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Further Refinements in Optimal Control

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A Beginner’s Guide to Dynamic Optimization in Economics

Part of the book series: Classroom Companion: Economics ((CCE))

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Abstract

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Chapter 9 introduces several considerations that sometimes arise in optimal control problems. These include the specification of transversality conditions, the presence of control parameters, and the situation of blocked intervals. The chapter also details the mathematical correspondence between present-value and current value Hamiltonians.

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Notes

  1. 1.

    Constraints at the start of the problem may also apply, although these are relatively rare in economics. For example, an initial cost incurred at the start of the problem could be incorporated as a negative salvage value.

  2. 2.

    Philosophers (and some economists) have pointed out that discounting tends to blur the line between efficiency concerns and equity concerns. Ethical debates regarding the appropriateness of discounting in economics, especially for social concerns, continue. Some of these arguments are summarized by Parfit (1984).

  3. 3.

    The practical implication is that in phase diagrams the \(dx/dt = 0\) locus does not shift over time. In a non-autonomous system, the \(dx/dt = 0\) locus shifts over time, making phase diagram analysis virtually impossible.

  4. 4.

    It is a somewhat technical point, but using a discount factor other than \({e^{ - rt}}\) (or the discrete time analogue) could lead to a situation in which the planner would wish to change the optimal plan over time. This problem, identified as dynamic inconsistency by Strotz (1955), is the reason for nearly universal use in economics today of exponential discounting.

References

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  • Parfit, D. (1984). Reasons and persons. Clarendon Press.

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  • Ramsey, F. (1928). A mathematical theory of savings. Economic Journal, 38(152), 543–559.

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  • Strotz, R. H. (1955). Myopia and inconsistency in dynamic utility maximization. Review of Economic Studies, 23, 165–180.

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  1. Department of Agricultural Economics, Purdue University, West Lafayette, IN, USA

    Gerald Shively

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Shively, G. (2025). Further Refinements in Optimal Control. In: A Beginner’s Guide to Dynamic Optimization in Economics. Classroom Companion: Economics. Springer, Cham. https://doi.org/10.1007/978-3-032-09374-5_9

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