How does it work?

To answer these questions, Ferreyra used what is known as a general equilibrium model to isolate the system's main agents. In Ferreyra's case, agents included families, schools, school districts, and the state government. Mathematical equations represent what these agents are seeking, under what circumstances they will pursue those goals, and how one agent's pursuit influences another's situation.

The thorny equation that determines a family's behavior, for example, includes varying interest in school quality, housing quality, schools' religious orientations, and consumption of all other goods. This equation is known as their utility function. Another equation represents families' budgetary constraints. Families' actions are ruled by their attempts to get the most of each aspect of the utility function within their budget. If they have a high desire for a good public school, they'll forgo a better house in a lesser school district. If they want a Catholic school, they'll put money into that instead of a mansion. If vouchers are available, a family may not have to worry about the school district they're in when house shopping.

Another equation determines school quality, which is a function of peer quality and dollars spent per student. These factors are, in turn, determined by other, ever-changing factors. In this simplified model, peer quality equals the average income of families whose kids attend the school, and public schools' spending is funded with property taxes, which the districts' families vote on.

Until all the families find their perfect situation and the housing markets settle—until the model reaches equilibrium—the families are on the move. And the computer model chugs away, tracking the fantastic interplay among the various equations.

"Imagine you dropped the cities' citizens from an airplane," Ferreyra says. "The wealthy who don't have an interest in religious education all move to the district with the best housing, and, because public-school spending depends partly on property taxes, they end up with public schools that spend the most per student. In the meantime, other people are trying to find the best place for them. Most people will not be happy where they land. And everyone is voting on taxes and changing the nature of all the districts and their public schools."

It's not just a matter of people finding their place; they're changing the nature of the place as they go. "The only thing that stays constant is housing quality. Everything else is moving parts," Ferreyra explains. "This is precisely why we want a general equilibrium model. It allows us to look at different markets—the housing and schools markets, in this case—and how they change as people move, choose schools, and vote on property taxes. With the model, we get to see how the markets change when a large fraction of families are involved—unlike current real-world programs, which happen at a much smaller scale."