Resource depletion, national income accounting, and the value of optimal dynamic programs

Under assumptions satisfied by many economic problems, I derive a fundamental new equation for the time rate of change of the optimal value function of any optimal control problem. This is then applied to Hotellingʹs model of the resource extracting firm. The precise differences between rent depreciation, and depletion charges are discovered, the flow and stock price appreciation rates are distinguished, and novel characterizations of mine value are derived. Most importantly, the correct contribution of mining to net national product (and to sustainable development) follows. In perfect foresight equilibrium, competitively managed mines can appreciate while being exhausted.