Two producers own finite resources of a commodity (for example mineral water). It costs nothing to produce, and there is an alternative technology (say, desalination) which allows production at a nonzero cost.
The producers compete in a Cournot market and the goal is to determine their optimal production strategies in a continuous-time Markov-perfect setting, including the option of producing using the alternative technology. The resulting differential game leads to a strongly nonlinear pair of coupled partial differential equations with some unexpected properties. I shall describe the model and our work on it (which is very much in progress, with many open issues). Joint work with Chris Harris (Cambridge), Ronnie Sircar (Princeton) and Jeff Dewynne (Oxford).
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