Berne and Stiefel (1984) developed a conceptual definition of vertical equity and then, in applications to district-level data in Michigan and New York, used weights for students of various types to adjust for differences among the students. They also assessed the effects of inter-district price-level differences.29 In Disparities in Public School District Spending: 1989-90 (Parrish et al., 1995), the authors used cost of education indexes, estimated by McMahon and Chang (1991) 30 for all districts in the United States, and weights, based on studies of special education students (weight equals 2.3), average federal Title I allocations for poor children (weight equals 2.0), and limited English proficient (LEP) students (weight equals 1.2). School finance legislation in the states has addressed vertical equity by weighting students according to needs or costs (37 states in 1993-94), by funding special needs programs categorically, or both (Gold et al., 1995).

Recent empirical work by several economists has resulted in cost indexes based on the economic definition of costs—resources that are consumed to produce given levels of outputs (Downes and Pogue, 1994; Duncombe et al., 1996; Ladd and Yinger, 1989, 1994). The researchers have estimated district-level cost functions that control (or statistically hold constant) outputs such as test scores and dropout rates (and sometimes efficiency levels as well).31 These empirically estimated cost functions then indicate the additional resources needed to achieve output levels due to external (or environmental) factors outside a district's control (such as percentages of students in poverty, with disabilities, with limited English proficiency, or living in single-parent households).32 The cost functions in turn can be converted to cost indexes, usually with a statewide average of 1.0, showing relative costs by district. Such indexes are empirical efforts to adjust expenditures for costs of obtaining given outputs and are thus useful in the measurement of vertical equity. The chapter by Duncombe and Yinger (Chapter 8 of this volume) is an example of this method.

Vertical equity is an appealing concept to many analysts because it takes into account differences among pupils and (implicitly) outputs. Federal legislation, such as Title I and the Individuals with Disabilities Education Act, is based on a vertical equity idea. Beyond some basic agreements that some children need more resources, however, there is little agreement on how much more. This absence of agreement goes back to the research findings on production functions, which have not been able to pinpoint exactly where and how more resources will result in more achievement.

Adequacy

A Nation at Risk (National Commission on Excellence in Education, 1983) changed the nature of the debate about the goals of public education, shifting it for several years from its preoccupation with equity to concern about achievement, especially achievement of U.S. students compared to those in other countries



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