equity and did not measure vertical equity or wealth neutrality. They concluded that reforms did improve (horizontal) equity.

Most recently, horizontal equity has been studied at the school level. Berne and Stiefel (1994) looked at general education funding in New York City elementary and middle schools. Rubenstein (1998), Moser (1996), and Sherman (1996) assessed the degree of horizontal equity in three other cities as part of a study of school-level budgeting in large cities.

The staying power of the concept of horizontal equity is unusual, especially since it could not reasonably be applied to outputs or outcomes. It is most useful as a concept for inputs that involve equally situated students. Such application requires that funding streams meant for compensatory or other purposes (such as gifted programs) be separated from streams meant for all students so that intra-group equality can be measured. To the extent that horizontal equity has been used extensively by analysts, lawyers, and legislators, it is important to evaluate the effects of trying to achieve it, as reviewed by Evans et al. in Chapter 3 of this volume.

Vertical equity, a children's concept that has been used both ex ante and ex post, specifies that differently situated children should be treated differently. Analogous to the challenge for users of horizontal equity, users of vertical equity must identify "differently situated" students. This identification is usually done, implicitly or explicitly, by identifying groups of students who differ in their needs for quality or use of inputs to achieve defined levels of outputs. Thus, in concept, vertical equity ties input equity to output equity. When inputs are "adjusted" for the costs of educating various groups of children, as is often done when vertical equity is measured, the adjustment is meant to indicate the amount of additional resources that are needed (higher costs that are incurred) to bring some students to given output levels. Such adjustments are empirically difficult to execute. For example, many might agree that learning disabled students need more resources, but how much more is not clear. One must define the outputs that are the goal (e.g., minimum competency, maximum potential, sufficient level, the point where the value of the marginal gain in output equals the marginal cost of resources), and all require knowledge of the quantitative relationship between inputs and outputs. Often we do not know enough about these quantitative relationships to know how to adjust resources.

The state and its districts have been the traditional units of analysis for vertical equity analyses. Now, as with horizontal equity, school-level data that allow students and funds to be separated into general education, special education, and other streams are resulting in an application of the concept at the school level, where the separate streams can be linked to the "differently situated" students.