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

We provide a complete study of existence and uniqueness (uniqueness up to multiples in the case $\mathbf{d} = \mathbf{0}$) of non-negative and non-trivial solutions $\mathbf{x}$ for the linear system $(\mathbf{I}- \mathbf{A})\mathbf{x} = \mathbf{d}$ with $\mathbf{A} \geq \mathbf{0}, \mathbf{d} \geq 0$ (which, in particular, applies to Leontief's Input Output Model). This study is done in terms of the block triangular form of the matrix $\mathbf{A}$ and is related to a directed graph associated to $\mathbf{A}$. 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 $(\mathbf{I} - \mathbf{A})\mathbf{x} = \mathbf{d}$.