www.elsevier.com/locate/econedurev
Multi-product total cost function for higher education: a case
of bible colleges
Rajindar K. Koshal
a,*, Manjulika Koshal
b, Ashok Gupta
c aDepartment of Economics, Ohio University, Athens, OH 45701-2979, USAbDepartment of Management Systems, Ohio University, Athens, OH 45701-2979, USA cDepartment of Marketing, Ohio University, Athens, OH 45701-2979, USA
Received 10 January 1999; accepted 21 January 2000
Abstract
This study empirically estimates a multi-product total cost function and output relationship for comprehensive univer-sities in the United States. Statistical results based on data for 184 Bible colleges suggest that there are both economies of scale and economies of scope in higher education. In addition, product-specific economies of scope do exist for all output levels and activities.2001 Elsevier Science Ltd. All rights reserved.
JEL classification: I22
Keywords: Economies of scale & scope; Bible education
1. Introduction
In the closing decades of the 19th century and the opening decades of the 20th century, liberalism by major theological seminaries paved the way for the establish-ment of the Bible institutions for religious education. Out of the movement of Bible institutes grew the notion of Bible colleges for higher education. By the 1970s Bible colleges gained a reputation as mainstream higher edu-cational institutions. According to Ferris and Enlow (1998), the American Association of Bible Colleges (AABC) won recognition and participation in the new council on post-secondary accreditation. However, this respectability and acceptance in the education world did not yield the expected growth during the 1980s. Accord-ing to Kallgren (1991), durAccord-ing the period 1979–1986, the overall enrollment in the Bible colleges declined by 15%.
* Corresponding author. Tel.: +1-740-593-2038; fax: + 1-740-593-0181.
E-mail addresses: [email protected] (R.K. Koshal), [email protected] (M. Koshal), [email protected] (A. Gupta).
0272-7757/01/$ - see front matter2001 Elsevier Science Ltd. All rights reserved. PII: S 0 2 7 2 - 7 7 5 7 ( 0 0 ) 0 0 0 1 6 - 9
This decline in demand resulted in the closing of numer-ous Bible colleges and some became Christian liberal arts colleges. A significant number of surviving Bible colleges experienced enrollment driven financial diffi-culties. This implies that there may be economies of scale and scope in the production of Bible education in the United States. The Bible colleges will be able to sur-vive if they are able to reap the benefits of economies of scale, as well as, of scope. Thus, the purpose of this study is to examine the existence of economies of scale, as well as scope for the Bible colleges.
be leaders of their churches and institutions with the fin-ancial and business sense needed to be “good stewards” of God’s resources. Bible colleges are committed to the education of students towards religious leadership and service in their respective institutions and in professions throughout the world.
To achieve efficiency in production, cost functions provide important information for producers. In the case of higher education, the question of economies of scale has been debated for over half of a century. The studies by Koshal and Koshal (1995), Nelson and Heverth (1992), de Groot, McMahon, and Volkwein (1991), Clot-felter, Charles, Ehrenberg, Getz, and Siegfried (1991), Getz, Siegfried, and Zhang (1991), Hoenack (1990), Cohn, Rhine, and Santos (1989), Brinkman (1990), Brinkman and Leslie (1986) and Friedman (1955) sider a single output approach as having conflicting con-clusions regarding economies of scale. In the 1970s, two studies (Verry & Layard, 1975; Verry & Davies, 1976) suggested the use of multiple-output total cost functions. Recent studies also suggest that institutions of higher education produce multiple products (Jimenez, 1986; Cohn et al., 1989; Cohn & Geske, 1990; Lloyd, Mor-gan, & Williams, 1993; Hashimoto & Cohn, 1997; Johnes, 1997; Koshal & Koshal, 1999). In addition, the quality of education, whether perceived or real, is differ-ent at differdiffer-ent institutions (Koshal & Koshal, 1995). For example, in terms of real or perceived quality, one year of education at Amherst College is not the same as one year of education at Huntington College. Therefore, the cost and output relationship must make adjustments for various outputs and for quality variations among insti-tutions. Thus, where data for quality are available, one needs to include a quality measure in a cost model. In addition to the single-output method, some of these stud-ies have other drawbacks. First, some used the assump-tion that all the instituassump-tions in a cross-secassump-tion analysis have the same objectives. It would be unreasonable to assume that the goal for all universities, as well as for four-year colleges, is to provide the same educational experience. One needs to account for such differences in relating cost to outputs. Second, most of the studies did not test their results for the presence of heteroscedastic-ity.
