New estimates of economies of scale and scope in higher education.

1. Introduction

The substantially greater than inflation increases in college tuition during the late 1980s and first half of the 1990s ignited considerable discussion of the costs of higher education by both academics and nonacademics. The general discussion covered such issues as how much tuition has risen, why college costs so much (Ehrenberg 2000), the extent to which tuition fully covers costs (Winston 1998; NACUBO 2002), and what colleges are doing to cut costs (Strosnider 1998). Indeed, concern over rapidly increasing tuition spurred Congress to establish a National Commission on the Cost of Higher Education in 1997; the Commission conducted a review of college costs and issued recommendations for holding costs down.

To economists, discussion of cost-cutting generally boils down to a simple question: What is the efficient organization of production? In a market economy, competitive pressures force profit-maximizing firms constantly to strive to produce more efficiently. Thus, information about the efficient organization of production can be deduced by observing organizations that survive and prosper (Stigler 1958). However, in the context of higher education, the answer is not so simple, for at least three reasons: (i) colleges are not profit-maximizing entities, thus market-driven pressures to minimize costs are, essentially, absent; relatedly, (ii) tuition/prices paid by students/customers do not cover the full cost of their educational experience (Winston 1998); and (iii) colleges typically produce multiple products, not just undergraduate education.

Supposing that the individuals who run institutes of higher education (IHEs) have an interest in minimizing costs, how should they structure production to achieve this? That is, what should they produce and how much of it should be produced? Should colleges specialize and produce only undergraduate education, or produce multiple outputs, as so many currently do? Should colleges be small or large, in terms of student enrollments and/or grant research?

These are complex questions in their own right. For example, take the question about the optimal mix of outputs to produce. Forgetting about the implications for revenues, there are a host of related empirical issues to investigate: What are the unit costs of producing different levels of only undergraduate education, graduate education, athletics, research, extension, or public services? What then, in comparison, are the unit costs of producing different levels of alternative combinations of two or more of these outputs? Knowing the answers to specific questions like these provides an essential foundation for informed decision making about the efficient organization of production in higher education.

Estimating the cost of producing academic outputs is complicated by the fact that many, if not most, IHEs produce multiple products. Typically, the products include undergraduate and/or graduate instruction and research. (1) In addition to these basic outputs, the state land-grant institutions also produce extension services. Many institutions also produce public services such as medical services, business assistance programs, museums of various sorts, theater productions, and the like. And, of course, IHEs produce both intramural and extramural athletics. Thus, for purposes of estimating unit costs, it is essential to treat IHEs as multiproduct "firms."

Further, it seems highly likely that the production of certain outputs affects the unit cost of producing others. For example, production of graduate instruction requires the administrators of an IHE to hire faculty with more extensive training and ability than is required to teach at the undergraduate level. Doctorally qualified faculty are more expensive to hire than non-doctorally qualified faculty, ceteris paribus. To the extent that the set of faculty providing graduate instruction and the set of faculty providing undergraduate instruction are mutually exclusive, the provision of the former has no cost spillover to the latter. However, if the graduate faculty also teach undergraduate courses, then the unit cost of providing undergraduate education will be higher at IHEs that produce both graduate and undergraduate education than at IHEs that produce only undergraduate education. On the other hand, to the extent that relatively low-paid graduate students are used to teach undergraduate courses, unit costs of the latter may actually be lower than one would find at a traditional, undergraduate education only institution. Likewise, the fact that an IHE has great athletic teams and/or facilities or strong art/music/theater programs may permit the institution to pay faculty lower salaries than would be the case in the absence of such facilities or programs.

There is evidence that higher education is indeed characterized by (dis)economies of scope. Using data from 1981-1982, Cohn, Rhine, and Santos (1989) estimated multiproduct cost functions for 1195 public IHEs and 692 private IHEs and found (p. 287) that at the mean levels of outputs in their samples there were "economies of scope in the private sector and diseconomies of scope in the public sector." They then investigated scale and scope economies for alternative multiples of the mean outputs, given fixed(at the mean)-proportion output bundles. Public IHEs were shown to have diseconomies of scope up to 150% of the mean output level but slowly increasing economies of scope at even larger output levels. Private IHEs were characterized by economies of scope at all output levels that increased much more rapidly with higher output levels than was estimated for the public IHEs.

