If a graduate student knows that they are on the lower end of the stipends offered to their cohort, and if they are performing as good or better than their other cohort members (as measured by grades, number of publications, number of current collaborations, etc.), then is it acceptable to ask for a higher stipend? If so, how would you go about this?
Yes, you can certainly ask that the department reconsider the amount of your stipend. However, you should also prepare for the case where they do not adjust it.
Normally, I would start by talking with your advisor, as she will most likely have to be on board for any increase to be successful. Then, I'd approach either the graduate student advisor orthe graduate student administrator for the department. There may already be rules and policies in place for how to request a stipend increase. If that doesn't work, the next steps are probably to contact the chair of the department and the dean for graduate students (or other appropriate officer at your institution).
However, depending on where your funding comes from, it may or may not be possible to get an increase. For instance, NSF or NIH stipends are normally set at a uniform level across a college or university. It's also possible that the students with higher stipends have outside funding which reduces the actual cost to the college. Finally, your advisor may not be able to afford a large stipend increase.
I'd recommend talking with your advisor or another mentor to see whether they have any information about how stipend decisions are made behind the scenes. The $6k disparity sounds high to me, and it's really awkward if it gets decided early on and then never adjusted, since how impressive someone seems on entering is imperfectly correlated with their accomplishments four years later. On the other hand, no department wants to individually negotiate each stipend on a yearly basis.
The first question is whether you are proposing an isolated fix for yourself (with the theory being that the current system is generally good, except for the poor result in your case), or a systemic change. If it's the latter, then it's probably best if you can get a faculty member to take up the cause, since they can try to build widespread faculty support in a less confrontational way than graduate students could. The best chances of change are probably if there's some clear unfairness or disparate impact. For example, if the current system is on average giving lower stipends to female students, not explicitly because of gender or performance but (let's say) because they negotiate less vigorously before signing, then I'd bet the administration will be eager to make changes. On the other hand, other large-scale changes may take more time and political finesse.
If you are just dealing with your own case, then you could still benefit from an advocate but can more easily try to make the case yourself. How tricky it is depends on what sort of argument you are making:
You might argue that your stipend was wrongly set from the beginning. This will be a difficult argument to win unless you can point to something objective, because the department will not want to set a precedent that might encourage half the grad students to try making the same argument. However, it can't hurt to try if you do it tactfully (nobody likes to be told that they screwed up in the past, so you should treat this gingerly).
You might argue that your initial stipend was reasonable, but the circumstances have changed dramatically. This may be your best shot, but again you need an objective argument that distinguishes you from the other students who would also like increased stipends.
Graduate students are not expected to have a lot of money. Just enough money to live, plus a beer or two at the weekend, is the general consensus in Biology. In my PhD intake, we were all placed on 50% contracts (approx. $1300), while the following year's intake all got a 60% contracts (approx. $1400). This came after years of student protests by current and former Max Plank graduates. And $100 a month goes a long way - at the very least I could get some health insurance, so I asked a few times if could be bumped up to the 60% contract, and the response from the Max Plank was a resounding "NO!"
But then one day I received an electricity bill for $2000, which was the sum of the last two years bills combined which I thought was included in the rent but now apparently had to pay. I went to the institute's financing department to ask for help or legal advice, but again I received a definite "not our problem."
After this I became quite poor. I sold what luxuries I had acquired over the last 10 years, I started working evenings as a freelance programmer, and I became quite depressed. I was "visibly poor" so to speak. Poor, depressed and very very tired.
That year my old contract was canceled, then a new contract given to me at 60% rate.
So I see the moral of this story as, you don't have to be nice to your sheep, but dead sheep are no use to anyone. Use the little leverage you have by saying to your institute "Pay me more money or I will have to divert my time and energy into something that does", and you will see the true value of your meat in the academic marketplace.