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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. 3 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. 4 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 5 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. 6 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. 7