The PED of an independent school education

We are looking to understand the PED of independent school fees. There is evidence of an independent education as a veblen good, and indeed the schools with the highest fees are those that are growing the fastest. The challenge is what does that mean for the the rest of them? At what point is a veblen good 'broken' and how do you come back from it?

We're looking for someone to research and write a paper on that topic for us to publish, preferably in September 2022.