These findings were derived from separate cost equations estimated for public and private IHEs, since the structural models for the two types of IHEs were found to differ significantly. However, since their data were for a single year only, Cohn, Rhine, and Santos suggest that estimations for additional years might improve our confidence in the conclusions. In this paper, we estimate multi-product cost functions for public and private IHEs using newly available data on IHE costs for 1996, employing the flexible, fixed-cost methodology employed by Cohn, Rhine, and Santos. We then investigate the extent to which production of undergraduate education, graduate education, and externally funded research are characterized by economies of scale and scope.

2. Methodology

Following in the tradition established by Baumol, Panzar, and Willig (1982) and developed specifically in the context of higher education by Cohn, Rhine, and Santos (1989), we estimate a multi-product cost function for IHEs. Our model is specified as a flexible fixed-cost quadratic (EFCQ) function, with a dummy variable [F.sub.i] that assumes a value of 1 (0) for (non)positive amounts of the output [Y.sub.i]:

[C.sub.i] = [a.sub.0] + [[SIGMA].sub.i] [a.sub.i][F.sub.i] + [[SIGMA].sub.i][b.sub.i][Y.sub.i] + (1/2) [[SIGMA].sub.i] [[SIGMA].sub.j] [c.sub.ij][Y.sub.i][Y.sub.j] + [[eta].sub.i].

[C.sub.i] refers to total expenditures by IHE i in 1996, [a.sub.0], the [a.sub.i]'s, the [b.sub.i]'s, and the [c.sub.ij]'s are scalars, and [[eta].sub.i] is the error term, which is assumed to be independently and identically distributed. Output produced includes undergraduate education (measured as full-time equivalents, in thousands), graduate education (full-time equivalents, in thousands), and research (measured as the sum of federal, state, local, and private grant dollars, in millions). The [F.sub.i] variables reflect differences across IHEs with respect to the fixed costs of producing different product sets.

Since our purpose was to update the Cohn, Rhine, and Santos estimates of economies of scale and scope in higher education using more recent data, we employed the same structural model that they used; that is, we included both linear and squared terms for the three output measures as well as the one factor price we had available (average faculty compensation). In addition, we included interaction terms between outputs and between the factor price and output measures. We estimated the multiple-output cost functions separately for public and private IHEs, since we independently reconfirmed the Cohn, Rhine, and Santos finding of structural differences between the public and private sectors. (2)

Economies of Scale and Scope

Based on our cost function estimation results, we calculated the impact on total cost of increasing production of all outputs simultaneously (ray economies of scale), the impact on total cost of increasing production of a single output holding production of other outputs constant at the sample means (product-specific economies of scale), and the degree to which complementarity among outputs generates lower per-unit costs when two or more outputs are produced simultaneously (economies of scope). Precise mathematical descriptions and discussion of these three concepts can be found in Baumol, Panzar, and Willig (1982, chapters 3 and 4).

3. Data

Our data come from the National Center for Education Statistics (NCES) 1995--1996 fiscal year surveys on THE finances, enrollments, and compensation. These surveys are part of the integrated postsecondary education data system (IPEDS), developed by and for the NCES. Prior to 1986, these institutions were surveyed under the higher education general information survey (HEGIS--the data source employed by Cohn, Rhine, and Santos). However, the IPEDS data are more extensive than HEGIS, since they not only include the schools surveyed under REGIS, they also include any other institutions that grant a bachelor's, master's, doctoral, or first professional degree and are eligible to participate in Title IV financial aid programs. Responses were received from 3520 of the 3965 IHEs surveyed. After omitting institutions with missing data on variables critical to our analysis, we had a usable sample of 2942 IHEs, of which 1492 were private and 1450 were public. Sample statistics for the variables used in our analysis are reported in Table 1.

