With the launching of this journal, the Pan-Pacific Association of Input-Output Studies seeks to expand its influence to a global community of scholars and to extend its scope beyond what it calls (in the journal brochure) ‘classical input-output economics’ by encouraging new approaches to understanding and analyzing relationships among the components of contemporary economic systems. Global financial crisis, urgent environmental challenges, and widespread social unrest are compelling reasons not only to refine our analytic frameworks but also explicitly to hone them for addressing specific critical problems, to collaborate across disciplines for inventing solution concepts to deal with them, and to put our databases, methods, and models to work for the express purpose of evaluating realistic and potentially promising, actionable alternatives. Not surprisingly, there is now a resurgence of research on compiling input-output databases and creating input-output models of the world economy for undertaking the analysis of such scenarios.
Leontief et al. () created the first input-output model of the world economy as well as the database to which it was applied. Today, there are freestanding input-output databases of the world economy that are available for use with a variety of models, including not only input-output models but also general equilibrium models. Several input-output databases that cover the entire world economy at substantial levels of regional and sectoral detail are in preparation; they include EXIOBASE (Tukker et al. ), Eora (Kanemoto et al. ), WIOD (WIOD ), and a widely-used one of earlier vintage, GTAP (Narayanan and Walmsley ). Among the input-output models, there is a convergence of regional and world modeling approaches and, especially, a sharing of common databases. Regional input-output analysis has, for decades, focused on multiplier analysis at the smaller than national level, the multipliers being derived from the inverse of a square matrix. By contrast, the input-output models of the world economy have been based on national spatial units and, while these models also used square matrices, it was for the purpose of deriving quantities of output and trade flows and changes in prices under alternative scenarios about the future rather than calculating multipliers. The emerging convergence is due in large measure to the common interest today in exchanges linking a variety of spatial units, such as river basins or urban areas and their hinterlands, in the context of both intra-country and multi-country regions.
This article proposes a solution to a particular problem that became apparent in the course of applying the World Trade Model (Duchin ), a linear programming input-output model of the world economy based on comparative advantage subject to resource constraints, to a new environmentally extended input-output database of the economy compiled from many sources at an unprecedented level of detail (EXIOBASE). While the problem is more general, it is readily explained by considering only the square (or ‘symmetric’) input-output tables representing the economies of the diverse countries and regions of the world. A typical requirement for databases of the world economy is that all tables have the same number n of sectors and that the sectors be defined in comparable ways. Comparability is required to assure the consistency between quantities of goods produced in and exported from the economies of origin and the corresponding imports into the destination economies. The implicit underlying assumption is that every region produces all n outputs, and each output is produced in a given region using a single, average input structure. Thus, if n sectors are to be included in the database, every region will be represented by an input-output matrix, A, with n columns and n rows. However, it is not evident what to put in those columns and rows if the original input-output table for this region does not identify the corresponding sectors, either because their products are not produced and possibly not even consumed in that region or because they were aggregated in that region’s table with other sectors.
In the database in question, we encountered columns and rows of 0 in these cases. This treatment works as a passive placeholder from a descriptive point of view in that it maintains the formal structure of n rows and n columns, and it appears unproblematic in that it does not affect row totals or column totals in the input-output flow table. However, it poses a problem for scenario analysis as a column of coefficients that are all zeroes makes the region appear to have a comparative advantage (because of very low, in fact zero, input costs) in precisely those sectors where it obviously does not. As a temporary fix, we replaced the column of coefficients by equally artificial, large entries (in contrast to the artificially low ones) to be sure the model did not interpret the region as a low-cost producer. This paper proposes a much more precise and parsimonious representation than the trick of fictive columns with extreme values and one that turns out to offer other advantages as well.
We make two assertions about a database with more than a handful of sectors, and contemporary world input-output databases aim at many dozens if not hundreds of sectors. First, at least some of the regions will not be equipped to produce some products, and second, for some sectors and regions, it is important to distinguish alternative technologies that both compete with each other for limited resources and may be simultaneously in use. Examples of the former are the extraction of petroleum or rare earth metals, which are obviously concentrated in certain regions and cannot, for physical reasons, be extracted in others even though the resource commodity may be imported and used there. Examples of multiple, potentially simultaneously utilized technological options that need to be distinguished include grain production on rainfed and on irrigated land or the generation of electric power using coal and nuclear fuels, since the options differ in virtually all attributes (namely cost structure, resource requirements, and environmental impact) except that they produce indistinguishable outputs. Naturally, all sectors whose outputs are tradable require that the sectoral output be defined similarly (in terms of a representative output or product mix) in all regions. In what follows, we assume a world economy with n sectors producing potentially tradable goods and services in some, but not necessarily all regions.
In an earlier paper, we introduced the rectangular choice-of-technology (RCOT) model (Duchin and Levine ) for allowing several choices of input-cost structures in a given sector and demonstrated its use both in a model of a single region and in the context of a world economy consisting of several regions. RCOT is a linear program with both a primal quantity model that solves for output, trade flows, and factor use, and a dual model that determines product prices and scarcity rents on fully utilized factors of production. This RCOT model assumed the same n sectors in all regions but allowed any number of alternative input structures for each sector in a given region. The RCOT model replaces the familiar square matrices A (the matrix of intermediate coefficients per unit of output) and I (the identity matrix) by rectangular matrices, and . In this paper, we generalize these rectangular matrices to accommodate the important case where a given region has zero options - rather than one or more than one - for producing the product of a particular sector, that is, it is not equipped to produce that output at all. In this case, there is no information for creating a column for this sector, and we will see that a column is not needed. The region may, however, require the use of the output in question, which then it must import; if it does not use the product either, the demand quantified in the row of intermediate inputs is legitimately and unproblematically zero. That is, whether or not the product is used in the region, the row for this sector need not be suppressed. The rectangularity of the regional matrices comes from the fact that each one has exactly n rows but may have a smaller or larger number of columns. The number of columns is reduced by each sector that does not produce in that region and increased for each sector that has more than one technology option. Note that even if the rectangular matrix happens to be square (i.e., with the same number of rows and columns), this does not assure a one-to-one relationship between rows and columns as in the standard A matrix. It may mean, for example, that one sector is not present in this economy, but another sector has two technological options.
In this paper, we extend the RCOT representation to include the absence of technological options for a sector (i.e., no capacity to produce), a true generalization of the original case (Duchin and Levine ) of 1 through s technological options for a sector to the case of 0 through s options. We also fully incorporate the extended RCOT model into the World Trade Model. The extended World Trade Model (WTM/RCOT) representation has the new feature that a region may use identifiable imported inputs that it cannot produce itself, the so-called non-competitive imports, that typically are vital for an economy such as resource commodities or capital goods. While non-competitive imports are often aggregated by money value into a single category in one-region input-output tables, a WTM/RCOT analysis permits them to be explicitly identified. An economy with many non-competitive imports will generally be represented by a rectangular input-output matrix with more rows than columns.
The rest of this paper is divided into four sections. In the following sections we describe the rectangular input-output matrices, and . Next, the algebra for the rectangular World Trade Model, with rectangular sector-by-technology matrices for all regions, is developed for the primal quantity model and illustrated by a numerical example for three regions with three sectors and four factors of production. The model and numerical solutions for the dual price model are also presented. Section 3 concludes.