This study explores different dimensions of economies of scale by estimating a multiple-product fixed total cost function for Bible colleges. The purpose of this study is to estimate for Bible colleges, the long run, multiple-product total cost function, and apply various diagnostic checks to the estimated results.
2. Model
In this study, we model our multiple-product cost function following the work of Baumol, Panzar, and
Wil-lig (1988), Mayo (1984), Cohn et al. (1989), de Groot et al. (1991), Nelson and Herverth (1992), Dundar and Lewis (1995), Hashimoto and Cohn (1997) and Koshal and Koshal (1999). Instead of a translog function, we assume that total cost (TC) of education output at a Bible college is represented by a flexible “fixed” cost quadratic function (FFCQ). The translog function is not suitable for this study as some of the colleges produce only one output. Thus, our function is of the following form:
TC5a01
O
where TC is the total cost of producing k products, a0is
the constant, and aiand the bijare the coefficients
asso-ciated with various output variables. Qiis the output of
the ith product. c1, c2are the coefficients of two dummy
variables DUand DG. DUhas a value of one if
undergrad-uate output is non-zero, otherwise it takes a value of zero. Similarly, DGis defined if graduate output is zero
or non-zero. Data are not available separately for M.A. and Ph.D. enrollment. To account for Ph.D. programs on cost, we introduce a dummy variable PHD. The coef-ficient d of the dummy variable PHD is for institutions that offer Ph.D. degrees. The PHD variable takes a value of one for institutions that offer the Ph.D. degree, other-wise, it takes a value of zero. V is a random error term. The cost function of Eq. (1) permits us to estimate both economies of scale and economies of scope.
Generally, Bible colleges for higher education produce two products: QU, undergraduate students and QG,
gradu-ate students. However, some of the Bible colleges pro-duce only one of these outputs. Following Baumol et al. (1988), Cohn and Geske (1990) and Hashimoto and Cohn (1997), we first define the average incremented cost (AIC) for undergraduate output as
AICU5
TC{QU,QG}−TC{0,QG}
QU (2)
where TC{QU,QG} is the total costs of producing QU
units of undergraduate students and QGunits of graduate
students. TC{0,QG} is the total cost when output for
pro-duct U is zero. Average incremental cost (AICG) for
pro-duct G is defined similarly. As in the case of a single product, the economies of scale are measured by the ratio of average to marginal costs. Economies of scale are said to exist if this ratio is greater than one. The product-specific economies of scale for product U is defined as
EU5AICMCU U
(3)
where MCU=∂TC/∂QUis the marginal cost of producing
product U. If EUis greater (smaller) than one, economies
U. Ray (overall) economies of scale (RE) may exist when the quantities of the product are increased pro-portionally. Ray economies of scale are defined as fol-lows:
RE5QUTC{QMC U,QG} U+QGMCG
. (4)
Ray economies (diseconomies) of scale are said to exist when RE is greater (less) than one.
In any production process, economies of scope are present when there are cost efficiencies to be gained by joint production of multiple products, rather than by being produced separately. The degree of global econ-omies (GE) of scope in the production of two products is defined as
GE5TC{QU,0}+TC{0,QTC{Q G}−TC{QU,QG} U,QG}
(5)
Global economies (diseconomies) of scope are said to exist if GE is greater (less) than zero.
3. Data
The data pertaining to Bible colleges in the United States for the academic year 1994-95 are collected from the U.S. Department of Education, National Center for Education Statistics (NCES), Integrated Post-Secondary Data System (IPEDS), 1995. There are 289 Bible col-leges in the United States. For some of the colcol-leges, values for the average salary for faculty are not available. Therefore, a complete set of data for this analysis is available only for 184 of these institutions. In higher education, as pointed out by Cohn et al. (1989), there is no consensus on the appropriate measures of output. For the purpose of our analysis, we assume Bible colleges produce two outputs: (i) the number of full-time equival-ent undergraduate studequival-ents (QU), and (ii) the number of
full-time equivalent graduate students (QG). QUincludes
all undergraduate students enrolled as freshmen, sopho-mores, juniors, and seniors. QG includes students
enrolled in masters-level and doctorate-level classes. We recognize that full-time equivalent enrollment (FTE) may not be the ideal measure of output, but from our perspective, it is an important improvement over just the absolute number of students used by some of the pre-vious studies. Thus, in the absence of any better data we have selected QUand QG in FTE units as our best
measurements of the two teaching outputs. The total cost variable used here (TC) is measured by a university’s current expenditure. Definitions of variables used in this study, along with some basic descriptive statistics, are presented in Table 1.