This split between public and private IHEs in our sample differs sharply from Cohn, Rhine, and Santos, whose sample consisted of a much smaller number of private institutions (692) and a slightly smaller number of public institutions (1195). Not surprisingly, we observe substantive differences between their data set and ours with respect to output and cost measures, differences that cannot be attributed to the 15 years that elapsed between their analysis (1980-1981) and ours (1995-1996). For example, 71% of the private IHEs in the Cohn, Rhine, and Santos sample reported externally funded research whereas only 25% of the private schools in our much larger sample did so. In our sample, the ratio of public to private THE (under)graduate student enrollment is (3.8 to 1)1.66 to 1, while the ratio of public to private THE spending on externally funded research is 3.05 to 1. Further, the ratio of public to private THE total expenditures in our sample is 2.08 to 1. By contrast, in the Cohn, Rhine, and Santos sample, the ratio of public to private THE (under)graduate student enrollment was (2.72 to 1)1.1 to 1, while the ratio of public to private THE spending on externally funded research was 1.1 to I. Finally, the ratio of public to private IHE total expenditures in the Cohn, Rhine, and Santos sample was 1.6 to 1. These differences suggest that the private IHEs in our sample are characterized by a larger number of small, private institutions with teaching missions than was true of the Cohn, Rhine, and Santos sample.

4. Results

Following Cohn, Rhine, and Santos, we specified total costs as a function of three outputs: fulltime equivalent (FTE) undergraduate student enrollment (UG); FTE graduate student enrollment (GR); and externally funded grant research (RES). (3) Accordingly, we estimated a three-output cost function using the FFCQ model for public and private institutions, respectively. Our estimated cost functions, which duplicate the structural models estimated by Cohn, Rhine, and Santos, are reported in Table 2.

The coefficients on the dummy variables, which reflect fixed costs (in millions of dollars), provide evidence of sizable and significant fixed costs of engaging in externally funded research activity and providing undergraduate instruction, especially among public institutions. Most of the nondummy variables in both equations are statistically significant at conventional levels. The superficially surprising exception to this is AVECOMP and COMPSQ in public IHEs. Whereas Cohn, Rhine, and Santos found labor input cost to have a positive but diminishing effect on total cost in both public and private IHEs, we find no significant relationship between AVECOMP/COMPSQ and total cost in public IHEs (we do find the familiar relationship for private IHEs).

Given the nonlinear structure of the model and the interaction terms, it is difficult to draw conclusions about the relationship between costs and outputs based on individual coefficients. Thus, we identify (in Table 3) the marginal effect of producing more of each output, evaluated at the sample means. We calculate that, in 1996, an additional dollar of externally funded research added $2.62 ($4.26) to total costs of public (private) universities. At the sample means, producing undergraduate education was cheaper, on the margin, for public institutions than private institutions ($5127 per additional student vs. $10,374), while the marginal cost of enrolling an additional graduate student was cheaper at private institutions than public institutions ($18,343.50 vs. $9998.50).

In addition, we calculated the F statistics for the null hypothesis that all of the compensation variables (the linear and squared terms plus the interaction terms) are statistically insignificant. For both the public (F = 36.49) and private (F 57.88) IHEs the F values permit us to reject the null hypothesis. To fully convince ourselves that there is a positive and significant relationship between faculty compensation and total costs in public IHEs, we calculated the total cost for public IHEs at the sample means for all variables ($61.027 million) and the 95% confidence interval ($58.414 million--$63.640 million). Then we recalculated total costs for levels of compensation 10% below the mean and 10% above the mean. When compensation is 10% below the mean compensation, total costs ($58.167 million) are statistically significantly less than the mean for total costs. Conversely, when compensation is 10% above the mean compensation, total costs are statistically significantly above the mean for total costs. Rela tively small changes in average faculty compensation lead to statistically significant changes in total costs, in the expected direction. Thus, we reiterate our description of the estimated coefficients on AVECOMP and COMPSQ as superficially surprising. In fact, the evidence indicates that there is a statistically significant, positive relationship between faculty compensation and total costs, for public institutions, as we would expect. (4)

Economies of Scale and Scope

In Table 4 we present our calculations of economies of scale and scope for public and private IHEs. These are based on the formulas identified in Cohn, Rhine, and Santos; the sample means reported in Table 1; and the estimated cost functions reported in Table 2. (5) At the mean levels of output and factor price, we find ray economies of scale for both public and private IHEs. Indeed, these economies exist at all levels of production examined up through 600% of the mean levels of output for public (private) institutions. Although the exact numbers are somewhat different, our results for private IHEs are highly consistent with Cohn, Rhine, and Santos, who also found ray economies of scale throughout the entire range of output levels considered (up to 600% of their sample means). However, with respect to public IHEs, Cohn, Rhine, and Santos found ray economies below and up to just over 100% of the mean levels of output, using data 15 years previous to ours, whereas we observe ray economies up through 600% of our sample means.

