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Bags, i.e. sets with duplicates, are often used to implement relations in database systems. In this paper we study the expressive power of algebras for manipulating bags. The algebra we present is a simple extension of the nested relation algebra. Our aim is to investigate how the use of bags in the language extends its expressive power, and increases its complexity. We consider two main issues, namely (i) the relationship between the depth of bag nesting and the expressive power, and (ii) the relationship between the algebraic operations, and their complexity and expressive power. We show that the bag algebra is more expressive than the nested relation algebra (at all levels of nesting), and that the difference may be subtle. We establish a hierarchy based on the structure of algebra expressions. This hierarchy is shown to be highly related to the properties of the powerset operator.