Harris and Ph.D. student Gabriel Olivier dug into this question in a recently published working paper that examined eight states that had expanded to near-universal eligibility.
Harris and Olivier found private school enrollment rose faster in those eight states than in non-choice states—but by only 3-4 percent more on average. The enrollment bump was most pronounced in small, non-Catholic religious schools—likely those that were already established, had the most room to grow, and had relatively low tuition that vouchers or education savings accounts worth about $7,000—close to the per-student amount provided in a number of states—would largely cover.