Untitled-2 406 College & Research Libraries September 2002 406 The Cost Function and Scale Economies in Academic Research Libraries Lewis Guodo Liu Lewis Guodo Liu is an Associate Professor in the Newman Library at the Baruch College of the City Univer- sity of New York; e-mail: lewis_liu@baruch.cuny.edu. The author would like to thank Dr. Richard E. Quandt, an emeritus professor of economics at Princeton University, and Dr. Robert M. Hayes, an emeritus professor of library and information science at UCLA, for their valuable comments and constructive suggestions on an earlier draft of this article. The author alone is responsible for all possible errors. This empirical research examined scale economies of academic research libraries that belong to the Association of Research Libraries and devel- oped a total cost function for estimating economies of scale. The author argues that libraries in general, and academic research libraries in par- ticular, are information provision organizations that provide multiproducts and multiservices and points out that some previous studies that used the production function have limitations due to the fact that this function only permits a single-output variable. This investigation incorporated a wide range of collections and service output variables into the total cost function. The regression results show that the R square of the cost func- tion model is .8142 and that the coefficients of three very important out- put variables (volumes held, serials, and group presentations) are sta- tistically significant at high confidence levels. The findings of this re- search show that the function coefficient is .93, indicating that slight economies of scale exist in academic research libraries. ibraries are important eco- nomic entities in modern soci- ety. To date, there are 9,046 public libraries, 3,685 academic libraries, 98,169 school libraries, 9,763 spe- cial libraries, 1,376 government libraries, and 335 armed forces libraries, with a to- tal of over 122,300 libraries in the United States.1 Improving efficiency and prevent- ing misallocation of resources are as im- portant in libraries as they are in other economic sectors of the national economy. One way to evaluate an organization’s efficiency is to examine whether scale economies exist in the organization. The concept of scale economies is rooted in economic studies of manufac- turing. Massive production of a product makes it possible for a firm to increase output and in the meantime reduce aver- age cost to the extent where the firm needs to hire more people, purchase more equipment, and rent more facilities. Economies of scale exist because of many factors. New technologies and specializa- tion of knowledge enable staff to work more efficiently. Purchasing a large quan- tity of supplies at a discount price also can save a bundle. Diseconomies of scale exist when output increases and average The Cost Function and Scale Economies in Academic Research Libraries 407 cost increases as well. Unskilled labor, outdated technologies, and poor manage- ment decisions can contribute to diseconomies of scale. Constant returns to scale occur when neither economies of scale nor diseconomies of scale exist. In many cases, researchers look at the rela- tionship between marginal cost and av- erage cost to determine whether excess capacity exists. If marginal cost (cost of producing an additional unit of output) is less than average cost, the firm is still better off producing additional output. Economic researchers have long been concerned with improving efficiencies of firms. Econometric models, such as pro- duction function, total cost function, av- erage cost function, and marginal cost function, have been developed to measure economies of scale and to improve effi- ciencies of firms as well as government agencies and nonprofit organizations. Although many of the early research ef- forts focused on manufacturing and in- dustries such as railroads, metals and machinery, aircraft, gas, coal, telephone industries, and so on, later research be- gan to focus on government agencies and educational institutions. 2–4 In the past, a few studies were conducted to examine economies of scale of libraries. Much of the previous research focused on public libraries. A handful of studies dealt with scale economies in academic libraries. Researchers wanted to know whether li- brary size makes a difference in terms of cost savings. They tested the assumption that larger libraries tend to operate more efficiently than smaller ones. Studies on scale economies have strong implications for public policy-making. The evidence that shows cost savings as a result of economies of scale can per- suade policy makers to make decisions to consolidate smaller libraries, although in many cases convenient service to local communities can be a more important factor than economic consideration.5 This investigation focused on academic re- search libraries that belong to the Asso- ciation of Research Libraries (ARL). The purpose of this investigation was to ex- amine whether economies of scale exist in academic research libraries. If scale economies do exist, academic research li- braries should find better ways to fully utilize their human, physical, and finan- cial resources and services to provide bet- ter and more information services for their institutions’ faculty and students. The missions and goals of academic research libraries are different from those of general academic libraries and public libraries.6 For example, academic research libraries have a stronger research