A Property of the Leontief Inverse and its Applications to Comparative Static Analysis

We arbitrarily choose a submatrix of order 2 in the Leontief inverse matrix B . If a main diagonal element in the submatrix is a main diagonal element in B , then the determinant of the submatrix is non-negative; in particular, if the two main diagonal elements in the submatrix are all the main diagonal elements in B , then the determinant of the submatrix is greater than or equal to 1. This property can be used in the comparative static analysis for the input-output model. For example, when the change only occurs in one sectoral final demand or one corresponding row of the input coefficient matrix, which leads to a change of this sectoral gross output, the equivalent conditions that the absolute value of the rate of change of this sectoral gross output is larger than or equal to all other sectoral change rates and larger than at least one sectoral rate of change are shown. It is impossible to derive the exact results for the comparative static analysis if the property is not applied. A previous fault confirms this point.