However, universities typically do not experience proportionate growth across all three outputs. That is, an IHE that has five times as many full-time equivalent undergraduate students as the sample mean likely does not also have five times as many FTE graduate students and five times as much extramural grant activity as those respective sample means. Growth occurs unevenly. Accordingly, the product-specific economies of scale may be of special interest.

For public IHEs, we observe that the economies of scale that characterize undergraduate education exhibit a consistent pattern of decline, with diseconomies appearing at approximately 70% of the mean level (roughly 3100 students). This finding is virtually identical to what Cohn, Rhine, and Santos reported for public IHEs. For private IHEs, we observe a pattern of declining then increasing economies of scale with respect to production of undergraduate education. These findings are substantially at odds with the work of Cohn, Rhine, and Santos, who found virtually no evidence of product-specific scale economies for undergraduate education for private If-IHEs.

We find product-specific economies that decline through 130% of the mean level-approximately 600 students-then increase with all levels of production of graduate education at public IHEs. We find no evidence of economies of scale in the production of graduate education at private IHEs. Our findings in this regard are substantially in agreement with Cohn, Rhine, and Santos, who found evidence of declining economies of scale at all levels of production in public institutions and no scale economies at any level of production in private IHEs.

Our estimated scale economies for externally funded research are inconsistent with the findings of Cohn, Rhine, and Santos. Only below 150% and above 350% of their mean level of research at public IHEs did Cohn, Rhine, and Santos report economies of scale. They found no evidence of scale economies involving research at private IHEs at any level of production. In contrast, we find economies of scale throughout the entire range of production considered, among both public and private IHEs. We observe that these economies decrease as the size of the IHE increases--a point that we will address presently.

We suspect that the differences between our findings and those of Cohn, Rhine, and Santos regarding private IHEs may be due, in part, to what we suggested previously to be a substantial difference between our sample of private schools and theirs. Specifically, we believe that our sample of private IHEs includes a large number of smaller schools that must have been excluded for some reason from the Cohn, Rhine, and Santos analysis. For example, the mean value of externally funded research in the Cohn, Rhine, and Santos sample from 1980 to 1981 (current dollars) was $2.64 million, whereas in our much larger sample taken in 1995-1996 the mean (current dollars) was $2.585 million. Yet this is a period of time during which there was considerable growth in execution of sponsored research at both public and private IHEs. While this growth is reflected in the substantially higher mean value for externally funded research in public IHEs in our sample ($7.89 million) than in the Cohn, Rhine, and Santos sample ($2.93 mi llion), it is not reflected in the means for private IHEs. Given the difference in the number of private IHEs in each sample (Cohn/Rhine/Santos = 692, Laband/Lentz = 1492), a likely explanation is that our sample contains a number of small schools engaged in little or no externally funded research, whereas the Cohn, Rhine, and Santos analysis excluded these schools. We are puzzled at this discrepancy because data on salaries, costs, and enrollments were available for nearly 1500 private IHEs at the time they conducted their research and they do not mention any filters they used that would have reduced their sample sizes.

Cohn, Rhine, and Santos only report scope economies for externally funded research (produced jointly with undergraduate and graduate education), finding economies at all ranges of production in private IHEs and above 150% of the mean level of research output for public IHEs. We find economies of scope for research at all levels of production in public IHEs, and at levels up through 400% of the sample mean for private IHEs.

We also report economies of scope between undergraduate education and the other two outputs at all levels of production considered in public IHEs and up through 250% of the sample mean for private IHEs. Finally, we find that graduate education is characterized by economies of scope with undergraduate education and research at all levels of production in public IHEs, and up to 100% of the sample mean in private IHEs. However, higher levels of production of FTE graduate education by private IHEs are characterized by diseconomies of scope with the other two outputs.

5. Concluding Comments

Despite some specific differences between our findings and those of Cohn, Rhine, and Santos (1989) with respect to the estimated cost functions and economies of scale and scope, our general conclusions are quite similar. Overall, our findings suggest the following: (i) there are significant structural differences in the cost structure of public versus private IHEs; (ii) public IHEs are characterized by ray economies of scale, scope economies, and, with the notable exception of undergraduate education, product-specific economies of scale for all outputs at all levels of production examined; (iii) private IHEs are characterized by ray economies of scale and product-specific economies of scale with respect to undergraduate education and research, at all levels of output examined; and (iv) private IHEs enjoy economies of scope beyond the sample means for the three outputs. But those economies of scope are exhausted quickly for graduate education, exhausted at 300% of the mean level of undergraduate education, and at 500% of the mean level of research.

