MODELS OF OPTIMUM USE OF UNCERTAIN RESERVES OF EXHAUSTIBLE NATURAL RESOURCE

The consumption of uncertain reserves of an exhaustible resource over a planning period is studied. In addition to reserves which are known at certain date other reserves become known as time passes. The stochastic discovery process is a discrete time absorbing Markov chain.;A time invariant utility function gives the value of consumption at each period and the optimization criterion is the expected value of the sum of discounted utilities over the planning horizon.;Dynamic programming formulations of three basic models are analyzed in order to establish properties of optimal value functions and of optimal policies.;In the first model the discovery process is autonomous. The optimal value function is shown to be increasing, concave, and differentiable. One of the optimality conditions can be interpreted as establishing that the expected price of the resource must grow at a rate equal to the exogenous interest rate, a stochastic form of Hotelling's basic result. Price movements after discoveries are discussed. Several bounds on the optimal policies are established. In particular, it is shown that if the derivative of the utility function is convex then optimal consumption is less than it would be if uncertain reserves were substituted by certain reserves equal to the mean of the probability distribution of total unknown reserves.;The exploration level is also a decision variable in the second model. In this model the speed of the discovery process can be controlled. It is shown that in this case the optimal value function does not need to be concave or differentiable. Optimality conditions under assumptions of differentiability are derived and it is shown that expected prices must grow at a rate equal to the interest rate. It is shown that for a wide range of forms of the functions which define the cost of exploration and the probability of discovery, the optimal exploration effort is either zero or at its maximum value. Sufficient conditions for positive exploration before exhaustion are examined.;The third model is a two goods economic growth model. The exhaustible resource is used in the production of capital and the discovery process is autonomous. The optimal value function is shown to be increasing, concave, and differentiable in the stocks of capital and of known reserves. The directions of change of optimal consumption and of optimal resource use as functions of the levels of the stocks of capital and of known reserves are analyzed.