This paper studies the optimal extraction of a non-renewable resource under uncertainty using a discrete-time approach in the spirit of the literature on precautionary savings. We find that boundedness of the utility function, in particular the assumption about U(0), gives very different results in the two settings which are often considered as equivalent. For a bounded utility function, we show that in a standard two-period setting, prudence is no longer sufficient to ensure a more conservationist extraction policy than under certainty. If on the other hand we increase the number of periods to infinity, we find that prudence is not anymore not anymore necessary to induce a more conservationist extraction policy and risk aversion is sufficient. These results highlight the importance of the specification of the utility function and its behavior at the point of origin.