Departmental productivity in American universities: Economies of scale and scope

Economics of Education Review. Vol. 14, No. 2, pp. 119-144, 1995

Copyright (~) 1995ElsevierScienceLtd Printed in Great Britain. All rights reserved 11272-7757/95$9.50 + 0.0(I

Pergamon

Departmental Productivity in American Universities: Economies of Scale and Scope HALIL DUNDAR* and DARRELL R. LEWlSt~ '~'Department of Educational Policy and Administration, University of Minnosota and -l-Department of Educational Policy and Administration, 136 Burton Hall, University of Minnesota, Minneapolis, MN 55455. U.S.A.

The study reported in this paper examines the departmental production and cost structures of a homogeneous sample of American public research universities in order to estimate their degrees of economies of scale and scope. Multiproduct cost functions are estimated for the departmental production of teaching and research in order to determine the most efficient level and product-mix for differing types of departments. Three clusters of departmental fields (i.e., the social, physical, and engineering sciences) are examined across 18 similar public research universities through the use of a four-output (i.e., undergraduate teaching, master level training, doctorate level instruction, and research productivity) quadratic cost function. At the department level, both ray- and product-specific economies of scale and global and product-specific economies of scope are estimated at differing levels for all outputs. [JEL 121] Abstract --

INTRODUCTION

production could be characterized by a single homogeneous output (measured almost always by MUCH RESEARCHHAS BEEN done in the area of cost , undergraduate student credit hours). James (1978), functions in the field of economics since the turn of for example, has noted that "'the failure in previous the centul:y. These studies have provided important studies Io adjust correctly for the substantial alloinformation in many sectors of the economy about cation to research and graduate training has appardecision making with respect to efficient resource ently meant that undergraduate costs have been allocation. On the other hand, estimating cost overstated, the social rate of return to underfunctions in higher education has been quite limited graduate costs have been overstated, [and] estimates until only recently. Spurred especially by the of productivity growth through time have been expansion of higher education during the 1961)s, understated" (p. 184). concern with increasing efficiency in higher eduThis limited attention to the multiproduct nature cation has generated a number of studies that of higher education has not been due to any lack of attempt to estimate the costs of higher education interest on the part of economists or higher cdu(see, for example, Adams, Hankins and Schroeder, cation policy analysts in examining multiproduct 1978: Brinkman, 1981; Brovender, 1974; Hoenack, cost functions. Rather, the lack of appropriate Weiler, Goodman and Pierro, 1986; Maynard, 1971; econometric models for explaining the nature of Smith, 1978; Southwick, 1969; Tierney, 19811; and multiproduct firms has been the foremost problem Vcrry and Davies, 1976). However, a major preventing the analysis of multiproduct firms and problem found within almost all these early studies organizations in both industry and non-profit in higher education has been their assumption that organizations such as higher education. Fortunately, [Manuscript received 27 January 1994; revision accepted for publication 8 June 1994.] 119

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recent developments in the field of economics relative to the theory of industrial organizations in their production processes permit an endogenous determination of the cost structures of multiproduct firms. Over the past decade an outpouring of empirical studies using multiproduct cost concepts have provided insights into the cost behavior and technology of multiproduct firms in a wide range of industries, including banking (Baumol, Panzar and Willig, 1988), transportation (Wang and Friedlaendcr, 1985), public utilities (Mayo, 1984), telecommunication (Panzar and Willig, 1981), petroleum (Shoesmith, 1988), and even hospital care (Fournier and Mitchell, 1992) among others. The early econometric modeling and use of quadratic equations for estimating multiproduct production and cost functions by Baumol and his colleagues (Bailey and Friedlander, 1982) was an especially important conceptual contribution, From this theoretical foundation developed in economics, three recent studies have examined the multiproduct nature of production and costs in higher education (Cohn, Rhine and Santos, 1989; De Groot, McMahon and Volkwein, 1991; Nelson and Heverth, 1992). Although the two studies by Cohen et al. (1989) and De Groot et al. (1991) represent major advances in our understanding of costs in higher education, especially in their attempts to estimate economies of scope, both studies used only aggregated institutional level data as their unit of analyses. Other cost studies in higher education have found that costs differ materially across academic departments and disciplines in relation to outputs (e.g., Berg and Hoenack, 1987; Carlson, 1972; Tierney, 1980: Verry and Davis, 1976) and that using only institutional level data often can be misleading. The most important problem seems to bc that different production technologies among academic disciplines may generate problems in analyzing departmental cost functions. For instance, results can be quite misleading if a single cost function is estimated for both chemistry and English departments because they have quite dissimilar production functions. Since aggregating institutional outputs may often yield unreliable conclusions concerning the costs of outputs and the existence of economics of scale and scope in higher education, Tierney (198(I), Verry (1987) and others consistently have recommended that separate analyses ought to bc conducted for each academic department. Although Nelson and

Heverth (1992) recently did focus on departmental data while examining the marginal costs of teaching outputs and searching for the possible existence of economies of scale, their study utilizes data from only a single university and consequently has limited generalizability. Purpose The study reported in this paper examines the departmental production and cost structures of a homogeneous sample of American universities in order to estimate the extent to which economies of scale and scope exist within their production of teaching and research activities. We use a fouroutput flexible quadratic cost function (for estimating the production effects of research, undergraduate teaching, master-level teaching, and doctorate-level teaching) to analyze the cost structures of 17 different departments found within three fields (i.e., the social sciences, physical sciences, and engineering) across 18 public research universities. Following the earlier work of Baumol et al. (1988) in industrial organizations and Cohen et al. (1989) in higher education, a quadratic cost function seems to best serve as the framework for analysis of cconomies of scale and scope within universities with multiproduct outcomes. Departments, as the fundamental organiz~tional units of colleges and universities, are relatively autonomous, make decisions on the curriculum, determine academic degree standards, recruit and promote faculty, and largely determine the tcchnology for the production of their output. Even if multiproduct cost analyses at the institutional level provide important insights about existing economies of scale and scope between institutions, as in the cases of prcviouis work by Cohen et al. (1989) and De Groot et al. (1991), such findings have less to offer policy and decision makers at the institutional level because of the differing production processes of each department. By contrast, the results of this study should enable institutional administrators and policy makers to compare the cost structures of different types of departments within American public universities. The study examines and reports on findings relative to a number of related questions. A first set of questions relates to whether there are ray- and product-specific economies of scale within and across individual departments. A second set examines whether there arc global and product-

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Departmental Productivity in A m e r i c a n Universities

specific economies of scope within and across individual departments. A r e t h e r e cost complemcntarities among the four types of outputs typically found within American universities (i.e., undergraduates, masters, doctorates, and departmental research) through their joint production? Should production be diversified, or should some types of departments concentrate on undergraduate teaching and others on graduate instruction, and/or on research for most cost-effective results'? What influences does the quality of a department and its products have upon its costs? Estimates are given as to the most efficient levels and mix of all outputs by type of departmental field,

METHODOLOGY In specifying a model for estimating the relationship between the costs and multiple outputs of higher education institutions, it is difficult to select one particular functional form as being most appropriate. Hybrid translog, quadratic, and constant elasticity of substitution forms all have been offered as promising multiproduct cost functions. Cohn and his colleagues (1989), for example, employcd a flexible fixed quadratic function as the basis for estimating cost functions for their sample of almost 1,900institt, tions in American higher education. On the other hand. De Groot et al. (1991) and Nelson and Nevcrt (1992) employed translog functions for estimating costs of research universities. Neverthcless, Baumo[, Panzer and Willig (1988), along with several others (('ohn etal., 1989; Mayo, 1984), recommend the use of a quadratic cost function for estimating scale and scope economies for most types of multiproduct organizations, They have argued that a quadratic cost function is most appropriate because it has bccn shown to comply most closely with the required features of a multiproduct production function, hs main shortcoming is the absence of any explicit theoretical foundation for using this form in preference to any other functional form (Baumol el a/.. 19,"{8, p. 453).

