A criterion commonly used to determine economic sustainability posits that the discounted welfare should not decrease over time. Resource management problems involving fluctuating stocks may be exacerbated by a lack of sustainable policies unless some violations of this condition are accepted. Moreover, sustainable policies may lead to the minimal welfare levels. To alleviate these problems, a new notion of sustainability is suggested. In this new form of sustainability, violations of nondecreasing welfare are accepted to a certain degree. The policies that maximize the discounted welfare under the constraint for the largest accepted violation for sustainability satisfy a dynamic programming type of fixed-point condition that can be effectively utilized for finding optimal policies. The new notion of sustainability can also be utilized in defining an index for measuring the degree of overall unsustainability of resource management problems and different harvesting policies.