Optimal Depletion of an Exhaustible Resource
Applied Mathematical Modelling, Vol. 3, No. 5, pp. 367-378, October 1979
Posted: 10 Jan 2008 Last revised: 31 Jan 2019
This paper considers an optimal control problem for the dynamics of an exhaustible natural resource model, the optimal control being the price over time to maximize the total present value of a parameterized social welfare function under the assumption that a substitute become available at a high enough price. Thus, the problem can be reinterpreted as one of the optimal phasing-in of an expensive substitute. Furthermore, the problem is constrained in the sense that the total consumption of the natural resource implied by a price trajectory via the assumed demand function must not exceed the available reserves. The problem is solved, using the maximum principle, for a complete range of the parameter reflecting the weights assigned to the consumer's surplus and the producer's surplus in the social welfare function. The results and algorithms obtained in the paper are illustrated by an example.
Keywords: Natural resources, exhaustible resources, consumer surplus, social welfare, economics of natural resources,optimal control, producer's surplus
JEL Classification: Q30, Q38
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