> "I can do it cheaper" implies that I can get my costs down lower.
Zero or negative net costs are lower than positive net costs. So "I'll do it for free" (or even, "I'll pay you for the opportunity to do it") are instances of "I can do it cheaper."
> "I can do it for free" means that I'm ignoring any of my costs.
No, it doesn't; assuming rationality it means that the net costs are nonpositive, not that they are ignored (which would be irrational.)
Of course, it could be the result of ignoring some or all costs, but so could "I can do it cheaper" when there is still some charge for the work. There's nothing special about the zero charge case.
No, it just means that monetary compensation is no longer the largest, or in this case of any consideration. There are other costs, such as time and effort, but there are also other profits, such as reputation and personal satisfaction. Compensation doesn't bottom out zero, it can also go negative. In this specific case, if there were two such people bidding against each other for the 18F contract, one or both of them might have been willing to pay a few dollars for the privilege of working on the project. I got a feeling from this post that the pair of them that did this contract would have had little problem throwing $10 at 18F for the chance to do the project.
Zero or negative net costs are lower than positive net costs. So "I'll do it for free" (or even, "I'll pay you for the opportunity to do it") are instances of "I can do it cheaper."
> "I can do it for free" means that I'm ignoring any of my costs.
No, it doesn't; assuming rationality it means that the net costs are nonpositive, not that they are ignored (which would be irrational.)
Of course, it could be the result of ignoring some or all costs, but so could "I can do it cheaper" when there is still some charge for the work. There's nothing special about the zero charge case.