orien- tation than general academic libraries that focus on teaching. To support their uni- versities’ mission for research, academic research libraries develop comprehensive collections that include a large number of research journals. Research journals con- sume a large portion of a research library’s budget. Compared with aca- demic research libraries, general aca- demic libraries have only a limited num- ber of research journals and general pub- lic libraries have virtually no research journals at all. This and other differences between academic research libraries and general academic libraries and public li- braries affect output factors in a cost func- tion and warrant a new investigation. Literature Review The previous empirical research pro- duced mixed results. Some researchers found evidence of increasing returns to scale. Others found evidence of decreas- ing returns to scale. Still others found evidence of constant returns to scale. The research on scale economies of libraries may be classified into three categories: studies that examined scale economies using the Cobb-Douglas production func- tion; studies that looked at scale econo- mies as well as elasticities of input sub- stitution using the translog cost function; Improving efficiency and preventing misallocation of resources are as important in libraries as they are in other economic sectors of the national economy. 408 College & Research Libraries September 2002 and studies that explored scale economies using general cost functions. Early Studies The cost study on libraries commissioned by the National Advisory Commission and conducted by William J. Baumol et al. in 1969 and another study by Stanley W. Black in the same year are probably the earliest studies on costs and econo- mies of scale of libraries.7,8 Baumol et al. studied various types of libraries and ex- amined total expenditures on staff and library materials, volumes owned and circulated, and growth rates. The report provided a library cost trend analysis and showed that library cost per capita and circulation unit cost decreased as public library size increased, indicating econo- mies of scale, but also pointed out that savings were not expected to be substan- tial.9 Baumol and Matityahu Marcus later studied costs of academic libraries, which led to the publication of a book in 1973.10 No production function or cost functions were dealt with in their book. Studies Using the Cobb-Douglas Produc- tion Function Stanley W. Black used the Cobb-Douglas production function for public libraries and treated circulation as the sole output and labor and book stock as inputs. The coefficients of the two observed variables were not statistically significant due to a high degree of multicollearity between them. As a result, he assumed that returns to scale were constant and was able to estimate labor and book stock elasticities, which were 0.833 and 0.167, respectively. Black’s study seemed to have a far-reach- ing influence on the later studies because the methodologies used by the later stud- ies tended to resemble his. Haynes C. Goddard studied more than a hundred public libraries in Indiana us- ing the Cobb-Douglas production func- tion.11 In his study, circulation was treated as output and book stock, labor, materi- als, and capital were treated as inputs. Labor was measured based on circulation staff hours, and materials were measured in terms of their values by annual expen- ditures on them. Capital was measured by furniture associated with library ser- vices and circulation, such as tables, chairs, card catalog, and so forth. He found that the function coefficient was 1.076, indicating slight increasing returns to scale. In addition, from a subsample, Goddard estimated that the marginal costs of circulation were lower than aver- age costs, an indication of excess capac- ity in the libraries.12 Robert M. Hayes used the Cobb-Dou- glas production function to study both public and academic libraries.13 He (1979) studied optimal use of labor and capital by applying the Cobb-Douglas produc- tion function to large public libraries in California as well as in Illinois, Ohio, Wis- consin, and Missouri. He regressed cir- culation, a measure of output, on capital measured by the size of collection and staff needed to maintain the facilities, and on labor measured by service staff. About 60 percent of the total variance in the de- pendent variable were accounted for by the Cobb-Douglas production function. Although Hayes used a single-output measure, he recognized that other output factors, such as reference service, could also be considered. However, he pointed out that reference statistics available at that time might not be consistent and re- liable. In his 1981 study on the use of li- brary collections as measured by circula- tion and in-house use, he concluded that circulation did not adequately measure the usage of library collections. In 1983, he and Harold Borko published an article examining the relationship between li- brary collections and faculty productiv- ity using the Cobb-Douglas production function. The findings showed that li- brary collections contribute significantly to faculty productivity as measured by faculty publications. Studies Using the Translog Cost Function Like Black, Larry DeBoer and Chritopher J. Hammond examined not only scale economies, but also substitution elastici- ties of inputs.14,15 However, they used a The Cost Function and Scale Economies in Academic Research Libraries 409 translog cost function instead of a produc- tion function. The translog cost function