We characterize the optimal behavior of a competitive extraction firm facing an arbitrary time profile of the resource price, when unit extraction costs depend on cumulative previous extraction but not on the quantity currently extracted. As is well known, the problem stated in its natural form has no solution. Spence and Starret[Spence-Starret 1975] suggested to use a most rapid approach, where arbitrary bounds constrain the rate of increase or decrease of the stock. We propose an alternative approach where finite extraction may take place at any date, the cost of which being the cumulative sum of marginal extraction costs. As in Spence-Starret, we reformulate the objective function but provide an economic interpretation of the new objective function in termes of user cost of natural resources and true instantaneous profits. We then provide a complete characterization of possible extraction paths.