Quality and Economies of Scale in Higher Education: A Semiparametric Smooth Coefficient Estimation

Fu, Tsu-Tan, Huang, Cliff J., Yang, Yung-Lieh, Contemporary Economic Policy

I. INTRODUCTION

Institutions of higher education comprise knowledge creation (research) and knowledge dissemination (teaching). Thus, the university's mission is to deliver quality undergraduate and graduate education and to expand the frontier of academic research. Numerous studies (Verry and Layard 1975; Cohn, Rhine, and Santos 1989; deGroot McMahon, and Volkin 1991; Nelson and Hevert 1992; Lloyd, Morgan, and Williams 1993; Dundar and Lewis 1995; Koshal and Koshal 1995, 1999) have investigated, as in the case of firms, the multiple-product cost function of producing the vector of outputs on undergraduate and graduate education and research. The cost structure of higher education is studied via the scale economies and the scope economies of the cost function. However, these measures of scale and scope economies may be elusive if the quality dimension of colleges and universities is not considered. Without taking into account the quality variation, in the short-run, a university can reduce the average cost of operation by lowering the quality. Universities with more congested educational facilities are substituting facility utilization for capital; universities with a higher student/faculty ratio and less research engagement are substituting quality of education for quantity of undergraduate and graduate enrollment. The consequence is an upward bias in estimating the economies of scale and scope. This conclusion is consistent with the early studies of Nelson and Hevert (1992) which showed that failure to control for class size in cost function may result in specification bias, and an economy of scale is evident if class size is allowed to expand.

Quality of higher education is multidimensional. Quality in instruction, faculty research, and the quality of educational environment are all significant factors determining the long-run university cost structure and the crucial determinants of the scale and scope economies of higher education. How and to what extent quality affects a university's operating cost depends on the goal and orientation of the institutions. Higher quality increases the university's cost of operation. But, in a large research-oriented university, a disproportional share of the budget often benefits the graduate, not the undergraduate program. It is unrealistic to assume that quality affects only the average cost and that the marginal costs of various educational programs are independent of quality. Quality is unlikely to be simply a neutral cost-shifting factor.

The purpose of this paper is to propose a university cost function and a semiparametric-estimating technique that treats quality as a nonneutral cost-shifting factor, which directly affects the marginal cost of outputs in undergraduate and graduate education. More precisely, the marginal costs of outputs are specified as a nonparametric smooth function of quality. A semiparametric specification of university cost is applied to 56 comprehensive and science/technology universities in Taiwan over the period 2000-2003.

II. SEMIPARAMETRIC COST FUNCTION AND ESTIMATION TECHNIQUE

Following the conventional multiproduct cost specification for higher education (Cohn, Rhine, and Santos 1989; deGroot, McMahon, and Volkin 1991; Lloyd, Morgan, and Williams 1993; Dundar and Lewis 1995; Koshal and Koshal 1995, 1999), the total cost (C) is specified as a quadratic function of outputs, undergraduate and graduate enrollments. However, the quadratic cost function is modified by allowing for the coefficients to be functions of quality in the following form:

(1) C = [[beta].sub.0](Q) + [k.summation over i=1] [[beta].sub.i](Q)[Y.sub.i] + (1/2) X [k.summation over i=1], [k.summation over j=1], [[beta].sub.ij](Q)[Y.sub.i][Y.sub.j] + [m.summation over i=1], [[alpha].sub.i][Z.sub.i] + [epsilon]

where C is the total cost of producing k educational outputs (Y), and [Z.sub.i] is the ith institutional characteristics. The quality Q is an index relating to teaching (e.g., faculty/student ratio, class size), to research (e.g., publication, grant awards), and to the learning environment (e.g., learning facility and living space). Following Li et al. (2002), the coefficients, [[beta].sub.i](Q) and [[beta].sub.ij](Q), of outputs are assumed to be unknown functions of the quality. When [[beta].sub.i](Q) = [[beta].sub.i] and [[beta].sub.ij](Q) = [[beta].sub.ij], the specification reduces to the conventional augmented cost function with quality as the cost-augmenting, neutral cost-shifting factor via the varying intercept [[beta].sub.0](Q). Koshal and Koshal (1995, 1999) specify a neutral cost-shifting specification with [[beta].sub.0](Q) being linear in the average Scholastic Aptitude Test of entering freshmen, whereas Dundar and Lewis (1995) use reputation ratings of programs as an additive to the total cost.

The advantage of specifying the total cost as a semi parametric smooth coefficient model allows the investigation of the profile of the cost-quality relation observed in the data without setting a prior parametric functional form of the relationship. (1) For example, the marginal cost of the ith output:

(2) [MC.sub.i] = [[beta].sub.i](Q) + [k.summation over j=1], [[beta].sub.ij](Q)[Y.sub.j]

varies with the quality. Quality improvement in faculty/student ratio, in research facilities, or in dormitory living may incur a differential marginal cost in undergraduate and graduate education. And the marginal cost (the shadow price) of quality varies directly with the quality:

(3) [MC.sub.Q] = [[beta].sub.0]'(Q) + [k.summation over i=1], [[beta].sub.i]'(Q)[Y.sub.i] + (1/2) X [k.summation over i=1], [k.summation over j=1], [[beta].sub.ij]'(Q)[Y.sub.i][Y.sub.j]

where [beta]'(Q) is derivative of the smooth coefficient functions. An increasing or even a decreasing marginal cost of quality is a distinct possibility. Furthermore, when the semiparametric cost function is estimated, the ray economies of scale (Baumol, Panzar, and Willig 1982; Cohn and Geske 1990; Hashimoto and Cohn 1997) are estimated as:

(4)RES = c / [k.summation over i=1] ([[gamma].sub.i] X [MC.sub.i])

The ray economies (diseconomies) of scale exist when RES is greater (less) than one. It is obvious that the ray economies of scale depend on quality, among other factors. The cost of expansion in two equal-size universities differs if the quality of the universities differs. The average cost of undergraduate expansion, measured by the average incremented cost (AIC), may differ from graduate expansion. The AIC of the ith ([Y.sub.i]) output is defined as:

(5) [AIC.sub.i] = (C - C.sub.-i)/[Y.sub.i]

where [C.sub.-i] is the total cost of producing all other outputs other than the ith output, that is, [Y.sub.i] = 0. The identification of the sources of ray economies of scale is measured by the product-specific economies of scale defined as:

(6) [PSE.sub.i] = [AIC.sub.i]/[MC.sub.i]

If [PSE.sub.i] is greater (less) than one, economies (diseconomies) of scale are said to exist for the ith ([Y.sub.i]) output. Again, the PSE depends on quality. Because the cost structure, marginal cost, and scale economies are all functions of quality Q, the semiparametric cost function …

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Publication information: Article title: Quality and Economies of Scale in Higher Education: A Semiparametric Smooth Coefficient Estimation. Contributors: Fu, Tsu-Tan - Author, Huang, Cliff J. - Author, Yang, Yung-Lieh - Author. Journal title: Contemporary Economic Policy. Volume: 29. Issue: 1 Publication date: January 2011. Page number: 138+. © 2003 Western Economic Association International. COPYRIGHT 2011 Gale Group.

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