can be used to deal with issues of the de- mand for inputs of libraries as well as scale economies. This method is flexible in approximating production technolo- gies in terms of input substitution elastisities. DeBoer used the translog cost function to examine economies of scale and input substitution elasticities of 194 Indiana public libraries. Book circulation was used as the output measure, and to- tal cost was treated as the dependent vari- able. The translog cost function required three input price variables for three in- puts (labor, books, and supplies/ser- vices). The labor price was calculated by total labor expenditures divided by total number of FTEs (full-time employees). He constructed a price index for supplies/ services based on wages and prices in each district county. The relative book prices were used in the estimated equa- tion. His findings showed that economies of scale existed for small libraries and constant returns to scale existed for large libraries. He also found that all produc- tion inputs were substitutes. Higher la- bor costs caused increases in purchases of supplies and services as well as books. Higher book prices led to an increase in purchases of supplies and services. But supplies and services were more respon- sive to higher labor prices than to book prices. Higher prices of supplies and ser- vices resulted in increases in purchases of both labor and books. In an analysis of the cost function for U.K. public libraries, Hammond also used the translog cost function to examine scale economies and input substitution elastici- ties. He found that increasing returns to scale and diseconomies of scope existed. Hammond also concluded that all pro- duction inputs were inelastic substitutes. What differentiated his study from some of the previous studies was that Hammond recognized the multiproduct nature of libraries and used annual book stock, audiovisual materials, and number of inquiries as outputs in the cost func- tion. Studies Using General Cost Functions Not all researchers were interested in studying substitution elasticities of in- puts. If the issue of the demand for in- puts is not the research concern, a gen- eral cost function is sufficient for studying scale economies. The research conducted by Kathleen Foley Feldstein, Michael D. Cooper, and Paul Kantor used the gen- eral cost function approach.16–18 A general cost function can be logarithmic or nonlogarithmic. Economic researchers often transform a cost function into a com- mon log form or a natural log form for the convenience of calculating the func- tion coefficient or for the convenience of developing a model that is a better fit of a data sample. Feldstein examined scale economies of public libraries using the national data and developed various cost functions. Be- cause she used circulation as a single-out- put measure, she was able to measure marginal costs from the total cost func- tion and average cost function using to- tal costs divided by circulation. She found that although library systems had small diseconomies of scale, some individual li- braries had economies of scale.19 Cooper examined whether economies of scale existed in public libraries as well as academic libraries. His 1979 study collected data from public libraries in California. He recognized the fact that libraries provide multiproducts and multiservices and re- gressed total expenditure on a number of output variables, including volumes added, volumes borrowed, volumes lent, reference transactions, and circulations. After testing five alternative models, lin- ear and nonlinear, logarithmic and nonlogarithmic, he concluded that the log-linear model was the best fit of the data. He found that the function coeffi- cient was slightly larger than 1 and ar- gued that it should be interpreted as con- stant returns to scale. Cooper’s 1983 study on academic libraries was similar to his 1979 study except that he added one more output variable—library hours opened. He found evidence of diseconomies of scale for two- and four-year public librar- 410 College & Research Libraries September 2002 ies and four-year private college and uni- versity libraries, and economies of scale for two-year private college libraries. But the R square of the model for two-year private college libraries was low (0.50). He warned that careful interpretation of this finding was necessary. Another noted author in the cost stud- ies on academic libraries is Kantor. In 1981, Kantor developed cost functions for library operations in scientific and tech- nical libraries. He also was aware of the multiproduct nature of libraries and thoughtfully included in-house material use, circulation, and reference queries as output variables. Kantor concluded that the best-fit model clearly demonstrated economies of scale. The previous studies used various econometric models. Some used the Cobb-Douglas production function, some used the translog cost function, and oth- ers used general cost functions. Where Cooper argued that “Both models (pro- duction function and cost function) are useful in determining whether scale economies exist,” Hayes, Cooper, Kantor, and Hammond recognized the multi-out- put nature of libraries and were able to incorporate various output variables into their studies whereas all the other re- searchers tended to use circulation as the sole indicator of output.20 DeBoer and Hammond used the translog cost func- tion to estimate input substitution elas- ticities in addition to scale economies. Goddard and Feldstein also examined the relationship between the marginal