A hard (theoretical) look at school choice, in the AER by Chris Avery and Parag Pathak

 What are some of the difficulties that might hamper school choice from achieving educational equality (or at least substantially reducing inequality)?  Here's a model by Chris Avery and Parag Pathak.  The theoretical intuitions of top experts in college and school assignments are the sort of thing that can keep you awake at night.  In a sentence, if school choice narrows the quality gap between the best and worst municipal schools, it may also narrow the gap in housing prices, and higher housing prices at the low end may drive poorer families to move to other school districts, just as lower quality at the high end drives richer families to suburbs with excellent schools. ("White flight" has been the subject of many papers, so the issue being raised here is that an improvement at the low end of school quality may also raise prices of less expensive housing and drive out poorer residents.)

The Distributional Consequences of Public School Choice  by Christopher Avery and Parag A. Pathak AMERICAN ECONOMIC REVIEW, VOL. 111, NO. 1, JANUARY 2021, (pp. 129-52)

"Abstract: School choice systems aspire to delink residential location and school assignments by allowing children to apply to schools outside of their neighborhood. However, choice programs also affect incentives to live in certain neighborhoods, and this feedback may undermine the goals of choice. We investigate this possibility by developing a model of public school and residential choice. School choice narrows the range between the highest and lowest quality schools compared to neighborhood assignment rules, and these changes in school quality are capitalized into equilibrium housing prices. This compressed distribution generates an ends-against-the-middle trade-off with school choice compared to neighborhood assignment. Paradoxically, even when choice results in improvement in the lowest-performing schools, the lowest type residents need not benefit."

"Our analysis contributes to a recent literature on school choice mechanisms, which has focused on the best way to assign pupils to schools given their residential location in a centralized assignment scheme. In particular, research has examined the best way to fine-tune socioeconomic or income-based criteria in choice systems. Cities have now experimented with complex school choice tie-breakers in an effort to achieve a stable balance (Kahlenberg 2003). 17 By incorporating feedback between residential and school choices, our model suggests that analysis of school assignment that does not account for possible residential resorting may lead to an incomplete understanding about the distributional consequences of school choice.

"A common rationale for school choice is to improve the quality of school options for disadvantaged students. But, our analysis shows that feedback from residential choice can undercut this approach, for if a school choice plan succeeds in narrowing the range between the lowest and highest quality schools, that change should compress the distribution of house prices in that town, thereby providing incentives for the lowest and highest types to exit from the town’s public schools. This intuition extends to the idealized case of a symmetric model of many towns and partisans, where each town adopts school choice and all schools within a given town have the same quality. Although there is an equilibrium in this idealized model where schools in all towns have the same quality, this equilibrium would likely be unstable, and instead we would expect to observe an equilibrium with differentiation of school qualities and housing prices across towns. That is, the within-town diversity observed in equilibrium under neighborhood assignment could be replicated in cross-town diversity under school choice.

A broader implication of our model is that systemic changes beyond the details of the school assignment system may be necessary to reduce inequalities in educational opportunities."