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The present value of resources with large discount rates. (English) Zbl 0881.49002
The authors have described a method to detect limit cycles for optimal control problems in the plane. This method has been applied to two specific examples from resource management: a taxation problem and an exploited system of predator-prey interaction which show that the limit cycles may grow as the discount rates decrease. The relation of the established results to theorems in optimal growth theory has also been discussed.

MSC:
49J15 Existence theories for optimal control problems involving ordinary differential equations
92D25 Population dynamics (general)
34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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[1] J.P. Aubin, H. Frankowska (1990), Set-Valued Analysis, Boston: Birkhäuser. · Zbl 0713.49021
[2] J. Benhabib, K. Nishimura (1979), The Hopf-Bifurcation and the Existence and Stability of Closed Orbits in Multisector Models of Optimal Growth, Journal of Economic Theory, 21:412–444. · Zbl 0427.90021 · doi:10.1016/0022-0531(79)90050-4
[3] M. Boldrin, L. Montrucchio (1986), On the Indeterminacy of Capital Accumulation Paths, Journal of Economic Theory, 40:26–39. · Zbl 0662.90021 · doi:10.1016/0022-0531(86)90005-0
[4] W.A. Brock, 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. · Zbl 0348.90018 · doi:10.1016/0022-0531(76)90031-4
[5] D. Cass, K. Shell (1976), The Structure and Stability of Competitive Systems, Journal of Economic Theory, 12:31–70. · Zbl 0348.90039 · doi:10.1016/0022-0531(76)90027-2
[6] C.W. Clark (1976), Mathematical Bioeconomics; The Optimal Management of Renewable Resources, New York: Wiley. · Zbl 0364.90002
[7] C.W. Clark (1985), Bioeconomic Modelling and Fishery Management, New York: Wiley-Interscience.
[8] C.W. Clark (1990), Mathematical Bioeconomics; The Optimal Management of Renewable Resources, second edition, New York: Wiley. · Zbl 0712.90018
[9] F. Clarke (1983), Optimization and Non-Smooth Analysis, New York: Wiley-Interscience. · Zbl 0582.49001
[10] M. Falcone (1987), A Numerical Approach to the Infinite Horizon Problem of Deterministic Control Theory, Applied Mathematics & Optimization, 15:1–13. · Zbl 0715.49023 · doi:10.1007/BF01442644
[11] M. Falcone (1991), Corrigenda: A Numerical Approach to the Infinite Horizon Problem of Deterministic Control Theory, Applied Mathematics & Optimization, 23:213–214. · Zbl 0715.49025 · doi:10.1007/BF01442399
[12] G. Feichtinger, V. Kaitala, A. Novak (1991), Stable Resource Employment and Limit Cycles in an Optimally Regulated Fishery, University of Technology, Vienna.
[13] O. Hajek (1979), Discontinous Differential Equations I, II, Journal of Differential Equations, 32:149–185. · Zbl 0365.34017 · doi:10.1016/0022-0396(79)90056-1
[14] P. Hartman (1982), Ordinary Differential Equations, Boston: Birkhäuser. · Zbl 0476.34002
[15] R. Luus (1990), Optimal Control by Dynamic Programming Using Systematic Reduction in Grid Size, Institutional Journal of Control, 51(5):995–1013. · Zbl 0703.49022 · doi:10.1080/00207179008934113
[16] M. Rauscher (1990), Environmental Resources by an Indebted Country, Journal of Institutional and Theoretical Economics, 146:500–517.
[17] W.J. Reed (1984), The Effects of Risk of Fire on the Optimal Rotation of a Forest, Journal of Environmental Economics and Management, 11:180–190. · Zbl 0535.90028 · doi:10.1016/0095-0696(84)90016-0
[18] R.T. Rockafellar (1976), Saddle Points of Hamiltonian Systems in Convex Lagrange Problems Having a Nonzero Discount Rate, Journal of Economic Theory, 12:71–113. · Zbl 0333.90007 · doi:10.1016/0022-0531(76)90028-4
[19] W. Semmler, M. Sieveking (1994), On the Optimal Exploitation of Interacting Resources, Journal of Economics, 59(1):23–49. · Zbl 0798.90018 · doi:10.1007/BF01225931
[20] M. Sieveking (1991), Stability of Limit Cycles, Preprint, University of Frankfurt.
[21] G. Sorger (1992), Minimum Impatience Theorems for Recursive Economic Models. Lecture Notes in Economics and Mathematical Systems, Volume 390. Heidelberg: Springer-Verlag. · Zbl 0787.90012
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