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I'm wondering to what extent money is a factor in phd admissions? For example, does an admissions committee limit the number of students admitted in a year depending on how much the department can afford that year? and approximately what percentage of the departments funds go to graduate funding(I am asking mostly in regards to US universities that rank in the top 20 for math.)?

And also, are qualified students who are less expensive more likely to get admitted because it wont cost the department much, if anything at all, since at many institutions certain groups receive their funding from the central administration rather than the department(such as women and minorities)?

Below is a source where the writer says

Even as faculty members on committees expressed philosophical commitment to diversity, Posselt observed financial motivations at play. Some universities offer extra funds for minority graduate students, so that a fellowship might be paid for from general university funds and not departmental funds. Where such incentives exist, they appear to have a strong impact, Posselt writes.

and in What do admission committees look for in a diversity essay?, Paul Garrett says:

being a woman in a STEM field (Science, Tech, Engineering, Math), or of ethnic origin other than northwestern European . . . opens certain money-pots to both the department and to the individual.

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    You have too many questions and most of them depend on the country and/or university. There are also a few misconceptions in your post regarding funding and costs of a PhD. – 101010111100 Jan 5 '18 at 19:38
  • I sort of deliberately asked many questions in my first paragraph not to elicit a response to each question but to get a better general understanding of how money factors into phd admissions. What misconceptions? – john fowles Jan 5 '18 at 19:41
  • "... since at many institutions certain groups (women and minorities) receive their funding from the central administration rather than the department." I have never heard of this practice, could you name such an institution? – astronat Jan 5 '18 at 19:42
  • I edited my answers with credible sources where I got the information from – john fowles Jan 5 '18 at 19:47
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    Anecdotally, I don't think "cost" enters too much into US graduate admissions decisions, with the possible exception of international students from certain countries. The additional "money-pots" might refer to designated university fellowships for URGs rather than standard grad funding. – Elizabeth Henning Jan 5 '18 at 20:25
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Top-ranked schools generally commit to offering accepted candidates a certain level of support during their graduate candidacies. Therefore, finances must play a role in how many students they can afford to admit: if you only have sufficient financial resources to support 30 students, and 40 enroll, you're in for a bit of trouble. Thus, the students who are most desirable are those who can provide their own funding, in part or in whole, via external or internal fellowships or "matching funds" from the department. This frees up money that can be used on other students. (I would imagine that in math, much of this funding is awarded in the form of teaching assistantships rather than RA-type positions.)

This makes deciding how many students to admit a difficult challenge. Top students will have competing offers, and thus universities need to admit enough students to make sure they fulfill their research and teaching needs, but need to not be overly generous in making offers, in case yields exceed expectations.

As far as diversity goes, I agree with the assessments in the article you linked to: diversity only comes to play after candidates have cleared the qualification bar.

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