The Hopf Bifurcation and Existence and Stability of Closed Orbits in Multisector Models of Optimal Economic Growth

  • Jess Benhabib
  • Kazuo Nishimura


The local and global stability of multisector optimal growth models has been extensively studied in the recent literature. Brock and Scheinkman (1976), Cass and Shell (1976), McKenzie (1976), and Scheinkman (1976) have established strong results about global stability that require a small rate of discount. Burmeister and Graham (1973), Araujo and Scheinkman (1977), Magill (1977), and Scheinkman (1978) have established conditions that yield stability conditions independently of the rate of discount.


Hopf Bifurcation Optimal Path Closed Orbit Characteristic Exponent Positive Real Part 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Allen, R. G. D. (1968), Mathematical Analysis for Economists, St. Martin’s Press, New York.Google Scholar
  2. Araujo, A. and J. A. Scheinkman (1977), “Smoothness, Comparative Dynamics and the Turnpike Property,” Econometrica,45,601–620.Google Scholar
  3. Benhabib, J. and K. Nishimura (1979), “On the Uniqueness of Steady States in an Economy with Heterogeneous Capital Goods,” International Economic Review,20,59–82.Google Scholar
  4. Benhabib, J. and K. Nishimura (1978), Multiple Equilibria and Stability in Growth Models, working paper.Google Scholar
  5. Benveniste, L. M. and J. A. Scheinkman (1979), “Differentiable Value Functions in Concave Dynamic Optimization Problems,” Econometrica,47,727–732.Google Scholar
  6. Brock, W. (1973), “Some Results on the Uniqueness of Steady States in Multisector Models of Optimum Growth when Future Utilities are Discounted,” International Economic Review, 14, 535–559.CrossRefGoogle Scholar
  7. Brock, W. A. and J. A. Scheinkman (1976), “Global Asymptotic Stability of Optimal Control Systems with Applications to the Theory of Economic Growth,” Journal of Economic Theory, 12, 164–190.CrossRefGoogle Scholar
  8. Burmeister, E. and A. R. Dobell (1970), Mathematical Theories of Economic Growth, Macmillan & Co., London.Google Scholar
  9. Burmeister, E. and D. Graham (1973), Price Expectations and Stability in Descriptive and Optimally Controlled Macro-Economic Models, J. E. E. Conference Publication No. 101, Institute of Electrical Engineers, London.Google Scholar
  10. Burmeister, E. and K. Kuga (1970), “The Factor Price Frontier in a Neoclassical Multi-sector Model,” International Economic Review, 11, 163–176.CrossRefGoogle Scholar
  11. Cass, D. and K. Shell (1976), “The Structure and Stability of Competitive Dynamical Systems,” Journal of Economic Theory, 12, 31–70.CrossRefGoogle Scholar
  12. Debreu, G. and I. N. Hernstein (1953), “Non-negative Square Matrices,” Econometrica, 21, 597–607.CrossRefGoogle Scholar
  13. Gantmacher, F. R. (1960), The Theory of Matrices, Vol. 1, 2, Chelsea, New York.Google Scholar
  14. Hartman, P. (1964), Ordinary Differential Equations, Wiley, New York.Google Scholar
  15. Hirota, M. and K. Kuga (1971), “On an Intrinsic Joint Production,” International Economic Review,12, 87–105.Google Scholar
  16. Hirsch, M. W. and S. Smale (1976), Differential Equations, Dynamic Systems, and Linear Algebra, Academic Press, New York.Google Scholar
  17. Hopf, E. (1976), Bifurcation of a Periodic Solution from a Stationary Solution of a System of Differential Equations, translated by L. N. Howard and N. Kopell, in The Hopf Bifurcation and Its Applications, ed. by Marsden, J. E. and M. McCracken, Springer-Verlag, New York, 163–194.Google Scholar
  18. Iooss, G. (1972), “Existence et stabilité de la solution périodique secondaire intervenant dans les problèmes d’évolution du type Navier-Stokes,”Archive for Rational Mechanics and Analysis, 49, 301–329.CrossRefGoogle Scholar
  19. Joseph, D. D. and D. H. Sattinger (1972), “Bifurcating Time–periodic Solutions and their Stability,” Archive for Rational Mechanics and Analysis, 45, 79–109.CrossRefGoogle Scholar
  20. Kelly, J. S. (1969), “Lancaster vs Samuelson on the Shape of the Neoclassical Transformation Surface,” Journal of Economic Theory, 1, 347–351.CrossRefGoogle Scholar
  21. Kurz, M. (1968), “The General Instability of a Class of Competitive Growth Processes,” Review of Economic Studies, 102, 155–174.CrossRefGoogle Scholar
  22. Kurz, M. (1978), “Optimal Economic Growth and Wealth Effects,” International Economic Review, 9, 348–357.CrossRefGoogle Scholar
  23. Lancaster, K. (1968), Mathematical Economics, Macmillian & Co., London.Google Scholar
  24. Liviatan, N. and P. A. Samuelson (1969), “Notes on Turnpikes: Stable and Unstable,” Journal of Economic Theory, 1, 454–475.CrossRefGoogle Scholar
  25. Magill, M. J. P. (1977), “Some New Results on the Local Stability of the Process of Capital Accumulation,” Journal of Economic Theory, 15, 174–210.CrossRefGoogle Scholar
  26. Marsden, E. J. (1976), and M. McCracken, The Hopf Bifurcation and its Applications, Springer-Verlag, New York.Google Scholar
  27. McKenzie, L. W. (1955), “Equality of Factor Prices in World Trade,” Econometrica, 23, 239–257.CrossRefGoogle Scholar
  28. McKenzie, L. W. (1976), “Turnpike Theory,” Econometrica, 44, 841–865.CrossRefGoogle Scholar
  29. Ruelle, D. and F. Tackens (1971), “On the Nature of Turbulence,” Communications in Mathematical Physics, 20, 167–192.CrossRefGoogle Scholar
  30. Pitchford, J. D. and S. J. Turnovsky (1977), Applications of Control Theory to Economic Analysis, North-Holland, New York.Google Scholar
  31. Samuelson, P. A. (1953–1954), “Prices of Factors and Goods in General Equilibrium,” Review of Economic Studies, 21, 1–20.Google Scholar
  32. Sattinger, D. H. (1973), Topics in Stability and Bifurcation Theory, Springer-Verlag, New York.Google Scholar
  33. Schmidt, D. S. (1976), Hopf Bifurcation Theorem and the Center Theorem of Liapunov, in The Hopf Bifurcation and Its Applications, ed. by Marsden, J. E. and M. McCracken, Springer-Verlag, New York, 95–103.Google Scholar
  34. Scheinkman, J. A. (1976), “On Optimal Steady States of n-Sector Growth Models when Utility is Discounted,” Journal of Economic Theory, 12, 11–30.CrossRefGoogle Scholar
  35. Scheinkman, J. A. (1978), “Stability of Separable Hamiltonians and Investment Theory,” Review of Economic Studies, 45, 559–570.CrossRefGoogle Scholar
  36. Whittaker, E. T. (1944), A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Dover, New York.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Department of EconomicsNew York UniversityNew YorkUSA
  2. 2.Institute of Economic ResearchKyoto UniversityKyotoJapan

Personalised recommendations