A limiting property for the powers of a non-negative, reducible matrix

Abstract Considering a dynamic multisector model, the behaviour of Ak is examined for a non-negative matrix A as k becomes large. It is well known that for a primitive matrix A, the matrix ( A λ ) k converges to yq ' ( q'y ) , where λ denotes the dominant eigenvalue, and where y and q' are the right and left eigenvector associated with λ. In this note, the same result is shown to hold, under certain conditions, when A is reducible with primitive diagonal block submatrices. Under weaker conditions A k ( e ′ A k e ) is proved to converge to yq ′ ( e ′ y )( q ′ e ) , where e denotes the summation vector. The results are interpreted in terms of dynamic multisector models and interindustry linkage indicators.

[1]  Erik Dietzenbacher,et al.  THE MEASUREMENT OF INTERINDUSTRY LINKAGES - KEY SECTORS IN THE NETHERLANDS , 1992 .

[2]  On the maximum eigenvalue of a reducible non-negative real matrix , 1973 .

[3]  A. E. Steenge.  Consistency and composite numeraires in joint production input-output analysis; an application of ideas of T.L. Saaty , 1987 .

[4]  M. Shubik,et al.  Convex structures and economic theory , 1968 .

[5]  Albert E. Steenge.  Saaty's consistency analysis: An application to problems in static and dynamic input-output models , 1986 .

[6]  Valerie Isham,et al.  Non‐Negative Matrices and Markov Chains , 1983 .

[7]  David Simpson,et al.  The Fundamental Structure of Input-Output Tables, An International Comparison , 1965 .

[8]  Hollis B. Chenery,et al.  International Comparisons of the Structure of Production , 1958 .

[9]  Bernhard Korte,et al.  Triangularizing input-output matrices and the structure of production☆ , 1971 .

[10]  L. Pasinetti.  Lectures in the Theory of Production , 1977 .

[11]  U. Krause.  Heterogeneous Labour and the Fundamental Marxian Theorem , 1981 .

[12]  Byung-Nak Song.  The production structure of the Korean economy : international and historical comparisons , 1977 .

[13]  Robert J. Plemmons,et al.  Nonnegative Matrices in the Mathematical Sciences , 1979, Classics in Applied Mathematics.

[14]  Yukio Fukui.  A More Powerful Method for Triangularizing Input-Output Matrices and the Similarity of Production Structures , 1986 .

[15]  P. Sraffa.  Production of commodities by means of commodities , 1960 .

[16]  K. Santhanam,et al.  A STUDY OF THE PRODUCTION STRUCTURE OF THE INDIAN ECONOMY: AN INTERNATIONAL COMPARISON , 1972 .

[17]  On the Complete Instability of Empirically Implemented Dynamic Leontief Models , 1990 .