revisiting economies of scale in higher education robert k ......the textbook depiction of economies...

1

Revisiting Economies of Scale in Higher Education

Robert K. Toutkoushian

Professor, Institute of Higher Education

University of Georgia

Draft: February 25, 2016

For presentation at the meeting of the Association for Education Finance and Policy (AEFP),

Denver, CO, March 17-19, 2016. I would like to thank Keith Allen for his helpful comments at

the early stages of this project.

2

Introduction

The cost of higher education services has always been an enduring topic for several

reasons. First and foremost, the notable rise over time in the prices charged to students and their

families for going to college has fueled concerns that the high cost of services is the driving force

behind this trend. If colleges are not providing services at their most efficient levels, the

argument goes, then some of this inefficiency is passed along to consumers in the form of higher

prices. Accordingly, one way to alleviate the pressure on students is to examine the spending

patterns and levels of institutions and determine if there is room for improvement. Higher

education costs are also a perennial policy topic due to the large subsidies given to higher

education, and the opportunity costs that accompany them. The higher education industry has

also come under fire for perceptions of its inefficiency and inability to produce outcomes at a

desired level.

Within this context, economists have conducted a number of studies to examine the cost

structure of colleges and universities (e.g., Bowen, 1980; James, 1978; Brinkman & Leslie,

1986). The focus of many of these studies is on whether there are economies and diseconomies

of scale in higher education.1 The concept of economies of scale holds that as an organization

produces more output, ceteris paribus, then up to some point its cost per unit of output would fall

because fixed costs are distributed over more units of output and the organization can take

advantage of the specialization of resources. A number of multi-campus institutions, and state

and university systems have undergone mergers in the hope of taking advantage of economies of

scale and thus provide educational services at a lower cost per student. It is also possible,

however, that an organization produces too much output and as a result costs per unit of output

rise as output increases beyond a certain point. This is known as diseconomies of scale.

1 These are also referred to in the literature as ray economies of scale or product-specific economies of scale.

3

The textbook depiction of economies and diseconomies of scale is shown in Figure 1. If

there are economies and diseconomies of scale in the organization, then the average cost and

marginal cost curves will both be U-shaped, meaning that they initially fall as output increases

and then eventually rise as output continues to increase. The lowest point along the average cost

curve is the output level at which economies of scale are exhausted. By definition the marginal

cost curve crosses the average cost curve at this minimum point when the curves are quadratic.

------------------------- Insert Figure 1 Here -----------------------

Overall, most empirical studies have found that economies of scale are present at some

level in higher education. However, studies have used a variety of approaches to measuring

economies of scale, and as a result have reached different conclusions as to where the average

cost-minimizing output level occurs. Some studies have treated colleges as single-product firms

that only provide instructional services, while others model colleges as multi-product firms. The

multi-product analogy applies particularly well to doctoral-granting institutions given that they

are heavily involved in producing undergraduate instruction, graduate instruction, and research.2

For multi-product firms, it is difficult to separate out the spending that is attributed to any

particular output. Other institutions such as 2-year colleges and many 4-year colleges that engage

in little or no graduate education and research can reasonably be treated as single-product firms.

Another important issue in economies of scale studies is the choice of functional form for

the cost curves. Empirical studies have either modeled total costs as a cubic or quadratic function

of output. Although the cubic total cost curve has the advantage of giving rise to quadratic (and

possibly U-shaped) average and marginal cost curves, it is more challenging to estimate. The

vast majority of studies have modeled total cost as a quadratic function of output.

2 Public service is usually omitted from consideration due to the lack of data on service outputs, and enrollments are

most often used as a measure of teaching outputs.

4

Another aspect of economies of scale studies is that they have used different approaches

to take into account the multi-product nature of colleges and universities. While some studies

have simply treated colleges as single-product firms, others have relaxed this assumption and

added other outputs to the regression model in linear, quadratic, of cubic form, and possibly

interacted with each other to examine economies of scope. Many studies have modified the cost

equation by adding dummy variables for the presence of each output, in recognition of the fact

that not all colleges produce each type of output and that there are fixed costs that go along with

the production of each output. Further complicating matters is that studies have used selected

combinations of functional form and method to examine multi-product firms. These variations in

cost functions make it more difficult to compare and contrast findings across studies, and may

have important implications for the conclusions reached about whether economies and

diseconomies of scale exist.

Putting aside methodological concerns for the moment, it is also important to obtain more

current evidence about economies of scale. Almost all of the studies on economies of scale and

scope in higher education were conducted using data from the 1970s to 1990s, and the few

studies that have appeared in more recent years examined cost structures for institutions outside

of the United States (e.g., Fu, Huang, & Tien, 2008; Lenton, 2008; Stevens, 2005). Given the

substantial changes that have occurred since this time in terms of how colleges compete and

function, and the state of the economy, there is a need for updated analyses of this topic.

In this study, I revisit the topic of economies of scale in US higher education. Using data

from the Delta Cost Project and IPEDS for the 2012-13 academic year, I focus on whether there

are economies and/or diseconomies of scale for 2-year (associate) institutions and 4-year

(bachelor, master) institutions. Because these institutions are less involved in research and

5

graduate education than 4-year doctoral institutions, I first examine whether there are economies

and diseconomies of scale for each group if they are treated as single-product organizations. I

then relax this assumption and test whether the results change when I treat master institutions as

multi-product firms. Throughout, I focus on whether the conclusions reached are sensitive to the

form of the cost function used in the analysis.

Review of Literature and Cost Study Methods

Background

There have been a number of efforts using these and other approaches to estimate cost

functions for institutions of higher education, and determine whether there are economies and

diseconomies of scale. Readers who are interested in the early literature on higher education

costs are referred to Russell (1954) and Witmer (1972). The first studies in higher education

documented relationships between credit hour production and average and marginal costs

(Stevens & Elliot, 1925; Reeves & Russell, 1935; Middlebrook, 1955; Moore, 1959).

Beginning in the 1960s, economies-of-scale studies began to rely on multivariate

statistical modeling to estimate cost functions. Thorough reviews of the literature on cost

equations prior to the 1990s can be found in Brinkman and Leslie (1986) and Brinkman (1990).

The late 1980s through the early 2000s saw a number of notable efforts to measure economies of

scale for the US (Getz, Siegfried, & Zhang, 1991; Koshal & Koshal, 1995; 1999; Laband &

Lentz, 2003; 2004; Paulsen, 1989; Toutkoushian, 1999). In addition, studies of economies of

scale began to appear outside of the US context as well (Fu, Huang, & Tien, 2008; Izadi, Johnes,

Oskrochi, & Crouchley, 2002; Stevens, 2005).

6

A cost function follows from the general optimization problem of an organization

(Pfouts, 1961; Bowen, 1980; Teece, 1982; Weldon, 1948). Economists have shown that the

relationship between inputs and outputs can either be expressed as the minimum cost needed to

produce a certain amount of output (“cost function”), or the maximum output that could be

generated from a given expenditure level (“production function”). The cost function (C)

therefore shows the relationship between the lowest total cost needed to produce different levels

of output (Q) given the prices of production inputs (P):

(1) C = C(Q, P)

Likewise, the average cost curve (AC) is defined as total cost divided by the level of output:

(2) AC = C / Q

and the marginal cost curve (MC) is the partial derivative of the total cost curve with respect to

output:

(3) MC = ∂C / ∂Q

The notion of economies and diseconomies of scale can be traced back to the 1800s and

the work of Mangoldt (1863). Basically, economies of scale holds that as an organization

increases its production of output, total costs rise at a decreasing rate. This pattern is thought to

arise at low levels of output because as output initially increases, fixed costs are distributed over

more output which in turn leads to lower average costs. In addition, economies of scale may be

enhanced as output rises if the organization can take advantage of the specialization of resources

to produce output more efficiently. However, if the organization becomes too large and produces

too much output given its resources, then total costs may eventually begin to rise at an increasing

rate due to inefficiencies in production. This is referred to as diseconomies of scale. As noted by

Brinkman (1990), economists usually assume that when there are economies and diseconomies

7

of scale, both the average cost and marginal cost curves will be quadratic (U-shaped) functions

of output as depicted earlier in Figure 1.

Approaches to Examining Economies of Scale

Economists have used both cubic and quadratic cost functions to model the relationship

between output and costs.3 Studies also differ in whether they treat postsecondary institutions as

single- or multi-product firms.

Cubic Cost Functions: Single-Output Firm. The cubic cost function for an

organization can be written as:

(4) 𝐶 = 𝛼 + 𝛽𝑄 + 𝛾𝑄2 + 𝛿𝑄3 + 𝑿′𝜽 + 𝑢

where X = non-output variables that may shift the total cost curve. If the institution only

produces one type of output, then average costs can be modeled as a quadratic function of output

as well as other regressors, as in: 4

(5) 𝐴𝐶 = 𝛽 + 𝛾𝑄 + 𝛿𝑄2 + 𝑿′𝜽 + 𝑢

Similarly, the marginal cost curve in this model would be written as:

(6) 𝑀𝐶 = 𝜕𝐶 𝜕𝑄⁄ = 𝛽 + 2𝛾𝑄 + 3𝛿𝑄2

3 The translog total cost function is of the form:

𝑙𝑛𝐶 = 𝛼 + 𝛽𝑙𝑛𝑄 + 𝛿𝑙𝑛𝑄2 + 𝑿′𝜸 + 𝑢

for a single-product firm, or:

𝑙𝑛𝐶 = 𝛼0 + ∑ 𝛽𝑗𝑙𝑛𝑄𝑗

3

𝑗=1

+ ∑ ∑ 𝛿𝑗𝑘𝑙𝑛𝑄𝑗𝑙𝑛𝑄𝑘

3

𝑘=1

+ 𝑿′𝜷 + 𝑢

3

𝑗=1

for a multi-product firm. The translog specification follows from a Cobb-Douglas cost function, and the resulting

coefficients represent the elasticity of total cost with regard to each output (deGroot, McMahon & Volkwein, 1991).

This approach is also used where researchers stochastic frontier analysis to model the minimum cost of producing

specific output levels given the state of technology (Izadi, Johnes, Oskrochi, & Crouchley, 2002; Stevens, 2005;

Titus & Eagan, 2016). 4 This regression equation is slightly different than the average cost curve found by dividing equation (4) by output

due to the omission of the fixed cost per unit of output term, and the coefficients θ will differ between the two

equations.

