Penobscot Valley: Prudent Investments Linking Our Towns
A Report on Education, Housing, and Capital Planning Opportunities

II. EDUCATION

Introduction - Economies of Scale in Education¹

Economies of scale is a fundamental economic principle. Cost per unit decreases as more units are produced when there are net economies of scale. To some extent there are economies of scale in just about every economic activity. Whether we are talking about building airplanes or baking cookies, it is almost always cost effective to produce more than one unit. Economies of scale are also usually limited, however. At some point diseconomies of scale are encountered. At some level of output, production bottlenecks and supervisory problems become increasingly severe and cost per unit begins to rise. Thus, cost-effectiveness is a tricky balancing act. To borrow from Goldilocks, some beds can be too large and some can be too small. 

In the case of public provision of education (and other public services) in the Penobscot Valley (and to an even greater extent in the rest of the state), it appears that some of our beds, i.e., our school districts and schools, are too small. Certainly there are important benefits of small schools. Teachers and children generally get to know each other better, thus raising children’s sense of belonging and security. Children’s relationships and social experiences generally can be expected to improve with smaller school size. Being able to walk to school is a big plus, in addition to saving on transportation costs. In an ideal world all our children would be a two-minute walk from their school. Moreover, competition between schools can be a healthy incentive for providing quality services. 

Small schools, however, come with a high price. We are paying a high cost for too much duplication of education services. Moreover, it is not just that we have to pay more (i.e., higher taxes) to educate our children, some of our children are also missing out on some educational opportunities. That is, having small schools and small school districts is costing us both in terms of taxes and in terms of quality. High-cost schooling might be an acceptable choice – if we were getting high-quality schools in return. Similarly, just-okay schools might be acceptable choice – if only the cost were okay. The data, however, indicate that the choice to have relatively small schools and school districts, on average, is causing us to have just-okay public education and at a relatively high cost.² 

It is important at the outset to stress three points. First, this report is not meant to condemn the effort and motivation of local school teachers and administrators. They are not the cause of the high-cost/average-quality problem. The problem is in our use of limited resources. That is, the issue is about using our resources more efficiently. Second, school size and school-district size are not the same as class size. We are not proposing decreases in the number of classes (i.e., we are not arguing for larger class sizes). We are proposing decreases in the number of schools, and especially, in the number of school districts. Third, the trend of rapidly rising costs of education is going to continue. Thus, unless we use our resources more efficiently, the problem of rising mill rates is going to continue.³  Furthermore, if we do not use our resources more efficiently, the quality of education that many of our young receive will lag further and further behind that in the rest of the country. 

School Size

Schools and school districts in the PV PILOT communities,⁴ as well as schools and school districts in Maine generally, are much smaller than in the rest of country on average. This is shown in Figure 1 and Figure 2. Figure 1 shows that Bangor is the only PV PILOT school district with more students than the national average, and S.A.D. 22 is the only other district with even half the national average. The other 12 school districts are more like the rest of the state. The average number of students in PV PILOT school districts is 1,097, which is only a little over a third of the national average, but well over the state average. Figure 2 indicates that most of the public schools in the PV PILOT area and in the state as a whole are considerably smaller than in the rest of the country. The average number of students is only about ^5/9ths of the national average. Moreover, this occurs despite the fact that Bangor High School is the largest school in the state (1,447 students in 2000-01). 

Moreover, future demographic changes forecasted recently by the Maine State Planning Office indicate that our schools and school districts will become even smaller unless there is consolidation of educational resources. As revealed in Figure 3, a rapid contraction of the school-aged population (ages 5 through 17) is expected in Maine and in most PV PILOT communities. The school-aged population in Maine is forecasted to shrink by almost 8 percent from 2000 to 2005, and by almost 13 percent over the 2000-10 decade. The reduction is expected to be even stronger in the combined PV PILOT area. The forecasted school-aged population in the PV PILIOT towns is over 9 percent lower in 2005 than in 2000, almost 15 percent lower in 2010 compared to 2000. These forecasts indicate that, unless there is significant consolidation, the fixed costs of providing education services (i.e., the costs of facilities operation, administration, etc.) will have to be spread over even fewer students in the near future. 