The Generalized Model The mLdtiproduct quadratic cost function employed in this study can be represented by a second-order approximation around the mean and is gcnerally written as

('(y) = c~. + E c~i(Yi - .vi) + (1/2) 55 \Z t

O~ii(Y,- Yi)(Yi-

t .91) +

/

E

where ('(y) is the total cost of y outputs, (~ is the constant or fixed cost parameter, the u(s and o~ii'S are coefficients, and ~ is a disturbance term. A problem with this model may arise because of the exclusion of quality aspects associated with the outputs. Since in the production of each output different combinations of inputs are used, the quality of the final product may be different cvcn if the quantity of output is the same. As such, in addition to the quantity of inputs, the quality of outputs in higher education becomes an important factor in determining cost structures. In a typical public research university the qualitics of tindergraduate and graduate teaching often differ by material degrees. Unfortunately, there arc no standard and accepted measures for the quality of outputs of higher education which enable us to directly control and measure for thc effects of any increase in costs. Instead, reputational ratings of the departments are usually used as proxies to present variations resulting from the quality of outputs. Accordingly, we added this measure of quality to our generalized model. Since aggregated department fields (i.e., the social, physical, and engineering sciences) will be examined within the context of the generalized model, the model could bc a ntis-specification because there might be differences in the production technology among even those departments that arc organized and analyzed as aggregated fields. Therefore departmental dummy variables needed to be identified within the model to control for the possible effect of any existing production technology differences between departlncnts. Allowing for a measure of quality and possible departmental differcnces, the final generalized model now can he written as ('(Y) = ~,, + \-t (~,(Y, - .v,) + (1/2) \_) \£ i I -- .('i)()': .f'i) + [3(_) + x 8il) ' + •

(~il(Yi

wherc [3 is the estimated coefficient l\)r quality rating and 81 are the estimated coefficients for thc dcpartmcntal dummy. In this model n o attempt is made to control for any price effect that might result from differing prices of the factor inputs. As noted by Dc Groot e/ al.. (1991) and others, the main resource inputs for

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departmental production in public universities are the faculty and administrative staff. Unfortunately, the data sets we were working with did not permit us to estimate average departmental wage rates for senior faculty. However, since the national labor market for faculty and administrators at such universities is very competitive and wages are recognized as being highly correlated with average productivity, we assumed that wages (corrected by productivity differences) were essentially constant across our relatively homogeneous sample of institutions and price measures were omitted in our cost function models. Our design and specification of the generalized model differ in several material ways from most earlier studies. First and foremost, the present study differs from almost all others in that our unit of analysis is a department while most earlier studies used institutions as their focus. A departmental approach is particularly important for the purpose of examining institutional cost structures because both tangible and intangible output differences exist between fields, especially at large research oriented public universities. Several previous studies have found that important differences exist in the costs and production between subject areas (Berg and Hoenack, 1987; Smith, 1978; Tierney, 198(/; Verry and Davies, 1976). Second, few previous studies have attempted to control for quality variation in outputs. This study not only focuses upon the department, but also controls for potential quality variations that might exist within the production of differing departments. Finally, because it can be argued that the underlying production technology facing each department even within related fields might be different, this potential problem is statistically tested by the use of department-specific dummy variables. Unlike any other reported work, the design for this study tests for any structural cost differences among departments that might arise from any underlying differences in production technology, Economies of Scale Economies of scale are conventionally assumed to exist if total costs increase proportionately less than output as production is expanded. In the case of a multiproduct setting Baumol and his colleagues (1988) make a distinction between two different types of economies of scale - - i.e., ray- and productspecific economies of scale,

Ray-economies of scale. First, a composition of output can be assumed to remain fixed while its size is allowed to vary. This form of scale economy, referred to as "'ray-economies of scale" by Baumol and his colleagues, is directly analogous to economies of scale in single-product firms, and measures overall economies of scale. Ray-economies of scale can be defined over the entire output set and noted as Sx =

C(y) YI'- lyiCi(Y)

where ci(y) = OC(y)/i~yi and represents the marginal cost of producing the ith output. Ray-economies (or diseconomies) of scale are said to exist if S x is greater (or less)than unity. S x can be interpreted as "'the elasticity of the outputs of the relevant cornposite output with respect to the cost needed to produce them" (Baumol et al., 1988, pp. 5(I-51). Product-specific economies of scale. Second, an elasticity of economies of scale can measure how costs change as both the output and composition of products change. This second dimension of economies of scale is referred to as "'product-specific economies of scale". Ray-economies of scale assume that output is expanded proportionally along a ray emanating from the origin. However, the magnitude of the multiproduct firm's operations may change through variation in the output of one product, holding the quantities of other products constant. Therefore, the product-specific expansion in a product set becomes an important feature of multiproduct cost concepts. The incremental cost of a multiproduct firm for producing an additional output i can be noted by IC(yi) = C(yx) - C(y.~ i)

where IC(y,) denotes the total cost of producing all of the multiproduct firm outputs except the ith one. From this concept the average incremental cost due to the additional production of the ith output can be noted by AlC(yi) = [C(y,,~,) - C(y,~_i)]/y i = IC(yi)/yi.

Product-specific returns to scale that are specific to a particular output now can be derived from the above specification as

Departmental Productivi O, in American Universities S;(y) = AIC(y;)/Ci(y). If in the production of teaching and research output there exist, for example, some product-specific fixed costs, then we would expect that product-specific economies of scale would also exist. Product-specific economies (or diseconomies) of scale are said to exist for the ith output when S, is greater (or less) than unity, Economies of Scope The presence or absence of complementarity between outputs in production becomes a crucial matter in the case of multiple product production. This concept of scope is quite new, appearing first in the works of Baumol (1977) and Panzar and Willig (1979), and only recently has been introduced as a complement to the older concept of "'economies of scale" in the theory and literature of industrial organizations and structures (Baumol, Panzar and Willing. 1988). It measures the cost savings accruing to firms producing two or more products jointly as against specializing in the production of a single output, The diversity of products within it single firm or organization, known as "'scope". may raise efficiency by providing cost advantages in a situation in which a single firm produces a given level of output for each product level spending less than a combination of specialized separate firms. Scope economies may arise when there exist some inputs (or forms of networking) which are shared in the production of two or more outputs. In the case of production within higher education institutions, for example, a university department can be viewed as a typical multiproduct organization because it produces multiple products (i.e.. undergraduate instruction, graduate tel, ching, research, and often public service as well) through the sharing and joint utilization of inputs such its faculty, graduate students, equipment, buildings, support staff, and the like. Scope economies can be said to arise when individual members of the academic staff diversify their activities rather than specialize in a single output. The concept of economies of scope can be examined its either global or product-specific economies of scope, Global economies of scope. Global economies of scope simply measure whether the cost of producing two or more products jointly will be cheaper than

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the cost of producing them separately. Complementarities in the production of a university's multiple outputs result in scope economies. Global economies (or diseconomies) of scope are said to be present when ('(y) -< (->)C(yx_,) + C(y,) where ('(y,) is the cost of producing the product set t. and C(y,v /) is the cost of producing all other products other than those in the product set t. The degree of economies of scope can be noted for a product set t by ,'s'(; = [C(y,) + C(yx ,) - C(y)]/C(y) where S¢; denotes the degree of global economies of scope. If S¢, -> O then cost advantages accrue for producing the output bundles jointly. Product-specific economies of scope. In addition to global economies of scope in the production of all outputs jointly at a production unit, there may be some cost advantages due to production of each output jointly with the other outputs. If this is the case, then the joint production of one of the outputs (e.g., undergraduate instruction) with the others is the most efficient way to produce rather than to produce the output independently. Thus, the degree of product-specific economies of scope (SC,) measures the proportional increase or decrease in costs resulting from producing all of the outputs except the ith one. Product-specific economies of scope can be noted as

SCi = [(C(y;) + C(yx_;) - C(y)]/C(y) where C(yv ,) represents the total costs of producing all the outputs jointly except the ith one. Product-specific economies of scope are said to exist if S()-> O, indicating cost advantages for producing the ith output jointly with the other products. Specification of Study Variables in the Model Total cost variables. The dependent variable for our total cost model includes all the costs for the departmental production of teaching and research. In this regard, the study followed procedures used in most other similar studies through the use of institutional expenditures as the costs of higher education. The private costs to students were

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excluded since the study focused on departmental and institutional cost comparisons. The typical costs of the departments included faculty and support salaries, computer and equipment expenditures, and unrestricted research funding. Non-departmental costs of central administration, library, and the other costs of non-departmental maintenance and operations were excluded+ Restricted research funding for externally sponsored research were also excluded. Thus, total departmental costs were the sum of costs for all departmental-based inputs employed for production of 3' in an academic year. The costs of departmental inputs in the study were specified as Ct = annual total faculty wages and fringe bentfits in the ith department; C_~ =annual salaries and fringe benefits of support staff in the ith department; C3 = annual expenditure for services and supplies in the ith department; C4 = annual expenditure for equipment in the ith department; and C5 = annual expenditure for computer in the ith department. At least two measurement problems exist within our departmental estimates of costs. The first relates to our exclusion of all central indirect costs. To fully account for all costs of production, assuming that all institutional production of outputs flow from individual departments, then all non-departmental indirect costs should also be properly prorated across all of the producing departments. This could possibly have been accomplished by estimating thc proration through, for example, student credit hour production. We elected, however, not to include such costs because our interest was largely focused upon estimating departmental differences and such indirect costs do not directly influence the production technology of departments. Our second measurement problem relates to treating all department equipment and computer expenditures its annualized costs. This, of course, is not conceptually accurate because such purchases have useful lives beyond one year. We elected to annualize such expenditures because we assumed that these expenditures would likely average out over time and that, in any case, such expenditures were only a small proportion of departmental costs and the extra effort required to acquire more precise esti-

mates probably would not materially affect our results. Output variables. This stud}, concentrated on thc teaching and research outputs of higher education. Although the importance of public service its an institutional output is recognized, the present study could find no reliable measure for this dimension. Student-credit hours at three teaching levels (i.e., undergraduate, master, and doctoral) were idcntified and used as proxies for teaching output for an academic year. The only reliable proxy available for measuring research output was the number of articles produced by each department during a representative pcriod of three years. The four outputs for the cost functions of this study were specified as 3't = annual undergraduate student-credit hours in the ith department: y: = annt, al master student-credit hours in the ith department: y; annual doctoral student-credit ho-trs in the ith department: and Y4 = the number of publications in the ilh department. The utilization of student credit hours as proxies for the teaching ot,tputs of each department is noteworthy. In most other studies dealing with educational outcomes the number of students or graduates havc bccn used as proxies for cducational outputs. Because these latter data arc specific to individuals who register in a particular department and do not measure departmental resources spent for other cross-departmental production of teaching, such departmental cost estimates are likely to be distortcd and materially overcstimatcd. Therefore, the student-credit hot, rs produced hy each department arc clearly a more accurate proxy for tcaching when measuring across departments. Although publication data were only collected for a three year period, it wits assumed that research productivit} in nlaturc departments generally have a similar pattern over time and that thrcc years is an adequate sample time period. Quality of output variables. In this study it was assumed that there might exist differences in the quality of outputs produced by differing departmcnts. To partially control for this possible effcct on

