Some applications of spectral theory of nonnegative matrices to input–output models

Abstract

We show the necessary and sufficient condition that a nonnegative matrix has a unique positive eigenvector, where the analytic expression displaying the linear relations between each remnant component and a basic characteristic subvector of the unique eigenvector is discovered when the nonnegative matrix is reducible. As a result, we infer the exact necessary and sufficient condition that the iteration matrix M −1N as a special nonnegative matrix has a unique positive eigenvector when M−N is an M-splitting, which is applied to the condition for the existence and uniqueness of a balanced growth solution for the Leontief dynamic input–output model. Previous work in the field did not clearly involve the uniqueness of the balanced growth solution. In this paper we develop the prior results. That is, we find the necessary and sufficient condition that the Leontief dynamic input–output model has a unique bal-anced growth solution. Finally, we obtain the necessary and sufficient condition for theexistence and uniqueness of both the balanced growth solution and the production prices system.