Is there any work on equality in a market versus friction?
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)