Looking for nonnegative solutions of a Leontief dynamic model. (English) Zbl 1044.15014

Silva, Manuela S.; de Lima, Teresa P.

Linear Algebra Appl. 364, 281-316 (2003).

In Section 2, the authors briefly review some useful terminology and results. In Section 3, using properties about nonnegative matrices they obtain sufficient conditions in order that the Leontief price model have nonnegative solutions, and then give many illustrative numerical examples. Finally, in Section 4, taking into account these results they draw some conclusions on the problem of nonnegativity of the solutions of the model.

Reviewer: T. Nono (Hiroshima)

MSC:

15B48	Positive matrices and their generalizations; cones of matrices
91B24	Microeconomic theory (price theory and economic markets)

Keywords:

descriptor systems; nonnegative matrices; rank factorization; general inverses; monotone matrix; Leontief price model; numerical examples

References:

[1]	Berman, A.; Plemmons, R. J., Inverses of nonnegative matrices, Linear Multilinear Algebra, 2, 161-172 (1974)
[2]	Berman, A.; Plemmons, R. J., Matrix group monotonicity, Proc. Am. Math. Soc., 46, 3, 355-359 (1972) · Zbl 0312.15002
[3]	Israel, A. B.; Greville, T. N.E., Generalized Inverses: Theory and Applications (1974), Wiley: Wiley New York · Zbl 0305.15001
[4]	Berman, A.; Plemmons, R. J., Nonnegative Matrices in the Mathematical Sciences (1979), New York: New York Academic Press · Zbl 0484.15016
[5]	Takayama, A., Mathematical Economics (1996), Cambridge University Press
[6]	Johnson, C. R., Inverse \(M\)-matrices, Linear Algebra Appl., 47, 195-216 (1982) · Zbl 0488.15011
[7]	Szyld, D. B., Conditions for the existence of a balanced growth solution for the Leontief dynamic input-output model, Econometrica, 53, 1411-1419 (1985) · Zbl 0585.90018
[8]	Valcher, E., Controllability and reachability criteria for discrete time positive systems, Int. J. Cont., 65, 3, 511-536 (1996) · Zbl 0873.93009
[9]	Zeng, L., Some applications of spectral theory of nonnegative matrices to input-output models, Linear Algebra Appl., 326, 205-218 (2001) · Zbl 0995.15017
[10]	Lewin, M.; Neumann, M., On the inverse \(M\)-matrix problem for (0,1)-matrices, Linear Algebra Appl., 30, 41-50 (1980) · Zbl 0434.05051
[11]	Jeter, M. W.; Pye, W. C., A note on nonnegative rank factorizations, Linear Algebra Appl., 38, 171-173 (1981) · Zbl 0485.15015
[12]	Bapat, R. B., Structure of a nonnegative regular matrix and its generalized inverses, Linear Algebra Appl., 268, 31-39 (1998) · Zbl 0885.15015
[13]	Kessler, O.; Berman, A., Matrices with a transitive graph and inverses \(M\)-matrices, Linear Algebra Appl., 71, 175-185 (1985) · Zbl 0606.05044