I am not quite getting the connection to saving money, unless you are trying to quickly produce compost for the garden instead of buying bagged compost?
Anyhow... the answer is "neither, because it totally depends on the AMOUNTS". The thing that controls speed of composting is the size of the particles (pretty comparable among all 3 materials) and the carbon to nitrogen ratio in your compostpile, i.e. how much poo in relation to how much bedding material.
The fastest route to reasonably-finished compost is to use methods that result in relatively little bedding amongst the poo in your pile. If you do the "clean the whole coop out every week (or whatever)" method, you virtually always end up with quite a lot too much bedding (too much C, too little N) for optimum composting. Using a droppings board and spot-cleaning makes it easier to achieve a lower C:N ratio and makes it more controllable-by-you.
All things being equal you do not need quite as much poo for a given amount of hay or straw as you do for a given amount of shavings, in order to achieve optimal C:N ratio. However IMO this is more or less academic because if you can control the am't of bedding material in your pile, you can cope quite well with ANY of those materials and they will compost about equally fast; and if you *don't* control the am't of bedding material in your pile, it really doesn't matter much which you use as ALL of them will compost more slowly.
(Note that grass or mix hay often has weed seeds in it, and you can't assume your whole pile will compost hot enough to kill them; also a reasonable argument (IMO) can be made that it is more appropriate to use NONfood things like straw or shavings as bedding, rather than hay which could be feeding someone's animals.)
Not sure if that answers your question?
Good luck, have fun,
Pat