Abstract
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.
Journal of Economic Theory 21, 421–444, 1979
We would like to thank Professor K.Lancaster and Professor L.McKenzie for their help and guidance at various stages of the paper. We are indebted to Professors W. A. Brock, E. Burmeister, D. Cass and K. J. P. Magill for valuable comments. Comments of an anonymous referee us helped us improve the paper and correct many mistakes.
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References
Allen, R. G. D. (1968), Mathematical Analysis for Economists, St. Martin’s Press, New York.
Araujo, A. and J. A. Scheinkman (1977), “Smoothness, Comparative Dynamics and the Turnpike Property,” Econometrica,45,601–620.
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.
Benhabib, J. and K. Nishimura (1978), Multiple Equilibria and Stability in Growth Models, working paper.
Benveniste, L. M. and J. A. Scheinkman (1979), “Differentiable Value Functions in Concave Dynamic Optimization Problems,” Econometrica,47,727–732.
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.
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.
Burmeister, E. and A. R. Dobell (1970), Mathematical Theories of Economic Growth, Macmillan & Co., London.
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.
Burmeister, E. and K. Kuga (1970), “The Factor Price Frontier in a Neoclassical Multi-sector Model,” International Economic Review, 11, 163–176.
Cass, D. and K. Shell (1976), “The Structure and Stability of Competitive Dynamical Systems,” Journal of Economic Theory, 12, 31–70.
Debreu, G. and I. N. Hernstein (1953), “Non-negative Square Matrices,” Econometrica, 21, 597–607.
Gantmacher, F. R. (1960), The Theory of Matrices, Vol. 1, 2, Chelsea, New York.
Hartman, P. (1964), Ordinary Differential Equations, Wiley, New York.
Hirota, M. and K. Kuga (1971), “On an Intrinsic Joint Production,” International Economic Review,12, 87–105.
Hirsch, M. W. and S. Smale (1976), Differential Equations, Dynamic Systems, and Linear Algebra, Academic Press, New York.
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.
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.
Joseph, D. D. and D. H. Sattinger (1972), “Bifurcating Time–periodic Solutions and their Stability,” Archive for Rational Mechanics and Analysis, 45, 79–109.
Kelly, J. S. (1969), “Lancaster vs Samuelson on the Shape of the Neoclassical Transformation Surface,” Journal of Economic Theory, 1, 347–351.
Kurz, M. (1968), “The General Instability of a Class of Competitive Growth Processes,” Review of Economic Studies, 102, 155–174.
Kurz, M. (1978), “Optimal Economic Growth and Wealth Effects,” International Economic Review, 9, 348–357.
Lancaster, K. (1968), Mathematical Economics, Macmillian & Co., London.
Liviatan, N. and P. A. Samuelson (1969), “Notes on Turnpikes: Stable and Unstable,” Journal of Economic Theory, 1, 454–475.
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.
Marsden, E. J. (1976), and M. McCracken, The Hopf Bifurcation and its Applications, Springer-Verlag, New York.
McKenzie, L. W. (1955), “Equality of Factor Prices in World Trade,” Econometrica, 23, 239–257.
McKenzie, L. W. (1976), “Turnpike Theory,” Econometrica, 44, 841–865.
Ruelle, D. and F. Tackens (1971), “On the Nature of Turbulence,” Communications in Mathematical Physics, 20, 167–192.
Pitchford, J. D. and S. J. Turnovsky (1977), Applications of Control Theory to Economic Analysis, North-Holland, New York.
Samuelson, P. A. (1953–1954), “Prices of Factors and Goods in General Equilibrium,” Review of Economic Studies, 21, 1–20.
Sattinger, D. H. (1973), Topics in Stability and Bifurcation Theory, Springer-Verlag, New York.
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.
Scheinkman, J. A. (1976), “On Optimal Steady States of n-Sector Growth Models when Utility is Discounted,” Journal of Economic Theory, 12, 11–30.
Scheinkman, J. A. (1978), “Stability of Separable Hamiltonians and Investment Theory,” Review of Economic Studies, 45, 559–570.
Whittaker, E. T. (1944), A Treatise on the Analytical Dynamics of Particles and Rigid Bodies, Dover, New York.
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Benhabib, J., Nishimura, K. (2012). The Hopf Bifurcation and Existence and Stability of Closed Orbits in Multisector Models of Optimal Economic Growth. In: Stachurski, J., Venditti, A., Yano, M. (eds) Nonlinear Dynamics in Equilibrium Models. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-22397-6_3
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