cost and average cost. There probably are a few reasons why the earlier research focused on public li- braries. First, the data on public libraries were readily available at the local, state, and national levels. Second, the policy in- centive for studying public libraries was stronger because consolidating smaller- sized libraries could lead to cost savings. Third, it was believed that production ac- tivities of public libraries could be mea- sured by a single-output indicator. It was convenient to use a production function model with a single-output measure. This Study This study argues that libraries in gen- eral, and academic research libraries in particular, are information provision or- ganizations providing multiproducts and multiservices. Their outputs are not ho- mogenous and cannot be simply mea- sured by a single-output indicator. Tra- ditional econometric methods, such as the production function, that can only be used to measure a single output are cer- tainly not a sufficient measure of the pro- duction of academic research libraries. The multiproduct and multiservice na- ture of libraries was not fully recognized by some researchers, and multi-output variables were not incorporated in some previous studies. Hayes, Cooper, Kantor, and Hammond are among the very few researchers who were able to incorporate some multi-output variables into their re- gressions. To be able to provide more accurate es- timates for library costs and economies of scale, this study took into account the multiproduct and multiservice nature of academic research libraries. This study is different from the previous studies in a number of ways. First, it dealt with the cost function and scale economies of aca- demic research libraries that belong to ARL. As stated in the introduction of this article, academic research libraries have different missions and goals from general academic libraries and public libraries. One of the important goals of academic research libraries is to support research in universities. This research orientation demands academic research libraries to have extensive scholarly journal subscrip- tions, which consume a substantial amount of library expenditures. This study treated serials as an independent- output variable. Second, this study treated general library collections as out- The value added to materials acquired and purchased by libraries lies in the fact that these materials can be readily accessed, retrieved, and used by library users. The Cost Function and Scale Economies in Academic Research Libraries 411 puts. A wide range of collections output variables were incorporated in the cost function. Third, this study added a new library service output variable to the cost function—group presentations by librar- ians. Such data were not available to the previous research. Fourth, this study treated circulation as one of the service outputs because circulation is only a ser- vice that helps users to check out readily available library items. Data, Measurements, and Model All the data used in this study were col- lected from the ARL 1999–2000 survey of 112 academic research libraries.21 The model in this study incorporated eleven independent variables and one depen- dent variable. Because not all libraries had the data related to all of these variables, eighty-nine academic research libraries were included in the regression analysis. A Multi-Output Measure versus a Single- Output Measure In the previous studies on public libraries, most of the researchers used circulation as the sole output measure. Circulation was used as a single measure of output for public libraries because it was believed that circulation could capture most of the us- age activities of public libraries and that the Cobb-Douglas production was conve- nient to measure library output. Others reasons were that data on some of the out- put variables at that time were unavailable. The data collected unsystematically were considered to be unreliable and inconsis- tent. In this study, the ARL data were used. The ARL has been gathering data system- atically from its members for many years, and its data set has been used widely by researchers, library administrators, and practicing librarians. The ARL data are believed to be reliable and comprehensive, although more detailed and more consis- tent data on academic research libraries need to be collected in the future. This study treated the library collections and various library services as outputs. It included various types of library collection materials and library services. Library col- lections include volumes, serials, maps, graphs, videos, and audios. Library ser- vices also are an important part of library outputs. They include reference service, library seminars and workshops, interli- brary loans (borrowed and lent), and cir- culation or information delivery service. Library collections were used as output measures for a number of reasons. The li- brary collections in this study were con- sidered to be final products of libraries. Although libraries do not directly create the contents of library collections, such as the contents of books and journal articles, and do not physically print these library materials, libraries do process them. The value added to materials acquired and purchased by libraries lies in the fact that these materials can be readily accessed, retrieved, and used by library users. Volumes Held versus Volumes Added as an Output Measure In this study, volumes held were used as an output measure instead of volumes added (which were used in two previous studies22) because