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doi: 10.3934/jimo.2021110
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Using optimal control to optimize the extraction rate of a durable non-renewable resource with a monopolistic primary supplier

Universitat Politècnica de Catalunya, Institute of Industrial and Control Engineering, Av. Diagonal, 647, 08028, Barcelona, Spain

* Corresponding author: Albert Corominas

Received  September 2020 Revised  February 2021 Early access June 2021

Fund Project: The first author is supported by Gen. de Catalunya through the AGAUR Proj. 2017 SGR 872

The problem dealt with in this paper is that of optimizing the path of the extraction rate (and, consequently, the price) for the monopolistic owner of the primary sources of a totally or partially durable non-renewable resource (such as precious metals or gemstones) in a continuous-time frame, assuming that there is an upper bound on the extraction rate and with an interest rate equal to zero. The durability of the resource implies that, unlike the case of non-durable resources, at any time there is a stock of already-used amounts of the resource that are still potentially reusable, in addition to the resource available in the ground for extraction. The problem is addressed using the Maximum Principle of Pontryagin in the framework of optimal control theory, which allows identifying the patterns that the optimal policies can adopt. In this framework, the Hamiltonian is linear in the control input, which implies a bang-bang control policy governed by a switching surface. There is an underlying geometry to the problem that determines the solutions. It is characterized by the switching surface, its time derivative, the intersection point (if any) and the bang-bang trajectories through this point.

Citation: Enric Fossas, Albert Corominas. Using optimal control to optimize the extraction rate of a durable non-renewable resource with a monopolistic primary supplier. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2021110
References:
[1]

J. I. Bulow, Durable-goods monopolists, Journal of Political Economy, 90 (1982), 314-332.  doi: 10.1086/261058.  Google Scholar

[2]

R. H. Coase, Durability and monopoly, Journal of Law and Economics, 15 (1972), 143-149.  doi: 10.1086/466731.  Google Scholar

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A. Corominas, Using discrete-time mathematical programming to optimise the extraction rate of a durable non-renewable resource with a single primary supplier, Oper. Res. Perspect., 4 (2017), 118-122.  doi: 10.1016/j.orp.2017.09.002.  Google Scholar

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L. C. Gray, Rent under the assumption of exhaustibility, Quarterly Journal of Economics, 28 (1914), 466-489.  doi: 10.2307/1884984.  Google Scholar

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H. Hotelling, The economics of exhaustible resources, Bulletin of Mathematical Biology, 53 (1991), 281-312.  doi: 10.1086/254195.  Google Scholar

[7]

D. E. Kirk, Optimal Control Theory. An Introduction, Dover Publications Inc, Mineola, 2004. Google Scholar

[8]

D. Levhari and R. S. Pindyck, The pricing of durable exhaustible resources, The Quarterly Journal of Economics, 96 (1981), 365-377.  doi: 10.2307/1882678.  Google Scholar

[9]

D. A. Malueg and J. L. Solow, On requiring the durable goods monopolist to sell, Economic Letters, 25 (1987), 283-288.  doi: 10.1016/0165-1765(87)90229-1.  Google Scholar

[10]

D. A. Malueg and J. L. Solow, A note on welfare in the durable-goods monopoly, Economica, New Series, 56 (1989), 523-527.  doi: 10.2307/2554327.  Google Scholar

[11]

D. A. Malueg and J. L. Solow, Exhaustibility and the durable goods monopolist, Mathematical and Computer Modelling, 10 (1988), 419-427.  doi: 10.1016/0895-7177(88)90031-3.  Google Scholar

[12]

D. A. Malueg and J. L. Solow, Monopoly production of durable exhaustible resources, Economica, New Series, 57 (1990), 29-47.  doi: 10.2307/2554079.  Google Scholar

[13]

M. B. Stewart, Monopoly and the intertemporal production of a durable extractable resource, Quarterly Journal of Economics, 94 (1980), 99-111.  doi: 10.2307/1884606.  Google Scholar

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V. Y. Suslow, Commitment and monopoly pricing in durable goods models, International Journal of Industrial Organization, 4 (1986), 451-460.  doi: 10.1016/0167-7187(86)90016-0.  Google Scholar

show all references

References:
[1]

J. I. Bulow, Durable-goods monopolists, Journal of Political Economy, 90 (1982), 314-332.  doi: 10.1086/261058.  Google Scholar

[2]

R. H. Coase, Durability and monopoly, Journal of Law and Economics, 15 (1972), 143-149.  doi: 10.1086/466731.  Google Scholar

[3]

A. Corominas, Using discrete-time mathematical programming to optimise the extraction rate of a durable non-renewable resource with a single primary supplier, Oper. Res. Perspect., 4 (2017), 118-122.  doi: 10.1016/j.orp.2017.09.002.  Google Scholar

[4]

A. F. Filippov, Differential Equations with Discontinuous Right-Hand Sides, Kluwer Academic Publishers, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[5]

L. C. Gray, Rent under the assumption of exhaustibility, Quarterly Journal of Economics, 28 (1914), 466-489.  doi: 10.2307/1884984.  Google Scholar

[6]

H. Hotelling, The economics of exhaustible resources, Bulletin of Mathematical Biology, 53 (1991), 281-312.  doi: 10.1086/254195.  Google Scholar

[7]

D. E. Kirk, Optimal Control Theory. An Introduction, Dover Publications Inc, Mineola, 2004. Google Scholar

[8]

D. Levhari and R. S. Pindyck, The pricing of durable exhaustible resources, The Quarterly Journal of Economics, 96 (1981), 365-377.  doi: 10.2307/1882678.  Google Scholar

[9]

D. A. Malueg and J. L. Solow, On requiring the durable goods monopolist to sell, Economic Letters, 25 (1987), 283-288.  doi: 10.1016/0165-1765(87)90229-1.  Google Scholar

[10]

D. A. Malueg and J. L. Solow, A note on welfare in the durable-goods monopoly, Economica, New Series, 56 (1989), 523-527.  doi: 10.2307/2554327.  Google Scholar

[11]

D. A. Malueg and J. L. Solow, Exhaustibility and the durable goods monopolist, Mathematical and Computer Modelling, 10 (1988), 419-427.  doi: 10.1016/0895-7177(88)90031-3.  Google Scholar

[12]

D. A. Malueg and J. L. Solow, Monopoly production of durable exhaustible resources, Economica, New Series, 57 (1990), 29-47.  doi: 10.2307/2554079.  Google Scholar

[13]

M. B. Stewart, Monopoly and the intertemporal production of a durable extractable resource, Quarterly Journal of Economics, 94 (1980), 99-111.  doi: 10.2307/1884606.  Google Scholar

[14]

V. Y. Suslow, Commitment and monopoly pricing in durable goods models, International Journal of Industrial Organization, 4 (1986), 451-460.  doi: 10.1016/0167-7187(86)90016-0.  Google Scholar

Figure 1.  The plane $ (q, \lambda_{2}) $
Figure 2.  Control strategies
Figure 3.  Optimal stock trajectories
Figure 4.  Optimal extraction rate (left) and stock trajectories (right)
Figure 5.  Optimal extraction rate (left) and stock trajectories (right)
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