This paper addresses the problem of determining an optimal rate of extraction of a stock of natural resource when the total resource available is uncertain. A distribution function of total resource available is given. Both discrete time and continuous time versions are considered. The generalized search optimization technique of W. R. Stromquist and the second author [Math. Oper. Res. 6, No. 4, 518-529 (1981; Zbl 0511.90087)] is employed to find necessary and sufficient conditions for a plan to be optimal. For continuous-time plans without constraints a result of G. C. Loury [Rev. Econ. Stud. 45, 621-636 (1978; Zbl 0435.90042)] is extended to a more general return function; a finite horizon is also allowed. The result has a nice economic interpretation: the discounted marginal return at time \(t\) of an optimal plan equals the expected discounted average return at the time the resource is exhausted (given that the resource lasts beyond time \(t\)), plus the marginal loss incurred from extracting when there is a higher level of depletion. Finally, in the discrete time case, an algorithm is designed to find the extraction plan and some numerical examples are presented.