Departmental Productivity in American Universities departmental costs, a quality variable relating to reputational ratings of departmental graduate programs was obtained from a national study on gradt.ate programs (Jones, Lmdzey and Coggcshall, 1982). The judgment of peers in each field on the effectiveness of departmental programs was used its a proxy for the quality of graduate education in the ith department, It is often commented that many public universities have considerable differentiation in the quality of their undergraduate and graduate products, Many public research universities, for example, may have an open-access policy at the undergraduate level, whilc cornpcting to obtain the best and most able students into their more exclusive graduate programs. This paradox often has resulted in more variation in the qt.ality of the prodt.cts at public universities than at private universities, To address this issue, it was originally assumed that institutional selectivity at the undergraduate level could be used as a proxy measure for the quality of the undergraduate teaching product. To test this assumption the study obtained undergraduate selectivity indexes of institutions from Barron's Educational Service (1986). Unfortunately, preliminary review of these undergraduate quality indexes indicated that they did not discriminate well across the institutions selected, Although Carlson (1072) used the Gourman undergraduate quality rating indicators, preliminary review indicated that this measure likewise did not discriminate well with the sample and data at hand. Moreover, both the Barron's and Gourman indicators have limited validity for these purposes (Webster, 1984), given that both largely measure the quality of inputs. In any case, the homogenous nature of undergraduate students at most of the institution~ in the sample did not allow for measuring variation m the selectivity of the institutions. As a consequence, the study did not use an tradergraduate selectivity index for its cost ftmction estimates.

Deparlmentai variables. Department-specific dummy variables were used within each of the 3 fields to identify types of departments in the study. It was expected that department-specific dummies controlled for any differences between departments that might arise from differing accounting procedures, definitions of outputs, and production technolog'+,

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DATA The major source of data for this study came from the statistics gathered every two to three years by the American Association of University Data Exchange ( A A U D E , 1990) from thirty-eight participating institutions. Data on expenditures (costs) and teaching output (student-credit hours at the threc instructional levels of undergraduate, master and doctorate programs) were drawn from the latest compIelc.AAUDE data set (i.e., 1985-1986). The A A U D E data provide large and reliable estimates on costs and enrollments at the departmental level from the leading research and doctoral granting universities in the nation. The subsamplc selected for the study consisted of all 18 public research universities that provided complete data for 1985-1986. In the original A A U D E sample, 12 other public universities reported incomplete data and 8 private universities were included. Since the final subsample covers man~ of the most distinguished public research universities in the United States (i.e., 17 of the 18 institutions within the sample were among the top 3(1 public research universities in the nation as estimated by Webster, 1985), it is believed that the results can be generalized to the national population of public research universities. Data on expenditures and enrollments from 17 types of constituent departments across three departmental fields (represented by the social sciences, engineering sciences, and the physical sciences) were used in the study. These departments m the social sciences included anthropology, economics, history, political science, sociology, geology and psychology. In engineering, departments were included from civil, chemical, electrical and mechanical engineering. The physical sciences were represented by departments in chemistry, computer sciences, geosciences, mathematics, biostatistics/ statistics, and physics. Empirical data with respect to measuring research output and the quality of graduate programs were taken from recent studies of the Conference Board of Associated Research Councils (Jones et al., 1982). Their estimates for research productivity in the form of article publications and their reputational studies on graduate programs represent extensive surveys of over 200 research and graduate training universities and examine thirty-one different types of departments found within five different

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field groups. The number of publications in journals associated with programs and the institutions with which the author was affiliated in the 1978-1981 period were reported and used by the present study as a proxy for departmental research productivity, For reputational ratings, each program was evaluated by an average of 90 survey respondents from other programs in the same field. The evaluators were asked to assess programs based on the accessibility of faculty, the curricula, the instructional and research facilities, the quality of graduate students, the performance of graduates, and other factors that were believed to contribute to a program's effectiveness. The ratings were calculated on a scale ranging from "extremely effective" to "'not effective", Undergraduate exclusivity data from Barron's Educational Service (1986) also were used in an attempt to measure the quality of undergraduate programs across the institutions, but the measure did not prove satisfactory or discriminating and was dropped from the model, Possible Data and Design Problems A number of potential problem areas emerge relative to the data and design of our model. First, several data and sampling problems may exist. The sample, for example, is drawn disproportionally from the most research-oriented public universities in the nation. Although this may be seen as a potential problem of selection bias, it is also one of the strengths of the study because a detailed examination of more homogeneous institutions in terms of their production technology (and quality) may provide important insights about the cost structures of these types of institutions, Nevertheless, exclusion of other A A U D E member universities which did not report the necessary data for the 1985-1986 academic year and other nonmember research oriented institutions may have caused us to miss some important institutional differences, Moreover, no attempt was made to control for and analyze the costs of inputs shared by all departments (e.g. central administration, libraries, buildings, computer centers, and the like) within a given university. This resulted not only because the focus of the study was on the costs of departmental structures, but also because it would have been problematic, in any case, to attempt with the limited data set available, Second, not all educational outcomes are clearly defined and measured in the study. The four most

commonly expected outputs of higher education institutions are undergraduate instruction, graduate education, research, and public services. However, there are typically no clear-cut measures to indicate the quality and quantity of all these outputs. Despite the fact that value-added measures for teaching and research outputs are most desired, it is very difficult, if not impossible, to obtain direct measures characterizing such outputs of higher education institutions across large numbers of similar institutions. Accordingly, we, like almost all other similar studies, employed only approximate or proxy variables for their measurement. We employed student credit hours and numbers of publications as our proxies for teaching and research. On the other hand, our use of quality ranking data drawn from a national study of graduate programs (Jones et at., 1982) was an advancement over most prior work, but it still may not accurately measure the true qualitative differences between departments. We employed no reliable measure for differentiating the quality of undergraduate studies; and our measure of publication rates may not be sensitive to qualitative differences. In addition to the largely known teaching and research products of public universities, such institutions also have outcomes in public service and outreach activities. Public service is particularly important for public research universities since most arc expected through their public land grant missions to transfer knowledge outside their institutions. Ideally, we would have liked to have had some index or proxy for the public service produced by our institutions, but these data arc not yet available. As a consequence, our model did not adequately account for such public service. Another problem area related to the fact that most higher cdt, cation institutions, as non-profit organizations, are m~t perceived to minimize their costs. This results from the fact that not only the quantity of outputs, but also their qualities arc desirable outcomes (James and Rosc-Ackerman, 1986). It has been argued that, rather than minimizing their costs, higher education institutions spend all their available revenue for the sake of increasing qt.ality and prestige (Galvin, 1980). Bowen (1981) defined this economic behavior of higher education institutions as a "'revenue theory of costs", at least in the short run. Although Brinkman (1990) has argued that this explanation for the cost behavior of higher education institutions has some validity, it is

Departmental Productivity in American Universities probably an overstatement without substantial empirical evidence. Similarly, Verry (1987) has observed that "the assumption that education institutions act as cost minimizers is not absurd; after all, the teachers and administrators in such institutions do have certain objectives and are constrained by limited resources, so that departures from cost minimization imply that whatever objectives arc being pursued will not be attained to the full extent possible'" (p. 408). l,astly, it should be noted that the utilization of quadratic models in the estimation of cost functions in higher education can be criticized for their lack of theoretical foundation in linking with production in higher education. The production function of departmental activities in higher education is still largely perceived as a "'black-box". It is not cxplicitly known which inputs and what kinds of technical requirements are necessary for producing optimum outputs in ordcr to derive a mathematic~dly sound cost function (Brinkman and l,eslie, 1986; Gilmore, 1990; Hopkins, 1990). In light of these difficulties in dealing with costs, a common approach has been to use statistically best fitting techniques in order to find an appropriate functional form (Brinkman, 1990). Prior to our use of the quadratic model, w,c first estimated a translog function but subsequently found that our quadratic functions resulted in higher R-squares and greatcr significance with several of the independent variablcs. The use of the specific quadratic model in this study appears to be thc "'best fit" for our purposes. RESULTS Through the four output quadratic model dcveloped for the study, we estimatcd several multiproduct cost functions using ordinary least squares for the social, physical, and engineering science departmental fields of our sample. This section reports tm the results of our estimates for these cost functions and the degrees of scale and scope economics for each field, Estimating the Cost Functions