volumes added only measure the costs of volumes added to collections for one time period, typically one year. The volumes-added approach may be appropriate in other studies, but for this study it was assumed that library users do not just use newly added vol- umes, they also use volumes purchased in the past. Maintaining existing volumes or entire book stock is an ongoing process and involves more staff time and effort and incurs more costs than newly added vol- umes and circulated items. The library collection management process generally includes assessing collections in terms of the needs of their patrons or communi- ties—identifying, selecting, acquiring, classifying, cataloging, shelving, or stor- ing all kinds of materials acquired and purchased by libraries. Many libraries as- sess their library collections in terms of age and subject strengths and weaknesses so that librarians can make adjustments to support teaching and research or to com- pare with other libraries for the collabora- tion purpose. Such an assessment requires 412 College & Research Libraries September 2002 searching and sorting entire collections and may take years to complete, but it is a necessary procedure to maintain relevant and useful collections. To put library col- lections in a library, library staff must go through this process. When library mate- rials are in place for use, library staff also need to frequently evaluate collections, weed out those that have few usable val- ues to make room for new purchased items, bind monthly and quarterly serials issues into annual volumes, repair dam- aged materials, replace missing items, and reshelve returned items. Other visible op- erating costs include electricity for lights and air conditioning. These costs are for entire collections, not just for volumes added in a year. The use of the volumes- added approach in this study might tend to have biased estimates on the costs nec- essary to maintain entire collections. In their recent study, Stephen R. Lawrence, Lynn Silipigni Connaway, and Keith H. Brigham made a convincing case that maintaining library collections is far more costly than one-time purchase costs using the ARL data.23 For example, they demonstrated that the unit cost for mono- graphs is $47.78, yet their life cycle costs are $343.03; the unit cost for serials is $590.97, yet their life cycle costs are $801.78; the unit cost for videos is $15.70, but their life cycle costs are $107.50; and the unit cost for government documents is 0 (free), but their life cycle costs are $55.40.24 Life cycle costs take into account ongoing expenses, including operating expenses, staff wages and salaries, building and facilities, and facility maintenance. All of these costs are for one purpose: to make library collections readily available for use. Because the vol- umes-added approach might produce bias estimates on costs of maintaining existing volumes, the volumes-held approach was used in this study. The use of volumes held as an output variable is similar to the use of book stock as an output variable. Volumes Held versus Circulation as an Output Measure The use of circulation as a single-output measure only considers the costs of those items that are checked out from libraries. But the fact is that whether library items are circulated is not very relevant from the total cost perspective because most of the costs of making them available al- ready have been incurred even before the items are checked out. The function of cir- culation service is simply to put readily available items or finished products in the hands of users. The cost of circulating a library item is only a small part of the to- tal cost of processing the item and repre- sents no more than the staff time and equipment involved in checking it out. Circulation was considered in this study as one of the service output measures. Another reason for using volumes held as one of output measures is that the use of circulation as an output measure for academic research libraries tends to ig- nore the fact that some parts of collections are not circulated and that users may use library materials in house. In-house use materials such as reference materials are not supposed to be checked out. The gen- eral library policy is that reference mate- rials must be used in the library. Many libraries do not have the financial and hu- man resources and a mechanism to con- sistently track the in-house use of refer- ence materials. Many academic research libraries also provide graduate students and faculty members with carrels where they can put the books for their learning, teaching, and research. Many users also use general library collections inside the library and do not check them out. Circu- lation records do not reflect the usage of these library resources. Volumes held as an output variable cover all in-house use of library volume materials. Serials as Output Academic research libraries have exten- sive scholarly journal collections that are important assets for learning, teaching, and research. Journals make up a large portion of serials. In general, journals in virtually every academic research library, such as reference materials, are not circu- lated items and are not recorded for use. Some libraries record current journal us- The Cost Function and Scale Economies in Academic Research Libraries 413 age but are unlikely to record usage of back issues on a regular basis because this would incur too much cost related to staff time and efforts. Some academic libraries track the use of current journals by requir- ing patrons not