Cost per Student 

Education in the PV PILOT communities, as well as education in Maine generally, costs more than in the rest of country on average. Figure 4 shows operating cost per student in the school districts in the PV PILOT area. The average operating cost per student in the PV PILOT area ($6,248) is practically the same as the state average ($6,233). Cost per student in Maine, however, is about 10 percent higher than for the country as a whole according to the latest data available from the U.S. Department of Education. This is shown in Figure 5.⁵ 

Figure 6 illustrates the rising cost of providing K-12 education. The cost of education rose significantly faster than the rate of inflation during the last decade. Even after removing the effect of inflation, per-student cost in Maine rose by an average of 2 percent per year over the nine academic years from 1992-93 to 2000-01. Moreover, public education costs rose particularly rapidly during the last three years of that period. In the latter three years per-student cost in Maine rose by an average of 4 percentage points per year more than the rate of inflation. Bradley was the only PV PILOT district that did not experience rising real cost per student over the nine-year period. 

The primary reason for the rising costs is that education is labor intensive, and average wages rose faster than inflation, particularly at the end of the last decade. That is, the ‘problem’ of rising education costs is mainly a consequence of increasing economic prosperity. In a labor-intensive area like education, as opposed to a capital-intensive area such as microchip manufacturing, technological advances do not offset rising wage rates. The implication of this is that we should expect that education costs will continue to rise - unless significant cost savings are found. A close examination of the costs of providing public education in Maine and in the Greater Bangor region reveals that there are indeed significant potential cost savings. 

A First Look at School Size and Cost per Student 

Net economies of scale exist when cost per student declines as the number of students increases. Net economies of scale are expected to occur at low numbers of students because the spreading of fixed cost over a larger number of students more than offsets the additional cost from a larger number of students. Net diseconomies of scale exist when cost per student rises as the number of students rises. This is expected to occur at high numbers of students because the additional cost of more students outweighs the spreading the fixed cost over more students. In other words, going from very low levels of students (imagine the cost per student of having a school for each student) to very high levels of students (imagine the cost per student of having only one school in the state); we can expect a U-shaped relationship between cost per student and the number of students. This expected relationship between school size and per-student cost is shown in Figure 7. It is not immediately clear, however, where our schools and school districts are in this relationship. Ideally our schools would be on the flat middle region of this relationship where cost per student is minimized (for some level of education quality). The data, however, suggests that our schools are on the declining portion of the relationship. 

Figure 8 plots average cost per student in each state in 1998-99 against its average number of students in each school district (comparable cost figures are only available for school districts rather than for schools).⁶ Although there is a considerable amount of variation in average per-student cost, Figure 8 suggests a U-shaped relationship. Moreover, Maine appears to be on the declining portion. The data shown in this figure are highly aggregated, though. It would be better to look at data from individual school districts to infer economies of scale and potential cost reductions from consolidation of school resources. Data of this sort are shown in Figure 9. 

Figure 9 also suggests a U-shaped relationship between cost per student and the number of students in Maine, although there again is considerable variation in cost per student across the districts.⁷ All of the 25 highest-cost districts (above $7,800 per student) are relatively small (all but one have less than 300 students). But there are also many small districts (so small that they appear very near the zero vertical axis) that have low per-student costs, which appears to contradict the hypothesis that there are significant increasing returns to scale in K-12 education. Closer inspection of the data, however, reveals that these data points do not contradict the hypothesis. Indeed, these cases provide further support for the hypothesis of increasing returns at low numbers of students. 

All of the 7 lowest-cost districts (below $4,850 per student), and 20 out of the 23 lowest cost districts (below $5,225 per student) pay other districts to educate at least some of their students. That is, by sending all or some (i.e., just the high schoolers) of their students to other districts, many of the very small districts are able to benefit from the economies of scale found in the larger school districts. One might initially think that tuitioning students to other districts would be relatively costly for these school districts. On average, however, this is not the case. For the small districts, tuitioning their students is not only cheaper than educating their students themselves, it is even cheaper than the average cost per student in the state. The average cost per student in Maine in 2000-01 was $6,233. The weighted average cost per student in school districts that tuition all of their students to other districts was only $5,889 – 5.5 percent less than the state average. 

