I am curious whether there are any systems that let parents choose teachers within a school — and let teachers admit students they will take. So they can make m

I am curious whether there are any systems that let parents choose teachers within a school — and let teachers admit students they will take. So they can make more if they are in demand — and that students that are "problems" will result in greater income for the teachers that take them. A kind of "choice-based" market system within a school.