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Economic Evolution and Structural Adjustment pp 291–312Cite as

A Multisector Model of the Trade Cycle

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Part of the book series: Lecture Notes in Economics and Mathematical Systems ((LNE,volume 293))

Abstract

The main purpose of this paper is to generalize to the multidimensional case certain results established for aggregate multiplier-accelerator models. The starting point of our investigation will be the well known dynamic input-output model:

$$x = Ax + B\dot x$$
(1)

where x ∈ Rn is a column vector indicating the levels of activity of the various sectors; A ∈ Rnxn and B ∈ Rnxn are matrices indicating, respectively, the flow and the stock input-output relations.

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Notes

  1. An interesting discussion of the priority of this result in bifurcation theory can be found in Arnold [1983], pp. 271-273. In this paper, however, we shall make use of the more common denomination of “Hopf bifurcation”. On this subject cf. Marsden and Mc. Cracken [1976] and Hassard, Kazarinoff and Wan [1981].

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  2. Cf. Marsden-Mc. Cracken, op. cit., pp. 63-83 Hassard-Kazarinoff-Wan, op. cit., pp. 14-24.

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  3. For an extensive discussion of this point, cf Medio [1979], Chapters II and V.

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  4. Cf. Gantmacher, [1966], vol 1, pp. 46-47.

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  5. We assume, in particular, that bii > 0 for some i=1, 2,...,n.

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  6. On the concept of D-stable matrices and related issues, see Magnani-Meriggi [1981], pp. 535-544.

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  7. An excellent and thorough discussion of this method can be found in Siljak [1978], on which this author has drawn heavily.

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  8. Cf. Gantmacher, op. cit., vol. 2, pp. 182-183.

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  9. Uniqueness of solution requires that no characteristic root is equal to zero, and that for no pair of characteristic values of μi and μj we have μi =-μj, (which we assume here).

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  10. Cf. Siljak, op. cit., pp. 39-40 and 96-99.

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  11. On the Lyapunov direct method, cf La Salle and Lefschetz [1961].

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  12. On the concept of “condition numbers”, see, for example, Stewart [1973], pp. 184-192.

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  13. A more rigorous statement of the conditions of the Hopf bifurcation may be found in Marsden-Mc. Cracken and Hassard-Kazarinoff-Wan, loc. cit..

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  14. The interested reader may consult Marsden-Mc. Cracken, cit., pp. 104-131 on this point, where an algorithm is provided for the determination of stability of periodic solutions. Unfortunately, in the general case, this involves estimating the sign of derivates up to at least third order, which may not be possible in an economic model on purely a priori grounds.

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References

  • Arnold, V.I., Geometrical Methods in the Theory of Ordinary Differential Equations, Springer-Verlag, New York, 1983.

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  • Gantmacher, Théori des matrices, Dunod, Paris, 1966.

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  • Hassard, B.D., Kazarinoff, N.D., Wan, Y.H., Theory and Applications of Hopf Bifurcation, Cambridge University Press, 1981.

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  • La Salle, J., Lefschetz, S., Stability by Lyapunov’s Direct Method, with Applications, Academic Press, New York, 1961.

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  • Magnani, U., Meriggi, M.R., “Characterization of K-Matrices”, in Castellani, G, Mazzoleni, P. (eds.), Mathematical Programming and its Economic Applications, Milano, 1981.

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  • Marsden, J.E., Mc. Cracken, M., The Hopf Bifurcation and Its Applications, Applied Mathematical Sciences, vol. 19, Springer-Verlag, 1976.

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  • Medio, A., Teoria nonlineare del ciclo economico, Il Mulino, Bologna, 1979. (The Italian version of an unpublished Ph.D. thesis, Cambridge, U.K., 1975).

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  • Siljak, D.D., Large-Scale Dynamic Systems, North-Holland, Amsterdam, 1978.

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  • Stewart, G.W., Introduction to Matrix Computations, Academic Press, New York, 1973.

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Authors
  1. A. Medio
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Editor information

Editors and Affiliations

  1. Department of Economics, University of Umeå, S-901 87, Umeå, Sweden

    David Batten  & Börje Johansson  & 

  2. Institute for Econometrics & Operations Research, Techn. Univ. of Vienna, Argentinierstraße 8, A-1040, Vienna, Austria

    John L. Casti

  3. CERUM (Centre for Regional Science Research), Sweden

    Börje Johansson

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© 1987 Springer-Verlag Berlin Heidelberg

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Medio, A. (1987). A Multisector Model of the Trade Cycle. In: Batten, D., Casti, J.L., Johansson, B. (eds) Economic Evolution and Structural Adjustment. Lecture Notes in Economics and Mathematical Systems, vol 293. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-02522-2_12

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  • DOI: https://doi.org/10.1007/978-3-662-02522-2_12

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-18183-5

  • Online ISBN: 978-3-662-02522-2

  • eBook Packages: Springer Book Archive

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