We close with several discussion items. First, as acknowledged by Cohn, Rhine, and Santos, it is possible that there are errors in measurement and/or specification that might bias the results. For example, the compensation data compiled by IPEDS were exclusive of the compensation of faculty at medical schools, so we know that the average compensation figures we used are not, in fact, truly representative. Second, it is hard to reconcile the observed product-specific diseconomies of scale in public IHEs with respect to undergraduate education with the observed product-specific economies of scale in private IHEs that increase at higher multiples of the mean undergraduate population. Indeed, it is somewhat problematic to reconcile the existence of so many public universities with large undergraduate student populations with the fact that product-specific economies of scale are exhausted so quickly (roughly 3100 students). With several times that number of undergraduate students, many of the large state universit ies are located in a region of substantial diseconomies of scale. One possible explanation of this apparent anomaly is that the increasing economies of scope observed in public IHEs between undergraduate education and research and graduate education overwhelm the product-specific diseconomy of scale. Another explanation, not grounded in cost efficiencies, is that state legislatures base appropriations to public universities on undergraduate student enrollment figures, such as FTEs.

We note that there are a number of very large universities in the United States, both public and private, that produce one or more of the three outputs at levels far above the sample means. For example, among private institutions, Stanford University produces externally funded research at a level that is 16,937% of the sample mean for private IHEs, and Brigham Young University produces undergraduate education at a level that is 2348% of the sample mean for private IHEs. Among public IHEs, the University of Wisconsin (Madison) produces research, undergraduate education, and graduate education at levels that were, in 1995-1996, 4848%, 573%, and 1976%, respectively, of the mean levels of our sample of public IHEs. For the University of Minnesota, these percentages were 3759%, 544%, and 1706%. Are there implications of such size for cost efficiency?

To shed light on this question, we determined the production levels for each of the three outputs at which the product-specific economies were exhausted. As noted previously, product-specific economies of scale with respect to undergraduate education in public IHEs play out very quickly. But the product-specific economies of scale for research (graduate education) in public IHEs do not disappear until production is at 65 (25) times the sample mean. In numbers, this means an IHE with approximately $513 million in external research funding (12,725 graduate students). So even the University of Wisconsin, at 49 (20) times the mean level of externally funded research (graduate education), falls well within the levels of production for those two outputs that enjoy product-specific economies of scale. Similarly with private IHEs, product-specific economies of scale are exhausted at 200 (85) (15) times the sample means (for externally funded research, graduate education, and undergraduate education, respectively). Th is means, for example, that Stanford's level of externally funded research that is nearly 170 times the sample mean for private IHEs still falls within the region characterized by economies of scale. With respect to undergraduate education, Brigham Young University, at 23 times the sample mean, is the only IHE operating nominally in a region of diseconomies of scale. (6)

We note that there may be substantial fixed costs but also substantial economies of scale and/or scope to production of certain types of research (e.g., medicine, veterinary medicine). If so, this may imply that different cost functions may be appropriate for different types of research. Given the added output of extension produced at state land-grant institutions, it may be that cost functions for the land-grant IHEs differ significantly from those for non--land-grant public IHEs.

Another fascinating, albeit unexplored, aspect of this work is that even though unit costs may be minimized at the previously identified levels of production of the various outputs, total revenues are not maximized. Since IHEs tend overwhelmingly to be not-for-profit organizations, cost-minimization is not an imperative (Ehrenberg 2000). Competition tends to take the form of being the best at everything, with expenditures following revenues (Bowen 1980; Winston 1999). One implication of this is, of course, that an external observer might find any number of unusual relationships between costs and outputs, such as production beyond the point where economies of scale are exhausted. In addition, with the relevant data on revenues, it would be possible to estimate functions that reveal both the direct and indirect effects of athletic success, student enrollments, and grant research on private donations. In the specific case of externally funded research, a plausible scenario is that prospective donors screen would -be recipient institutions on the basis of how much externally funded research they are engaged in, not the unit cost of engaging in that research.

