Is there any work on equality in a market versus friction? This is a question about educational economics.

There is a phenomenon when studying countries like the USA and Chile (that have implemented voucher systems) that reducing the friction in a "marketplace" in education (e.g. choosing a school) results in a reduction in equality. It results in stratification of students into schools that match with social class.

It seems fairly obvious to me that this is a phenomenon that I'm sure has been studied or talked about in economics in a general way - I just don't even know where to look.

I'm looking for research into the idea that where there is some kind of inelastic good (e.g. health, housing, education) then reducing friction in a market will always have an effect of reducing equality.

Has anybody studied this? Any pointers to literature? Or if I'm just wrong in my observation of this being "fairly obvious" then I'd love to hear it.
The key theoretical work, at least in the area of education, comes from Milton Friedman and his &quot;Theory of School Choice&quot;. Important references are:

Friedman, M. (1962), The Role of Government in Education. In Capitalism and Freedom. Chicago: University of Chicago Press.
Friedman, M. and Friedman, R. (1980). Free to choose. New York: Harcourt
Brace Jovanovich.

Check here for a not so old review of the literature on School Choice. Here is an interesting analysis of the Chilean education system, in the light of Friedman's ideas. Friedman had the idea that competition would increase quality across the whole spectrum of schools:

the quality of    all    schooling    would    rise    so    much    that    even    the    worst,    while    it    might    be
relatively    lower    on    the    scale,    would    be    better    in    absolute    quality”    (Friedman    and    Friedman 1980,    170)