Tablcsl,2and3rcportonthevariablesincludcd within thc study and their estimated coefficients for our three models for erich of the social, physical, and engineering scicncc fields. In our first model the quadratic cost function is estimated with only the four outputs. In the second model a quality rating

127

variable is added to the original quadratic model. In the third model department-specific dummy variables are added to the cost function to account for the possibility of variations in costs arising from possible differences in departmental production across departments within a givcn field. Social sciences. Table 1 prcscnts the coefficient cstimatcs for each variable in the thrcc models from data supplied by 119 individual social scicncc departments. All three models havc reasonably high cxphmatory power with adjusted R-square terms accounting for 70% or more of the variations in total departmental costs. Estimation of the basic quadratic equation for Model 1 provides information with respect to the prospective prcscnce of economits of scale and scope within all the social science departments. The results reported through Model 1, for example, imply that there is a uniform set-up cost for most social science departmcnts. In a multiproduct cost function, economics of sealc will be dependent upon the magnitude of the fixed costs (a,). The presence of negative estimates for any quadratic terms found within the models of Tablc 1 suggest lhat for the production of those terms the cost function is likely concave and generative of economies of scale. For example, w'e note that both master instruction and rcsearch production are negative and suggcstive of scale economies. The interactive terms found within all the models in Table 1 also ,.zive us early insight into the prospects for whether there might also exist cost complementaritics ~md economies of scope. The negative coefficients for the interaction between undergraduate and doctoral training, for cxample, clearly suggest the eost-cffectivencss of their joint production. Model 2 was estimated to account for the possiblc cffect of the quality of outputs on total departmental costs. However, when the quality variable is introduccd we do not find a significant relationship between thc quality of programs on departmental production and costs. This finding is in contrast to an earlier finding by De Groot and his colleagues (1991) wherein they' found that quality had a statistically significant impact on total instittttional costs. The insignificance of output quality effects on costs in the present study likely result from the relatively homogeneous nature of our subsamplc of public rose,itch universities.

UND ( U n d e r g r a d u a t e SCH )': M A ST (Master Level SCH)" DOCT (Doctorate Level S C H ) RES (Published Articles) UNDSQ ( U n d e r g r a d u a t e SCH S q u a r e d ) ; MASTSQ (Master SCH Squared)$ DOCTSQ (Doctorate SCH Squared):i: RESSQ (Published Articles Squared) UND'=MAST ( U n d e r g r a d u a t e SCH x Master S C H ) ; UND'DOCT (UndergraduateSCH x DoctonteSCH)~: UND'RES ( U n d e r g r a d u a t e SCH x Published Art.):i:

Constant

Variables symbol (descriptions)

21.38 (11.521 1.10 (1.51) 0.86 (I).921 711.65 (54.98) 589,11110.011 (667,000.00) 2531.10 (6796.75) 1595.96 (3784.93) 7.98 (13.114) 27,995.03 141,798.35) 26,007.99 145.519.721 1944.86 (2661.56)

--

Mean+ (standard deviation) 1.388,113 :~ (24.67) 17.77" 13.21 ) 245.84 ~ (4.39) 33.88 (11.44) 5291 ':' (4.54) t1.1111113295 111.64) -0.114697 =:: ( 2.83) 0.18871 ..... (2.48) 36.117 (-1.61) -11.1t1171192 ( 11.951 11.1/221163..... ( 2.29) -0.01169 (-0.05)

Model 1 (t-statistics) 1.5311.972 ~' (7.07) 16.661 (2.88) 245.83':' (4.38) 52.16 (11.64) 5768" (4.24 0.1X)113680 (I).71 - 11.114561 :' (-2.72 0.181118 :~.... (2.35 39.07': . . . ( 1.711 -1t.11116936 (-0.93 -0.(122455 .... ( 2.32 -0.0077 ( 11.t16)

Model 2 (t-statistics)

Table I. Estimated parameters for the quadratic cost functions in social sciences

.

1,315,794':' (3.91) 6.5t0 10.91 ) 253.63':' (4.37) 78.85 (0.93) 6292 =' 13.611) 0,0007443 (1.39) 0.11448 l" (-2,56) 0.18819': .... (2.32) . 43.31 ~::;: (-1.84) -1t.I1117711 ( 1,1t21 - 0 . 0 2 5 7 6 ...... (-2.56) -1t,11239 (-0.17)

Model 3 (t-statistics)

"~ ~"

~i ,~

1324.18 {682.47}

-

IIg 21,73 I}.71

---

-

-

-

-

119 211.21 0.70

-

--

-

-

--

---

--

--

--

--

--

--

--

0.(18244 (1.61) 1.3977 { 1.611) 2.164 { I. 19) -76.831 (-0.681 --

1M)85{16...... (I.(17) 1.4189 ( I .(,2) I. 961 ( 1.091 --

--

l 119,79 12159.541 94.99 (167.93) 93.93 (183.96) 1.79 ({L42)

Notes: + A l l m e a n units ol +m e u s u r e :ire t h l ) u s a n d s e x c e p l q u a l i l y rating. :i:SCH = Stud,2nt C r e d i t H o u r s . : Significant at the 1% level or b e t t e r . 2-tailed test. Significant at the lit!,, level or b e t t e r . 2 - t a i l e d lest.

MAST DOCT (MaMer S C | I × Dt}ct~}latc SCH):i: M AST;:: R ES ( M a s t e r S C H x P u b l i s h e d Art. ):i: D O C T " R ES ( D o c t o r a t e S C H x P u b l i s h e d Art. ):): OUAL (Quality Rating) DECON ( D e p a r t m e n t s of Economics} DGEO {D e p a r t m c n l s {}t' G c { } g r a p h y ) DHIST { D e p a r t m e n t s {}1 H i s t o r y ) DPOL ( D e p a r t m e n t s of Pt}litical Science) DPSYC { D e p a r l m e n t s of P s y c h o l o g y ) DSOC ( D e p a r t m e n t s nt S{wiology} Costs {'['ota[ D e p a r t m e n t a l E x p e n d i t u r e s ) N F-statistic A d i. R :

-

19 }5.56 I}.72

-

343,365 {2. Ill} 144.__1 ({1.95) 3{17,{178 11.53} 121{I ({I.{H } --

{ I. 21 )

(I .47} 185,237

278.843

(1.(}8915 .... {I .77) I. 27811 (1.46) 2.3411 { 1.261 -65.175 (-~1.45}

% ~-

F ~" ",

~.

~ ~' ~. '~ ~" 2~

.~

t~

UND:'RES

U N D :' D O C T

UND::MAST

RESSQ

DOCTSQ

M A S SQ

UNDSQ

RES

DOCT

M A ST

UND

Constant

Variables

23.88 (21.62) 1.81 (2.20) 1.54 (1.96) 82.53 (75.11) 1,030,[100.011 (1,920,[X1(1.00) 8120.85 (29,272.76) 6184.41 (19,926.93) 12.39 (22.93) 55,297.48 (124,[100.00) 49,217. I 1) (911,59[).81) 2298.[)[) (2875.55)

--

Mean; ( s t a n d a r d deviation)

(2[).23) 45.39* (6.18) 150.08 ':'':' 12.21) - 104.05 ( - 1.29) 12,8[)4" * (6.44) 11.[111026[13 ( - 1.41) - 0.006100 (-[).85) 0.04787" (3.44) -20.73 ( - 1.33) 0.000741 ([/.48) 0.11[13186 ( 1.0(1) 0.1176 ( I. 13)

2.415,852':'

Model 1 (t-statistics) 2, t03,995" (3.71/) 46.82" (6.00) 141.28 ':' 12.01) - 114.73 ( - 1.38) 12,156':' (5.271 -0.11(102951 ( - 1.51) - 11.005136 (-0.69) 0.04777* (3.42) - 18.08 ( - 1. I 1 ) 0.1X)0734 ([I.47) 0.[X13443 (1.117) 0.1049 (0.98)

Model 2 (t-statistics)

Table 2. E s t i m a t e d p a r a m e t e r s for the quadratic cost f u n c t i o n s in physical sciences

786.616 ( 1.061 43.74':' (4.35) 168.95 ':' ':' (2.22) - 143.19 .... ( - 1.68) 7473" * (2.36) -0.00(10727 (-11.36) - 0.[)/18958 ( - 1.17) 0.04185 :~' (2.98) 1.85 111.10) -[).000738 (0.46) 0.[)[t 1875 (0.60) 0.1171 (1.[19)

Model 3 (t-statistics)

~

~.