Why do these tuitioning districts get such a good deal? Are the receiving school districts being benevolent to the smaller districts? Perhaps, but probably not. The larger districts have their own children and taxpayers to consider. More likely, the larger districts benefit by accepting students from other districts at a tuition rate below their average cost per student. How is this possible? Economies of scale. The cost of the additional students is less than their overall average cost per student. These districts benefit from having more students share their costly infrastructure. In other words, both the sending and receiving districts can share in the cost savings from moving from points on the downward-sloping part of the curve shown in Figure 7 to a point on or nearer the flat part of the curve. 

Figure 10 clearly shows the benefit from tuitioning students out of the smallest districts.⁸ The complete-tuitioning districts generally lie below and to the left of the K-8 districts, which generally lie below and to the left of the K-12 districts. Tuitioning students out of very small districts, i.e., consolidating educational resources, reduces the cost per student. The State thus already benefits from some consolidation of school resources. Indeed, consolidation of school resources has been occurring in the state for decades. There does, however, appear to be room for more consolidation. Moreover, one cannot help but wonder if there is any reasonable justification for having the extra bureaucracy from 56 school districts that do not operate any schools and tuition all of their students (with an average of 38 students in each). 

Figure 11 and Figure 12, unlike Figure 9, compare like with like. That is, they show how cost per student varies with school district size for K-8 districts only and K-12 districts only. The economies of scale are clearly visible in these charts. Moreover, many of Maine’s schools appear to be on the downward-sloping part of the relationship between cost per student and school district size. That is, there appears to be significant cost savings from greater consolidation of educational resources. 

Transportation Cost per Student 

It is reasonable to suspect that there is much more to the cost story than simply reducing the number of schools and school districts to achieve significant cost savings (indeed, Figures 11 and 12 reveal considerable variation in per-student cost that is not due to variation in school district size). For example, the suggested cost savings could be severely constrained by geography and transportation costs. Certainly transportation costs would become prohibitive at some level of consolidation. At the current level of consolidation (or lack of it), though, transportation costs do not appear to be an important constraint on economies of scale.

In fact, the Maine Department of Education believes that transportation costs could be significantly reduced by further consolidation of school districts. After recently exploring transportation costs in some detail, they have come to the conclusion that there is an inefficient level of duplication in school bus transportation in the state. This conclusion is based on the finding that transportation cost per student per mile is smaller the larger the number of students. Greater consolidation of school transportation resources is expected to reduce overall bus costs through more efficient routing of buses, through less duplication of transportation infrastructure, maintenance, and overhead costs, and through volume-discount purchasing. 

School Quality 

It is possible, although perhaps not likely, that the cost savings from larger schools and schools districts come from a reduction in quality. It seems more reasonable to expect that at least part of the cost savings will be put into the quality of instruction. For instance, casual observation indicates that students in larger schools have more educational choices. Small schools cannot feasibly offer a full range of curricular and extracurricular options. Although the quality of educational services cannot be quantified with any degree of precision, it is worth examining some crude measures. The readily-available crude measures are for individual schools (as opposed to school districts as examined earlier). 

Figure 13 plots the percentage of the school staff with graduate degrees against school size. Staff with greater credentials can presumably provide better services on average. Although there is a great deal of variation in the ratio across schools, there is also a clear positive correlation between the percentage of graduate staff and the number of students (the correlation coefficient is 0.25). Of the 82 schools with less than 80 students (and without missing data), 36 do not have any staff with graduate degrees (i.e., there is an overlap of many data points near the origin on the horizontal axis). Evidently larger schools do indeed use some their cost savings from economies of scale to hire relatively more staff with higher qualifications. 

Figures 14, 15, and 16 plot average scores from the Maine Education Assessment against the number of students in the school.⁹ Again there is a lot of variation in average tests across schools, but there is also a positive correlation between the 4^th, 8^th, and 11^th grade average test scores and the number of students (the correlation coefficients are 0.16, 0.11, and 0.23).¹⁰ Again, the evidence suggests that economies of scale enable the larger schools to provide higher quality instruction (as well as at a lower cost). 

Figure 17 plots the percentage of graduating seniors that intend to further their education against school size. Perhaps surprisingly, there is essentially no correlation between these variable (the correlation coefficient is 0.03, but is not statistically different from zero). 

