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Nonnegative Square Matrices: Irreducibility, Reducibility, Primitivity and Some Economic Applications

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  • Giorgio Giorgi

    (University of Pavia)

Abstract

We give an overview of the various characterizations of irreducibility, reducibility and primitivity for nonnegative square matrices. Some comments are made for the so-called "Gantmacher normal form" and for a turnpike theorem, due to Achmanov (1984). In addition, we discuss a proposal of Manara (1968) for the reducibility of a Sraffa model with joint production.

Suggested Citation

  • Giorgio Giorgi, 2019. "Nonnegative Square Matrices: Irreducibility, Reducibility, Primitivity and Some Economic Applications," DEM Working Papers Series 175, University of Pavia, Department of Economics and Management.
    • Handle: RePEc:pav:demwpp:demwp0175
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    File URL: http://dem-web.unipv.it/web/docs/dipeco/quad/ps/RePEc/pav/demwpp/DEMWP0175.pdf
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    References listed on IDEAS

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    1. Murata, Yasuo, 1977. "Mathematics for Stability and Optimization of Economic Systems," Elsevier Monographs, Elsevier, edition 1, number 9780125112505 edited by Shell, Karl.
    2. Christian Bidard, 2004. "Prices, Reproduction, Scarcity," Post-Print hal-03094334, HAL.
    3. Enrico Zaghini, 1967. "On Non-Basic Commodities," Swiss Journal of Economics and Statistics (SJES), Swiss Society of Economics and Statistics (SSES), vol. 103(II), pages 257-266, June.
    4. Bidard, Christian & Woods, John E, 1989. "Taxes, Lands and Non-basics in Joint Production," Oxford Economic Papers, Oxford University Press, vol. 41(4), pages 802-812, October.
    5. Takayama,Akira, 1985. "Mathematical Economics," Cambridge Books, Cambridge University Press, number 9780521314985.
    6. Dana, Rose-Anne & Florenzano, Monique & Levan, Cuong & Levy, Dominique, 1989. "Asymptotic properties of a Leontief economy," Journal of Economic Dynamics and Control, Elsevier, vol. 13(4), pages 553-568, October.
    7. Kurz,Heinz D. & Salvadori,Neri, 1997. "Theory of Production," Cambridge Books, Cambridge University Press, number 9780521588676.
    8. Suzumura, Kotaro, 1983. "Perron-Frobenius Theorem on Non-Negative Square Matrices : An Elementary Proof," Hitotsubashi Journal of Economics, Hitotsubashi University, vol. 24(2), pages 137-141, December.
    9. Bidard,Christian, 2011. "Prices, Reproduction, Scarcity," Cambridge Books, Cambridge University Press, number 9780521180191.
    10. Dana, Rose-Anne & Florenzano, Monique & Le Van, Cuong & Levy, Dominique, 1989. "Production prices and general equilibrium prices : A long-run property of a Leontief economy," Journal of Mathematical Economics, Elsevier, vol. 18(3), pages 263-280, June.
    11. József Móczár, 1995. "Reducible Von Neumann Models And Uniqueness," Metroeconomica, Wiley Blackwell, vol. 46(1), pages 1-15, February.
    12. Szyld, Daniel B, 1985. "Conditions for the Existence of a Balanced Growth Solution for the Leontief Dynamic Input-Output Model," Econometrica, Econometric Society, vol. 53(6), pages 1411-1419, November.
    13. Dana, Rose-Anne & Florenzano, Monique & Le Van, Cuong & Lévy, Dominique, 1984. "Production prices and general equilibrium prices," CEPREMAP Working Papers (Couverture Orange) 8422, CEPREMAP.
    14. Morishima, M. & Fujimoto, T., 1974. "The Frobenius theorem, its Solow-Samuelson extension and the Kuhn-Tucker theorem," Journal of Mathematical Economics, Elsevier, vol. 1(2), pages 199-205, August.
    15. G. R. Blakley & W. F. Gossling, 1967. "The Existence, Uniqueness and Stability of the Standard System," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 34(4), pages 427-431.
    16. Steedman, Ian, 1977. "Basics, Non-Basics and Joint Production," Economic Journal, Royal Economic Society, vol. 87(346), pages 324-328, June.
    17. Hukukane Nikaidô, 1962. "Some Dynamic Phenomena in the Leontief Model of Reversely Lagged Type," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 29(4), pages 313-323.
    18. Alexander P. Abramov, 2014. "Balanced and Cyclical Growth in Models of Decentralized Economy," Lecture Notes in Economics and Mathematical Systems, Springer, edition 127, number 978-3-319-07917-2, December.
    19. Ulrich Krause, 1981. "Heterogeneous Labour and the Fundamental Marxian Theorem," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 48(1), pages 173-178.
    20. Fuad Aleskerov & Hasan Ersel & Dmitri Piontkovski, 2011. "Linear Algebra for Economists," Springer Texts in Business and Economics, Springer, number 978-3-642-20570-5, August.
    21. Moczar, Jozsef, 1991. "Irreducible balanced and unbalanced growth paths : Business cycles and structural changes," Structural Change and Economic Dynamics, Elsevier, vol. 2(1), pages 159-176, June.
    22. Ilaria Ossella, 1999. "A turnpike property of optimal programs for a class of simple linear models of production," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 14(3), pages 597-607.
    23. Weil, Roman L, 1970. "Solutions to the Decomposable von Neumann Model," Econometrica, Econometric Society, vol. 38(2), pages 276-280, March.
    24. Roy Radner, 1961. "Prices and the Turnpike: III. Paths of Economic Growth that are Optimal with Regard only to Final States: A Turnpike Theorem," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 28(2), pages 98-104.
    25. József Móczár, 1997. "Non-uniqueness Through Duality in the Von Neumann Growth Models," Metroeconomica, Wiley Blackwell, vol. 48(3), pages 280-299, October.
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