A Note on the Comparison of Two Rows (Columns) of the Inverse of an M-matrix; An Elaboration of Metzler's Mehod
YOSHIO KIMURA, Keiko Nakayama · Studies in Regional Science · 2002
IntroductionMetzler (1951) investigated the effects of taxes and subsidies on prices in an input-output sys tem (henceforth, Metzler's problem, for short).Later, Atsumi (1981) extended and elaborated Metzler (1951).However, a close scrutiny of Metzler's problem reveals that the mathematical essence of the problem lies in the comparison of two rows of the inverse of an M-matrix (defined later).To this point, the former approached by the manipulation on determinants, while the latter by his new elementary method based directly on the fact that the Leontief inverse equals the sum of an identity matrix and the product of the input-output coefficients matrix and the Leontief inverse.Speaking from another point of view, the plentiful contents of Atsumi's mathematical result (Atsumi (1981; THEOREM, pp.33-34)) rely widely on the special form of Leontief matrix (an identity matrix-a nonnegative square matrix) and Leontief matrix is a special species of M-matrices.On the other hand, Metzler's analytical method is of wider applicability than Atsumi's.Therefore, it would be of some value to inspect to what extent Atsumi's theorem remains valid for M-matrices in general, along the line of Metzler's thought.