In this study, we are not able to control for the quality of education for each institution since we are unable to
Table 1
Definition of variable and summary statistics
Variable Description Mean/(standard deviation) n=184 TC Total cost in thousands of 3,455,426
dollars (3,679,105)
QU Undergraduate FTE in 164
thousands (269)
QG Graduate FTE in thousands 129
(286) Q2
U (Undergraduate FTE)2 98,684
(384,552) Q2
G (Graduate FTE)2 97,576
(555,711) CSIZE Students per teacher 20
(13) FS Average faculty salary in 33,137
dollars (12,503)
FS2 Average of square of faculty 3354
salary in square dollars (2115)
QU·QG 6 Number of colleges without 71 undergraduate students
Number of colleges without 59 graduate degree
Number of colleges offering 83 Ph.D.
find any publication that lists quality measure like SAT or the quality index for colleges by US News and World
Reports. It has been established that to purchase a higher
quality educational experience, students/parents are wil-ling to pay more at institutions with higher average SAT scores. A number of studies imply that the average SAT score signals the quality of an institution to prospective students and their future employers (Koshal, Koshal, Boyd, & Levine, 1994; Koshal & Koshal, 1995). There-fore, our results may be somewhat biased due to the omission of the quality variable.
4. Statistical results
Using the above data and applying the multiple regression technique, we first estimate the above quad-ratic, multi-product total cost function. The results of this analysis are summarized in Table 2 as Eq. (6).
Table 2
Summary of regression resultsa
Equation (6) TC Equation (7) TC Equation (8) TC
Intercept 2,439,871 (3.76)*** 973,656 (0.73) 1,530,332 (1.64)*
QU 6529.6527 (3.30)*** 12,882 (3.60)*** 14,121 (5.52)***
QG 9642.8756 (4.20)*** 223,571 (2.99)*** 11,873 (1.95)**
(QU)2 21.8522 (1.19) 23.1842 (2.88)*** 22.2527 (2.92)***
(QG)2 21.0458 (0.99) 20.6007 (0.63) 22.1504 (3.06)***
QU∗QG 21.4285 (0.38) 18.5761 (3.36)*** 4.3340 (1.08)
FS — 17.2462 (0.29) 24.3990 (0.58)
FS2 — 0.0005 (0.67) 0.0008 (1.48)
QU∗FS — 20.1952 (2.02)** 20.1856 (2.74)***
QG∗FS — 0.6874 (4.10)*** 20.0687 (0.52)
CSIZE — — 268,617 (5.08)***
PHD 1,175,005 (2.16)** 831,992 (1.71)* 760,507 (2.24)**
DU 21,795,395 (3.01)*** 21,287,861 (2.25)** 21,077,150 (2.58)***
DG 2610,254 (0.98) 2141,259 (0.24) 21,060,292 (2.59)
Dummy (for three largest colleges) — — 15,375,310 (12.22)***
Adj-R2 0.4921 0.5981 0.8054
(F-ratio) {23.16}*** (23.70)*** {55.11}***
n 184 184 184
a Notes: where applicable one tail test is applied; the values in parentheses below the coefficients are t-values, Adj-R2 is the
coefficient of determination adjusted for the degrees of freedom. *Denotes 10% level of significance; **denotes 5% level of signifi-cance; ***denotes 1% level of significance.
producing undergraduate, and graduate student output. Yet, as explained below, this equation does not tell the entire story about the relationship between cost and out-put.
