Personally I wouldn't; but then I also tend not to negotiate / just roll over and take what I'm offered since generally I care much more about the interestingness of the work & the people I'd be surrounded by each day (though I'm getting to the age where I may start a family; so hopefully my subconscious' priorities will change a bit in my wallet's favour).
However, I suspect that many people do, as this avoids being seen as uncooperative, and suggests that people were willing to pay you more so presumably you're worth it. I doubt anyone would check (likely it's illegal to), so this seems a no-brainer for those ethically unencumbered by this option.