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A More Powerful Method for Triangularizing Input-Output Matrices and the Similarity of Production Structures

Yukio Fukui

Abstract

Studies in the production structure of any national economy, as revealed in its input-output table, have been closely related to the question of the interindustrial dependence or hierarchical structures of productive sector leading from primary to final production. In particular, the notion of hierarchy provides a useful tool when one wants to draw inferences on the structural change or international difference of industrial structures. The standard technique for studying this notion is to triangularize the input-output table by interchanging sectors in order to maximize the entries below the main diagonal. In a perfect triangularized table, the entries above the main diagonal should be zero. For example, such a strong one-way interdependence relation as cotton-textiles-clothing can easily establish the hierarchy. Due to the existence of circular relations like coal-steel-mining equipment-coal, it is not possible to triangularize the input-output table perfectly. This paper extends and revises the previous triangulation method which is based on a permutation theorem deriving from the interchange of adjoining two industrial groups, into that among vhree industrial groups (Theorem 2). The new algorithm based on Theorem 2 is demonstrated by actually computing the suboptimal orderings for the four input-output tables for such more developed countries (MDC's) as the United States, Italy, Norway, Japan, and the two tables for such less developed countries (LDC's) as India and Korea. The empirical results suggest that sectors can be arranged in a similar hierarchical order among MDC's and LDC's. Transport Equipment, Machinery, Apparel, Leather and Products, Grain Mill Products, and Processed Foods, which link directly to the final demand, are recorded as higher-order sectors. Trade, Transport, and services are lower-order ones, and Energy sectors such as Electric Power, Coal Products, Coal Mining, Petroleum, Petroleum Products and Natural Gas are the lowest. Consequently, this paper provides some more evidence in support of the similarity of hierarchical structures of production among these countries.