~" ,~

1324.18 (682.47)

-. 98 34.09 0.82

--

--

__

--

__

--

-fl.(13939 ((I.82) 1.479 { 1.43) 11.2542 { -(}.36) --

__

--

3541.03 (8038.94) 163.57 {243.{)1 ) 217.14 (22.93) 1.86 {{}.35}

Notes: - A l l m e a n units of m e a s u r e are in t h o u s a n d s e x c e p t q u a l i t y rating. ;:: Significant at the 1% level or b e t t e r , 2-tailed test. : :::S i g n i f i c a n t at the I{VY,, le,:el or b e t t e r . 2-tailed test.

N F-statistic Adj. R 2

( DeparinlcnI~, of PhysicM Cost

D PHYS

( D e p a r t m e n t s of G e o l o g y }

DGEO

( D e p a r t m e n t s of C o m p u t e r Scion. ) DMATH ( D e p a r t m e n t s of M a t h c n m t i c s ) DCH E M { D e p a r t m e n t s of C h e m i s t r y )

DCOPM

Q U A l,

DOCI:::RES

MAST'::RES

MASI" DOCT

.

. 98 31.58 11.82

--

--

--

--

.

-0.03883 0.810 1.544 (1.48) -1}.2663 { -11.38) 166,278 ((1.56) -(

98 26.78 0.84

(2.(}2) 258,241} ((}.63) I, I (17,663 ::' t2.88) 3{}1,286 ((}.88) 854,169;:: ;: (1.99) --

IL004I)7 (0.118) 1.259 {I .21 ) -11 . 3~9"} ... ( (}.49) 493.47 I (I .5(}) h33,576;:

U.

~-" ~.

~,~ B~"

~, ~~. ~. '~ ~"

"-~

"~ -', ~%

U N D ~:D O C T

UND :MAST

RESSO

I)()('TSO

MASSO

UNDSQ

RES

DOCT

MAST

UND

Constant

Variables

9.09 (7.37) 2.34 (2.59 1.17 ( 1.63 21.28 ( 1t,~.84 136.1~00.~1(1 ( I 8S.(lIIO.O0 1[ .946.24 (_,~,4_11._4 3967.31 (I 7.595.22 (~.83 (2.2o 29,966.51 (4~.2S2.71 I (i ,592.41 {42.535.71

Mean(standard deviation) 2,152.746 '~' (17.32) 57.84':' (2.84) .~'.~.(~. . . "~' . ~::~ (3.16) 525.60::: (4.69) 12.~73 (1.39) (I.001944 (~1.77) -(I.08386 ~ (-3.66) 0.06198 (lt.87} 792.00 ( - 1.14) I~.1H25(~1~ ( 1.29) 0.(1(~756 (0.40)

Model 3

I, 139.539 (I.62) 71 .(1(I~ (2.83) _'~10.,,'~ (2.73) 422.70::: (3.35)

4t)51 (0.49} 0.0027()3 (1 .~13) 0.(17136 : (-3.(17) {).l)6t,~{l~ (I .00) 277. Ill (0.40) I).[){)7,'.;211 ({I.,gl) I).()l 127 ((I.60)

~(188 (0.51 ().l)(l()1148 (0.45 (I.08525 ~ ( -3.77 0.07281 (I .114 -600.9 ( 0.90, 1~.[)13153 ( - 1.37 0.Ill ,~_8-~' {0.81)

(t-statistics)

1.304.281 ~:.... (2.33) 65.36: (3.17 2311.93::' (3.27 433. I': (3.45

Model 2

(t-statistics)

Model I

(t-statistics)

"Fable 3. Estimated p a r a m e t e r s for the quadratic cost functions in engineering

~. =

= %

t~

~ a" :.4

U, b,a

Notes: -; All m e a n u n i t s o f m e a s u r e a r c m t h o u s a n d s e x c e p t q u a l i t y r a t i n g . :::Significant at t h e 1% level o r b e t t e r , 2 - t a i l e d test. ::'::Significant at t h e IO% l e v e l or b e t t e r , 2 - t a i l e d test.

N F-statistic Adj. R 2

1957.34 (1399.74)

--01 29.63 0.87

-++1 28.67 11.87

--

283.3~5

(I.32)

1.34 (-0.81) o. 13672 ..... (1.69) 15.09 ...... (2.33) - 15.(111...... (1.72) 423.(~113

188,22~) ( (I.59) 179.548 (0.75) --61 .~q . . . .~q 11.88

-I).845 (0.51) 1). 150L)O...... (1.84) 19.434 (3.09) - 14.545 ( 1.65) 441.1711 (I.55) --

I)EI+E(_" (Departmentsol Electrical Eng.) DMECH ( Departments of Mechanical kng. ) Cost

--

1.061 (11.63) 1). 161)58 ...... (1.93) IS.95 (2.97) 12.11 (-1.37) --

(I.32)

RES,

280.6t1 i529.41) 4508.98 (10.306.74) 79.23 ( 159.51 ) 49.78 (192.78) 1.8~ (11.37) --

(Dcpartmcnts ol Civil E n g i n e e r i n g )

I)CIVII.

O U A 1_

DOC'I

M ASF: RES

MAST: DOCT

UND:RES

U.

~.

Q .~

~.

2; ~. '~ ~"

',~ ,~

"~-

134

Economics of Education Review

Model 3 in Table 1 was estimated to take into account the possible existence of structural differences in the production functions of individual departments within the social sciences. If department-specific variables indicate the existence of statistically different cost structures across departments within a given field (such as the social sciences), then an estimated quadratic model for the aggregated field will tell us little about the presence of multiproduct economies within the aggregated fields. Dichotomous departmental variables were entered for departments of economics, geology, history, political science, psychology, and sociology, Departments of anthropology were omitted. When these departmental variables were introduced m Model 3, t-tests on the coefficients of the additional quality and departmental dummy variables in the model were shown to be statistically insignificant for all the departments except history. There appear to be few differences in the underlying technology facing differing social science departments. Consequently, analyses of the degrees of economies of scale and scope in the social science departmental production of outputs based upon the use and results of thc original basic quadratic model were assumed to be appropriate for this study,

Physical sciences. The three models described above were also estimated for 98 physical science departments and are presented in Table 2. The physical science subsample contained departments in computer science, mathematics, chemistry. geology, and physics, with departments of statistics as the omitted variable. Not only were all three models for the physical sciences statistically significant, but each had relatively high explanatory power with adjusted R-squares of 82% or above. As with the case of the social sciences, adding quality and department-specific variables to the physical science field of departments did not add much to the explanatory power of the model, nor did they materially change the significance and signs of the coefficients. However, in the case of the physical sciences there do appear to be significant differences in departmental costs between departments of computer science, chemistry and physics and the other fields within the physical sciences. Thus, because of this variation in costs between departmerits, any subsequent findings based on Model 1, especially with regards to prospective economics of

scope in this field, will need to be appropriately tempered. In examining the coefficients of these models within the physical sciences, scale economies can bc said to possibly exist for UND, MAST and RES because of the negativc signs of their quadratic forms. Howcvcr, several of thc coefficients werc not statistically significant, indicating that it is promature and difficult to suggest much about the extent of scale economics with these models alone. Engineering sciences. The engineering departm ents were similarly examined with our thrcc models and are reported in Table 3. There were 61 departments in the engineering subsamplc from civil, electrical and mechanical engineering departments, with chemical engineering placed in the intercept for Model 3. As with the other two fields, the results for engineering continuc to indicate significance for the models and with high cxplanatory power. As with the other field, inclusion of quality and subfield departmental variables do not appear to affect the explanatory power, significance. or signs of the coefficients in the basic model. Moreover. departmental costs within the engineering field were not significantly different across most subfield departments, except for the lower costs with electrical engineering. Economies of scale were apparent with only the quadratic terms for master instruction and rcscarch, and then only master teaching was significant.

EstimatingMuitiproduct Economies Are there economies of scale and scope in the departmental production of teaching and research outputs in the three departmental fields of this study? In the following sections we estimate the average and marginal costs of our subsample fields along with more closelx examining their prospects for scale and scope economies. Average incremental and marginal costs. Estimates of averagc incremental (A1C) and marginal (MC) costs for each type of ot, tput at the sample output means in each of the three departmental fields utilizing our basic model arc presented in Table 4. From our reported results we can draw several important inferences with respect to costs of ot, tputs across all three fields. First, research outputs clearly have the highest costs among the four outputs across all three fields, followed next by

Departmental Productivi O' in American Universities master, undergraduate and doctorate levels of teaching in the social and physical science departments and by the costs of doctorate, master and undergraduate level teaching outputs in engineering fields. Generally, the lowest costs for all outputs were found to be in the social sciences. Conversely, the highest costs were generally found to be within engineering, Somewhat surprisingly, the costs of providing instruction at advanced levels (particularly at the docto,al level) were not necessarily the highest in cach field. This can be partly explained by the dual role of many advanced level graduate students as an input for the production of teaching and research and as an output for graduate education. The effect of increased enrollments on departmental costs clearly depends on the ways that graduate students are utilized. A n o t h e r important observation is the relatively Im~ cost estimates for doctoral level SCH production in the social and physical science fields, since it is generally expected that doctorate level instruction will have the highest costs among all teaching outputs. On the other hand, m engineering departmcnts only the high costs of research ot, tputs exceeded those of doctoral level teaching, suggesting relatively high costs for doctoral education in engineering departments. An important difference in engineering could be associated with their utilization of graduate students in research rather than in teaching as in the social and physical sciences (see,

135

for example, Hauptman, 1986). lf graduatcstudents are extensively utilized as research assistants in engineering departments, we would expect that the cost complementarity between the two products would also vary according to their utilization by the departments. As we note in the following section, this cost complementarity does indeed appear to exist within engineering.

