Input - output cross analysis: a theoretical account

In this paper we deal with extractions of one sector (region) or a number of sectors (regions) from an (interregional) input - output system. When a certain number of sectors is taken out of the input - output matrix one obtains a cross. It is shown that the Leontief-inverse for a cross can be decomposed into the product of three matrices, OUT*INTRA*IN, where INTRA represents all intracross economic interactions, IN represents all effects from the rest of economy upon the cross, and OUT represents all effects from the cross upon the rest of economy. Furthermore, we present a general scheme of additive as well as multiplicative decompositions of the Leontief-inverse, reflecting the hierarchical decomposition of the matrix of input coefficients into the sum of crosses. These decompositions provide us with the means to find new insights into efficient aggregation, importance of regions, and issues of industrial complexes in inter alia input - output analysis.