From a working paper by Dan Otto on the economies of scale in Iowa’s K-12 schools:

First, expenditures per student generally rise as district sizes fall below about 750 students. Given no additional knowledge about educational processes, it is possible there are economies of scale relative to school district size for these districts. It also is possible small districts have a tendency to spend more on education per student simply because they believe it is a valuable investment...The scatter diagram of reported data has a general shape similar to the letter “J” lying on its back. This is not conducive to linear regression, where every increase in district size would result in a fixed decrease in expenditures per student. Such a relationship would imply constant economies of scale throughout the range of possible school district sizes. That obviously is not the case here....

The scatter diagrams of per student expenditures relative to district enrollments for these two categories are shown as Figures 4 and 5. Both of these plots have shapes similar to the total expenditure per student plot from Figure 3. These plots give good reason to believe that a reciprocal relationship exists between both instruction and administration with respect to district enrollment size...

Looking at instruction expenditures and administrative services expenditures as a percent of total expenditures relative to district enrollment provides some additional insight. If increasing perstudent expenditures indicate that smaller districts inherently placed more value on educational investments in students, instructional expenditures as a percent of total budgets also would consistently rise among districts with smaller enrollments. A look at Figure 6 shows this is not the case. While there is significant variance in this measure across the entire range of district enrollments, there is not immediate visual evidence that smaller districts spend a consistently larger portion of their budgets on instruction. This would tend to go against arguments that small district expenditures per student are higher due to a cultural predisposition to invest more on education in smaller environments...

It is clear expenditures per student rise at an accelerating rate as enrollment sizes fall below about 750 students. It also is clear expenditures per student are relatively constant at enrollment levels above 1,000 students. In fact, it is relatively easy to fashion a simple statistical relationship between district enrollments and total expenditures per student that accounts for nearly 40 percent of the expenditure per student variation across these districts. While we cannot be certain this is due to economies of scale among the smaller districts, data on instructional and administrative expenditures as percents of total expenditures tend to support this conclusion.