8

One appealing feature of this approach is that the average and marginal cost curves that

follow from (3) are quadratic functions of output and thus have the U-shaped functional form

that go along with economies and diseconomies of scale (provided δ > 0 and γ < 0). In this case,

economies of scale exist when average costs fall as output rises (𝜕𝐴𝐶 𝜕𝑄⁄ < 0) and

diseconomies of scale occur when average costs rise as output rises (𝜕𝐴𝐶 𝜕𝑄⁄ > 0). Average

costs are minimized at the output level where 𝜕𝐴𝐶 𝜕𝑄⁄ = 0. In the average cost curve shown in

equation (5), the average cost-minimizing output is found by solving for 𝑄𝑚𝑖𝑛 = −𝛾 2𝛿⁄ .

Cubic Cost Functions: Multi-Output Firm. When the firm produces multiple outputs,

however, there is no direct analogy to average costs as in equation (5) because total costs cannot

be easily apportioned between these outputs. Researchers have attempted to address this by

defining average cost relative to a specific output (such as undergraduate instruction) and then

adding the other outputs to the right-hand side of the average cost equation as shifters. In the case

of higher education, for example, the average cost curve for the j-th output (ACj) may be

expressed as follows for the multi-product firm:

(7) 𝐴𝐶𝑗 = 𝛽𝑗 + 𝛾𝑗𝑄𝑗 + 𝛿𝑗𝑄𝑗2 + ∑ 𝛾𝑘𝑘≠𝑗 𝑄𝑘 + ∑ 𝛿𝑘𝑘≠𝑗 𝑄𝑘

2 + 𝑿′𝜷 + 𝑢

where 𝐴𝐶𝑗 = 𝐶/𝑄𝑗. Economies and diseconomies of scale for each output can then be assessed

as before by determining whether the change in average costs for the j-th output as more output

is produced (𝜕𝐴𝐶𝑗 𝜕𝑄𝑗⁄ = 𝛾𝑗 + 2𝛿𝑗𝑄𝑗) is positive, negative, or zero.

This approach preserves the notion that average and marginal costs are U-shaped curves

with average costs initially falling as output rises and then eventually increasing as output rises.

Note, however, that the average cost curves in this method can only be estimated for institutions

that produce positive levels of the output in question.

9

Quadratic Cost Functions: Single-Product Firm. The more frequently-used approach

for examining economies and diseconomies of scale in higher education is to model total costs as

a quadratic function of output. Cohn, Rhine and Santos (1989) showed that when the firm

produces a single type of output, the quadratic cost function can be specified as follows:

(8) 𝐶 = 𝛼0 + 𝛼1𝑄 + 𝛼2𝑄2 + 𝛼3𝑃 + 𝛼4𝑃2 + 𝛼5(𝑄𝑥𝑃) + 𝑿′𝜷 + 𝑢

where P = input price (typically average faculty salary), and (QxP) = interaction of output and

input price. Average cost in the single-product case is then found by dividing total cost by

output:

(9) 𝐴𝐶 = 𝐶 𝑄⁄ = 𝛼1 + 2𝛼2𝑄 + 𝛼5𝑃 + (1

𝑄) [𝛼3𝑃 + 𝛼4𝑃2 + 𝑿′𝜷]

The resulting average cost curve is typically shown as an L-shaped curve as in Figure 2:

------------------------------ Insert Figure 2 Here -------------------------------

Economies of scale are said to occur as long as average costs are falling as output rises,

and vice-versa. The change in average costs as output rises, holding all else constant, is as

follows:

(10) 𝜕𝐴𝐶 𝜕𝑄⁄ = 2𝛼2 − (1

𝑄2) [𝛼3�̅� + 𝛼4�̅�2 + �̅�′𝜷]

From (10), it can be seen that average costs will fall at a decreasing rate and can only become

negative when the second part of the equation exceeds the first. This may or may not occur

depending on the parameter estimates in the total cost equation. Average costs are estimated by

inserting values for Q, P, and X’ into the total cost curve and then dividing by output. Typically,

the means for P and X’ are used for this purpose, and then average costs are predicted for

different values of output.

Quadratic Cost Functions: Multi-Product Firm. The quadratic total cost function has

also been used in instances where the firm produces multiple outputs. Baumol, Panzar and Willig

10

(1982) introduced the flexible fixed cost quadratic (FFCQ) function to examine the cost structure

of multi-product firms5. The FFCQ function is written as:

(11) 𝐶 = 𝛼0 + ∑ 𝛼𝑗𝐷𝑗3𝑗=1 + ∑ 𝛽𝑗𝑄𝑗

4𝑗=1 + (

1

2) ∑ ∑ 𝛾𝑗𝑘𝑄𝑗𝑄𝑘

4𝑘=1 + 𝑿′𝜷 + 𝑢4

𝑗=1

where all variables are defined as before and Dj = dummy variable for whether the j-th output is

produced by the institution.6 The inclusion of the dummy variables for whether each of the

outputs is produced is a strength of the FFCQ approach because not all institutions produce all

three outputs, and they capture fixed costs associated with the outputs. Each output and factor

price is therefore entered in linear and quadratic form in the cost function, and is interacted with

all of the other outputs and factor prices.

The parameters from the FFCQ equation can then be used to assess economies of scale

by comparing the estimated average cost of producing each output -- referred to as average

incremental cost (AICj) -- to its marginal cost. The general form of the average incremental cost

calculation is:

(12) 𝐴𝐼𝐶𝑗∗ = (𝐶𝑗

∗ − 𝐶−𝑗)/𝑄𝑗∗

where 𝐶𝑗∗ = estimated cost of producing 𝑄𝑗

∗ units of the j-th output and the mean levels of all

other outputs, and C-j = estimated cost of producing all but the j-th output.

To see how this is done for a three-output firm, the estimated total cost of producing 𝑄1∗

units of the first output and the mean levels of the other outputs is:

(13.1) 𝐶1∗ = 𝐶(𝑄1

∗, �̅�2, �̅�3, �̅�)

where �̅�𝑗 = average of the j-th output.7 Likewise, the estimated total cost of producing all but the

first output is:

5 Readers are also referred to the earlier work by Panzar and Willig (1977). 6 It is common for studies to treat factor prices (P) in the same manner as outputs in the FFCQ function. 7 Estimated total cost is affected by all of the linear, quadratic, and interaction terms involving Q1.

11

(13.2) 𝐶−1 = 𝐶(0, �̅�2, �̅�3, �̅�)

The resulting average incremental cost of the first output is then calculated as follows:

(14) 𝐴𝐼𝐶1∗ = (𝐶1

∗ − 𝐶−1)/𝑄1∗

The marginal cost of the j-th output is found by taking the partial derivative of the total cost

function in equation (11):

(15) 𝑀𝐶𝑗∗ = 𝛼𝑗 + 𝛽𝑗 + 𝛾𝑗𝑗𝑄𝑗

∗ + (1

2) ∑ 𝛾𝑗𝑘�̅�𝑘𝑘≠𝑗

where the interaction coefficients (𝛾𝑗𝑘) are multiplied by the means for prices and the other

outputs.

At the average cost minimizing level of output it must be true that AICj = MCj because

the marginal cost curve crosses the average cost curve at its minimum. Therefore, the ratio:

(16) 𝑆𝑗∗ = AI𝐶𝑗

∗ / M𝐶𝑗∗

is used to examine economies and diseconomies of scale. When 𝑆𝑗∗ > 1, the average incremental

cost exceeds marginal cost and the institution is said to be operating in the economies of scale

portion of its production function for this output. Likewise, when the ratio is less than one, it

indicates that there are diseconomies of scale. One possible depiction of the FFCQ approach is

shown in Figure 3. When the firm’s output is below Q*, there are economies of scale because

AIC > MC, and vice-versa when output exceeds Q*.

------------------------------- Insert Figure 3 Here ------------------------------

Despite its advantages, there are two disadvantages to the FFCQ approach. First, it is

more difficult to assess economies and diseconomies of scale with the FFCQ function because

the researcher must use the parameters of the model and the means of the variables to simulate

costs at different output levels. The second, and perhaps more important, concern is that the

quadratic cost curve will, by definition, give rise to non-quadratic average and marginal cost

12

curves for each output. In most instances average incremental costs and marginal costs will

either fall or rise over the entire range of output levels, whereas quadratic average and marginal

cost curves would allow for the possibility of changes in direction as output rises. There is also

no guarantee that under this approach the firm will exhibit economies of scale followed by

diseconomies of scale. It is possible, for example, that the model will result in either economies

or diseconomies of scale throughout the entire range of output, or even show diseconomies of

scale at low output levels followed by economies of scale at higher output levels.

Finally, some researchers have used a hybrid approach between these alternatives.

Laband and Lentz (2004), for example, combined a cubic cost function with the flexible fixed

cost approach and the Baumol, Panzar and Willig (1982) method to calculate economies and

diseconomies of scale. Other studies have likewise used the flexible fixed cost function approach

in a double-log (or translog) cost function.

Data and Methodology

Data

The primary dataset that I used in this study is the Delta Cost Project (DCP). The DCP

contains selected institution-level data assembled from the various surveys reported to the federal

government through the Integrated Postsecondary Education Data System (IPEDS). One of the

main advantages of the DCP data is that financial data have been reconciled between public and

private institutions, making it easier to directly compare and contrast the two sectors. I omitted

from the sample all groups of institutions that aggregate their financial data and report it for only

one campus (Jaquette, 2016). I also restricted the analysis to public and private not-for-profit

institutions at the associate, bachelor, or master levels. By focusing on non-doctoral institutions,

13

the cost models should be less susceptible to problems with multi-product outputs because they

are not as involved as doctoral-granting institutions in producing research and graduate

education. I will test this assumption, however, for the bachelor- and master-level institutions in

the sample. After eliminating a few specialized institutions and other institutions without

financial data on the variables in question, the final sample consisted of 779 associate-level

institutions, 378 bachelor-level institutions, and 438 master-level institutions.

Output variables. For the purpose of treating colleges and universities as single-product

firms, I defined output as the total full-time equivalent enrollments at the undergraduate and

graduate levels (QFTE). Subsequently for the models where institutions are viewed as multi-

product firms I followed the standard convention and defined three separate output variables for

graduate and undergraduate headcounts (Qg and Qu) and grant dollars as a proxy measure for

research output (QR).8

Dependent variables. The dependent variables represent measures of total cost and cost

per unit of output. Total cost included all expenditure categories -- operating, nonoperating, and

capital costs – at the institution.9 I then defined four different measures of average cost. The first

measure is cost per full-time equivalent student (ACFTE), which was calculated as total cost

divided by the number of FTE students (graduate + undergraduate). This measure was used in

the models where institutions were treated as if they were single-product firms. In recognition of

the multi-product nature of higher education, I then identified three alternative measures of

average cost where I divided total cost by either the number of undergraduate students (ACU), the

number of graduate students (ACG), or the research dollars brought into the institution (ACR).

