Existence and uniqueness of solutions for Leontief's Input–Output Model, graph theory and sensitivity analysis

Abstract

We provide a complete study of existence and uniqueness (uniqueness up to multiples in the case d = 0 ) of non-negative and non-trivial solutions x for the linear system ( I − A ) x = d with A ≥ 0 , d ≥ 0 (which, in particular, applies to Leontief's Input–Output Model). This study is done in terms of the block triangular form of the matrix A and is related to a directed graph associated to A . In particular, this study of existence and uniqueness up to multiples of solutions provides a framework that allows us to rigorously perform a sensitivity analysis of the normalized solutions of the linear system ( I − A ) x = d .