The Properties Of The Coefficient Matrices Of The Input-output Model

This paper mainly focuses on the properties of the coefficient matrices of the input-output model. As a very important model in economic research, the properties of the coefficient matrices in input-output model gain more and more attentions. This paper mainly discusses the properties of the coefficient matrices of the input-output model. At first, some basic properties of the input-output matrices are listed and a brief introduction to some wonderful results done by former researchers is given. After that we mainly discuss the properties of input-output matrices when the direct consumption matrix A is in some special kinds. In this paper, four kinds of special matrices will be considered. They are symmetric matrices, arrow-like matrices, Toeplitz matrices and Jacobi matrices.It comes out that the spectral radius of the directly consumption matrix A is no bigger than one, and that the Leontief inverse matrix B is positive definite when A is symmetric. Moreover, if A is a symmetric arrow-like matrix we have a more direct estimation for the eigenvalues of A and B. Thus a simple way to judge whether A is positive definite or not under this circumstance is presented. In addition, an expression for the Cholesky decomposition of matrix I - A, which is very efficient for solving the model, is demonstrated afterwards. Then it turns to discuss other two different circumstances when the direct consumption matrix A is Toeplitz matrices and Jacobi matrices respectively. The properties of the eigenvalue of matrices A and B are covered. A simplified Trench-Zohar algorithm is given to calculate the Leontief inverse matrix B when A is a Toeplitz matrix.At last, some numerical examples are given to validate the theoretical results. The numerical results further prove the above discussion about the properties of the Input-Output coefficient matrices. It verifies that both the formula to do Cholesky decomposition with I - A when A is a symmetric arrow-like matrix and the simplified Trench-Zohar algorithm to get the Leontief inverse matrix B when A is a symmetric Toeplitz matrix are validated.