8 Research dollars include revenues from federal sources net of Pell grants, plus state and local grant contracts. It

should be noted, however, that not all of these revenues may have been used for research purposes. 9 I also repeated the analysis using a narrower definition of total cost that includes only education and general

(E&G) costs, and found very similar results to those reported here.

14

Control variables. I created a number of control variables that may lead to shifts in the

cost curves. I included a variable in the models for input prices based on the average salary of

full-time faculty. Geographic measures for the region of the country and whether the institution

was located in an urban or rural area were used to capture possible cost-of-living differences

across institutions. I relied on two variables – the 75th percentile of SAT mathematics scores of

students and the percentage of applicants who were admitted -- to represent the quality of

students at an institution since the cost of educating students may vary with their academic

quality. Finally, I considered a number of institutional variables that the literature suggests also

affect institutional costs. These factors include the percentage of graduates in STEM fields

(science, engineering, mathematics) where instructional costs are typically higher, the percentage

of students enrolled part-time, the percentage of students at the graduate level, whether the

institution is public or private, and the extent to which students take online courses.

Table 1 contains the means for selected variables in the study broken down by type of

institution. Because associate-level institutions are not engaged in graduate education or

research, these particular measures are not reported in the table for 2-year institutions. The

average cost per FTE student at associate institutions ($13,488) was less than half as large as at

bachelor institutions ($28,165) and lower than master-level institutions ($20,909). The results for

bachelor and master institutions show that average cost per graduate student were substantially

higher than for undergraduate students due to the smaller numbers of graduate students at most

institutions in the sample. Keep in mind, however, that these are not truly “average costs”

because the numerators include spending on all outputs produced by the institution.

-------------------------- Insert Table 1 Here ---------------------------

15

Methods

Using these data, I estimated a series of cost equations to determine whether there are

economies of scale within each institution type. Each regression model was estimated by

ordinary least squares. To help account for possible heteroscedasticity, robust standard errors

were used in each model. I began with a quadratic average cost model where I treated institutions

as single-product firms that produce only instruction as measured by total FTE enrollments:

(17.1) 𝐴𝐶𝐹𝑇𝐸 = 𝛼0 + 𝛼1𝑄𝐹𝑇𝐸 + 𝛼2𝑄𝐹𝑇𝐸2 + 휀

(17.2) 𝐴𝐶𝐹𝑇𝐸 = 𝛼0 + 𝛼1𝑄𝐹𝑇𝐸 + 𝛼2𝑄𝐹𝑇𝐸2 + 𝑿′𝜷 + 휀

When 𝛼1 < 0, this is evidence of economies of scale, and similarly 𝛼2 > 0 indicates that there are

eventually diseconomies of scale. The average cost minimizing output level in these situations is

calculated as 𝑄𝐹𝑇𝐸∗ = −𝛼1 2𝛼2⁄ . Equation (17.1) focuses on economies of scale ignoring the

effects of other factors that may shift the average cost curve up or down. The second equation

(17.2) examines the sensitivity of conclusions about economies and diseconomies of scale to the

consideration of other cost shifters.

I then used the quadratic (total) cost function approach to examine economies and

diseconomies of scale in the case of a single-product firm. The cost equations were specified as

shown in equation (8), where QFTE is the single output, P = average faculty salary, and the

remaining variables in X are the same as used in equation (17.2). The results from this model

were then used to find the predicted average and marginal costs at selected enrollment levels, and

examine the trend in average and marginal costs to determine if there are economies and/or

diseconomies of scale.

In the next step, I treated bachelor and master institutions as multi-product organizations.

The first set of models relied on the quadratic average cost function, and economies and

16

diseconomies of scale were then assessed by the estimated coefficients on the linear and

quadratic terms for the output variable of interest. I also used the FFCQ function to assess

economies and diseconomies of scale, and then combined a cubic total cost curve with the

flexible fixed cost method to determine if the results were sensitive to this functional form.

Results

Single-Product Firms

In Table 2, I present the findings from the regression model where institutions were

viewed as single-product firms producing education. The dependent variable in each model is

cost per FTE student, and the results are reported separately for associate, bachelor, and master

institutions. The first model for each group only includes controls for the level of output and

squared output. The second model adds control variables for average faculty salaries, geographic

location, degree of urbanicity, the percentage of part-time students, the percentage of students

who have not taken distance education courses, whether the institution is public or private, and

the percentage of degrees awarded in STEM fields. The second models for bachelor and master

institutions also include control variables for the institution’s acceptance rate and the 75th

percentile of SAT-math scores.

------------------------------- Insert Table 2 Here ----------------------------

The results for the more fully-specified average cost models show that there is evidence

of both economies and diseconomies of scale for all three types of institutions. Average costs in

these models were minimized at about 25,000 FTE students for 2-year associate institutions,

9,900 for bachelor institutions, and 22,000 for master institutions. The different average cost-

minimizing output levels for bachelor and master institutions most likely reflects the fact that

17

many bachelor institutions are smaller, private 4-year institutions. With regard to average cost

shifters, average faculty salaries had a positive and significant effect on average costs.

Interestingly, the location of an institution as represented by geography and urbanicity had little

impact on its average costs even though the cost-of-living is substantially higher along the east

and west coasts and in urban areas. Average costs were higher at associate and master

institutions where more students have not taken distance education courses. At 4-year

institutions, average costs fell as the percentage of part-time students increase and as the

percentage of STEM degrees fell. Finally, there is some evidence that the quality of student

inputs matters in that 4-year institutions with higher acceptance rates and lower average SAT

scores (bachelor institutions only) had lower predicted costs.

In Table 3, I show the findings from the quadratic cost equation where the dependent

variable is total cost. Recall that the models presented here presume that colleges and universities

are single-product firms. In this model, total costs are a function of FTE enrollments and squared

enrollments, average salary and average salary squared, the interaction of FTE enrollments and

average salary, and other regressors that may shift the total cost curve. The models account for

89% to 93% of variations in total costs for the three groups of institutions. Because the sign of

the coefficient for the squared output variable was negative, total costs increased with

enrollments at a decreasing rate and then eventually would fall as enrollments continued to rise

past the maximum value.


The findings from Table 3 were then used in Table 4 to estimate the average costs for

each institution type at selected enrollment levels.10 The enrollment levels used for the

simulations varied from a low of 50% of average FTE students to a high of 600% of average

10 Details of the calculations are shown in Tables A1 – A3 in the Appendix.

18

FTE enrollments. It can be seen here that due to the quadratic cost function and the resulting

signs and magnitudes of the coefficients, average and marginal costs fell as enrollments rose

throughout the output range considered here. Accordingly, there were economies of scale for

each type of institution up to FTE enrollments of six times the average using this functional

form. In fact, due to the signs and magnitudes of the coefficient estimates in the cost equation,

the change in average costs will approach but never equal zero in this model specification,

leading to the conclusion that there are no diseconomies of scale.


Multi-Product Firms

In Table 5, I present the findings from the quadratic average cost models where bachelor

institutions were treated as multi-product firms. In the first column, the dependent variable is

cost per undergraduate student, and similarly columns two and three examine cost per graduate

student and cost per research dollar, respectively. Economies and diseconomies of scale were

assessed by finding the output level at which the average cost function was minimized, holding

constant the other two outputs as well as the regressors in X. The coefficients for the other output

measures are then interpreted as average cost shifters (e.g., the coefficients for graduate students

in the first column show how the cost per undergraduate student changes as graduate enrollments

change). Because not all bachelor institutions were involved in teaching graduate students or

conducting research, note that the sample sizes for the regression models for these outputs fell to

247 and 337 respectively.


For bachelor-level institutions, the results from the quadratic average costs models show

that they exhibited both economies and diseconomies of scale for all three output measures

19

because the coefficients on the linear output measures were negative and the coefficients on the

squared output measures were positive. Setting the partial derivatives for average cost equal to

zero and solving for each output revealed that costs per undergraduate student were minimized at

about 14,000 undergraduate students, costs per graduate student were minimized at 900 graduate

students, and costs per research dollar were minimized at $23 million research dollars. There

were substantial changes in the estimated coefficients for the other control variables across

models due to the different scale used for average cost. In particular, the coefficients were

particularly large in the second column due to the relatively small numbers of graduate students

enrolled at bachelor-level institutions.

The analysis for bachelor institutions was then repeated for master-level institutions, and

the findings are shown in Table 6.


In these models, I again found evidence of economies and diseconomies of scale for both

undergraduate and graduate education, but not for research dollars. The enrollment levels at

which average costs per student were minimized were higher for master institutions than for

bachelor institutions (35,200 for undergraduates and 6,900 for graduates). The model for cost per

research dollar, however, did not fit the data well (R2 = 0.04) and had no control variables

exhibiting significant relationships at even the 10% significance level.

In Tables 7 and 8 I summarize the findings for economies and diseconomies of scale in

the multi-product case for bachelor and master institutions using the FFCQ function. Results are

shown for undergraduate instruction, graduate instruction, and research dollars at four different

assumed values for each output (50%, 100%, 200%, and 300% of the means). For each output,

the first row contains the estimated average incremental costs, the second row shows the

20

estimated marginal costs, and the third row provides the ratio of AIC to MC, which is used to

assess economies and diseconomies of scale.

-------------------------- Insert Tables 7 and 8 Here --------------------------

Beginning with bachelor institutions, the FFCQ equation reveals that there were

economies of scale in undergraduate instruction throughout the range of values considered here.

Subsequent calculations showed that there were diseconomies of scale in undergraduate

instruction for enrollments levels below 20% of the mean (~ 400 students). Graduate education

changed from diseconomies of scale to economies of scale once graduate enrollments exceeded

170% of the mean (~240 students). Likewise, research changed from diseconomies of scale to

economies of scale once it exceeded 300% of the mean (~$6.9 million). These results are

somewhat at odds with the notion of economies of scale in that it is normally the case that

production costs are high at low output levels and thus economies of scale can occur at low

output levels. The results from the FFCQ equation, however, suggest that the opposite is true for

bachelor institutions.

