Download Estimating a Multiple Output Cost Function for Higher Education

Estimating a Multiple Output Cost Function for Higher Education Institutions
Rajaram Gana and Michael F. Middaugh
Institutional Research and Planning, University of Delaware, Newark, Delaware
Presented at the Association for Institutional Research national meeting, Boston, MA, May 1995.
Abstract
A three-output flexible fixed quadratic cost function for higher education institutions (HEIs) is
estimated using data, by academic discipline, for the 1991-92 academic year and the 1992 fiscal
year. The outputs used are undergraduate and graduate enrollment, and sponsored research
monies. The input price used is average faculty compensation. Results indicate the existence of
economies of scale and scope.
Estimating a Multiple Output Cost Function for Higher Education Institutions
Introduction
A question of special interest (see Hoenack and Collins, 1990) is whether economies of
scale and scope (Baumol, Panzar, and Willig, 1988) exist for the instruction and research outputs
produced by higher education institutions (HEIs). In this present study, following Cohn, Rhine,
and Santos (1989), a three-output flexible fixed cost quadratic (FFCQ) function is estimated in
order to find out if such economies do exist for a sample of HEIs selected by Middaugh (1994).
Middaugh collected data, by academic discipline, on 101 HEIs across the nation. This
data covers the 1991-92 academic year and the 1992 fiscal year. The 101 institutions comprise a
sample belonging either to the National Association of State Colleges and Land Grant
Universities or to the American Association of State Colleges and Universities. The data covers
38 disciplines. These disciplines are grouped into 7 broader curricular groupings. These
groupings are Humanities, Fine Arts, Natural and Physical Sciences, Mathematics and Computer
Science, Behavioral and Social Sciences, Business, and Preprofessional. The 101 institutions
included 58 Carnegie Comprehensive, 22 Doctoral, 16 Research, and 5 Liberal Arts institutions.
After missing data are eliminated, the sample size is 1,696.
Methodology
Total educational expenditures is the regressand. The three outputs used are
undergraduate full-time equivalent (FTE) enrollment, graduate FTE enrollment, and sponsored
research monies. Average faculty compensation is the input price used.
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Ridge regression (Hoerl and Kennard, 1970) is used to estimate the cost function. Here
the coefficients are estimated by adding a constant k  0 to each of the variances in the variancecovariance matrix of the regressors. This procedure makes the estimated FFCQ function less
sensitive to the effects of multicollinearity and outliers (Saccucci, 1985). Multicollinearity is
considered a problem whenever variance inflation factors (VIF) exceed 10. Saccucci's results are
outlined below.
Suppose m out of n observations are variance-inflated outliers (VIOs). Each of
these m observations is assumed to have variance 2w, where w > 1 is a constant, and each of the
remaining n - m observations is assumed to have variance 2. Let Xm be the sub-matrix of X, the
matrix of regressors, containing the m outliers. Saccucci showed that the mean square error
(MSE) of the ridge estimated coefficients under the assumption of VIOs is equal to the MSE of
the ridge estimated coefficients under the assumption of no outliers plus 2(w - 1) times the sum
of the diagonal elements of the matrix (XTX + kI)-1XmTXm(XTX + kI)-1, where I is the identity
matrix and T is the transpose operator. He showed that this additional MSE for the ridge
estimator decreases (monotonically) with k, and that there always exists a k > 0 for which the
MSE of the ridge estimated coefficients under the assumption of VIOs is less than the least
squares estimated coefficients under the assumption of VIOs.
The FFCQ function includes 0-1 dummy variables for the discipline groupings, for
positive amounts of research and graduate outputs, for type of academic year (semester or
quarter), and for employing faculty whose primary job is not teaching (for example, teaching
assistants). These dummy variables are used to control all influences (such as fixed-cost
differences) from these factors on the cost function.
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Results
The least squares (k = 0) estimated FFCQ function (which is unstable), t-ratios, VIF, and
the corresponding ridge (k = 0.1) coefficients are given below. The resultant R2 and F-ratios are
0.9329 and 968.283, respectively.
