Matrix sensitivity, error analysis and internal/external multiregional multipliers

Sherman and Morrison (1949, 1950) provided essential insights into the properties of sensitivity and error analysis in input-output systems through procedures for specifying the effects of changes in one coefficient or changes in one row or column on the rest of the system. Sonis and Hewings (1989, 1992) extended this work and generalized it to include a whole range of changes in direct inputs using the concept of a field of influence. The present paper considers the matricial changes in which sub-matrices of the matrix of direct coefficients change simultaneously; these sub-matrices may be considered to reflect different economic substructures. In this way, the block-generalization of the Sherman-Morrison and Sonis-Hewings formulae are possible through the development and extension of Miyazawa's (1976) notion of internal and external multi-region multipliers. In this way, the matricial decompositions of the matrix of fields of influence can be established.