Cost eomplementarities. With respect to the joint supply of outputs, the measurement of any interaction betwccn thc outputs should provide some important insights into the possible existence of cost complementarities in the joint production of the outputs. For a twice continuously differentiable cost function, cost complcmentarities between two products i and j arc said to be present at v' !I (',i - ifi('(y')/i)y,Oyi ~ 0. i :4-j. The results reported in Table 5 illustrate the signs of c o s t c o m p l c m e n t a r i t i e s b e t w e c n the four outputs for each of the three departmental fields from ()tit study. While positive signs imply weak complementarities, negative signs (indicating reductions in costs) imply possible cost complementarities and the possible existence of joint supply effects ill the production of mulitiplc outputs. The sign of the cost-complementarity between outpt, t Yi and output Yi is indicated by the sign of (', (id-('(y)/ig',i~y,). For the fot, r-output nmdel reported in Table 5 wc

"Fable 4. Comparison of average incremental and marginal costs of individual outputs at the output mean Field',

Social sciences Ph\sica[ sciences Engineering

M(',

A/(',

M(',,

AI(',

-6. I4 ."~S ~4 . 23.55

2.60 . 55.14 32.39

_4_._0 "~ "~ "~ . 223.55 (}05.55

.41(',

216.30 "IS.()0 5117.40

M(',

I-..7_ ~ "~

-413)5

&~.fl>~ ~ 748.73

-46.72 785.( .r,

71~,5.'~1 . . . 171)54.53 24475.70

Note: A/(" = Average costs: M(" = Marginal costs: y~ = Undergraduate student credit hours: y, student credit hours: y~ = Doctorate level student credil hours: va = Published articles. "Fable 5. ('osl complemcntarities between departmental Fields

vly:

viv~

vl v

Social sciences Phxsical sciences F_nginccrmg

+ -

+ +

+

M('

A/(',,

=

. 5S61 ~0 Ifllt)9.00 I (~(149.()1) MaslLcr level

outputs

y ,v~

vev4

5 3'a

~

+ + +

+

Notes: The signs of cost-complementarity between output y, and output Yi are indicated by the sign of (',i = (fie(, v)/ av,0v, . and taken from the results reported in Tables I, 2 and 3. For o u r f o u r - o u t p u t model we have v . v~. y~ and y~ = LIND. MAST. DOCT and RES, respectively.

136

E c o n o m i c s o f Education R e v i e w

have 3,~, Y2, Y3 and Y4 = UND, MAST, DOCT, and RES, respectively, An important question is whether the utilization of doctoral students in the joint production of other outputs results in any efficicncies. For example, it is commonly understood that departments in the social sciences utilize large numbers of graduate students to teach their undergraduates, while departments in the engineering fields often utilize their graduate students more as research assistants (Hauptman, 1986:. Malancy, 1988). Do any of these academic departments achieve cost savings due to such joint production of outputs? Although such a theory has long been argued, there is little empirical support regarding cost complcmentaritics of such utilization of graduate ' students in the various disciplines. A study by Dc Groot and his colleagues (1991) argued that such cost complementarity existed between graduate education and undergraduate education when viewed with aggregated institutional level data. However, little is known about the joint supply effects that might actually exist for differing disciplincs and departmental fields across similar types of institutions. This question can bc directly addressed by examining the interaction terms of the cost functions found in Tables 1, 2 and 3 for each discipline or set of departmental fields in our study, Utilization of doctoral students as undergraduate teaching assistants is indeed a widespread phcnomcnon in most research universities, but its "'costeffectiveness" surprisingly has never been systematically examined at the departmental level in field specific terms. The negative sign of y~y~ in Table 5, for example, illustrates that there does exist a joint supply effect for producing undcrgraduatc and doctoral students in thc social sciences. Producing these two outputs jointly provides cost savings for departments in the social sciences. These findings are consistent with similar findings from previous studies ut the institutional level (Cohn et al., 1989: Dc Groot el al., 1991). De Groot el al. (1991) explained that such cost savings were obtained by employing graduate stt.dents as teaching assistants at relatively low prices (p. 427). However, the present study did not find such cost complementarity between undergraduate teaching and doctoral teaching in the physical sciences and engineering fields, suggesting that these departments do not usc graduatc students as much as social science departmcnts in the production of undergraduate teaching,

Regarding the general assumption that graduate education is complementary to research, as measured by the interaction term (i.e., Y3Y4) for DOCT and RES in Table 5, our findings indicate that such complementarity does not exist with respect to costs between the two outputs in social science departments. The sign is both positive and statistically significant. On the other hand, such complementarity was found in the physical sciences and engineering. These findings are important because it is often assumed that the marginal cost of producing one product decreases when the quantity of the other product is increased. The reason for this assumption is that faculty can use graduate students as research assistants. Hoenack et al. (1986)even make the argument that higher quality students may enhance the research productivity of the faculty due to student participation in their research activities (p. 345). Although both Cohn et al. (1989) and Dc Groot et al. (1991) reported similar findings of cost complementarity between production of research and graduate instruction at the institutional level, they did not differentiate by fields. The findings in this study indicate that such cost complementarity is field specific and not true across allfiehls. In social science departments faculty apparently prefer to produce research more by themselves and to utilize fewer graduate students in the production of their research output. The engineering or physical science departments, on the other hand, employ closer collaboration between faculty and graduate students in their research production. Production of doctoral students and research output jointly in these two latter fields results in cost advantages. Table 5 also provides some evidence with respect to the joint production of undergraduate teaching and research (i.e. YIY4). Although there was no statistical significance associated with any relationship between undergraduate teaching and research, the negative signs of the interaction terms between undergraduate instruction and research output in the social sciences and engineering suggest that there may be some weak (nonsignificant) complementary with respect to these costs. These findings are contrary to the general expectation that there would be no complementarity between the two outputs. Ray- and product-specific economies of scale. From the estimated coefficients of the cost functions reported in Tables 1-3, we computed the degrees of

Departmental Productivity in American Universities both ray- and product-specific economies of scale at given points of production for the output sets. Table 6 reports on these estimates for the academic departments in our three disciplinary fields producing at their sample output means. Since a multiproduct cost function behaves best at the point of mean approximation in an output sample, the sample output means are used to calculate our returns to scale. Ray-economies of scale for the social sciences, physical sciences and engineering departments all were estimated to be highly positive and greater than one (i.e., 3.917, 1.689 and 2.654, respectively, at their sample output means). These findings suggest that ray-economics of scale exist for the "'typical" department operating at the sample product mean and that cost advantages would accrue from producing more output in fixed proportions. Such academic departments producing at their output means have clear incentives to expand their production of outputs to exploit existing potential scale economies, While wc can report that product-specific economits of scale in the social and physical sciences exist for all their products in Table 6, productspecific economies of scale in engineering departmcnts exist for only master level teaching and research outputs. Product-specific economies of scale for outputs of doctoral and undergraduate levels of teaching are most prommnced in the social science departments. Increasing these outputs independently from other outputs in the social sciences will result m an increase of efficiency. However, in engineering departments we do not find any apparent product-specific economies of scale for the outputs of undergraduate and doctoral levels of teaching. Although there can bc economies associated with ray- and product-specific scales of operations, there also may, bc economies associated with the cornposition of outputs, which measures whether there arc cost adwmtages associated with the simultancous production of many products. As is evident

137

from our discussions on cost complementarities, production in higher education typically exhibits joint production since a number of products arc produced jointly and the costs of production arc not allocated based on the types of outputs (e.g., the number of students at each level and research). Thus, it is often argued thai one advantage of American higher education compared with the European higher education model is the presence of cost advantagcs clue to the joint production of teaching and rcscarch output. Consequently, it is desirable to estimate the degrees of economics of scope that measure the effects of joint production upon costs. This can be empirically tested by examining both global and product-specific coonomits of scope in the production of teaching and research outputs. Global economies of scope. Global economics of scope in higher education production suggest tha! a single institution or department can produce a given btmdle of output (i.e., UND, MAST, DOCT, and RES) in a less costly manner than can specialized teaching or research institutions, each of the latter agencies producing smaller outpt, t levels with the same proportions. In the four-product case of this study, global economics of scope exist with respect to the product scts o f y , , 3'2, y ~ , a n d y 4 ( i . e . , UND, MAST, DOCT, and RES, respectively) if

C(yt, Ye, Y~, Y4) <- C(v , 0, 0, 0) + ('(0..w, O, O) + ('(0, 0, 3'> 0 ) + ('((), 0, 0. y4). The degree of global economics of scope ,S), for our study can be noted as