Turning to master institutions (Table 8), I found that there were economies of scale for all

three output measures over the entire ranges considered. At the mean output levels, for example,

average incremental costs for undergraduates were 12% higher than marginal costs, 102% higher

for graduate students, and almost three times higher for research grants. For undergraduate

students, the ratio of AIC to MC increased as output increased. With regard to graduate education

and research, the ratios of AIC to MC at first fell as output increased, but eventually started to

rise once graduate enrollments and research exceeded 300% of their means.

Finally, in Tables 9 and 10 I report the summary statistics on AIC, MC, and S for bachelor

and master institutions when I used a flexible fixed cost cubic function. Beginning with bachelor

21

institutions, I found that institutions were operating in the economies of scale range of output up

to about 16,300 students. Likewise, there were economies of scale in graduate education up to

about 180 students, and economies of scale in research up to about $11.4 million. Turning to

master institutions, the results from the flexible fixed cost cubic function reveal that

undergraduate and graduate education were always within the economies of scale portions of

production. In research, however, master institutions exhibited economies of scale up to about

$12 million, followed by a period of diseconomies of scale up to $30 million, and then

economies of scale for remaining increases in output.

-------------------------- Insert Tables 9 and 10 Here --------------------------

Summary and Discussion

The main goal of this paper was to provide updated estimates of economies and

diseconomies of scale in higher education for associate, bachelor, and master institutions. In

doing so, decisions had to be made as to whether to treat colleges and universities as single- or

multi-product firms, and what functional form to use for the cost equation. My findings showed

that when institutions are treated as single-product firms that primarily educate students, there

are economies of scale throughout large ranges of enrollments. The cubic cost function approach

revealed that for associate institutions, for example, there were economies of scale up to about

25,000 students, which is more than five times their average FTE enrollment levels. The

quadratic cost function, on the other hand, showed economies of scale persisting over all levels

of enrollments. This result follows from the fact that a quadratic total cost function will lead to

an average cost curve that falls at a decreasing rate as output rises…but will not switch direction.

22

When bachelor and master institutions were treated as multi-product firms, however, the

findings differed somewhat across methods. The approach where costs per type of output

followed from a quadratic average cost curve were largely consistent with the single-product

firm results in that both economies and diseconomies of scale were found in most cases, with the

cost-minimizing output levels being notably above the mean output levels. The quadratic total

cost method revealed economies of scale for all three outputs at master-level institutions;

however, the findings for bachelor-level institutions were more challenging to interpret. The

FFCQ approach showed economies of scale over the entire range of undergraduates, but for

graduate education and research the results are a bit counterintuitive in that they suggested that

there were at first diseconomies of scale followed by economies of scale. When I used a cubic

total cost function instead of the quadratic total cost function, the findings were more comparable

to what I found using the quadratic average cost curve approach in that there were initially

economies of scale followed by diseconomies of scale for graduate education and research.

My examination here highlights the fact that conclusions regarding economies and

diseconomies of scale in higher education depend crucially on the method chosen to do the

assessment. The cubic cost function approach has two main advantages: (1) the cost function is

consistent with the textbook depiction of U-shaped average and marginal cost curves; and (2) the

determination of economies and diseconomies of scale is fairly straightforward application of

derivatives. In contrast, the FFCQ function gives rise to linear marginal cost curves, and the

calculations are sensitive to the particular assumptions used by the researcher when simulating

average and marginal costs. At a minimum, researchers need to be as transparent as possible in

showing how their calculations were done using the FFCQ method.

23

References

Baumol, W., Panzar, J., & Willig, D. (1982). Contestable markets and the theory of industry

structure. New York: Harcourt and Brace Jovanovich.

Bowen, H. (1980). The costs of higher education. San Francisco: Jossey-Bass Publishers.

Brinkman, P. (1990). Higher education cost functions. In S. Hoenack & E. Collins (Eds.), The

economics of American universities: Management, operations, and fiscal environment.

New York: State University of New York Press.

Brinkman, P., & Leslie, L. (1986). Economies of scale in higher education: Sixty years of

research. Review of Higher Education, 10, 1-28.

Cohn, E., Rhine,S., & Santos, M. (1989). Institutions of higher education as multi-product firms:

Economies of scale and scope. The Review of Economics and Statistics, 71, 284-290.

deGroot, H., McMahon, W., & Volkwein, J. (1991). The cost structure of American research

universities. The Review of Economics and Statistics, 73, 424-431.

Fu, T., Huang, C., & Tien, F. (2008). University cost structure in Taiwan. Contemporary

Economic Policy, 26, 651-662.

Getz, M., Siegfried, J., & Zhang, H. (1991). Estimating economies of scale in higher education.

Economics Letters, 37, 203-208.

Izadi, H., Johnes, G., Oskrochi, R., & Crouchley, R. (2002). Stochastic frontier estimation of a

CES cost function: The case of higher education in Britain. Economics of Education

Review, 21, 63-71.

James, E. (1978). Product mix and cost disaggregation: A reinterpretation of the economics of

higher education. Journal of Human Resources, 13, 157-186.

Johnes, G., & Johnes, J. (2009). Higher education institutions’ costs and efficiency: Taking the

decomposition a further step. Economics of Education Review, 28, 107-113.

Koshal, R., & Koshal, M. (1995). Quality and economies of scale in higher education. Applied

Economics, 27, 773-778.

Koshal, R., & Koshal, M. (1999). Economies of scale and scope in higher education: a case of

comprehensive universities. Economics of Education Review, 18, 269-277.

Laband, D., & Lentz, B. (2003). New estimates of economies of scale and scope in higher

education. Southern Economic Journal, 70, 172-183.

24

Laband, D., & Lentz, B. (2004). Do costs differ between for-profit and not-for-profit producers

of higher education? Research in Higher Education, 45, 429-441.

Lenton, P. (2008). The cost structure of higher education in further education colleges in

England. Economics of Education Review, 27, 471-482.

Mangoldt, H. (1863). The exchange ratio of goods. Translated by E. Henderson from Grundriss

der volkswirtschaftslehre, Stuttgart: Engelhorn. In International Economic Papers, No.

11, (pp.32-59). London: Macmillan, 1962.

Middlebrook, W., et al. (1955). California and western conference cost and statistical study.

Berkeley: University of California.

Moore, F. (1959). Economies of scale: Some statistical evidence. Quarterly Journal of

Economics, 73, 232–245.

Panzar, J., & Willig, R. (1977). Economies of scale in multi-output production. The Quarterly

Journal of Economics, 91, 481-493.

Paulsen, M., (1989). Estimating instructional cost functions at small independent colleges.

Journal of Education Finance, 15, 53-66.

Pfouts, R. (1961). The theory of cost and production in the multi-product firm. Econometrica,

29, 650-658.

Reeves, R., & Russell, J. (1935). The evaluation of higher institutions, finance, 7, Chicago:

University of Chicago Press.

Robst, J. (2000). Do state appropriations influence cost efficiency in public higher education?

Applied Economics Letters, 7, 715-719.

Robst, J. (2001). Cost efficiency in public higher education institutions. The Journal of Higher

Education, 72, 730-750.

Russell, J. (1954). The finance of higher education. Chicago: University of Chicago.

Sav, G. (2004). Higher education costs and scale and scope economies. Applied Economics, 36,

607-614.

Stevens, P. (2005). A stochastic frontier analysis of English and Welsh universities. Education

Economics, 13, 355-374.

Stevens, E., & Elliott, E. (1925). Unit costs of higher education, publications of the educational

finance inquiry, 13, New York: Macmillan.

http://msuweb.montclair.edu/~lebelp/MooreEcsScaleQJE1959.pdf

http://en.wikipedia.org/wiki/Quarterly_Journal_of_Economics

http://en.wikipedia.org/wiki/Quarterly_Journal_of_Economics

25

Teece, D. (1982). Towards an economic theory of the multiproduct firm. Journal of Economic

Behavior and Organization, 3, 39-63.

Tirivayi, N., van den Brink, H., & Groot, W. (2014). Size and economies of scale in higher

education and the implications for mergers. UNI-MERIT Working Paper Series, paper

#2014-066.

Titus, M., & Eagan, K. (2016). Examining production efficiency in higher education: The utility

of stochastic frontier analysis. Forthcoming, Higher Education: Handbook of Theory and

Research.

Toutkoushian, R. (1999). The value of cost functions for policymaking and institutional

research. Research in Higher Education, 40, 1-16.

Weldon, J. (1948). The multi-product firm. Canadian Journal of Economics and Political

Science, 14, 176-190.

Witmer, D. (1972). Cost studies in higher education. Review of Educational Research, 42, 99-

127.

26

Figure 1: Average and Marginal Cost from Cubic Total Cost Function

Cost Per Unit of

j-th Output

ACj

Output

---- Economies of Scale ------- - Diseconomies of Scale --

MCj

27

Figure 2: Average Cost from Quadratic Total Cost Function

AC

Output (Q)

Cost Per Unit of

j-th Output

28

Figure 3: Economies and Diseconomies of Scale – Flexible Fixed Cost Quadratic Function

Cost Per Unit of

j-th Output

Output (Q)

Economies of Scale Q* ---- Diseconomies of Scale -----

AICj

MCj

29

Table 1: Means for Selected Variables by Institution Type

Variable Associate Bachelor Master

Dependent Variables

Cost per FTE Student $13,488 $28,165 $20,909

Cost per UG Student ----- $29,316 $24,258

Cost per G Student ----- $1,417,106 $148,867

Cost per Research $ ----- $120.5 $207.5

Explanatory Variables

FTE Students 4,631 1,898 5,750

Undergrad Students (U) ----- 1,897 5,438

Grad Students (G) ----- 155 1,197

Research $ (R) ----- $2.29 million $7.96 million

% Grad Students ----- 6.9% 20.8%

% Part Time Students 54.5% 14.4% 26.4%

Avg. Faculty Salary $55,085 $59,770 $64,299

Acceptance Rate ----- 61.5% 66.4%

SAT Math 75th Percentile Score ----- 575.6 563.0

Public Institution 91.4% 12.2% 47.0%

% Students No DE Classes 69.9% 88.6% 80.0%

% STEM Degrees 15.7% 26.4% 19.3%

Region 1 5.4% 5.0% 7.5%

Region 2 12.1% 15.6% 20.1%

Region 3 13.9% 16.7% 16.9%

Region 4 9.8% 15.9% 9.6%

Region 6 12.1% 4.8% 8.0%

Region 7 4.0% 2.4% 1.8%

Region 8 13.9% 6.9% 11.4%

Urban 32.1% 33.9% 47.0%

Rural 23.6% 9.3% 3.0%

Any Grad ----- 65.3% 99.8%

Any Research ----- 42.3% 74.7%

UG x G ----- 39.6 944.1

UG x R ----- 81.7 826.6

G x R ----- 4.3 139.1

UG x SAL ----- 1182.2 3666.1

G x SAL ----- 95.4 808.1

R x SAL ----- 144.9 538.1

Sample Size 779 378 438 Notes: All data were retrieved from the Delta Cost Project except for the variable “% Students in No DE Classes”

(IPEDS). All data are for the 2013 academic year. Institutions were omitted from the sample if they (a) had a

medical school, (b) were a for-profit institution, (c) had average faculty salaries below $25,000 or above $250,000,

(d) had fewer than 100 students, or (e) reported data to IPEDS for multiple institutions (“Parent-child” problem).