Thus, the evidence, albeit crude, does not indicate that cost reductions from larger schools come at the expense of education quality. Indeed, the evidence suggests that larger schools are able to use some of their cost savings to provide better instruction. An important implication of this is that expenditures per student are not the whole story for judging the fairness of educational opportunities. Equality of spending per student does not necessarily imply equality of education quality and opportunity when there are significant differences in economies of scale across school districts. The evidence suggests that, even if spending per student were the same across every school district, students in smaller school districts essentially have less educational resources and opportunities. 

Naturally it is possible that larger schools have lower levels of some unmeasured, yet important, aspects of education quality. To the extent that this is the case, then the loss of these benefits from smaller schools needs to be weighed against the estimated cost savings presented below. 

Statistical Analysis 

The previous charts showing correlations suggest significant potential cost savings from greater consolidation of educational resources. It would be more useful, however, to have a more specific idea of the extent of these cost savings. Rough estimates of the potential cost savings can be calculated using regression analysis. That is, an equation that best fits the data shown in Figure 10 can be estimated. 

The data and the theory discussed earlier indicate that the relationship between cost per student (abbreviated as C) and the number of students (abbreviated as S) is nonlinear: 

C_i = a + bS_i + gS_i² . 

The subscript i denotes the individual school districts (i.e., the values of C and S are different in different districts), and a, b, g are parameters to be estimated. The data and theory also indicate that K-8 school districts benefit from the economies of scale in the districts where they send their high school students. Thus, a dummy variable for the K-8 districts (abbreviated as D, where D = 1 for K-8 districts and D = 0 otherwise) is added to the equation to be estimated: 

C_i = a + bS_i + gS_i² + dD_i. 

d is another parameter to be estimated. There is also clearly a large amount of unexplained variation (abbreviate as e) in cost per student (due to differences in efficiencies, quality of instruction, etc.). Thus, the regression equation is

 C_i = a + bS_i + gS_i² + dD_i + e_i. 

As discussed earlier, the number of students in school districts that tuition all their students gives a misleading indication of the school sizes where their students attend. Thus, the regression equation is estimated using data from the school districts where teaching occurs (i.e., for the K-8 and K-12 districts only). 

The equation that that best matches the (K-8 and K-12) data shown in Figure 10 is 

C^{^}_i = 7,505.53 – 1.07457S_i + 0.000159S_i² – 408.56D_i. 

The coefficient estimates on S and S² indicate that the relationship between cost per student and size of school district is indeed U-shaped.¹¹ The coefficient estimate on D indicates that cost per student in K-8 school districts is $409 lower on average than the K-12 districts after controlling for S and S² (although their per-student cost is $242 higher than for K-12 schools when not controlling for district size). The estimated equation (for both K-8 and K-12 districts) is shown in Figure 18 along with the data points.¹² Clearly there is a considerable amount of unexplained variation across districts in cost per student. But 12.7 percent of all the variation in cost per student is explained by only three variables: the number of students, its square, and the tuitioning of 9-12 students. School district size clearly affects per-student cost. This suggests significant cost savings from moving to more cost-effective sizes. 

The estimated school district size that achieves minimum cost per student (denoted as S^*) is 3,378 students (S^* = b^{^} /2g^{^}). Only 9 of the State’s 261 school districts are this large. Thus, there appears to be substantial potential cost savings from greater consolidation of educational resources in Maine. 

Some Illustrative Estimates of Cost Savings 

The estimated per-student cost curve shown in Figure 18 is nonlinear. Therefore the implied potential cost savings from consolidation depend on levels of consolidation. The likely cost savings are clearly greater when going from, say, 500 to 1500 students per school district, than from, say, 2000 to 3000 students per district. The cost curve is relatively steep at low numbers of students, but relatively flat near S^* . Thus, a few illustrative examples of cost savings from reducing duplication of educational services are shown below.¹³

Four points are worth making before turning to these illustrative cases. First, the estimated regression equation indicates the average potential cost savings. We cannot expect the same cost savings in every case (even if the level of consolidation were the same). The actual cost savings would probably be smaller in some circumstances, but larger in others. Second, there are at least some physical limits to the potential cost savings. To take an extreme example, according to the estimates Augusta and Orono could merge to form a school district with 3,358 students, which is very close to the cost-minimizing size. The 80 mile distance between the towns, though, obviously makes such a merger absurd. To the extent that there are severe physical limits, the estimated cost savings shown below overstate the likely actual cost savings. Third, the estimated cost savings are implicitly long-run cost savings. Clearly there would be some adjustment costs from school and school-district realignment. The estimates below do not account for these costs; hence there is another reason why the estimated cost savings overstate likely actual cost savings to some extent. Fourth, the estimates do not take differences in educational quality into account. If some the cost savings from economies of scale are used to increase school quality, as suggested by the evidence on teacher qualifications and average test scores, then the estimates below understate the true cost savings from consolidation. 

