Economists have long advocated that competition improves the efficiency of an industry. Elementary education, however, is typically a state monopoly, and where private education is offered, it is typically much more expensive as not subsidized. While there are locations where voucher programs allow parents to use the subsidy elsewhere, it is hard to find examples where public systems compete with each other.
David Card, Martin Dooley and Abigail Payne look at an example that comes close to this: Ontario, Canada, has a parallel state school system for Catholics. The later can choose where to send their children, and thus the spatial variation in the proportion of Catholics in the population can provide interesting insights in the efficiency of local schools.
The general idea is that in areas with a larger share of Catholics, non-catholic schools has larger incentives to improve to attract these children, especially if parents are not particularly attached to religious education. Using detailed data at the school and child level, this study finds that competition does indeed improve performance as measured by student achievement. One more example highlighting that school choice is good.