I don't have recently written books and studies to support what I'm about to say about rising costs, just some experience with my own university's budgetary choices. I think Mark's statement about administrators plays a role, though not a determining one. And I don't believe that competition to get into the best schools is significant. Unless you are one of the handful of very prestigious institutions, the competition is all in the other direction.
Over the last 20 years there has been a substantial change to the fixed costs of running universities. To give an example that is not determinative, but also not trivial, when I was hired, 30 years ago before the age of PCs, one was given an office (a shared office) and a desk. If you wanted a typewriter, you brought your own in. Now all tenure-line faculty members have their own offices, each one equipped with an up to date PC. They are updated every 3-5 years. The University has a very complicated campus network that supplies us with our own clouds, plus various teaching programs such as Blackboard. One can order books from our library and a consortium library through a library program, and if one orders articles, they are emailed to you. Students are guaranteed access to computers because they are expected to be reachable through email, Blackboard, etc. I am only scratching the surface of all this. Needless to say, we have increased our IT employees from 0 to multitudes in those years since all this stuff demands considerable tech support. All of that is a substantial addition to the University budget that never even existed 30 years ago.
Faculty salaries went up a lot in the 70s and 80s. After that, they mostly kept up with cost of living. But the cost of research faculty has gone way up. At research universities, we teach fewer courses and have more leave time through access to various granting organizations. Time away from teaching is the direct cost of research, and it is costly. Tuition, at teaching colleges and community colleges at which faculty teach 4 large courses a semester is much less expensive (and frequently, though it pains me to say it, the education one gets there, though basic, is quite strong).
Any of you who were undergraduates 30 years ago or more can also tell where some money is going by walking into student housing and cafeterias. The mass produced mystery meat dinners of our nostalgic memories no longer exist. The eateries aren't three star restaurants, but they are the equal of most budget restaurants. Living conditions have also improved considerably over my freshman dorm, which had little over a minimum security prison cell.
I won't even start on the cost of bringing classrooms up to date for the expectations of current teaching methods. Suffice it to say that one thing one sees less of is audio-visual people since one's computer, plugged into the classroom can do far more than they do.
Multiply all this by everywhere you look (and especially science labs, VLM). The expectations on institutions that charge the kind of tuitions talked about here are pretty high. I don't know if it's worth it in terms of what students get educationally, but the only way to change it is to change the expectation of what goes on in the student experience of costly private universities.