Why are Ph.D admission numbers low? 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?
Money is only a part of equation in academia. Other constrains are the following:


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. 
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.
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.