A New Matrix Theorem: Interpretation in Terms of Internal Trade Structure and Implications for Dynamic Systems

Economic systems often are described in matrix form as x = Mx. We present a new theorem for systems of this type where M is square, nonnegative and indecomposable. The theorem discloses the existence of additional economic relations that have not been discussed in the literature up to now, and gives further insight in the economic processes described by these systems. As examples of the relevance of the theorem we focus on static and dynamic closed Input-Output (I-O) models. We show that the theorem is directly relevant for I-O models formulated in terms of difference or differential equations. In the special case of the dynamic Leontief model the system’s behavior is shown to depend on the properties of matrix M = A + C where A and C are the matrices of intermediate and capital coefficients, respectively. In this case, C is small relative to A and a perturbation result can be employed which leads directly to a statement on the system’s eigenvalues. This immediately suggests a solution to the well-known problem of the instability of the dynamic Leontief model.