2,238 Students to S^* 

The weighted-average school district size in Maine in 2000-01 is 2,238 students. The average school district size is 754 (1,018 when excluding the districts that tuition all their students), but most of the students in Maine are in the larger districts. Thus, the average school size for Maine’s public school students is the weighted average. The estimated long-run annual cost saving from moving from a school district of this size to cost-minimizing size is 

C^{^}₂₂₃₈ - C^{^}₃₃₇₈ = 7,505.53 – 1.07457(2,238) + 0.000159(2,238)² 
                          - 7,505.53 + 1.07457(3,378) - 0.000159(3,378)²
                         = $206.74. 

The estimated cost saving of $207 per student represents 3.5 percent of the estimated cost per student at S = 2,238 (3.3 percent of the actual average operating cost per student in the State). Moreover, because the cost-size relationship is nonlinear, this understates the potential cost savings of moving all of Maine’s school districts to the cost-minimizing size (subject to the caveats above). This estimate would be the potential cost savings for the State if all school districts in Maine had 2,238 students. Obviously some districts have more, and many have less, but, because of the nonlinearity, the greater cost savings from the smaller schools more than offsets the lower cost savings from the larger schools. 

1,018 Students to S^* 

As noted above, the average teaching school district in Maine has 1,018 students. The estimated cost savings from moving the average teaching school district to the cost-minimizing size are 

C^{^}₁₀₁₈ - C^{^}₃₃₇₈ = 7,505.53 – 1.07457(1,018) + 0.000159(1,018)²
                          - 7,505.53 + 1.07457(3,378) - 0.000159(3,378)² 
                         = $885.92. 

The estimated cost savings of $886 per student is 13.5 percent of the estimated cost per student at S = 1,018 (and 14.2 percent of the actual average operating cost per student in Maine). 

Merger of Brewer, Dedham, Orrington, & S.A.D. 63 

Brewer, Dedham, Orrington, and the communities in S.A.D. 63 (Clifton, Eddington, and Holden) recently considered, but apparently rejected, merging into a single school district which would have had 3,347 students (remarkably close to S^* ) in 2000-01. The estimated long-run costs of that decision are 

(1,414C^{^}₁₄₁₄ + 276C^{^}₂₇₆^K-8 + 641C^{^}₆₄₁^K-8 + 1016C^{^}₁₀₁₆^K-8)/3347 - C^{^}₃₃₄₇ = 
                             1,414(7,505.53 - 1.07457(1,414) + 0.000159(1,414)²)/3347
                              + 276(7,505.53 - 1.07457(276) + 0.000159(276)² - 408.56)/3347 
                              + 641(7,505.53 - 1.07457(641) + 0.000159(641)² - 408.56)/3347
                              + 1,016(7,505.53 - 1.07457(1,016) + 0.000159(1,016)² - 408.56)/3347 
                              - 7,505.53 + 1.07457(2,430) - 0.000159(2,430)² 
                              = $646.88. 

The estimated cost savings of $647 per student per year is 10.2 percent of the estimated weighted-average cost per student in these school districts (and 10.2 percent of the actual weighted-average cost per student in these districts). Evidently the decision not to consolidate these districts is very costly. 

Conclusion 

The rough estimates just presented are just that, rough estimates. They should not be interpreted as any more than that. This initial examination of the data cries out for more in-depth analysis. The primary reason why further study is warranted is that this initial study suggests that the stakes are very high. 

For example, the estimated potential long-run cost savings from one possible consolidation of educational resources within the PV PILOT area is about 10 percent. In 2000-01, 72.8 percent of local property taxes in Maine were used for K-12 education (and this ratio has been rising). The (unweighted) average was 72.5 percent in 1999 in the six towns affected by the proposed consolidation. Thus, if education costs can be reduced by 10 percent, then property tax rates can potentially be reduced by about 7 percent. That is, potentially the (unweighted) average mill rate in the six towns could be reduced from 15.4 percent (in 1999) to 14.3 percent. This means about $110 of potential taxpayer savings per year per $100,000 of property tax evaluation. 

