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A New Matrix Theorem: Interpretation in Terms of Internal Trade Structure and Implications for Dynamic Systems


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  • Albert E. Steenge


  • Mark J. P. M. Thissen


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    Economic systems often are described in matrix form as x=Mx. We present a new theorem for systems of this type where M is square, nonnegative and indecomposable. The theorem discloses the existence of additional economic relations that have not been discussed in the literature up to now, and gives further insight in the economic processes described by these systems. As examples of the relevance of the theorem we focus on static and dynamic closed Input-Output (I-O) models. We show that the theorem is directly relevant for I-O models formulated in terms of difference or differential equations. In the special case of the dynamic Leontief model the system’s behavior is shown to depend on the properties of matrix M=A + C where A and C are the matrices of intermediate and capital coefficients, respectively. In this case, C is small relative to A and a perturbation result can be employed which leads directly to a statement on the system’s eigenvalues. This immediately suggests a solution to the well-known problem of the instability of the dynamic Leontief model. Copyright Springer-Verlag Wien 2005

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    Bibliographic Info

    Article provided by Springer in its journal Journal of Economics - Zeitschrift für Nationalökonomie.

    Volume (Year): 84 (2005)
    Issue (Month): 1 (02)
    Pages: 71-94

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    Handle: RePEc:kap:jeczfn:v:84:y:2005:i:1:p:71-94

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    Keywords: leverage matrix; stability analysis; input-output analysis; dynamic input-output models; C62; C67; D57;

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    Cited by:
    1. Daniele SCHILIRO, 2012. "Structural Change And Models Of Structural Analysis: Theories, Principles And Methods," Journal of Advanced Research in Law and Economics, ASERS Publishing, vol. 0(2), pages 31-49, December.
    2. Schilirò, Daniele, 2009. "Structural models and structural change: analytical principles and methodological issues," MPRA Paper 24480, University Library of Munich, Germany.
    3. Theodore Mariolis & Lefteris Tsoulfidis, 2012. "On Brody’S Conjecture: Facts And Figures From The Us Economy," Discussion Paper Series 2012_06, Department of Economics, University of Macedonia, revised May 2012.
    4. Mariolis, Theodore & Tsoulfidis, Lefteris, 2010. "Eigenvalue distribution and the production price-profit rate relationship in linear single-product systems: theory and empirical evidence," MPRA Paper 43716, University Library of Munich, Germany.
    5. Iliadi, Fotoula & Mariolis, Theodore & Soklis, George & Tsoulfidis, Lefteris, 2012. "Bienenfeld’s approximation of production prices and eigenvalue distribution: some more evidence from five European economies," MPRA Paper 36282, University Library of Munich, Germany.
    6. Faye Duchin & Albert E. Steenge, 2007. "Mathematical Models in Input-Output Economics," Rensselaer Working Papers in Economics 0703, Rensselaer Polytechnic Institute, Department of Economics.
    7. Schilirò, Daniele, 2007. "Theories and models of structural dynamics: an ‘ideal’ general framework ?," MPRA Paper 38256, University Library of Munich, Germany.


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