A finite group of agents share a (one output) production function. A cost sharing rule allocates the total cost among the users for every conceivable profile of output demands. We investigate the space of possible cost sharing rules from an axiomatic perspective. We provide two characterizations of average cost pricing, one based on the axioms of Additivity and Monotonicity (both with respect to the cost function), and the other based on the axioms of Additivity and a version of Consistency. We also provide a characterization of serial cost sharing based, essentially, on Additivity and a Free Lunch axiom.