30

Table 2: Quadratic Average Cost Models and Economies of Scale for Single-Product Firms

Associate Institutions Bachelor Institutions Master Institutions

Variable (1) (2) (1) (2) (1) (2)

FTE Students -72.38*** -71.25*** -8.89 -494.70*** -51.74*** -84.04***

(10.43) (12.13) (84.06) (82.96) (16.07) (19.48)

FTE Students Squared 0.14*** 0.14*** -0.83 2.50*** 0.11* 0.19**

(0.04) (0.04) (0.59) (0.53) (0.06) (0.07)

Avg. Faculty Salary ----- 59.70** ----- 520.61*** ----- 207.46***

(18.56) (45.21) (53.18)

New England ----- 394.95 ----- 2030.43 ----- 406.54

(693.85) (2360.87) (1003.86)

Mideast ----- 992.89+ ----- -255.11 ----- 300.87

(599.83) (1168.31) (785.07)

Great Lakes ----- 1441.83** ----- -1449.98 ----- -581.16

(518.55) (955.45) (723.29)

Plains ----- 650.08 ----- -1975.10* ----- -1303.31

(559.28) (960.25) (875.65)

Southwest ----- 161.67 ----- 75.64 ----- 872.32

(460.13) (2050.89) (772.73)

Rocky Mountains ----- 2017.82* ----- -2748.60 ----- -371.71

(885.13) (2191.44) (1388.74)

(Table continues)

31

Associate Institutions Bachelor Institutions Master Institutions

Variable (1) (2) (1) (2) (1) (2)

Far West ----- 2368.22* ----- -300.29 ----- -1322.48

(1126.09) (2534.72) (824.27)

% Part-Time Students ----- -25.86 ----- -158.12** ----- -116.56***

(22.73) (48.28) (29.15)

% No Distance Ed ----- 22.93* ----- 17.10 ----- 34.03*

(10.43) (28.77) (15.25)

Public Institution ----- -3876.10*** ----- 1788.25 ----- -1434.01+

(1028.10) (1783.10) (848.30)

% STEM Degrees ----- 18.28 ----- 63.85* ----- 77.99***

(21.95) (32.31) (15.99)

Acceptance Rate ----- ----- ----- -107.02*** ----- -43.46*

(28.20) (17.02)

SAT Score ----- ----- ----- 38.75*** ----- -11.52

(8.81) (14.73)

Constant 16173.35*** 15011.13*** 28803.24*** -1.2e+04* 23267.58*** 19463.03**

(476.17) (1399.29) (1450.84) (4776.75) (790.77) (7078.61)

Sample Size 779 779 378 378 438 438

R2 0.13 0.23 0.01 0.76 0.05 0.42

Economies of Scale? Yes Yes No Yes Yes Yes

Diseconomies of Scale? Yes Yes No Yes Yes Yes

Minimum AC 25,850 25,446 ----- 9,894 23,518 22,116 Notes: Robust standard errors are shown in parentheses. Dependent variable is expenditure per FTE student. Model (2) includes control variables for urbanicity

and percent graduate students (bachelor and master). Reference category for region is Southeast. + p<.10, * p<.05, ** p<.01, *** p<.001.

32

Table 3: Quadratic Total Cost Functions for Single-Product Firms

Institution Type:


FTE Students 584.72*** 1409.12* 1921.16***

(72.29) (561.09) (171.28)

FTE Students Squared -0.27** -7.91*** -1.11+

(0.09) (0.98) (0.59)

Avg. Faculty Salary 440.70 -1502.21** -2171.43*

(368.26) (496.56) (914.16)

Avg. Salary Squared -4.98 17.17*** 22.10**

(3.51) (4.35) (7.16)

FTE Students x Avg. Salary 4.96*** 14.06 0.28

(0.85) (9.36) (2.67)

New England 3932.05* 4471.19 -476.95

(1567.05) (5162.70) (4658.71)

Mideast 7842.87*** 194.90 3021.73

(1571.08) (3103.25) (4089.18)

Great Lakes 11268.14*** -628.87 -3492.23

(1982.95) (2237.72) (4395.04)

Plains 1142.24 -3266.71+ -7521.13+

(1186.54) (1806.00) (4129.23)

Southwest -2305.82 -476.39 435.24

(1648.96) (2448.86) (3657.68)

Rocky Mountains 1555.81 -9213.51* -1453.26

(2483.08) (3602.92) (8315.98)

Far West 7642.17*** -1.3e+04** -1.1e+04*

(2075.19) (4251.11) (4589.43)

% Part-Time Students -150.38*** -469.24*** -476.51***

(31.90) (96.42) (108.52)

(Table continues)

33

Institution Type:


% No Distance Ed -44.32 57.88 187.09*

(36.02) (53.05) (89.15)

Public Institution 6285.19*** -732.20 -8228.36

(1589.04) (3210.82) (6317.15)

% STEM Degrees 61.01* -54.99 494.96*

(27.04) (109.22) (231.66)

Urban 3133.01* 1334.03 1312.74

(1311.28) (1445.81) (2672.41)

Rural -1585.48 -1876.38 4641.15

(1109.16) (2355.96) (4078.16)

Acceptance Rate ----- -194.98*** -262.04**

(49.76) (80.05)

SAT Score ----- 89.73*** 17.11

(16.44) (39.66)

Pct Graduate Students ----- 64.40 290.06**

(99.53) (102.02)

Constant -127.04 196.33 48459.08

(10006.63) (14379.73) (34200.84)

Sample Size 779 378 438

R2 0.93 0.89 0.93 Notes: Robust standard errors are shown in parentheses. Dependent variable is total expenditures. Reference

category for region is Southeast. Reference category for urbanicity is suburban and town. + p<.10, * p<.05, **

p<.01, *** p<.001.

34

Table 4: Predicted Average and Marginal Costs Using FFCQ Function for Single-Product

Firm

Pct of Mean FTE

Enrollment

Predicted Average Cost

Associate Bachelor Master

50% $11,969 $33,679 $21,960

100% $10,179 $26,961 $20,198

200% $9,190 $22,476 $18,839

300% $8,777 $19,980 $17,961

400% $8,508 $17,981 $17,202

500% $8,269 $16,181 $16,492

600% $8,113 $14,480 $15,806

Pct of Mean FTE

Enrollment

Predicted Marginal Cost

Associate Bachelor Master

50% $8,452 $20,994 $18,756

100% $8,327 $19,492 $18,118

200% $8,076 $16,489 $16,842

300% $7,826 $13,486 $15,565

400% $7,575 $10,483 $14,289

500% $7,325 $7,480 $13,013

600% $7.074 $4,476 $11,736

Mean FTE Enrollment 4,631 1,898 5,750 Notes: Details of predicted average cost calculations are shown in Tables A1 to A3 in the Appendix. Values were

obtained by substituting FTE student figures into the estimated FFCQ functions shown in Table 3.

35

Table 5: Quadratic Average Cost Models at Bachelor Institutions for Multi-Product Firm

Output Measure:

Variable

Undergraduate

Students

Graduate Students

Research Dollars

Undergrad Students -623.28*** 56814.55 3.05

(115.52) (37916.27) (3.15)

Undergrads Squared 2.22*** -184.54 -0.01

(0.51) (138.45) (0.01)

Graduate Students 985.41* -1.3e+06+ -6.25

(411.74) (7.0e+05) (10.60)

Grads Squared -30.87 75661.88+ -0.62

(30.07) (40155.93) (0.77)

Research Dollars 1066.48*** 2.3e+05 -30.86***

(257.48) (2.5e+05) (8.10)

Research Squared -22.70*** -6997.52 0.67**

(6.84) (7445.01) (0.26)

Avg. Faculty Salary 558.30*** 1.1e+05+ 5.28*

(57.49) (62591.66) (2.25)

New England 2175.95 -1.4e+06 61.50

(2599.61) (1.1e+06) (101.87)

Mideast -733.67 -1.6e+05 -70.83*

(1284.13) (6.4e+05) (31.94)

Great Lakes -1777.50+ 6.7e+05 62.35

(1044.75) (4.5e+05) (104.39)

Plains -2433.38* 5.1e+05 -38.54+

(997.63) (4.0e+05) (20.79)

Southwest -1296.63 7.0e+05 -74.73*

(2068.14) (6.0e+05) (34.37)

Rocky Mountains -3064.48 -2.6e+05 -3.39

(2506.89) (1.1e+06) (37.47)

(Table continues)

36

Output Measure:

Variable

Undergraduate

Students

Graduate Students

Research Dollars

Far West 3397.55 5.3e+06 143.34

(6130.40) (5.4e+06) (167.70)

Acceptance Rate -41.27 -1.6e+04 -0.35

(60.59) (12807.68) (0.78)

SAT Score 54.93*** 372.89 -1.08*

(10.83) (3714.03) (0.54)

% Part-Time Students -153.08*** 12241.50 4.76+

(37.00) (26780.36) (2.69)

% No Distance Ed 25.30 -2.2e+04 3.62*

(32.15) (16894.05) (1.63)

Public Institution -1321.86 -3.0e+06 -111.45+

(1767.84) (2.2e+06) (65.12)

% STEM Degrees 14.48 9005.02 -1.92+

(36.69) (14127.42) (1.04)

Constant -2.6e+04*** -2.3e+06 144.06

(6167.85) (3.1e+06) (103.30)


R2 0.66 0.16 0.10

Economies of Scale? Yes Yes Yes

Diseconomies of Scale? Yes Yes Yes

Minimum AC 14,038 891 $23.0 million Notes: Robust standard errors are shown in parentheses. Dependent variables are expenditure per undergraduate

student, expenditure per graduate student, and expenditure per research dollar. Each model also contains control

variables for degree of urbanicity. Reference category for region is Southeast. + p<.10, * p<.05, ** p<.01, ***

p<.001.

