In this paper, we study a dissipativity property which was recently used in several results on economic model predictive control to ensure optimal operation of a system at steady-state as well as stability. In particular, we first investigate whether this dissipativity property is not only sufficient, but also necessary for optimal steady-state operation. In the most general case, this is not true; nevertheless, under an additional controllability assumption, we show that dissipativity is in fact necessary. Second, we provide a robustness analysis of the dissipativity property with respect to changes in the constraint set, which can result in a change in the considered supply rate.