Blending Positive Matrix Pencils with Economic Models

Abstract

We consider E(x)x(k + 1) = F(k)x(k), $k \in Z^+_0$ , where E(k) and F(k) are real square matrices of order n, not necessarily invertible. Assuming the regularity of the matrix pencils $\lambda E(x) - F(k)x(k), k \in Z^+_0$ and the existence of a nonzero common eigenvector of the family of n × n real matrices { Ê $(k) = [\alpha_kE(k) - F(k)]^{-1} E(k), \alpha_k \in R, k \in Z^+_0$}, we will obtain a solution to the above descriptor system. We also analyse a particular case related with a positive equation — the closed dynamic Leontief model.