We know they had an economy of some sort (Valorum provides an image in their answer with evidence), and we also know that there must have been some form of capitalism on Earth around/up to the time of departure (Buy n Large had bought most of the world's countries and ran pretty much everything). This is given further credence by the Axiom being the "jewel of the fleet", which could imply that the wealthy could afford access to better ships than others.
This is merely speculation, but given that robots do virtually everything (and BNL already owns everything), I would venture to say that anything remotely capitalist post-departure is out. Combine this with the fact that they do use money, and we're looking at an economy that likely gives a monthly wage to everyone, who can then choose what to spend it on (to prevent people from just hoarding everything they want for "free").
I'm sure there's a name for this kind of future hypothetical economy, but I haven't found it in my searching. It has aspects of post-scarcity combined with basic income of some sort, but it's not socialist/communist, as there is a sense of property/ownership, the resources aren't communally managed, and there is no labor involved.