These potential tax savings, however, are probably significantly less than the actual tax savings that these towns would see. Much of the cost savings would be passed on the State rather than kept within the district. The State’s school funding formula subsidizes the small districts, and hence unintentionally subsidies sprawl. That is, the data show that, despite the fact that small school districts generally have significantly higher per-student costs, small districts also generally have significantly lower property tax rates for education. This unintentional subsidy to sprawl from the school funding formula is an issue that merits further research. 

Although costs have received the lion’s share of the emphasis in the study (for the simple reason that they are quantifiable), it is important to keep in mind that costs are only half of the story. The quality of the instruction that our children receive is also at issue here. Reducing unnecessary duplication of infrastructure is also important because it can help provide more opportunities for our kids. Larger schools and school districts can facilitate more opportunities for taking advanced placement courses, vocational courses, as well as recreational courses. 

A final implication of this study is that the data on K-12 education are probably indicative of all local public service provision. That is, it is unlikely that our provision of education is dramatically different from our provision of public services generally. It is reasonable to expect that similar cost savings and quality improvements can be found throughout local service provision.

Endnotes

1. This report is based on research conducted by Philip Trostel of the Margaret Chase Smith Center for Public Policy, University of Maine, Orono, "Potential Efficiency Gains from Consolidation of Educational Resources in PV PILOT Communities," October 6, 2002. Philip would like to thank Walter Harris, Judith Lucarelli, and David Silvernail for sharing their knowledge on the subject, and Ewa Kleczyk for research assistance. Contact the author for additional results.

2. This evaluation may seem rather harsh to some readers familiar with assessments such as Maine having “the highest performing K-12 education system” (National Education Goals Panel, 1999) and the “biggest bang for its education buck” (Forbes Magazine, 1997). These studies (as well the widely-publicized high average scores of Maine students on national standardized exams), however, fail to account for Maine being the least ethnically diverse state in the country. After accounting for this fact, the performance of Maine schools is only average. For further discussion on this issue see Philip A. Trostel, “Workforce Development in Maine: Held Back by the Lack of Higher Education”, Margaret Chase Smith Center for Public Policy Technical Report, forthcoming 2002. 

3. Moreover, this problem is compounded by Maine’s current budget shortfall and already high level of state and local taxes.

4. The PV PILOT towns are Bangor, Bradley, Brewer, Eddington, Glenburn, Hampden, Hermon, Holden, Kenduskeag, Milford, Old Town, Orono, Orrington, Veazie, and the Penobscot Indian Nation on Indian Island. Hampden is in S.A.D. 22 (with Newburgh and Winterport), Eddington and Holden are in S.A.D. 63 (with Clifton), and Kenduskeag is in S.A.D. 64 (with Bradford, Corinth, Hudson, and Stetson).

5. Costs shown in Figures 3 and 4 are not directly comparable because of differences in the way that the Maine and U.S. Departments of Education calculate costs per student. The U.S. Department of Education figure includes all State spending on public education, not just that which can be attributed to individual school districts.

6. Hawaii with a single school district of 185,860 students is omitted from Figure 7 to avoid distorting its scale.

7. Data for the whole state is emphasized rather than for just the PV PILOT communities because the PV PILOT school districts are too few (14) from which to draw reliable inferences.

8. The largest school district (Portland) is omitted to make this chart easier to read by substantially reducing its scale.

9. To streamline the discussion these test scores are the average of the average scores on the reading, writing, and mathematics tests. 

10. The positive correlation between the 8^th grade average test score and the number of students is not quite statistically significant, however. This correlation is different from zero with only 87 percent statistical confidence. The correlations shown in Figures 13, 14, and 16 are different from zero with at least 98 percent statistical confidence.

11. The other pertinent technical details of this regression are: N = 205, R² = 0.127, t_a = 31.58, t_b = 5.06, t_g = 4.04, and t_d = 1.74. 

12. The Portland school district is again omitted to make the chart easier to read.

13. Other examples can be computed by request to philip.trostel@maine.edu.

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