37

Table 6: Quadratic Average Cost Models at Master Institutions – Multi Product Firm

Output Measure:

Variable

Undergraduate

Students

Graduate

Students

Research Dollars

Undergrad Students -225.34*** -2433.97 6.67

(24.94) (2705.51) (6.63)

Undergrads Squared 0.32*** 7.64 -0.03

(0.08) (7.11) (0.03)

Graduate Students 497.34*** -9723.93*** -5.82

(81.37) (1157.29) (6.80)

Grads Squared -1.37** 70.11*** 0.05

(0.50) (11.47) (0.05)

Research Dollars 322.99*** 2803.53 -37.53

(76.24) (1986.61) (27.78)

Research Squared -3.63** -19.16 0.52

(1.31) (20.95) (0.42)

Avg. Faculty Salary 234.46** 499.86 -14.10

(78.53) (665.82) (12.29)

New England 1221.31 12303.11 6.43

(1645.22) (23316.23) (200.48)

Mideast -423.81 529.49 64.40

(998.84) (20212.87) (244.95)

Great Lakes -472.07 1.1e+05 -130.15

(968.04) (90700.95) (141.33)

Plains -1849.60+ -2.6e+04 -181.22

(1073.34) (21709.94) (141.39)

Southwest 120.20 -8337.50 -90.46

(975.03) (20640.53) (135.92)

Rocky Mountains -1526.78 -5.4e+04 -9.90

(1713.98) (48401.40) (188.64)

(Table continues)

38

Output Measure:

Variable

Undergraduate

Students

Graduate

Students

Research Dollars

Far West -356.30 -6473.30 1279.88

(1224.22) (21191.70) (1414.23)

Acceptance Rate -38.39+ 1085.14 -1.50

(19.64) (1649.05) (3.04)

SAT Score 6.27 758.70+ 0.29

(21.42) (418.22) (0.84)

% Part-Time Students -191.73*** -2657.18*** -4.08

(37.61) (649.96) (4.41)

% No Distance Ed 30.11 1148.94 -5.21

(25.34) (997.67) (7.17)

Public Institution -3457.07* 1.7e+05 -220.93

(1425.26) (1.3e+05) (195.76)

% STEM Degrees 33.04+ 431.46 -4.60

(18.66) (596.53) (5.12)

Constant 14481.17 -3.2e+05 1555.87

(9997.41) (3.5e+05) (1423.99)


R2 0.58 0.15 0.04

Economies of Scale? Yes Yes No

Diseconomies of Scale? Yes Yes No

Minimum AC 35,209 6,935 ----- Notes: Robust standard errors are shown in parentheses. Dependent variables are expenditure per undergraduate

student, expenditure per graduate student, and expenditure per research dollar. Each model includes additional

controls for degree of urbanicity. Reference category for region is Southeast. + p<.10, * p<.05, ** p<.01, *** p<.001

39

Table 7: Flexible Fixed Cost Quadratic Results – Multi-Product Firm: Bachelor

Institutions

Output

Metric

Output Level

50% 100% 200% 300% 400%

Undergraduate

Education

AIC $19,278 $19,051 $18,597 $18,143 $17,688

MC $19,051 $18,597 $17,688 $16,780 $15,872

S 1.01 1.02 1.05 1.08 1.11

Graduate

Education

AIC $15,353 $17,583 $17,966 $17,444 $16,696

MC $20,300 $19,325 $17,375 $15,425 $13,475

S 0.76 0.91 1.03 1.13 1.24

Research

Dollars

AIC $1.03 $1.22 $1.30 $1.31 $1.30

MC $1.43 $1.40 $1.35 $1.30 $1.25

S 0.72 0.87 0.96 1.00 1.04 Notes: AIC = average incremental cost, MC = marginal cost, S = ratio of AIC to MC. Details of calculations can be

found in Tables A4 – A6 in the Appendix.

Table 8: Flexible Fixed Cost Quadratic Results – Multi-Product Firm: Master Institutions

Output

Metric

Output Level

50% 100% 200% 300% 400%

Undergraduate

Education

AIC $15,555 $14,746 $13,127 $11,508 $9,890

MC $14,746 $13,127 $9,890 $6,652 $3,415

S 1.06 1.12 1.33 1.73 2.90

Graduate

Education

AIC $63,518 $42,641 $31,268 $26,646 $23,713

MC $22,386 $21,140 $18,649 $16,158 $13,667

S 2.84 2.02 1.68 1.65 1.74

Research

Dollars

AIC $10.66 $6.14 $3.79 $2.92 $2.43

MC $1.69 $1.56 $1.31 $1.07 $0.82



40

Table 9: Flexible Fixed Cost Cubic Results – Multi-Product Firm: Bachelor Institutions

Output

Metric

Output Level

50% 100% 200% 300% 400%

Undergraduate

Education

AIC $26,329 $24,001 $19,787 $16,164 $13,130

MC $23,927 $19,493 $11,950 $6,178 $2,175

S 1.10 1.23 1.66 2.62 6.04

Graduate

Education

AIC $13,848 $10,450 $11,670 $13,942 $15,820

MC $4,778 $9,217 $16,125 $20,408 $22,066

S 2.90 1.13 0.72 0.68 0.72

Research

Dollars

AIC $3.85 $2.35 $1.69 $1.54 $1.49

MC $0.78 $0.91 $1.14 $1.31 $1.43



Table 10: Flexible Fixed Cost Cubic Results – Multi-Product Firm: Master Institutions

Output

Metric

Output Level

50% 100% 200% 300% 400%

Undergraduate

Education

AIC $16,470 $15,442 $13,445 $11,525 $9,685

MC $15,432 $13,405 $9,528 $5,886 $2,479

S 1.07 1.15 1.41 1.96 3.91

Graduate

Education

AIC $63,618 $42,402 $31,143 $26,749 $24,015

MC $21,591 $20,770 $18,959 $16,924 $14,665

S 2.95 2.04 1.64 1.58 1.64

Research

Dollars

AIC $1.64 $1.31 $1.28 $1.33 $1.34

MC $0.84 $1.08 $1.39 $1.44 $1.25



41

Appendix:

Table A1: Quadratic Cost Function Calculations for Associate Institutions – Single-Product Firms

Predicted Cost as Percentage of Mean FTE:

Variable Mean Beta 50% 100% 200% 300% 400% 500% 600%

FTE Students 46.31 584.72 13539 27079 54157 81236 108314 135393 162471

FTE Students Squared 4616.85 -0.27 -145 -580 -2320 -5221 -9281 -14502 -20882

Avg Salary 55.09 440.70 24276 24276 24276 24276 24276 24276 24276

Avg Salary Squared 3183.41 -4.98 -15859 -15859 -15859 -15859 -15859 -15859 -15859

Students x Salary 4031.77 4.96 6321 12642 25285 37927 50569 63212 75854

New England 0.05 3932.05 212 212 212 212 212 212 212

Mideast 0.12 7842.87 946 946 946 946 946 946 946

Great Lakes 0.14 11268.14 1562 1562 1562 1562 1562 1562 1562

Plains 0.10 1142.24 111 111 111 111 111 111 111

Southwest 0.12 -2305.82 -278 -278 -278 -278 -278 -278 -278

Rocky Mountains 0.04 1555.81 62 62 62 62 62 62 62

Far West 0.14 7642.17 1060 1060 1060 1060 1060 1060 1060

% PT Student 54.53 -150.38 -8201 -8201 -8201 -8201 -8201 -8201 -8201

% No Distance Ed 69.95 -44.32 -3100 -3100 -3100 -3100 -3100 -3100 -3100

Public Institution 0.91 6285.19 5745 5745 5745 5745 5745 5745 5745

% STEM Degrees 15.70 61.01 958 958 958 958 958 958 958

Urban 0.32 3133.01 1005 1005 1005 1005 1005 1005 1005

Rural 0.24 -1585.48 -374 -374 -374 -374 -374 -374 -374

Constant 1.00 -127.04 -127 -127 -127 -127 -127 -127 -127

Total Cost (C) = $27.7m $47.1m $85.1m $121.9m $157.6m $192.1m $225.4m

Enrollment (Q) = 2,316 4,631 9,262 13,893 18,524 23,155 27,786

Average Cost (AC) = $11,969 $10,179 $9,190 $8,777 $8,508 $8,296 $8,113

Marginal Cost (MC) = $8,452 $8,327 $8,076 $7,826 $7,575 $7,325 $7,074

42

Table A2: Quadratic Cost Function Calculations for Bachelor Institutions – Single-Product Firms

Predicted Cost as Percentage of Mean FTE:


FTE Students 18.98 1409.12 13374 26748 53496 80244 106992 133741 160489


Avg Salary 59.77 -1502.21 -89787 -89787 -89787 -89787 -89787 -89787 -89787

Avg Salary Squared 3797.58 17.17 65193 65193 65193 65193 65193 65193 65193


New England 0.05 4471.19 225 225 225 225 225 225 225

Mideast 0.16 194.90 30 30 30 30 30 30 30

Great Lakes 0.17 -628.87 -105 -105 -105 -105 -105 -105 -105

Plains 0.16 -3266.71 -519 -519 -519 -519 -519 -519 -519

Southwest 0.05 -476.39 -23 -23 -23 -23 -23 -23 -23

Rocky Mountains 0.02 -9213.51 -219 -219 -219 -219 -219 -219 -219

Far West 0.07 -12523.78 -861 -861 -861 -861 -861 -861 -861

Acceptance Rate 61.48 -194.98 -11987 -11987 -11987 -11987 -11987 -11987 -11987

SAT Score 575.56 89.73 51644 51644 51644 51644 51644 51644 51644

% Grad Student 6.86 64.40 442 442 442 442 442 442 442

% PT Student 14.42 -469.24 -6768 -6768 -6768 -6768 -6768 -6768 -6768

% No Distance Ed 88.60 57.88 5128 5128 5128 5128 5128 5128 5128

Public Institution 0.12 -732.20 -89 -89 -89 -89 -89 -89 -89

% STEM Degrees 26.41 -54.99 -1452 -1452 -1452 -1452 -1452 -1452 -1452

Urban 0.34 1334.03 452 452 452 452 452 452 452

Rural 0.09 -1876.38 -174 -174 -174 -174 -174 -174 -174

Constant 1.00 196.33 196 196 196 196 196 196 196


Enrollment (Q) = 949 1,898 3,796 5,695 7,593 9,491 11,389

Average Cost (AC) = $33,679 $26,961 $22,476 $19,980 $17,981 $16,181 $14,480

Marginal Cost (MC) = $20,994 $19,492 $16,489 $13,486 $10,483 $7,480 $4,476

43

Table A3: Quadratic Cost Function Calculations for Master Institutions – Single-Product Firms