Jimenez (1986) and Hashimoto & Cohn (1997) sug-gest that including an index of input prices in the total cost function is important, since in a cross-section data set, the institutions are usually located all over the coun-try. Each university may face different factor costs that would then influence total cost. The Bible colleges in our sample are situated across the country. Therefore, we proxy the index of input prices by the average faculty salary. As faculty and staff salaries constitute a high pro-portion of the total cost of education they are the most predominant factor cost. We re-estimate our cost func-tion including average faculty salary plus instrucfunc-tional material per faculty member (FS). The results are sum-marized in Eq. (7) in Table 2. Since institutions can con-trol total cost by varying the number of students per fac-ulty member, we estimate Eq. (8) by adding the number of students per faculty (CSIZE). Also included in Eq. (8) is a dummy for three institutions - Jewish Theological Seminary of America (NY), Fuller Theological Seminary of California (CA) and Princeton Theological Seminary (NJ). These institutes are the largest and located in expensive cities of the United States. To test for the pres-ence of heteroscedasticity, we apply the RESET test (Ramsey, 1969) to the residuals of Eqs. (6) and (7). The test statistic indicates the presence of heteroscedasticity. Therefore, the coefficients of these Eqs. (6) and (7) are
unbiased but inefficient, thus invalidating the test of sig-nificance. However, the RESET test suggests that the residuals of Eq. (8) are homoscedastic. Therefore, the coefficients of Eq. (8) are unbiased and efficient. The discussions that follow use the results of Eq. (8). In the following paragraphs, we also provide various calcu-lations of economies of scale and scope for Eq. (8).
In Table 3, we provide the marginal cost of each duct for different levels of output along with the pro-portional output ray. These marginal cost values are
cal-Table 3
Marginal cost estimates (in dollars)a
% of mean MCU MCG MCG/MCU
output ratio
50 7879.17 9676.20 1.23
75 7833.35 9716.05 1.24
100 7787.52 9755.90 1.25
125 7741.70 9795.75 1.27
150 7695.88 9835.60 1.28
175 7650.06 9875.44 1.29
200 7604.23 9915.29 1.30
225 7558.41 9955.14 1.32
250 7512.59 9994.99 1.33
275 7466.77 10034.84 1.34
300 7420.94 10074.69 1.36
a Notes: MC
U, marginal cost of undergraduate education;
culated at the mean values of the relevant variables. An examination of the estimates in Table 3 suggest that for all levels of output, the marginal cost of a FTE graduate is higher than that for a FTE undergraduate. This is con-sistent with the cost structure of higher education in the United States.
The marginal cost for undergraduate FTE declines as the output level increases. On the other hand, the mar-ginal cost of a FTE graduate increases as the output level increases. For example, the marginal cost for graduate FTE is US$9676.20 at the 50% level of mean output and US$10,074.69 for a 300% level of mean output.
For the total cost function as formulated above, the estimated mean value of the output’s cross marginal cost (CMC) is defined as follows:
CMCUG5
cross marginal products, are given by the coefficients of the corresponding interaction terms in Eq. (8), Table 2. A negative sign of the coefficients would imply that there is cost complementarity. A review of the results in Table 2 suggests that for Bible institutions of higher education, the coefficient of interaction term for (QU·FS) and
(QG·FS) have a negative sign. This implies cost
comp-lementarity among undergraduate output and faculty sal-ary, and graduate output and faculty salary. However, only the coeffient on (QU·FS) is statistically significant.
The coefficient of interaction term (QU·QG) is positive
impling substitutability between undergraduate and graduate output.
A major shortcoming with the multiple product case is that there is no direct analogy to the the “average cost” concept in the single output case (Cohn et al., 1989; Hashimoto and Cohn, 1997). However, as discussed earl-ier, the nearest analogy is provided by the average incremental cost (AIC). These values are listed in Table 4. As is obvious from the marginal cost values for all output levels, the values of average incremental cost for a FTE graduate is higher than the average incremental cost for a FTE undergraduate. An examination of the results in Table 4 suggests that the AIC for a FTE under-graduate, as well as for a FTE under-graduate, increases as out-put level increases.
The estimates for the values of ray and product-spe-cific economies of scale are summarized in Table 5. Reviewing the values in Table 5, one realizes that ray economies apply at all output levels. The level of ray economies increases as output level increases. The results also indicate that product-specific economies of
Table 4
Average cost estimates (in dollars)a
% of mean AICU AICG AICG/AICU
output
50 6283.69 7537.87 1.20
75 7517.41 9164.46 1.22
100 8157.63 10032.22 1.23
125 8560.44 10596.45 1.24
150 8844.55 11008.92 1.24
175 9060.82 11334.66 1.25
200 9234.71 11606.19 1.26
225 9380.33 11841.60 1.26
250 9506.17 12051.71 1.27
275 9617.62 12243.42 1.27
300 9718.28 12421.33 1.28
a Notes: AIC
U, average incremental cost of undergraduate
education; AICG, average incremental cost of graduate
edu-cation.