('(YL, ()- (), 0) + ('(0, Y2, 0, 0) + ('(l), I), y~, ()) + ('((), 0, (), 3'4) C(TI, Y2, ),'~, 3'4) ('(Yl, Y2, Y.~,YD Estimatcd degrees of both global and productspecific economics of scope calculated at the sample

Table 6. Comparison ol degrees of economics o1 scale at 111,2output 111ciin5. F,conomics of scale Prt~duct-spccific ccononlics [:iclds

Ray economics

UNI)

MAST

I)()(T

RES

Social sciences Physical sciences Engineering

3.917 1.689 2.645

2.345 I.(15b 0727

1. 119 1.025 I. 193

2.949 1.78t) 0,953

1.217 1.34S 1.525

Economics o f Education Review

138

means for each departmental field are presented in Table 7. The results indicate that for all of the departments there appear to be marked economies of joint production from combining the production of teaching and research. For typical academic departments producing all four outputs at public research universities, we can estimate that there are global economies of scope in all three departmental fields. The &;s in all three fields are estimated to be positive and the costs of producing UND, MAST, DOCT, and RES simultaneously are smaller than the costs of producing them separately. These findings at a departmental level of analysis for the production of teaching and research are consistent with earlier findings at the institutional level reported by Cohn etal. (1989) and De Groot et al. (1991). It is interesting to note that Verry and Layard (1975) and Verry and Davies (1976) found no statistical evidence of economies of scope in their analysis of the joint supply effect of production for teaching andresearchoutputsin British universities. Nelson and Heverth (1992) presented largely inconclusive results with respect to their examination for the possible existence of economies of scope in a single American university, Product-specific economies of scope. Since the departments examined in this study produced various levels of differing outputs and appeared to have product-specific economies of scale associated with most of these outputs in all three fields (with undergraduate instruction and doctoral training in engineering being the only exceptions), it is important to ask whether economies of scope are also associated with the production of each type of output. Suppose, for example, that some departments were to specialize and only produce research (i.e., RES) and others were to produce only a combination of all teaching outputs, could we expect there would be cost savings from such specialization? The answer, of course, could be found if we

were to control for such a setting in the real world, but such settings do not exist within most public universities within the United States. Alternatively, the question can be addressed by estimating the product-specific economies of scope associated with the production of research output. For example, in the case of the four outputs of this study, product-specific economies of scope associated with yl exist if

C(yl, y> y~, Y4) <- C(yj, 0, 0, 0) + C((}, y_,, y~, y~) and the degree of product-specific scope economics for our study can be noted as SC,q =

C(yl, 0, 0, 0) + C(0, Y2, Y3, Y4) - C(yl, y:, y.> Y4) C(yl, y,, y3, y4) where sC, q represents the economies of scope for YL. Cost advantages accrue to the department producing y~ jointly with another product if SCvl >- O. Table 7 reports findings for the estimated product-specific economies of scope for the typical departments in all three fields at their sample output mean. We find that all the estimates for productspecific economies of scope are positive, implying that there are cost savings to producing each output simultaneously with the other three outputs in all three fields. These findings largely result from the persistence of common departmental fixed costs in the departmental production of teaching and research. Although the presence of several cost complementarities were previously puzzling because of the unexpected signs of the interaction terms, the existence of economies of scope suggests that there are indeed complementarities in the production of teaching and research output jointly. A particularly important finding for policy purposes is the exist-

Table 7. Comparisons of economies of scope at the output means

Fields

Global economics

UND

Social science Physical science Engineering

(I. [25 1.348 1.2X3

0.899 0.404 0.566

Economies of scope Product-specific economies MAST DOCT 0.620 0.451 0.230

(I.726 0.485 0.353

RES 0.531 0.396 0.381

Departmental Productivity in American Universities ence of product-specific economies of scope for the production of research output jointly with all the other teaching outputs. This suggests that. at least at the output mean for our sample of departments, it would be advantageous to produce research output simultaneously with the three instructional outputs, An important shortcoming of all these findings is the fact that these estimates are completed for only the sample output mean. Although these findings have some policy implications for "'typical" acadcmic departments at the sample output mean, they do not allow us to predict the extent to which economics of scale or scope might be prescnt at other levels of production in smaller or larger departments. This requires the examination and analysis of a range of differing production levels. Degrees of Economies of Scale and Scope al Differing Levels of Production Our previous analysis focuses on the cost behavior of "'typical departments" operating around sample means. Since these departments have wide variability in their size and range of production, it is useful to analyze the cost behavior of departments at the extremes of the sample and estimate their degrees of scale anti scope economies. For this purpose we simulated various levcls of outputs, ranging from l0 to 400% of the sample output means. The results are presented in Table S and reportcd in three subsections representing the three departmental fields of study (i.e., the social, physical, and engineering sciences). We recalculated thc multiproduct economies of scalc and scope for thc departmental production of teaching and research outputs at various levels of production using the estimated rcsults of Model 1 from Tables 1, 2, and 3. Ray-economies of scale. Ray-economies of scale were earlier reported to be greater than one and highly positive at the sample mean for all the departments within all three fields. Now, from Table 8 wc discover that ray-economies of scale have also appeared to be greater than onc and highly positive for all the departments at the lower levcls of their output means. This finding suggcsts that increasing output levels proportionally for small departmems would increase efficiency in the production of teaching and research, assuming, of course, that we hold the quality levels of the outputs constant,

139

Table 8 also indicates the existence of diseconomies of scale at some upper departmental level of production. Ray-diseconomies of scale appear to emerge at around 3tX)% of the sample output means for most departments regardless of field. The principal implication from such a finding is clear: The reduction in size of some very large depart ments in public research universities would likel+~ lead to increased cost savings. This is obviously a controversial finding since most conventional thinking abouthigher education suggests that there would continue to be economies of scale even at very large output levels, assuming that the quality of outputs does not change.

Product-specific economies of scale. Table 8 also presents estimates for product-specific economies of scale at various levels of production for the three fields. Our estimates indicate largely mixed results across the three fields. In the production of social science departments, for example, product-specific scale economies for both teaching and research appear to exist at all levels of production within the relevant ranges examined. Increasing production of these outputs at all levels independent from the other outputs would result in an increase in efficiency. In the case of undergraduate teaching in the social sciences our findings indicate product-specific diseconomies of scale at lower levels of output, with economies of scale for undergraduate instruction appearing at around the output sample mean and extending to all applicable higher levels of production. As a consequence, in low production (i.e., at less than the sample output mean) social science departments any increase in production of undergraduate teaching should be undertaken only with a similar increase in other products. This dependence of undergraduate teaching on other outputs to increase cost-effectiveness can be explained partly bv the relationship between the production of doctoral and undergraduate teaching outputs. In fact, at the lower levels of doctoral output (e.g., at 10% of the output mean) there are also productspecific discconomics of scale. Thus, these outputs would need to be increased simultaneously and proportionately in order to increase product-specific scale cconomies in the social sciences. The cost behavior of the physical sciences profiles a different picture with respect to the presence of product-specific economies of scale. We found

I 1.371 2.777 1.6:49 1.319 1.13() 1).93(~ ().:435

~ "~ I _4.__ 4.97 t) 2.645 I.:491 1.523 1. 165 ().991

I0% 50 l()0 O u t p u t mc,m 1511 21111 30o 40(1

111",, 50 11)() ( ) u t p u t m e a n 150 2()0 3(1(I 40(1 11.9:44 0.91)2 11.727 . 11.~2¢~ 1.542 2.433 I.:404

1.006 1.03(1 I.II56 1.077 I .()95 1. 124 1.147

0.977 0.767 2.345 1.412 1.306 1.243 I.__0 -'-'

UND

1.03:4 I. 133 1. 193 1.227 1.248 "~ 1._7,', ] .2t)()

1.1)03 1.015 1.1125 1.1133 1.113:4 1.1)47 1.052

1.010 .056 .119 .193 .277 .494 .:412

MAST

().t)t)3 ().972 11.953 ().t)411 0.931t (I.916 ().9()7

1.037 1.244 1.7:49 4.1)6~ 5.957 0.629 -(I.17:4

0.689 I 1. 132 2.949 2.533 2.38:4 -1 -) _._67 2.214

DOCT

Idngincering D e p a r t m e n t s 1.1166 2.744 1.296 1.951 3 ' 1.525 1._:43 11.8__ 1.706 '~'~ I.:454 11.4:43 2.():41) 11.1t211 _._44 0.2:40

11.917 11.69:4 (D.566 11.5(1:4 0.4:47 0.4:49 11.514

11.911 0.63:4 0.4114 () . ~'~ ... 0.106 -(I.1181 -0.213 '

Physical Science D e p a r t m e n t s ~ 1.006 _.73:4 1.029 1.963 1.348 I .()~_ -'~ I.II71 11.939 1.0:47 1L645 1. 112 -0.249 1.129 ().()(16

