Since the main purpose of this study is to explore the relationship between district size and spending, a variety of different functions of size are discussed here and then compared with one another in the following section, “Empirical Results and Analysis.”

The simplest model is of course a linear one, where ƒ(Size) = Size and the enrollment term thus reduces to b * Size. This linear model assumes that increasing district size either always increases spending (if the coefficient b is positive) or always decreases it (if b is negative), and that the rate of change in district spending due to a change in enrollment is constant across the entire range of enrollments (because the slope of a straight line is constant along its length).

For the sake of completeness, the author ran a regression on a version of the model with that simple linear function of Size, but a linear function is not in fact consistent with any of the plausible theories about how enrollment and district spending might be related. Any theory of that relationship must begin with a rapid drop in per-pupil spending from very tiny districts up to those of a few hundred students, simply because of the fixed overhead cost of a principal’s or secretary’s salary, not to mention a superintendent’s. These costs obviously weigh much more heavily on a district with 10 students (and such small districts do exist) than on one with several hundred students.

For districts of more than a few hundred students, differing theories about the relationship between district size and spending diverge both from one another and from a simple linear function of Size. If we assume that district officials seek to be as efficient as possible and are successful in their efforts, then per-pupil spending should continue to fall off as Size grows, but at a decelerating rate, possibly even hitting a plateau beyond which no further efficiency gains are realized. That’s because economies of scale would be greatest when going from extremely tiny districts to medium-size districts. This is a nonlinear relationship — the slope of the line changes as district size changes.

An appropriate function to embody this efficient school spending thesis is ƒ(Size) = ln (Size), the natural logarithm of Size, with a negative coefficient b expected in the regression’s size term, b * ln (Size). This would allow the cost-saving effect of increased district size to be greater when moving from tiny districts to medium sized districts than when moving from large to very large districts.

According to public choice theory, however, school officials would be inclined to grow their budgets rather than economize. Under this theory, initial savings that come from sharing fixed costs among a greater number of pupils would be overwhelmed by district officials’ self-interest once districts reach a certain size. As a district becomes increasingly large, complex and removed from the everyday oversight of community members, administrators might well find it easier to expand district staff and spending. So, under public choice, the correlation between spending and enrollment should eventually become positive once a certain district size is reached.

There is a further reason to expect that very large districts might be less efficient. In “agency shop” states, where all or most teachers are required to pay union dues whether or not they belong to the union, unions in larger districts will invariably have more resources to employ in their efforts to raise salaries, decrease workloads and eliminate competition for their members. To the extent that such union action is effective, it will raise district spending and perhaps also decrease student performance, since artificial barriers to entry into the teaching profession may exclude potentially talented candidates. As it happens, a highly sophisticated 1997 study by Harvard professor Caroline Minter Hoxby[17] found this to be the case, concluding, “[T]eachers’ unions are primarily rent seeking, raising school budgets and school inputs but lowering student achievement by decreasing the productivity of inputs.”

Public choice theory thus suggests that per-pupil spending should fall steeply when moving from tiny to small districts, but then gradually reverse course and begin to rise — steeply at first, but flattening out as district size becomes very large and taxpayers’ resources are stretched thin. This spending curve, shaped essentially like a checkmark, should be noticeable in agency shop states such as Michigan.

Correctly specifying this potential Size equation for the public choice view of bureaucratic behavior is not straightforward. Most researchers trying to determine if increased district size results in lower spending for small districts but in higher spending for already-large districts use a quadratic function of the form:

*f(Size) = a * Size + Size ^{2}*,

where a is an unknown coefficient to be determined by regression.[*] As a basis for comparison to the existing literature, the author performed a regression using this quadratic function of district size, but the simple U-shaped curve it describes is distinctly different from the checkmark shape that would seem to follow from public choice theory.

A better fit for the relationship between enrollment and spending predicted by public choice is given by the following equation:

*f(Size) = ln ^{2} (Size) − a * ln (Size)*

If a is a moderately sized positive number, this equation produces the checkmark-shaped curve needed to model the public choice view of how size and spending should be related.

A graphic illustration of the three nonlinear functions of enrollment appears in **Graphic 2**. These curves were obtained by regressing each of the functions against per-pupil spending in the absence of controls. (This figure is meant only to clarify the preceding discussion about the contrasting shapes of the curves. The complete models with the controls incorporated will differ somewhat.)

**Graphic 2: Functions of District Size (Without Controls)**

[*] Note that in the discussion that follows, the author used the same letters to refer to the unknown quantities in several different equations. Readers should not assume that an ‘a’ constant in one equation represents the same quantity as does an ‘a’ constant in a different equation.