Finally, to the extent that administrative costs can be separated out from total costs, one can employ this methodology to estimate the impact of externally funded research on administrative costs--with obvious implications for the setting of indirect cost recovery rates. With relevant data, one also could estimate cost functions with additional outputs such as athletics or extension (at land-grant institutions). This would permit us to improve our understanding of not only the impact of athletics on the total costs of IHEs, but also the impact on factor prices such as nonfaculty compensation and on factor quality. It is possible, for example, that having a great football program permits an THE to attract higher quality staff at a discount, compensation wise, to what they would have to pay these individuals to locate at an THE with a mediocre football team. These quality issues, which are obscured in our analysis by the single factor price variable in which quality is implicitly assumed to be constant across faculty, are important ones for future researchers to explore.

Table 1

Variable Descriptions and Sample Statistics

                                                      Public
Variable
Symbol    Description                              Mean        SD

TC        Total IHE expenditures (millions       65.483   131.223
           of $)
AVECOMP   Average annual salary plus fringe     53,247    11,955
           benefits for nonmedical faculty
COMPSQ    Average annual compensation squared  2978.049  1365.621
           (millions)
RESDUM    = 1 if research > 0; = 0 otherwise      0.424     0.494
UGDUM     = 1 if undergraduate enrollment >       0.995     0.069
           0; = 0 otherwise
GRADDUM   = 1 if graduate enrollment > 0; 0       0.335     0.479
           otherwise
RES       Research output (millions of            7.890    32.639
           federal, state, local, and private
           grant $)
RESSQ     Research output squared (trillions)  1126.797  7997.497
UG        Full-time equivalent (PIE)              4.413     4.733
           undergraduate student enrollment
           (thousands)
UGSQ      FTE undergraduate student              41.856   100.780
           enrollment squared (millions)
GRAD      FTE graduate student enrollment         0.509     1.332
           (thousands)
GRADSQ    FTE graduate student enrollment         2.032     9.763
           squared (millions)
RESUG     FTE undergraduate enrollment X        126.317   708.503
           research output (billions)
RESGRAD   FTE graduate enrollment X research     39.650   247.544
           output (billions)
GRADUG    FTE undergraduate enrollment X FTE      6.961    27.801
           graduate enrollment (millions)
COMPRES   Faculty compensation X research       571.868  2501.853
           output (billions)
COMPUG    Faculty compensation X FTE            262.376   335.593
           undergraduate enrollment
           (millions)
COMPGRAD  Faculty compensation X FTE graduate    34.832    98.673
           enrollment (millions)

                Private
Variable
Symbol        Mean        SD

TC          31.514    96.375

AVECOMP    45,567    17,431

COMPSQ    2379.961  1874.443

RESDUM       0.249     0.433
UGDUM        0.890     0.313

GRADDUM      0.537     0.499

RES          2.585    20.222


RESSQ      415.339  6074.500
UG           1.159     1.635


UGSQ         4.016    23.258

GRAD         0.307     0.956

GRADSQ       1.007     7.978

RESUG       15.467   140.914

RESGRAD     14.348   158.807

GRADUG       1.287     8.341

COMPRES    226.293  1988.766

COMPUG      63.922   118.645


COMPGRAD    21.197    81.825

Table 2

Three-Output Quadratic Cost Function Estimates

Variable            Public Institutions  Private Institutions

Intercept                 5.2486               -1.6945
                         (9.6683)              (2.0953)
RESDUM                    5.7110 ***            3.9123 ***
                         (1.7826)              (1.0629)
GRADDUM                   0.3522               -0.8290
                         (2.1799)              (0.9075)
AVECOMP                  -0.0670                0.1885 **
                         (0.360 1)             (0.0873)
COMPSQ                    0.0017               -0.0015 *
                         (0.0033)              (0.0009)
RES                       2.8095 ***            5.8325 ***
                         (0.3317)              (0.2857)
RESSQ                    -0.0025 ***           -0.0030 ***
                         (0.0005)              (0.0004)
UG                        0. 1783              -0.3553
                         (1.2064)              (1.2285)
UGSQ                      0.1208 ***           -0.4073 ***
                         (0.0316)              (0.0331)
GRAD                     -5.9125               -4.3117
                         (7.9503)              (3.6649)
GRADSQ                   -0.7061               -0.1841
                         (0.5821)              (0.3328)
RESUG                    -0.0462 ***            0.0294
                         (0.0056)              (0.0205)
RESGRAD                   0.1319 ***            0.1322 ***
                         (0.0277)              (0.0190)
GRADUG                   -0.2888                0.8422 ***
                         (0.2043)              (0.2813)
COMPRES                  -0.0002               -0.0358 ***
                         (0.0048)              (0.0039)
COMPUG                    0.0826 ***            0.2489 ***
                         (0.0195)              (0.0222)
COMPGRAD                  0.4734 ***            0.2876 ***
                         (0.1161)              (0.0597)

