(I−AD)−1ˆ f =τSGVC i +τCGVC i =τGVC i . C =    2112 0.5 0.33 0.33 0.33 0.5 0.5 0.167 0.33 1 1.5 2 2    f=    4 2 3 5    . x =C1+fx=    10 3.5 4.5 11.5    . v T C=1 T (I−A)=�0.6 0.0476 0.2222 0.5942� , A=Cˆ x−1=   0.2 0.2857 0.2222 0.1739 0.05 0.09523 0.0740 0.0289 0.05 0.1428 0.0370 0.0289 0.1 0.4285 0.4444 0.1739   , ADom =   0.2 0.2857 0 0 0.05 0.09523 0 0 0 0 0.0370 0.0289 0 0 0.4444 0.1739    ACB =   0 0 0.2222 0.1739 0 0 0.0740 0.0289 0.05 0.1428 0 0 0.1 0.4285 0 0    . 28 The decimal numbers are truncated on the fourth digit. Page 35 of 37 Knezetal. Economic Structures (2021) 10:13 Value chain tree matrices are calculated for each country-sector in the following manner: For each type of value chain (DVC, GVC, NVC,...) we have 4 matrices, each covering all the value chain paths of each country-sector (we have 4 in our example) that conform to our value chain criteria. W =ˆ x−1(I−A)−1ˆ f=    0.5461 0.1337 0.1555 0.1645 0.1044 0.6788 0.1224 0.0942 0.0820 0.1047 0.7391 0.0740 0.09126 0.1433 0.1913 0.5740    , Z=ˆ vC(I−A)−1=   0.8192 0.4013 0.3110 0.1974 0.0043 0.0565 0.0068 0.0031 0.0205 0.0523 0.2463 0.0148 0.1559 0.4896 0.4357 0.7845    . τi=Z�ei⊗�ei T W τ 1=   0.4474 0.1095 0.1274 0.1348 0.0023 0.0005 0.0006 0.0007 0.0112 0.0027 0.0031 0.0033 0.0851 0.0208 0.0242 0.0256   ,τ2=   0.0419 0.2724 0.0491 0.0378 0.0059 0.0383 0.0069 0.0053 0.0054 0.03556 0.0064 0.0049 0.0511 0.3324 0.0599 0.0461   , τ 3=   0.0255 0.0325 0.2299 0.0230 0.0005 0.0007 0.0050 0.0005 0.0202 0.0258 0.1820 0.0182 0.0357 0.0456 0.3220 0.0322    ,τ4=   0.0180 0.0283 0.0377 0.1133 0.0002 0.0004 0.0006 0.0018 0.0013 0.0021 0.0028 0.0084 0.0716 0.1124 0.1501 0.4504   . τ DVC 1=    0.1502 0.0616 0 0 0.0017 0.0002 0 0 0 0 00 0 0 00    τDVC 2=    0.0194 0.1556 0 0 0.0043 0.0073 0 0 0 0 00 0 0 00    τ DVC 3=    00 0 0 00 0 0 0 0 0.0169 0.0096 0 0 0.2374 0.0138    τDVC 4=    00 0 0 00 0 0 0 0 0.0012 0.0044 0 0 0.1083 0.1327    τ GVC 1=    0.0571 0.0479 0.1274 0.1348 0.0006 0.0003 0.0006 0.0007 0.0112 0.0027 0.0031 0.0033 0.0851 0.0208 0.0242 0.0256    τGVC 2=    0.0224 0.1167 0.0491 0.0378 0.0015 90.0038 0.0069 0.0053 0.0054 0.0355 0.0064 0.0049 0.0511 0.3324 0.0599 0.0461    τ GVC 3=    0.0255 0.0325 0.2299 0.0230 0.0005 0.0007 0.0050 0.0005 0.0202 0.0258 0.0170 0.0085 0.0357 0.0456 0.0846 0.0183    τGVC 4=    0.0180 0.0283 0.0377 0.1133 0.0002 0.0004 0.0006 0.0018 0.0013 0.0021 0.0016 0.0040 0.0716 0.1124 0.0417 0.0592    τ NVC 1=    0.24 0 0 0 0 000 0 000 0 000    τNVC 1=    0000 0 0.0272 0 0 0000 0000    τ NVC 3=    00 0 0 00 0 0 0 0 0.1481 0 00 0 0    τNVC 4=    000 0 000 0 000 0 0 0 0 0.2583    Page 36 of 37 Knezetal. Economic Structures (2021) 10:13 The value chain participation shares are obtained by summation of all elements of the value chain tree matrices: Acknowledgements The authors thank the editor and all reviewers for their comments and suggestions that helped improve this article. Authors’ Contributions KK contributed the methodological derivations and empirical results, all three authors contributed to the literature review, the discussion of the results and extensive proofreading. Funding The authors of this article acknowledge the financial support received from the Slovenian Research Agency (research core funding No. P5-0177 and No. 52075). Availability of data and materials The datasets analysed during the current study are available at http:// www. wiod. org. Declarations Ethics approval and consent to participate Not applicable. Consent for publication Not applicable. Competing interests The authors declare that they have no competing interests. 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