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I am applying to a few Ph.D programs in mathematics and I found that most schools only admit about 5% of applicants. I know most programs fund students; some schools give stipends to all incoming students; some schools give stipends to only a fraction of students, but no one pays out of pocket.

Why not admit a larger percentage and have those pay their way through?

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    Think of the students you would attract with such a policy. Mostly people who couldn't get funding anywhere, i.e. relatively weak students. – user37208 Nov 18 '15 at 21:30
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    no one pays out of pocket --- Is there any evidence to justify this claim? Looks a bit anecdotal to me. – Dmitry Savostyanov Nov 18 '15 at 21:36
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    @DmitrySavostyanov I'm speaking about most programs. Not all. Basically, top programs. I have been told some weaker programs do have students pay. – Al Jebr Nov 18 '15 at 21:51
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    I'm not sure that "most schools" admit such a low percentage. Most top schools, maybe. Our admittance rate is much higher, and I remember someone telling me their acceptance rate is approximately 100%. – Kimball Nov 19 '15 at 1:55
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    Some might say that PhD admissions are too high, particularly with an eye on the potential job market within academia that these students are presumably aiming to compete in. There are many more graduating PhDs now than could become professors today, and pretending that new opportunities will come along in the future is arguably running a pyramid scheme. (Not saying I agree, but the viewpoint is out there and there is something to those views.) – E.P. Nov 19 '15 at 2:07
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Money is only a part of equation in academia. Other constrains are the following:

  1. Supervision: A PhD student should be supervised, which requires a commitment of a Professor and often one or two co-supervisors. Supervising a PhD project is very time-demanding process, and even the most experienced professors can not efficiently manage ten or more PhD students without compromising the quality.
  2. Space: A PhD student needs time in a lab, access to computing resources, or at least a desk allocated for them. Funny enough, even the latter can be a real issue and a good reason why universities can not hire twice as many PhD students as they like.
  3. Success rate: Universities are keen to improve their position in various ratings, especially international. The success rate of PhD programs is a very important factor, which directly impacts the funding which university gets from funding bodies and (sometimes) government. Hiring a student who is happy to pay for their tuition is of course nice; however, if they do not complete the program, more money can be lost in the following years as the university goes down the ratings.

This can explain why the numbers of PhD students are not as important as their skills, excellence, and commitment.

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    I like this answer! One thing I might add would be funding - it is true that a self-paying PhD student would not cost as much as one who had a fellowship, but for hard sciences, the cost of a stipend and tuition is often a small part of the overall cost of the research program. If there is not enough grant money to fund projects for students, then even if they pay for themselves there will not be any opportunities. I think this explains why humanities programs often do have self-paying graduate students, but science programs do not. – thomij Nov 19 '15 at 14:56
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As with several other questions on this site, my suggestion is to flip your question around, and ask yourself "What would the department gain by admitting those students?"

Given your suggestion that these students pay their own way, presumably your answer is "Money". But I'd suggest to you that PhD students are an intensely inefficient way to generate income for a department for a number of reasons:

  • PhD classes are small. At my former institution, even for the classes that were required for PhD students, the class size was easily dwarfed by most undergraduate classes. The ratio of students paying tuition to instructors being paid to teach isn't particularly favorable.
  • This ratio becomes worse when, as @DmitrySavostyanov points out, you factor in that PhD students, to be successful, require more than just admission. They need a faculty advisor, and that faculty advisor has to actually do things, rather than just sign off on your course plan for the year. They'll need a committee. They may or may not need resources to actually work on their projects, and these resources are considerably above what might be needed for an undergraduate.
  • It also becomes worse when you keep in mind that at many institutions, tuition scales by the number of classes you take. So as PhD students enter the "post-coursework" period of their career (and the resources they demand increases) they're actually paying less tuition, not more.

Beyond that, as some people have noted, failing to get a funded position anywhere is a pretty strong signaling mechanism that, regardless of how much they might think this is a good idea, the applicant may want to consider other options. Departments care about their reputation - both for ego reasons, and very practical reasons like "This effects our faculty and student's job prospects, funding opportunities, etc." Following a path where you admit tons of students who have a higher probability of being unqualified risks being thought of as a "diploma mill", and that's...pretty much a disaster for a department's reputation.

There's also simple self-interest - while in the short term flooding the pipeline with PhD students might be somewhat beneficial, in the long term there's no benefit to having a number of graduates well above equilibrium.

It should be noted however that in several fields, Masters degrees, which are much less resource intensive than PhDs, are indeed used as income generating programs.

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I currently have a weak PhD student, & I can tell you that it is extremely stressful for all involved, student & supervisors alike. Admitting weak students will either reduce the quality of the PhDs, or require the supervisors to invest an unsustainable amount of time in training the student or, as is currently the case for my student, result in a very confused, stressed, tired student who is barely hanging on, barely understanding, but not wanting to give up after the initial time investment, not to mention moving countries.

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Why doesn't an artisan's guild accept any talented artisan from off the street? There isn't enough demand for their trade to keep all their members working if they can't keep a grip on their numbers.

While the amount of funding a program can support certainly factors in, some would argue that even current levels of Ph.D. students are too big. If you think about each student a mentor spits out being a replicant, how many times can a prof replicated before there are no more positions? Hence -- the vicious postdoc circle.

Until we get a grip on how best to get these graduates into non-academic careers, the numbers need to stay somewhat low.

Everything else sort of aligns with this. The number of universities, the space that they have, the funding available, all have reached some sort of equilibrium related to the size of the research body society is willing to support. To have more practitioners than can be supported is a bad outcome.

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