N                           1450                  1492
Adjusted [R.sup.2]        0.9680                0.9763

Numbers in parentheses are standard errors.

* Significant at the 10% or better, two-tailed test.

** Significant at the 5% level or better, two-tailed test.

*** Significant at the 1% level or better, two-tailed test.

Table 3

The Marginal Impact of Increasing Each Output at the Sample Means

                          Public Institutions  Private Institutions

Research                            2.62                 4.26
Undergraduate enrollment         5127.45            10,374.70
Graduate enrollment            18,343.50              9998.50

Table 4

Degree of Scale and Scope Economies for Alternative Fixed-Proportion
Output Bundles

                                      Product- Specific
                                          Economies

Percentage of        Ray     Undergraduate  Graduate
Output Means      Economies    Education    Education  Research

Public colleges
and universities
       10           3.465        1.667        1.366     3.030
       50           1.488        1.065        1.084     1.412
      100           1.239        0.951        1.057     1.213
      150           1.154        0.892        1.056     1.150
      200           1.110        0.851        1.060     1.120
      250           1.083        0.819        1.068     1.104
      300           1.064        0.793        1.077     1.094
      400           1.038        0.752        1.100     1.086
      500           1.021        0.722        1.125     1.085
      600           1.008        0.699        1.154     1.087
Private colleges
and universities
       10           3.457        1.133       -1.675     5.678
       50           1.495        1.049        0.465     1.938
      100           1.251        1.059        0.735     1.471
      150           1.171        1.081        0.827     1.316
      200           1.132        1.108        0.875     1.239
      250           1.109        1.138        0.904     1.193
      300           1.094        1.173        0.925     1.163
      400           1.078        1.255        0.953     1.126
      500           1.069        1.362        0.973     1.105
      600           1.065        1.506        0.988     1.091

                              Economies of
                                 Scope

Percentage of     Undergraduate  Graduate
Output Means        Education    Education  Research

Public colleges
and universities
       10             0.377        0.376     0.377
       50             0.189        0.175     0.181
      100             0.139        0.106     0.121
      150             0.130        0.077     0.101
      200             0.134        0.062     0.094
      250             0.144        0.052     0.094
      300             0.158        0.046     0.096
      400             0.190        0.039     0.107
      500             0.226        0.0355    0.120
      600             0.262        0.033     0.136
Private colleges
and universities
       10             0.522        0.047     0.371
       50             0.167        0.017     0.170
      100             0.091        0.011     0.097
      150             0.055       -0.010     0.064
      200             0.031       -0.020     0.044
      250             0.013       -0.029     0.030
      300            -0.001       -0.038     0.019
      400            -0.026       -0.055     0.003
      500            -0.047       -0.071    -0.011
      600            -0.066       -0.088    -0.022

Received April 2002; accepted September 2002.

(1.) As was pointed out by a thoughtful reviewer, the quality of the outputs produced (and the inputs used to produce them) is not homogeneous across IHEs. That is, the average quality of the graduate students in economics produced by the University of Pennsylvania differs significantly from the average quality of graduate students in economics produced by Auburn University. Likewise, the average quality of faculty and the average faculty compensation differs significantly between the two institutions. Unfortunately, we do not (and cannot) control for such differences.

(2.) The Chow test statistic F[11, 1832] = 39.41, with P < 0.001.

(3.) It is not clear how Cohn. Rhine, and Santos determined their measures of full-time equivalent (under)graduate enrollments, since we are unaware of any standardized procedure for convening part-time enrollments to full-time enrollments. We simply assumed that a part-time student was equivalent to one-third of a full-time student, the practice at Aubum University and, we understand, certain other universities. Subject to data availability, we believe that a good measure of equivalency could be constracted by examining the fraction of total tuition revenues generated by each population of students.

