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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

    ()

Abstract

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

Suggested Citation

  • Albert E. Steenge & Mark J. P. M. Thissen, 2005. "A New Matrix Theorem: Interpretation in Terms of Internal Trade Structure and Implications for Dynamic Systems," Journal of Economics, Springer, vol. 84(1), pages 71-94, February.
  • Handle: RePEc:kap:jeczfn:v:84:y:2005:i:1:p:71-94
    DOI: 10.1007/s00712-004-0094-6
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    Citations

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    Cited by:

    1. Schilirò, Daniele, 2012. "Structural change and models of structural analysis: theories, principles and methods," MPRA Paper 41817, University Library of Munich, Germany.
    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. Theodore Mariolis, 2015. "Norm Bounds and A Homographic Approximation for the Wage–Profit Curve," Metroeconomica, Wiley Blackwell, vol. 66(2), pages 263-283, May.
    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.
    8. 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.

    More about this item

    Keywords

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

    JEL classification:

    • C62 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Existence and Stability Conditions of Equilibrium
    • C67 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Input-Output Models
    • D57 - Microeconomics - - General Equilibrium and Disequilibrium - - - Input-Output Tables and Analysis

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