What is it based on?

The model provides a plausible depiction of reality through the previously described equations, representing families, schools, districts, and states. Some data for these equations are provided. The data establish things like the income distribution among families in each of the 20 cities she looked at, the number of Catholic families, housing quality in the different neighborhoods and districts, and how much money per student the state gives to public schools. For this information, Ferreyra relied on several sources including the 1990 census, a general social survey that tracks religious preferences, and a guide to school district spending.

Other parameters of the equations are missing. Some of them correspond to more ethereal traits that drive families—how much a family values a religious education or how much they value house quality. Ferreyra's task was first to quantify these traits.

To determine the values, Ferreyra tried thousands of sets of plausible hypothetical parameter values, running the model of each city to equilibrium for each set of values. The data that came out after the model reached equilibrium included new figures for property values, school spending, and income—as well as some data that were not provided, such as the number of students in private nonreligious schools and the number of students in Catholic schools.

Ferreyra compared these data to real-world statistical data to determine the quality of the hypothetical values she had inserted to represent the parameters. Choosing values that yielded equilibrium conditions that best matched the real-world data, this process gave her a glimpse of families' thinking and the situations that produced varying levels of school quality.

Then, using those optimal values, she added features to the model to examine the same economic system, now with vouchers. Having already quantified the unknown parameters, she could be confident that the resulting picture of families using vouchers would be accurate.

Ferreyra's model is the progeny of a model built in the mid-1990s by Tom Nechyba. Nechyba began the model as part of his dissertation at the University of Rochester, introducing schools to the mix later. Ferreyra added the functions that track families' religious interest, represent the distinction between Catholic and nonreligious private schools, and incorporate vouchers.

These additions are significant because of the prominence of religious schools. According to Ferreyra, about 85 percent of enrollment in private schools in grades 9 through 12 is in religious schools. About 60 percent of that enrollment is in Catholic schools alone.

"Maria fleshes out that it's not all about peer quality or dollars or a school quality hierarchy. Yes you care about peers, yes money, but many also care about religion. And they care about all these things to varying degrees," says Nechyba, now an associate professor of economics and public policy studies at Duke University.

Ferreyra also estimated the model instead of simply calibrating it. Calibrating would have involved looking at a single city. It would have still allowed her to derive optimal values for the model's parameters and to compare the values' implications to real-world statistics, but it would have also lacked scope. By estimating the model and looking at 20 cities, she proved the reliability of the model in a more general way.

"In estimating," Ferreyra says, "you use more data, search more exhaustively for the optimal parameter values, and provide a measure of how good the final parameter values are. You get a much broader sense of quality. And since those values are used to simulate voucher experiments, we want to ensure their quality as much as we can."