Is there evidence that elite private schools are cheaper than state schools for a significant portion (or better yet, majority) of the population?
Given a median household income of ~$50k in the US, I'd say yes, elite schools are almost unequivocally cheaper across the board than state schools. Common misconception held by most.
Part of this is Age: "Households headed by persons between the ages of 45 and 54 had a median household income of $61,111 and a mean household income of $77,634." http://en.wikipedia.org/wiki/Household_income_in_the_United_...
And part of this is education: http://en.wikipedia.org/wiki/File:Income_Education_91_to_03.... with a higher percentage of people with a collage degree sending their children to collage.