(4.) We conducted this same exercise for private IHEs, with similar results. The predicted value of total costs at the sample means of the explanatory variables is $33023 million; the 95% confidence interval is $31.097-$34.949 million. The estimated total cost when average faculty compensation falls (rises) 10% is $30.87 ($35177) million. Both estimates are outside the 95% confidence interval, indicating that the changes in average compensation lead to statistically significant changes in total cost.

(5.) It is difficult to determine the extent to which our calculation of economies of scale and scope matches the procedure followed by Cohn, Rhine, and Santos. Since there are squared terms of the outputs in the regression model, the derivatives that are essential to the calculation of economies of scale and scope for the Y outputs take the form d[Y.sub.i] = [b.sub.1] + 2[b.sub.2][Y*.sub.i] + [c.sub.i][Y*.sub.j] where [b.sub.1] is the estimated coefficient of the linear term of [Y.sub.i], [b.sub.2] is the estimated coefficient of the squared term of [Y.sub.i], [Y*.sub.i] is the mean value of [Y.sub.i] the [c.sub.i] are the estimated coefficients on the interactive [Y.sub.i][Y.sub.j] terms, and [Y*.sub.j] is the mean value of [Y.sub.j]. The economy of scale/scope calculations are made at various multiples (or fractions) of the sample means of the three outputs. It is easy to confuse taking a fraction of the squared mean value of the output in question with squaring the fractioned mean value of that output. Th e two procedures yield very different results

at output levels other than 100% of the sample mean. It is the latter calculation that correctly defines the relevant derivatives.

(6.) One technical issue that we worry about is the inclusiveness of costs in the numbers reported in IPEDS. For example, a university hospital may have its own budget that is completely separate from the university budget Cenain inputs, such at faculty, may be included, costwise, by the hospital, yet redound to the benefit of the university. Thus, the faculty cost for these individuals is not reported by the institution. However, because these faculty provide teaching services for the university the institution may be observed to provide teaching at relatively tow cost. While this may look like an economy of scale, in fact that interpretation is problematic. To check our own work in this regard, we omitted from our sample IHEs with hospitals and reestimated the model, finding continued evidence of economies of scale. Our point, however, is that there are a host of technical higher education accounting practices like this that need to be taken into account for us to have real confidence in the interpretations drawn from any of the statistical work in this genre.

References

Baumol, William J., John C. Panzar, and Robert D. Willig. 1982. Contestable markets and the theory of industry structure. New York: Harcourt Brace Jovanovich, Inc.

Bowen, Howard R. 1980. The costs of higher education. San Francisco, CA: Jossey Bass.

Cohn, Elchanan, Sherrie L. W. Rhine, and Maria C. Santos. 1989. Institutions of higher education as multi-product firms: Economies of scale and scope. Review of Economics and Statistics 71:284-90.

Ehrenberg, Ronald G. 2000. Tuition rising: Why college costs so much. Cambridge, MA: Harvard University Press.

National Association of College and University Business Officers. 2002. Explaining college costs. Washington, DC: NACUBO.

Stigler, George J. 1958. The economies of scale. Journal of Law and Economics 1:54-71.

Strosnider, Kim. 1998. Private colleges in Ohio are collaborating to cut costs. Chronicle of Higher Education, 29 May, pp. A41-A42.

Winston, Gordon C. 1998. Economic research now shows that higher education is not just another business. Chronicle of Higher Education, 27 March, p. 86.

Winston, Gordon C. 1999. Subsidies, hierarchy and peers: The awkward economics of higher education. Journal of Economic Perspectives 13:13-36.

David N. Laband *

Bemard F. Lentz +

* Forest Policy Center, School of Forestry and Wildlife Sciences, Auburn University, 202 M. White Smith Halt, Auburn, AL 36849, USA; E-mail [e-mail]; corresponding author.

+ Institutional Research and Analysis, 3401 Walnut Street, Suite 352B, University of Pennsylvania, Philadelphia, PA 19104, USA.

Laband gratefully acknowledges financial support in the form of a McIntire-Stennis grant awarded through the School of Forestry and Wildlife Sciences at Auburn University. We appreciate the helpful comments of the reviewer. The usual caveat applies.