A Long-Term Mathematical Model for Mining Industries

Yves Achdou (), Pierre-Noël Giraud (), Jean-Michel Lasry and Pierre Louis Lions ()
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Yves Achdou: LJLL - Laboratoire Jacques-Louis Lions - UPMC - Université Pierre et Marie Curie - Paris 6 - UPD7 - Université Paris Diderot - Paris 7 - CNRS - Centre National de la Recherche Scientifique
Pierre-Noël Giraud: CERNA i3 - Centre d'économie industrielle i3 - Mines Paris - PSL (École nationale supérieure des mines de Paris) - PSL - Université Paris Sciences et Lettres - I3 - Institut interdisciplinaire de l’innovation - CNRS - Centre National de la Recherche Scientifique
Jean-Michel Lasry: CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique
Pierre Louis Lions: CEREMADE - CEntre de REcherches en MAthématiques de la DEcision - Université Paris Dauphine-PSL - PSL - Université Paris Sciences et Lettres - CNRS - Centre National de la Recherche Scientifique, Collège de France - Chaire Équations aux dérivées partielles et applications - CdF (institution) - Collège de France

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Abstract: A parcimonious long term model is proposed for a mining industry. Knowing the dynamics of the global reserve, the strategy of each production unit consists of an optimal control problem with two controls, first the flux invested into prospection and the building of new extraction facilities, second the production rate. In turn, the dynamics of the global reserve depends on the individual strategies of the producers, so the models leads to an equilibrium, which is described by low dimensional systems of partial differential equations. The dimen-sionality depends on the number of technologies that a mining producer can choose. In some cases, the systems may be reduced to a Hamilton-Jacobi equation which is degenerate at the boundary and whose right hand side may blow up at the boundary. A mathematical analysis is supplied. Then numerical simulations for models with one or two technologies are described. In particular, a numerical calibration of the model in order to fit the historical data is carried out.

Keywords: Heterogeneous agents model; Mean field games; Master equation; Hamilton Jacobi equations; Viscosity solution; Model calibration (search for similar items in EconPapers)
Date: 2016-12-01
Note: View the original document on HAL open archive server: https://hal.science/hal-01412551v1
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Citations: View citations in EconPapers (1)

Published in Applied Mathematics and Optimization, 2016, 74 (3), pp.579-618. ⟨10.1007/s00245-016-9390-0⟩

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Persistent link: https://EconPapers.repec.org/RePEc:hal:journl:hal-01412551

DOI: 10.1007/s00245-016-9390-0

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