UND

11.911 0.5:41 11.23() 11.111",0 -0.3113 -tl.6:41 --().961

0.912 11.655 0.451 0.316 11.219 (~.119(1 11.1101~

(L930 0.745 0.620 11.552 0.514 0.501 0.601

MAST

11.913 1t.624 11.353 11.14:4 11.1tl3 ~ [ (1._, 11.419

11.913 0.666 O.4:45 (k374 0.31111 o . _~1 _ ~ 11.163

11.932 0.776 0.726 0.76:4 0.:483 1.398 2.:447

DOCT

1

1t.914 0.634 11.3:41 11. 191"~ 11.053 - 0 . 153 () -~t),q

-().251

O.Ol

(1.911 11.635 1t.396 11.221 (1.087

0.929 I).719 0,531 0.369 11.202 -0.255 1 . _ )-~t _ -~

RES

P r o d u c t - s p e c i f i c e c o n o m i e s of scope

0.934 0.827 [I.:499 1.121 1.4:4:4 2.:464 6.517

Global economies of s c o p e

Social Science D e p a r t m e n t s 1.023 0.(}18 I. 114 0.037 1.217 0.125 1.310 0.256 1.396 0.437 1.545 I.{)61 1.672 2.65:4

RES

P r o d u c t - s p e c i f i c e c o n o m i e s of scale

Note: LIND = U n d e r g r a d u a t e s t u d e n t c'rcdit hc, urs; M A S T = M a s t e r level s t u d e n t c r e d i t hours: I ) O ( ' T = D o c t o r a t e level s t u d e n t credit botH's: R E S = T h e n u m b e r of articles.

14.833 4.590 3.917 5.135 1:4.913 I.:414 -11.44:4

Ray economies of scale

I 0'~,'~, 50 I00 O u t p u t m e a n 150 200 ~00 41)0

P e r c e n t a g e of output means

T a b l e 8. D e g r e e s of e c o n o m i e s of scale a n d scope for a l t e r n a t i v e o u t p u t b u n d l e s

~." ,~ t":l ~. = ~"~ ~. "

"

4a

Delmrtmemal Productfi'i O' in American Umversities evidence of product-specific economics of scale at all ranges of production in undergraduate teaching, master teaching, and research outputs. The production of doctoral teaching, on the other hand, indicates product-specific diseconomies of scale arising after production exceeds around 200% of the output mean. Engineering departments also have differing cost structures with respect to product-specific economits of scale. In these departments we find that product-specific economies of scale exist at all levels of production only in the production of master level teaching and research outputs. Undergraduate teaching, on the other hand, indicates diseconomies of scale at the lower levels of depamnental ot, tpt.t with economics of scale only appearing at around 200% of the output sample means. We do not find any product-specific economics of scale at any level of output for the production of doctoral level teaching m engineering, Global economies of scope. In our earlier analysis we found both global and product-specific cconorates of scope at the sample outpt.t means aooss all three of our departmental fields, indicating the clear existence of efficiencies resulting from the joint production of teaching and research outputs. However, the question is what happens when departmcnts get smaller or larger? Do the}' still share the same inputs for the joint production of teaching and rescarch, or do they simply hire nex~ inputs for the new levels of outputs? The simulated results in Table 8 gives us some insight into these questions for ot, r departmental fields. In all three fields there appear to be consistcnt global economies of scope at all levels of production below approximately 300% of the samplc means, suggesting again that there arc sharcable inputs m the utilization of resources for the joint production of teaching and research outputs in most fields. Thc apparent existence of these global economies of scope suggests that for a typical department operating within 200 to 300% of the sample mean. the cost of producing all teaching and research outputs simultaneously is less than the cost of producing them separately. On the other hand, global diseconomies of scope do begin to appear at some point beyond 300% of the mean for both the physical sciences and engineering. Apparently, at somc point of vcrv high production most departments hire ncx~ inputs to prodt, ce specific outputs,

141

The implications of such findings is that at very high levels of outputs most departments will begin to obtain new efficiencies from producing teaching and research outputs separately. Product-specific economies of scope. Although we previously found product-specific economies of scopc at the sample output means for all four products across our three fields, when we examine this effect more closely by varying the levels of output we find somewhat differing results. As in the case of product-specific economies of scale, when we vary output levels we find economies of scope differences across the departmental fields, In social science departments, for example, we note that economies of scope are particularly high for undergraduate teaching and there exist increasing economies of scope at higher output levels. Similarly, product-specific economies of scope exist for both mastcr and doctoral at all relevant levels of instruction and for most relevant levels of research production, suggesting that it would be highly advantagcous to produce the three teaching outputs and research jointly. In the production of physical science outputs, we found product-specific economies of scope for all outptHs at all levels of output up through 200% of the output means. Diseconomies of scope set in only for undergraduate instruction and research at around 3(t0% of the means. In engineering, product-specific economies of scope are clear for all outputs up through the sample means, and for undergraduate instruction throughout all the relevant levels of production. On the other hand, product-specific diseconomies of scope set in for engineering departments for both forms of graduate education at around 150 to 200% of the samplc means and at around 300% of the means for research products. CONCLUSIONS Our principal concern in this study was to examine whether economies of scale and scope existed in public research universities at the departmental level, and if they did exist, to what extent and with what effect. Results from our estimates using multiproduct quadratic cost functions suggest a number of important and practical implications for policy makers at both institutional and system levels.

142

Economics of Education Review

First, our results indicate that it is possible to examine the cost structures of American universities and their departments through the use of a multiproduct quadratic cost model. Our model had high explanatory power and illustrated material differences in structural form (i.e., production technology) for producing teaching and research outputs across differing disciplinary fields. Inclusion of departmental quality and subfield department measures did not appear to affect the explanatory power of the basic multiproduct quadratic model, Although the quality of departments in our sample did not have a statistically significant effect on the costs of academic departmental production, this does not necessarily suggest the insignificance of output quality. Rather, it should be interpreted as more likely resulting from our relatively homogeneous sample of universities, since all the institutions in the sample are comparable public research universities, Second, our findings suggest a number of generalizations about costs within public universities. The findings include the propositions that (a) average incremental and marginal costs are generally highest for research outputs and lowest for undergraduate instruction, (b) of the fields we examined, the social sciences generally have the lowest costs across almost all categories of outputs with engineering departments generally having the highest costs, (c) although costs generally follow levels of instruction, there are some notable exceptions, and (d) most departmental outputs have shared resource use and costs, Although conventional wisdom believes, and most previous cost studies have found, that the costs of instruction will necessarily increase by level (e.g., graduate education costs more than undergraduate with doctoral education costing the most), the findings in our study indicate that these assumptions do not hold for every field. Advanced education is not always the most expensive across all departments. We found, for example, that in the social sciences doctoral training was less costly at the margin than master level education. Again, we found that production technology differs across field groups, Our findings indicate that the estimated departmental-based marginal costs of research output may be much lower than previous estimates (De Groot et al., 1991). On the other hand, our findings also

argue that excluding the costs of departmental research output from any institutional cost study will necessarily result in incorrectly estimating the costs of other outputs due to their shared costs. Studies focusing only on the instructional costs of higher education as they might design financing and tuition policies for universities, for example, need to be especially careful. By the same token, our inability to find measurable dimensions for departmental service outputs and our omission of these costs t, ndoubtcdly has resulted in some overestimation of our instruction and research costs. If, for example, a department might have a larger service component in its mission and production of outcomes, these activities would clearly use resources that we have otherwise charged to instruction and research production. Third, our results indicate the existence of both economies of scale and scope at departmental levels, and that these economies differ by field group but not necessarily by departments within field groups. Most departments within public research institutions will enjoy efficiencies with the expansion of both their teaching and research outpt, ts at most relevant levels of production due to the presence of rayeconomics of scale. While all departments in the study operating at an output range smaller than the sample output mean will enjoy cost savings duc to an increasc in the level of their outputs, rayeconomies of scale do exhaust at some point with large increases in levels of production. Productspecific economics of scale also exist at most relevant levels of production for both teaching and research outputs across most fields, but for engineering we did not find such economics for the outputs of undergraduate teaching at several low levels and for doctoral teaching at any level. Global economies of scope were found across all the field groups for most relevant levels of production. Only when the joint production of all outputs was expanded to over 300% of the typical department mean in the physical science and engineering fields did global economics of scope turn negative. Similarly, product-specific economics of scope were found for all outputs across all fields for their .joint production up to at least 3(10% of the sample means in all cases, except for graduate education in engineering. Findings in the study clearly indicate that there are cost advantages associated with the joint production of departmental teaching and research.

D e p a r t m e n t a l P r o d u c t i v i t y in A m e r i c a n Universities

These economies of scope arise from the joint utilization of faculty, administrators, support staff, and equipment and services for the production of both teaching and research outputs. However, global economies of scope are ultimately exhausted at some point at very high levels of output production. Nevertheless, the findings in this study support the commonly understood (but seldom cmpirically verified) notion that the departmental

143

costs of producing undergraduate teaching along with graduate education, and the costs of producing graduate education along with departmental research, are significantly less than producing them separately. Nourishing specialized and free standing research centers outside and independent of departmental instruction will likely result in less efficiency than in the joint production of the same outputs in American public universities.

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