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Endogenous cycles in discrete symmetric multisector optimal growth models

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Abstract

This paper discusses the asymptotic stability of the steady state in discrete symmetric multisector optimal growth models. Using a variational method, we provide a new proposition which gives some conditions ensuring the local saddle point property. A characterization of the bound above which the steady state is locally unstable is also proposed in terms of the indirect utility function concavity properties. On this basis, some sufficient conditions for the existence of competitive cycles are stated. We thus prove the existence of a Flip bifurcation.

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References

  1. Cass, D., andShell, K.,Editors,The Hamiltonian Approach to Dynamic Economics, Academic Press, New York, New York, 1976.

    Google Scholar 

  2. McKenzie, L.,Optimal Economic Growth, Turnpike Theorems, and Comparative Dynamics, Handbook of Mathematical Economics, Vol. 3, Edited by K. Arrow and M. Intriligator, North Holland, Amsterdam, Holland, pp. 1281–1355, 1986.

    Google Scholar 

  3. Sutherland, W. A.,On Optimal Development in a Multisectoral Economy: The Discounted Case, Review of Economic Studies, Vol. 46, pp. 585–589, 1970.

    Google Scholar 

  4. Samuelson, P.,Optimality of Profit, Including Prices under Ideal Planning, Proceeding of the National Academy of Sciences, USA, Vol. 70, pp. 2109–2111, 1973.

    Google Scholar 

  5. McKenzie, L.,Turnpike Theory, Discounted Utility, and the Von Neumann Facet, Journal of Economic Theory, Vol. 30, pp. 330–352, 1983.

    Google Scholar 

  6. Boldrin, M., andMontrucchio, L.,On the Indeterminacy of Capital Accumulation Paths, Journal of Economic Theory, Vol. 40, pp. 26–39, 1986.

    Google Scholar 

  7. Montrucchio, L.,The Occurrence of Erratic Fluctuations in Models of Optimization over Infinite Horizon, Growth Cycles and Multisectoral Economics: The Goodwin Tradition, Edited by G. Ricci and K. Velupillai,Springer Verlag, Berlin, Germany, pp. 83–92, 1988.

    Google Scholar 

  8. Sorger, G.,On the Optimality of Given Feedback Controls, Journal of Optimization Theory and Applications, Vol. 65, pp. 321–329, 1990.

    Google Scholar 

  9. Benhabib, J., andNishimura, K.,The Hopf Bifurcation and the Existence and Stability of Closed Orbits in Multisector Models of Optimal Economic Growth, Journal of Economic Theory, Vol. 21, pp. 421–444, 1979.

    Google Scholar 

  10. Benhabib, J., andNishimura, K.,Competitive Equilibrium Cycles, Journal of Economic Theory, Vol. 35, pp. 284–306, 1985.

    Google Scholar 

  11. Benhabib, J., andNishimura, K.,On Endogenous Cycles in Discrete-Time Optimal Growth Models, Optimal Control Theory and Economic Analysis 3, Edited by G. Feichtinger, North Holland, Amsterdam, Holland, pp. 3–20, 1988.

    Google Scholar 

  12. Medio, A.,Oscillations in Optimal Growth Models, Journal of Economic Behavior and Organization, Vol. 8, pp. 413–427, 1987.

    Google Scholar 

  13. Benhabib, J., andRustichini, A.,Equilibrium Cycling with Small Discounting, Journal of Economic Theory, Vol. 52, pp. 423–432, 1990.

    Google Scholar 

  14. Cartigny, P., andVenditti, A.,Turnpike Theory: Some New Results on the Saddle-Point Property of Equilibria and on the Existence of Endogeneous Cycles, Journal of Economic Dynamics and Control, Vol. 18, pp. 957–974, 1994.

    Google Scholar 

  15. Magill, J. P. M.,Some New Results on the Local Stability of the Process of Capital Accumulation, Journal of Economic Theory, Vol. 15, pp. 174–210, 1977.

    Google Scholar 

  16. Scheinkman, J. A.,Stability of Separable Hamiltonians and Investment Theory, Review of Economic Studies, Vol. 7, pp. 34–41, 1978.

    Google Scholar 

  17. Magill, J. P.M., andScheinkman, J. A.,Stability of Regular Equilibria and the Correspondence Principle for Symmetric Variational Problems, International Economic Review, Vol. 20, pp. 297–315, 1979.