Table 5
Economies of scalea
% of output ERAY EU EG
mean
50 1.19 0.80 0.78
75 1.46 0.96 0.94
100 1.75 1.05 1.03
125 2.02 1.11 1.08
150 2.31 1.15 1.12
175 2.59 1.18 1.15
200 2.87 1.21 1.17
225 3.15 1.24 1.19
250 3.44 1.27 1.21
275 3.72 1.29 1.22
300 4.00 1.31 1.23
a Notes: E
RAY, ray economies of scale; EU, undergraduate
education economies of scale; EG, graduate education
econom-ies of scale.
scale for both undergraduate, as well as graduate, edu-cation exist only at 100% or more of mean output.
Table 6
Economies of scopea
% GES ESU ESG
50 0.62 0.67 0.62
75 0.49 0.51 0.49
100 0.41 0.41 0.41
125 0.34 0.33 0.34
150 0.29 0.26 0.29
175 0.25 0.21 0.25
200 0.21 0.17 0.21
225 0.18 0.13 0.18
250 0.15 0.10 0.15
275 0.13 0.07 0.13
300 0.11 0.05 0.11
a Notes: GE
G, global economies of scope; ESU, economies
of scope for undergraduate education; ESG, economies of scope
for graduate education.
5. Conclusions
Using data for 184 Bible colleges in the United States, this paper shows that it is possible to obtain estimates of economies of scale, product-specific economies of scale, ray economies of scale, and global economies of scope. The main findings of our study are:
1. Holding all other things equal, the overall total cost is affected by class size. For example, on average, if class size is increased by one student (FTE), ceteris paribus, the overall total cost would decrease by US$68,617.
2. Bible colleges which offer a Ph.D. degree will, on average, have US$760,507 in additional costs. 3. In our study, the marginal cost of graduate teaching
is higher than that of undergraduate teaching. This is an obvious conclusion, however, because our results for Bible colleges imply that the ratio (MCG/MCU) is
much smaller compared to some of the previous stud-ies of higher education (Nelson & Heverth, 1992; Hashimoto & Cohn, 1997). Our ratio for Bible col-leges varies between 1.23 and 1.36. These ratios are quite similar to the results obtained by Koshal and Koshal (1999) for the comprehensive universities in the United States.
4. Our results indicate that at all levels of output under consideration, ray economies of scale exist for Bible colleges in the United States. These results are quite similar to the findings by Cohn et al. (1989), Dundar and Lewis (1995), Hashimoto and Cohn (1997) and Koshal and Koshal (1999).
5. Contrary to previous results, our statistical estimates for product-specific economies of scale for under-graduate and under-graduate education indicate clear-cut product specific economies of scale only at and
beyond the mean value of output. In earlier studies, the results for undergraduate and graduate product-specific economies of scale have varied.
6. For Bible colleges, our results indicate that global economies of scope exist across the entire range of output. Our results are, therefore, different from the findings of the previous studies. This study of com-prehensive universities by Koshal and Koshal (1999) implies that for undergraduate and graduate edu-cation, product-specific economies and diseconomies of scope exist depending on the output level, as well as the type of institution. Regarding scope economies, similar conclusions were reached in earlier studies by Dundar and Lewis (1995), Cohn et al. (1989), Nelson and Heverth (1992) and Hashimoto and Cohn (1997).
Overall, our results suggest that Bible colleges in the United States can reap benefits from both scale and scope economies. Large Bible colleges appear to be more cost efficient. Thus for Bible colleges to remain financially viable, they must keep attracting students to reap the benefit of economies of scale and scope. Of course, beyond some level of output, inefficiencies may exist, but based on the results in this study, we are not able to pinpoint any optimum level of output. In the future, with the availability of more refined data, that is, data on “quality” of inputs, many of the limitations of this study could be overcome and researchers might be able to esti-mate an optimum size for Bible colleges.
Acknowledgements
The authors are thankful to Asim Rijal, graduate associate, and Robert D. Welch and Caryn Thomas, PACE students, for their help in the preparation of this paper. They are also grateful to Vinita K. Kennedy and two anonymous referees for their suggestions on an earl-ier version of this paper. Any omissions are the responsi-bility of the authors.
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