Abstract
A Hamiltonian dynamical system (HDS) naturally arises in the standard control problem involving optimization over time. On this ground alone, a systematic study of the basic structure of the general HDS should be extremely useful for mathematical control theory. Applications of HDS theory extend beyond models involving optimization. The classic studies of such systems were motivated by problems in celestial mechanics. While much of the analysis of my lecture will be motivated by the normative (optimizing) model of macroeconomic growth, I will show in passing how HDS theory may be useful in analyzing many positive (“nonoptimizing”) models of macroeconomic growth.
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References
W. A. Brock & J. Scheinkman, “Global Asymptotic Stability of Bounded Trajectories Generated by Modified Hamiltonian Dynamical Systems: A Study in the Theory of Optimal Growth,” Journal of Economic Theory Symposium, forthcoming.
D. Cass, “Duality: A Symmetric Approach from the Economist’s Vantage Point,” Journal of Economic Theory, 7 (1974), 272–295.
D. Cass, paper on Hamiltonian duality, in preparation for a Buffalo conference volume.
D. Cass & K. Shell, “The Structure and Stability of Competitive Dynamical Systems,” Journal of Economic Theory Symposium, forthcoming.
C. Caton & K. Shell, “An Exercise in the Theory of Heterogeneous Capital Accumulation,” Review of Economic Studies, 38 (1971), 13–22.
F. H. Hahn, “On Some Equilibrium Paths” in (J. A. Mirrlees & N. Stern, editors) Models of Economic Growth, London: The Macmillan Press Limited, 1973, 193–211.
L. Lau, “A Characterization of the Normalized Restricted Profit Function,” Journal of Economic Theory Symposium, forthcoming.
E. Malinvaud, “Capital Accumulation and Efficient Allocation of Resources,” Econometrica, 21 (1953), 233–268.
L. W. McKenzie, “Accumulation Programs of Maximum Utility and the von Neumann Facet,” in Value, Capital, and Growth, (J. Wolfe, ed.), Edinburgh University Press, 1968, 353–383.
H. Pollard, Mathematical Introduction to Celestial Mechanics, Englewood Cliffs, New Jersey: Prentice-Hall, Inc., 1966.
R. Radner, “Paths of Economic Growth that Are Optimal with Regard Only to Final States: A Turnpike Theorem,” Review of Economic Studies, 28 (1961), 98–104.
R. T. Rockafellar, “Saddle Points of Hamiltonian Systems in Convex Problems of Lagrange,” Journal of Mathematical Analysis and Applications, forthcoming.
R. T. Rockafellar, paper in preparation for Journal of Economic Theory Symposium, forthcoming.
P. A. Samuelson, “A Catenary Turnpike Theorem Involving Consumption and the Golden Rule,” American Economic Review, 55. (1965), 486–496.
K. Shell (editor), Essays on the Theory of Optimal Economic Growth, Cambridge, Mass.: MIT Press, 1967.
K. Shell, “Applications of Pontryagin’s Maximum Principle to Economics” in H. W. Kuhn & G. P. Szegö (eds.), Mathematical Systems Theory and Economics I, Berlin-Heidelberg-New York: Springer-Verlag, 1969, 241–292.
K. Shell, “On Competitive Dynamical Systems” in H. W. Kuhn & G. P. Szegö (eds.), Differential Games and Related Topics, Amsterdam: North-Holland Publishing Co., 1971, 449–476.
K. Shell, M. Sidrauski, & J. E. Stiglitz, “Capital Gains, Income, and Saving,” Review of Economic Studies, 36 (1969), 15–26.
K. Shell & J. E. Stiglitz, “The Allocation of Investment in a Dynamic Economy,” Quarterly Journal of Economics, 81 (1967), 592–609.
M. L. Weitzman, “Duality Theory for Infinite Horizon Convex Models,” Management Science, 19 (1973), 783–789.
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© 1975 D. Reidel Publishing Company, Dordrecht-Holland
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Shell, K. (1975). The Theory of Hamiltonian Dynamical Systems, and an Application to Economics. In: Grote, J.D. (eds) The Theory and Application of Differential Games. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-1804-3_17
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DOI: https://doi.org/10.1007/978-94-010-1804-3_17
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