Variable Description
Estimate
t
VIF
Ridge
Intercept
Undergraduate enrollment
Graduate enrollment
Sponsored research monies
Undergraduates squared
Graduates squared
Research monies squared
819,243
83.502
-4,800
0.6155
-0.9133
-5.8924
-13E-11
1.5
.22
-3.7
3.1
-15
-13
-.02
0
81
128
108
15
26
10
-184,240
906.2
2,170
0.4660
-0.1138
-1.089
0.000
Undergraduates  graduates
4.16471
12
39
0.3537
Undergraduates  research monies
8.11E-5
2.2
10
0.0002
Graduates  research monies
Average faculty wage
Average faculty wage squared
6.67E-4
-3,218.9
89.6179
7.4
-.16
.47
17
126
130
0.0003
10,010
106.7
Wage  undergraduates
49.5161
7.1
98
19.08
Wage  graduates
157.798
7.1
136
44.04
Wage  research monies
Research output dummy
Graduate output dummy
Supplemental faculty dummy
Semester dummy
Humanities
Fine Arts
Natural and Physical Sciences
Math and Computer Science
Behavioral and Social Science
Business
0.0095
209,488
32,755
-1.22E5
84,142
-9.13E5
-5.48E5
-6.72E5
-8.06E5
-1.09E6
-8.64E5
2.9
4.8
.78
-3.0
1.7
-13
-7.4
-9.4
-10
-16
-12
112
1.4
1.2
1.1
1.1
3.1
2.6
3.0
2.2
3.8
2.2
0.009
2.5E5
7.3E4
-8.9E4
3.0E4
-4.7E5
-1.7E5
-2.1E5
-4.1E5
-6.2E5
-4.2E5
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The mean levels of undergraduate and graduate enrollments and research monies are 379,
51, and $170,767.19, respectively. The mean level of faculty compensation is $53,930. The
economies of scale and scope (calculated using the ridge estimated coefficients) for several
multiples of the mean levels of outputs is shown below. The wage variable is always set equal to
its mean. Note, since the (ridge) estimated coefficient of the square of sponsored research
monies is 0, the research output exhibits constant product-specific economies of scale.
Multiple of output
means
0.1
0.5
1.0
1.5
2.0
3.0
3.5
5.0
6.0
7.0
Ray Economies
Product-Specific Economies Scope
Undergraduates Graduates
1.002
1.001 0.855
1.011
1.006 0.537
1.023
1.012 0.360
1.034
1.018 0.263
1.046
1.024 0.200
1.071
1.034 0.120
1.083
1.040 0.091
1.122
1.056 0.027
1.150
1.067 -0.006
1.178
1.077 -0.035
6.902
2.176
1.584
1.386
1.286
1.185
1.156
1.101
1.079
1.062
Concluding Remarks
The results indicate that economies of scale and scope appear to exist up to 500% of the
mean output levels. Scope economies appear to be exhausted at 600% of the mean output levels,
but ray and product-specific economies appear to continue to exist. These results suggest that
HEIs producing all three outputs may be more efficient than HEIs specializing in producing
some, but not all, of these outputs.
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References
Baumol, W.J., Panzar, J.C., and Willig, R.D. (1988). Contestable Markets and the Theory of
Industry Structure, Harcourt Brace Jovanovich, New York.
Cohn, E., Rhine, S.L.W., and Santos, M.C. (1989). Institutions of Higher Education as MultiProduct Firms: Economies of Scale and Scope, The Review of Economics and Statistics.
Hoenack, S.A. and Collins, E.L. (1990). The Economics of American Universities, State
University of New York Press, Albany, New York.
Hoerl, A.E. and Kennard, R.W. (1970). Ridge Regression: Biased Estimation of Nonorthogonal
Problems, Technometrics.
Middaugh, M.F. (1994). Interinstitutional Comparison of Instructional Costs and Productivity
by Academic Discipline: A National Study, paper presented at the Association for
Institutional Research annual forum, New Orleans, Louisiana.
Saccucci, M.S. (1985). The Effect of Variance-Inflated Outliers on Least Squares and Ridge
Regression, unpublished Ph.D. dissertation, University of Delaware, Newark, Delaware.
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