    Google Scholar 

  18. Magill, J. P. M.,The Origin of Cyclical Motion in Dynamic Economic Models, Journal of Economic Dynamics and Control, Vol. 1, pp. 199–218, 1979.

    Google Scholar 

  19. Dasgupta, S.,A Local Analysis of Stability and Regularity of Stationary States in Discrete Symmetric Optimal Capital Accumulation Models, Journal of Economic Theory, Vol. 36, pp. 302–318, 1985.

    Google Scholar 

  20. Araujo, A., andScheinkman, J. A.,Smoothness, Comparative Dynamics, and the Turnpike Property, Econometrica, Vol. 45, pp. 601–620, 1977.

    Google Scholar 

  21. Montrucchio, L.,A New Turnpike Theorem for Discounted Programs, Economic Theory, 1984.

  22. Brock, W.,Some Results on the Uniqueness of the Steady States in Multisector Models of Optimum Growth when Future Utilities Are Discounted, International Economic Review, Vol. 14, pp. 535–559, 1973.

    Google Scholar 

  23. Levhari, D., andLiviatan, N.,On Stability in the Saddle-Point Sense, Journal of Economic Theory, Vol. 4, pp. 88–93, 1972.

    Google Scholar 

  24. Santos, M. S.,Smoothness of the Policy Function in Discrete-Time Economic Models, Econometrica, Vol. 59, pp. 1365–1382, 1991.

    Google Scholar 

  25. Scheinkman, J. A.,On Optimal Steady States of n-Sector Growth Models When Utility Is Discounted, Journal of Economic Theory, Vol. 12, pp. 11–30, 1976.

    Google Scholar 

  26. Rockafellar, R. T.,Saddle Points of Hamiltonian Systems in Convex Lagrange Problems Having a Nonzero Discount Rate, Journal of Economic Theory, Vol. 12, pp. 71–113, 1976.

    Google Scholar 

  27. Vial, J. P.,Strong and Weak Convexity of Sets and Functions, Mathematical Operations Research, Vol. 8, pp. 231–257, 1983.

    Google Scholar 

  28. Bougeard, M., andPenot, J. P.,Approximation and Decomposition Properties of Some Classes of Locally DC Functions, Mathematical Programming, Vol. 41, pp. 195–228, 1988.

    Google Scholar 

  29. Penot, J. P., andVolle, M.,On Strongly Convex and Paraconvex Dualities, Lecture Notes in Economics and Mathematical Systems, Springer Verlag, Berlin, Germany, Vol. 345, pp. 188–218, 1989.

    Google Scholar 

  30. Guckenheimer, J., andHolmes, P.,Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields, Springer Verlag, New York, New York, 1986.

    Google Scholar 

  31. Ruelle, D.,Elements of Differentiable Dynamics and Bifurcation Theory, Academic Press, London, England, 1989.

    Google Scholar 

  32. Boldrin, M.,Paths of Capital Accumulation in Two-Sector Models, Economic Complexity: Chaos, Sunspots, Bubbles, and Nonlinearity, Edite by W. Barnett, J. Geweke, and K. Shell, Cambridge University Press, Cambridge, England, pp. 231–252, 1989.

    Google Scholar 

  33. Boldrin, M., andDeneckere, R.,Sources of Complex Dynamics in Two-Sector Growth Models, Journal of Economic Dynamics and Control, Vol. 14, pp. 627–653, 1990.

    Google Scholar 

  34. Amir, R. Sensitivity Analysis of Multi-Sector Optimal Economic Dynamics, CORE Discussion Paper No. 9106, 1991.

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Author information

Authors and Affiliations

  1. Groupement de Recherche en Economie Quantitative d'Aix-Marseille, Université d'Aix-Marseille II, Marseille, France

    P. Cartigny (Professor)

  2. Groupement de Recherche en Economie Quantitative d'Aix-Marseille, Université d'Aix-Marseille II, Marseille, France

    A. Venditti (Research Fellow)

Authors
  1. P. Cartigny
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  2. A. Venditti
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Additional information

Communicated by G. Leitmann

The authors would like to thank J. M. Grandmont, P. Michel, and L. Montrucchio for helpful discussions which greatly improved the exposition of the paper. The present work also benefited from comments received during a presentation at the Meetings of the Society for Economic Dynamics and Control, UCLA, June 30–July 2, 1994.

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Cartigny, P., Venditti, A. Endogenous cycles in discrete symmetric multisector optimal growth models. J Optim Theory Appl 86, 17–36 (1995). https://doi.org/10.1007/BF02193459

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