Earlier quoted context omitted.
Long term this may indeed happen but in the short term, it seems unlikely. Like with school vouchers, if I ran a private for-profit school and knew my students suddenly had an extra $X,000 to spend on tuition, my short term move would understandably be to raise tuition at least some percentage of that amount. If the market responds in a couple of years by forcing me to dial it back, so be it, but in the meantime I've…
In this context "long term" is the time it takes for the market to respond by building more schools or housing or whatever, which generally takes a low single digit number of years (like one). To which the response then becomes, so what? And that's assuming your argument is true, which in general it isn't. If you raise prices then some of your customers will go elsewhere. That's why sellers don't raise prices already…
Is it the case that existing private schools have much excess capacity? Frequently the argument for private schools is smaller class sizes; a rapid influx of new students without new, excellent teachers defeats the purpose. Presumably new classrooms would need to be constructed and so on.
If you receive a $5,000 voucher and your private school jacks up tuition by let's say $1,000, it is unlikely many parents are going to disrupt their routines and children's routines to reclaim that amount, especially since they are feeling relief from a net $4K tuition break. Tuition has effectively decreased, so why rock the boat?