Predicted Cost at Selected FTE Enrollments:


FTE Students 57.50 1921.16 55237 110474 220949 331423 441897 552371 662846


Avg Salary 64.30 -2171.43 -139620 -139620 -139620 -139620 -139620 -139620 -139620

Avg Salary Squared 4266.37 22.10 94290 94290 94290 94290 94290 94290 94290


New England 0.08 -476.95 -36 -36 -36 -36 -36 -36 -36

Mideast 0.20 3021.73 607 607 607 607 607 607 607

Great Lakes 0.17 -3492.23 -590 -590 -590 -590 -590 -590 -590

Plains 0.10 -7521.13 -721 -721 -721 -721 -721 -721 -721

Southwest 0.08 435.24 35 35 35 35 35 35 35

Rocky Mountains 0.02 -1453.26 -27 -27 -27 -27 -27 -27 -27

Far West 0.11 -10768.66 -1229 -1229 -1229 -1229 -1229 -1229 -1229

Acceptance Rate 66.41 -262.04 -17403 -17403 -17403 -17403 -17403 -17403 -17403

SAT Score 563.00 17.11 9631 9631 9631 9631 9631 9631 9631

% Grad Student 20.82 290.06 6039 6039 6039 6039 6039 6039 6039

% PT Student 26.37 -476.51 -12563 -12563 -12563 -12563 -12563 -12563 -12563

% No Distance Ed 80.00 187.09 14967 14967 14967 14967 14967 14967 14967

Public Institution 0.47 -8228.36 -3870 -3870 -3870 -3870 -3870 -3870 -3870

% STEM Degrees 19.33 494.96 9568 9568 9568 9568 9568 9568 9568

Urban 0.47 1312.74 617 617 617 617 617 617 617

Rural 0.03 4641.16 138 138 138 138 138 138 138

Constant 1.00 48459.08 48459 48459 48459 48459 48459 48459 48459


Enrollment (Q) = 2,875 5,750 11,501 17,251 23,002 28,752 34,502

Average Cost (AC) = $21,960 $20,198 $18,839 $17,961 $17,202 $16,492 $15,806

Marginal Cost (MC) = $18,756 $18,118 $16,842 $15,565 $14,289 $13,013 $11,736

44

Table A4: FFCQ Calculations of Economies of Scale for Bachelor Institutions:

Undergraduates

Predicted Costs at Selected UG Enrollments:

Variable Mean Beta 0% 50% 100% 200% 300% 400%

Undergrads 18.97 139.08 0 1319 2638 5276 7914 10552

UG Squared 639.29 -2.39 0 -215 -861 -3445 -7752 -13781

Grads 1.55 1104.04 1711 1711 1711 1711 1711 1711

G Squared 9.19 -62.91 -578 -578 -578 -578 -578 -578

Research 2.29 1716.06 3936 3936 3936 3936 3936 3936

Res Squared 25.79 -10.73 -277 -277 -277 -277 -277 -277

Avg Salary 59.77 -1243.86 -74345 -74345 -74345 -74345 -74345 -74345

Sal Squared 3797.58 12.61 47875 47875 47875 47875 47875 47875

UG x Res 81.75 -10.31 0 -224 -448 -897 -1345 -1794

G x Res 4.28 23.48 101 101 101 101 101 101

UG x G 39.59 -33.71 0 -495 -991 -1982 -2973 -3963

UG x Sal 1182.16 31.58 0 17898 35795 71591 107386 143182

G x Sal 95.37 26.92 2567 2567 2567 2567 2567 2567

Res x Salary 144.89 -1.76 -255 -255 -255 -255 -255 -255

Any Research 0.42 3779.87 1600 1600 1600 1600 1600 1600

Any Grad 0.65 -421.15 -275 -275 -275 -275 -275 -275

New England 0.05 4337.03 218 218 218 218 218 218

Mideast 0.16 -1179.61 -184 -184 -184 -184 -184 -184

Great Lakes 0.17 -624.85 -104 -104 -104 -104 -104 -104

Plains 0.16 -4683.47 -743 -743 -743 -743 -743 -743

Southwest 0.05 -2880.24 -137 -137 -137 -137 -137 -137

Rocky Mts 0.02 -7709.03 -184 -184 -184 -184 -184 -184

Far West 0.07 -12654.17 -870 -870 -870 -870 -870 -870

Acceptance Rate 61.48 -154.79 -9516 -9516 -9516 -9516 -9516 -9516

SAT Score 575.56 130.95 75372 75372 75372 75372 75372 75372

% PT Students 14.42 -441.11 -6362 -6362 -6362 -6362 -6362 -6362

% No Dist Ed 88.60 -11.83 -1048 -1048 -1048 -1048 -1048 -1048

Public Inst 0.12 -11127.56 -1354 -1354 -1354 -1354 -1354 -1354

% STEM Deg 26.41 -117.00 -3090 -3090 -3090 -3090 -3090 -3090

Urban 0.34 507.13 172 172 172 172 172 172

Rural 0.09 -1227.98 -114 -114 -114 -114 -114 -114

Constant 1.00 -13547.71 -13548 -13548 -13548 -13548 -13548 -13548

Total Cost (C) = $20.6m $38.8m $56.7m $91.1m $123.8m $154.8m

Undergrad (QU) = 0 948 1,897 3,793 5,690 7,587

AICU =

$19,278 $19,051 $18,597 $18,143 $17,688

MCU =

$19,051 $18,597 $17,688 $16,780 $15,872

SU =

1.01 1.02 1.05 1.08 1.11

45

Table A5: FFCQ Calculations of Economies of Scale for Bachelor Institutions: Graduates

Predicted Costs at Selected Grad Enrollments:


Undergrads 18.97 139.08 2638 2638 2638 2638 2638 2638

UG Squared 639.29 -2.39 -1531 -1531 -1531 -1531 -1531 -1531

Grads 1.55 1104.04 0 856 1711 3422 5134 6845

G Squared 9.19 -62.91 0 -38 -151 -604 -1360 -2418

Research 2.29 1716.06 3936 3936 3936 3936 3936 3936

Res Squared 25.79 -10.73 -277 -277 -277 -277 -277 -277

Avg Salary 59.77 -1243.86 -74345 -74345 -74345 -74345 -74345 -74345

Sal Squared 3797.58 12.61 47875 47875 47875 47875 47875 47875

UG x Res 81.75 -10.31 -843 -843 -843 -843 -843 -843

G x Res 4.28 23.48 0 42 83 167 250 334

UG x G 39.59 -33.71 0 -495 -991 -1982 -2973 -3963

UG x Sal 1182.16 31.58 37328 37328 37328 37328 37328 37328

G x Sal 95.37 26.92 0 1247 2494 4987 7481 9975

Res x Salary 144.89 -1.76 -255 -255 -255 -255 -255 -255

Any Research 0.42 3779.87 1600 1600 1600 1600 1600 1600

Any Grad 0.65 -421.15 0 -421 -421 -421 -421 -421

New England 0.05 4337.03 218 218 218 218 218 218

Mideast 0.16 -1179.61 -184 -184 -184 -184 -184 -184

Great Lakes 0.17 -624.85 -104 -104 -104 -104 -104 -104

Plains 0.16 -4683.47 -743 -743 -743 -743 -743 -743

Southwest 0.05 -2880.24 -137 -137 -137 -137 -137 -137

Rocky Mts 0.02 -7709.03 -184 -184 -184 -184 -184 -184

Far West 0.07 -12654.17 -870 -870 -870 -870 -870 -870

Acceptance Rate 61.48 -154.79 -9516 -9516 -9516 -9516 -9516 -9516

SAT Score 575.56 130.95 75372 75372 75372 75372 75372 75372

% PT Students 14.42 -441.11 -6362 -6362 -6362 -6362 -6362 -6362

% No Dist Ed 88.60 -11.83 -1048 -1048 -1048 -1048 -1048 -1048

Public Inst 0.12 -11127.56 -1354 -1354 -1354 -1354 -1354 -1354

% STEM Deg 26.41 -117.00 -3090 -3090 -3090 -3090 -3090 -3090

Urban 0.34 507.13 172 172 172 172 172 172

Rural 0.09 -1227.98 -114 -114 -114 -114 -114 -114

Constant 1.00 -13547.71 -13548 -13548 -13548 -13548 -13548 -13548


Graduate (QG) = 0 77 155 310 465 620

AICG =

$15,353 $17,583 $17,966 $17,444 $16,696

MCG =

$20,300 $19,325 $17,375 $15,425 $13,475

SG =

0.76 0.91 1.03 1.13 1.24

46

Table A6: FFCQ Calculations of Economies of Scale for Bachelor Institutions: Research

Predicted Costs at Selected Research Dollars:


Undergrads 18.97 139.08 2638 2638 2638 2638 2638 2638

UG Squared 639.29 -2.39 -1531 -1531 -1531 -1531 -1531 -1531

Grads 1.55 1104.04 1711 1711 1711 1711 1711 1711

G Squared 9.19 -62.91 -578 -578 -578 -578 -578 -578

Research 2.29 1716.06 0 1968 3936 7872 11809 15745

Res Squared 25.79 -10.73 0 -14 -56 -226 -508 -903

Avg Salary 59.77 -1243.86 -74345 -74345 -74345 -74345 -74345 -74345

Sal Squared 3797.58 12.61 47875 47875 47875 47875 47875 47875

UG x Res 81.75 -10.31 0 -224 -448 -897 -1345 -1794

G x Res 4.28 23.48 0 42 83 167 250 334

UG x G 39.59 -33.71 -1334 -1334 -1334 -1334 -1334 -1334

UG x Sal 1182.16 31.58 37328 37328 37328 37328 37328 37328

G x Sal 95.37 26.92 2567 2567 2567 2567 2567 2567

Res x Salary 144.89 -1.76 0 -121 -242 -483 -725 -967

Any Research 0.42 3779.87 0 1600 1600 1600 1600 1600

Any Grad 0.65 -421.15 -275 -275 -275 -275 -275 -275

New England 0.05 4337.03 218 218 218 218 218 218

Mideast 0.16 -1179.61 -184 -184 -184 -184 -184 -184

Great Lakes 0.17 -624.85 -104 -104 -104 -104 -104 -104

Plains 0.16 -4683.47 -743 -743 -743 -743 -743 -743

Southwest 0.05 -2880.24 -137 -137 -137 -137 -137 -137

Rocky Mts 0.02 -7709.03 -184 -184 -184 -184 -184 -184

Far West 0.07 -12654.17 -870 -870 -870 -870 -870 -870

Accept Rate 61.48 -154.79 -9516 -9516 -9516 -9516 -9516 -9516

SAT Score 575.56 130.95 75372 75372 75372 75372 75372 75372

% PT Students 14.42 -441.11 -6362 -6362 -6362 -6362 -6362 -6362

% No Dist Ed 88.60 -11.83 -1048 -1048 -1048 -1048 -1048 -1048

Public Inst 0.12 -11127.56 -1354 -1354 -1354 -1354 -1354 -1354

% STEM Deg 26.41 -117.00 -3090 -3090 -3090 -3090 -3090 -3090

Urban 0.34 507.13 172 172 172 172 172 172

Rural 0.09 -1227.98 -114 -114 -114 -114 -114 -114

Constant 1.00 -13547.71 -13548 -13548 -13548 -13548 -13548 -13548


Research (QR) = 0 $1.15m $2.29m $4.59m $6.88m $9.18m

AICR =

$1.03 $1.22 $1.30 $1.31 $1.30

MCR =

$1.43 1.40 $1.35 $1.30 $1.25

SR =

0.72 0.87 0.96 1.00 1.04

47

Table A7: FFCQ Calculations of Economies of Scale for Master Institutions:

Undergraduates

Predicted Costs at Selected UG Enrollments:


Undergrads 54.38 1394.33 0 37909 75817 151634 227452 303269

UG Squared 5530.15 -2.98 0 -2200 -8802 -35206 -79214 -140826

Grads 11.97 510.16 6104 6104 6104 6104 6104 6104

G Squared 276.93 -10.41 -2883 -2883 -2883 -2883 -2883 -2883

Research 7.96 2160.52 17201 17201 17201 17201 17201 17201

Res Squared 174.94 -15.61 -2730 -2730 -2730 -2730 -2730 -2730

Avg Salary 64.30 -1908.68 -122725 -122725 -122725 -122725 -122725 -122725

Sal Squared 4266.37 16.69 71227 71227 71227 71227 71227 71227

UG x Res 826.60 12.24 0 2649 5297 10595 15892 21190

G x Res 139.09 -38.49 -5353 -5353 -5353 -5353 -5353 -5353

UG x G 944.07 3.42 0 1114 2227 4455 6682 8909

UG x Sal 3666.05 1.61 0 2820 5639 11278 16917 22556

G x Sal 808.07 30.69 24799 24799 24799 24799 24799 24799

Res x Salary 538.14 -8.62 -4637 -4637 -4637 -4637 -4637 -4637

Any Research 0.75 1200.50 896 896 896 896 896 896

Any Grad 1.00 24235.80 24180 24180 24180 24180 24180 24180

New England 0.08 4053.77 305 305 305 305 305 305

Mideast 0.20 3122.58 627 627 627 627 627 627

Great Lakes 0.17 2269.62 383 383 383 383 383 383

Plains 0.10 -5324.59 -511 -511 -511 -511 -511 -511

Southwest 0.08 605.53 48 48 48 48 48 48

Rocky Mts 0.02 -9287.30 -170 -170 -170 -170 -170 -170

Far West 0.11 -9937.85 -1134 -1134 -1134 -1134 -1134 -1134

Accept Rate 66.41 -284.52 -18896 -18896 -18896 -18896 -18896 -18896

SAT Score 563.00 111.99 63052 63052 63052 63052 63052 63052

% PT Students 26.37 -833.95 -21987 -21987 -21987 -21987 -21987 -21987

% No Dist Ed 80.00 205.03 16402 16402 16402 16402 16402 16402

Public Inst 0.47 -14758.47 -6941 -6941 -6941 -6941 -6941 -6941

% STEM Deg 19.33 319.47 6176 6176 6176 6176 6176 6176

Urban 0.47 -1121.50 -527 -527 -527 -527 -527 -527

Rural 0.03 5727.09 170 170 170 170 170 170

Constant 1.00 -7839.59 -7840 -7840 -7840 -7840 -7840 -7840


Undergrads (QU) = 0 2,719 5,438 10,875 16,313 21,750

AICU =

$15,555 $14,746 $13,127 $11,508 $9,890

MCU =

$14,746 $13,127 $9,890 $6,652 $3,415

SU =

1.06 1.12 1.33 1.73 2.90

48

Table A8: FFCQ Calculations of Economies of Scale for Master Institutions: Graduates

Predicted Costs at Selected Grad Enrollments:


Undergrads 54.38 1394.33 75817 75817 75817 75817 75817 75817

UG Squared 5530.15 -2.98 -16463 -16463 -16463 -16463 -16463 -16463

Grads 11.97 510.16 0 3052 6104 12209 18313 24418

G Squared 276.93 -10.41 0 -373 -1490 -5962 -13414 -23847

Research 7.96 2160.52 17201 17201 17201 17201 17201 17201

Res Squared 174.94 -15.61 -2730 -2730 -2730 -2730 -2730 -2730

Avg Salary 64.30 -1908.68 -122725 -122725 -122725 -122725 -122725 -122725

Sal Squared 4266.37 16.69 71227 71227 71227 71227 71227 71227

UG x Res 826.60 12.24 10115 10115 10115 10115 10115 10115

G x Res 139.09 -38.49 0 -1833 -3666 -7333 -10999 -14666

UG x G 944.07 3.42 0 1114 2227 4455 6682 8909

UG x Sal 3666.05 1.61 5913 5913 5913 5913 5913 5913

G x Sal 808.07 30.69 0 11806 23611 47223 70834 94445

Res x Salary 538.14 -8.62 -4637 -4637 -4637 -4637 -4637 -4637

Any Research 0.75 1200.50 896 896 896 896 896 896

Any Grad 1.00 24235.80 0 24236 24236 24236 24236 24236

New England 0.08 4053.77 305 305 305 305 305 305

Mideast 0.20 3122.58 627 627 627 627 627 627

Great Lakes 0.17 2269.62 383 383 383 383 383 383

Plains 0.10 -5324.59 -511 -511 -511 -511 -511 -511

Southwest 0.08 605.53 48 48 48 48 48 48

Rocky Mts 0.02 -9287.30 -170 -170 -170 -170 -170 -170

Far West 0.11 -9937.85 -1134 -1134 -1134 -1134 -1134 -1134

Accept Rate 66.41 -284.52 -18896 -18896 -18896 -18896 -18896 -18896

SAT Score 563.00 111.99 63052 63052 63052 63052 63052 63052

% PT Students 26.37 -833.95 -21987 -21987 -21987 -21987 -21987 -21987

% No Dist Ed 80.00 205.03 16402 16402 16402 16402 16402 16402

Public Inst 0.47 -14758.47 -6941 -6941 -6941 -6941 -6941 -6941

% STEM Deg 19.33 319.47 6176 6176 6176 6176 6176 6176

Urban 0.47 -1121.50 -527 -527 -527 -527 -527 -527

Rural 0.03 5727.09 170 170 170 170 170 170

Constant 1.00 -7839.59 -7840 -7840 -7840 -7840 -7840 -7840


Graduates (QG) = 0 598 1,197 2,393 3,590 4,786

AICG =

$63,518 $42,641 $31,268 $26,646 $23,713

MCG =

$22,386 $21,140 $18,649 $16,158 $13,667

SG =

2.84 2.02 1.68 1.65 1.74

49

Table A9: FFCQ Calculations of Economies of Scale for Master Institutions: Research

Predicted Costs at Selected Research Dollars:


Undergrads 54.38 1394.33 75817 75817 75817 75817 75817 75817

UG Squared 5530.15 -2.98 -16463 -16463 -16463 -16463 -16463 -16463

Grads 11.97 510.16 6104 6104 6104 6104 6104 6104

G Squared 276.93 -10.41 -2883 -2883 -2883 -2883 -2883 -2883

Research 7.96 2160.52 0 8600 17201 34402 51603 68803

Res Squared 174.94 -15.61 0 -247 -989 -3957 -8903 -15828

Avg Salary 64.30 -1908.68 -122725 -122725 -122725 -122725 -122725 -122725

Sal Squared 4266.37 16.69 71227 71227 71227 71227 71227 71227

UG x Res 826.60 12.24 0 2649 5297 10595 15892 21190

G x Res 139.09 -38.49 0 -1833 -3666 -7333 -10999 -14666

UG x G 944.07 3.42 3232 3232 3232 3232 3232 3232

UG x Sal 3666.05 1.61 5913 5913 5913 5913 5913 5913

G x Sal 808.07 30.69 24799 24799 24799 24799 24799 24799

Res x Salary 538.14 -8.62 0 -2205 -4411 -8822 -13233 -17644

Any Research 0.75 1200.50 0 896 896 896 896 896

Any Grad 1.00 24235.80 24180 24180 24180 24180 24180 24180

New England 0.08 4053.77 305 305 305 305 305 305

Mideast 0.20 3122.58 627 627 627 627 627 627

Great Lakes 0.17 2269.62 383 383 383 383 383 383

Plains 0.10 -5324.59 -511 -511 -511 -511 -511 -511

Southwest 0.08 605.53 48 48 48 48 48 48

Rocky Mts 0.02 -9287.30 -170 -170 -170 -170 -170 -170

Far West 0.11 -9937.85 -1134 -1134 -1134 -1134 -1134 -1134

Accept Rate 66.41 -284.52 -18896 -18896 -18896 -18896 -18896 -18896

SAT Score 563.00 111.99 63052 63052 63052 63052 63052 63052

% PT Students 26.37 -833.95 -21987 -21987 -21987 -21987 -21987 -21987

% No Dist Ed 80.00 205.03 16402 16402 16402 16402 16402 16402

Public Inst 0.47 -14758.47 -6941 -6941 -6941 -6941 -6941 -6941

% STEM Deg 19.33 319.47 6176 6176 6176 6176 6176 6176

Urban 0.47 -1121.50 -527 -527 -527 -527 -527 -527

Rural 0.03 5727.09 170 170 170 170 170 170

Constant 1.00 -7839.59 -7840 -7840 -7840 -7840 -7840 -7840


Research (QR) = 0 $3.98m $7.96m $15.92m $23.88m $31.85m

AICR =

$10.66 $6.14 $3.79 $2.92 $2.43

MCR =

$1.69 $1.56 $1.31 $1.07 $0.82

SR =

6.32 3.93 2.88 2.74 2.97

revisiting economies of scale in higher education robert k ......the textbook depiction of economies...

Documents