to put them back on the shelves so that library staff can record which journals have been used and how often. To track the use of back issues of tens of thousands of serials is extremely difficult, if not impossible. Back issues of journals are used a great deal by both fac- ulty and students because they contain accumulated research findings in the past and are indispensable for research and teaching. To study scale economies of aca- demic research libraries without includ- ing serials would miss a critical part of academic research library output. Serials collections consume a large portion of academic research libraries’ budgets and were treated as an important output vari- able in this study. Other Library Collections as Output In addition to volumes held and serials, other library collections, such as audios, videos, maps, and graphs, were treated as library outputs because they are dif- ferent kinds of materials from volumes and serials. The prices of these materials differ from those of volumes and serials and incur costs related to staff time and library equipment in order to make them readily available for use. In general, li- brary materials such as large-sized maps and some audio and video items may well be in-house use materials. Circulation records do not record the use of such li- brary collections. Although it is true that only a small portion of library collections is used at a certain time period, this does not necessarily mean that only that por- tion incurs costs. Library Services as Output Libraries provide various types of ser- vices including reference, instruction in the forms of workshops and seminars, borrowing items through interlibrary loan (ILL) for local users and lending items through ILL to external users, and circulation. These services incur costs in terms of staff time and facilities and equipment necessary to carry out these activities. Reference service can be mea- sured by reference transactions, and li- brary workshops and seminars are mea- sured by library group presentations. ILL and circulation data are also readily avail- able from the ARL data set. Library Costs Total library expenditures were used as a measure of total library costs for materi- als, staff, binding, and other operating ac- tivities. Library materials expenditures consist of costs for monographs and seri- als; materials such as maps and audiovi- sual items; and items other than materi- als such as bibliographic utilities. Total salary expenditures include those for pro- fessional staff, nonprofessional staff, and student assistants. The Model The general form of total cost function is the following: TC= f (V, S, U, D, M, G, P, R, B, L, C) TC, the total cost, is a function of a wide range of library outputs. The letters V, S, U, D, M, G, P, R, B, L, and C represent library collections and service outputs and can be written as: 1. TC=A Vβ1 Sβ2 Uβ3 Dβ4 Mβ5 Gβ6 Pβ7 Rβ8 Bβ9 Lβ10 Cβ11 Taking the natural log of both sides produces the following cost equation: 2. lnTC I = ln A+ β 1 lnV i + β 2 lnS i +β 3 lnUi+ β4 lnDi + β5 ln Mi+ β6 lnGi+ β7 lnPi + β8 lnRi +β9 ln Bi+ β10 lnLi + β11 lnCi+εi Where: i indexes individual institutions (i=1,…,N), N is the total number of observations, TC is the total cost, A is the constant, β1, β2, β3,…,β11 are the coefficients, ε is the statistical noise or the error term, It is specified that: A >0, β 1 >0, β 2 >0,…,β 11 >0 Library collections outputs: V is the total number of volumes held; 414 College & Research Libraries September 2002 S is the total number of serials; U is the total number of audio items; D is the total number of video items; M is the total number of maps; G is the total number of graphs. Library service outputs: P is the total number of group presen- tations; R is the total number of reference trans- actions; B is the total number of interlibrary loans borrowed; L is the total number of interlibrary loans lent; C is the total number of circulated items. Findings and Analysis Table 1 shows that the R square of the model is .8142, indicating that over 80 percent of the variance of the dependent variable can be explained by the model. The R square is high enough not to re- ject the model. The t statistics show that the coefficients of volumes held and se- rials, very important parts of library col- lections, are statistically significant at a very high level (.0001 and .0003, respec- tively). The coefficient of group presen- tations, part of library services, is also statistically significant. The coefficients of other variables are not statistically sig- nificant except for the coefficient of maps with a negative sign, which does not make any sense and needs to be disre- garded. As Hammond pointed out, “it is not practical to include all the identi- fiable dimensions of output. In addition, some dimensions may not be easily quantified.”25 With three variables that measure the important part of libraries’ collections and service outputs, the re- gression results should be considered satisfactory. To measure scale economies, the total coefficients of the independent variables are summed. If the function coefficient is larger than 1, diseconomies of scale exist. If the function coefficient is smaller than 1, economies of scale exist. If the function coefficient equals 1, constant returns to scale exist. The function coefficient (B 1 +B 2 +B 3 ,…,+B 11 ) is found to be .928597, or .93, TABLE 1 Regression Results, Dependent Variable is TC Variable Parameter Standard t Statistics Sig. Level Estimate Error INTERCEPT 5.134733 0.96568951 5.317 0.0001 In V 0.515291 0.10586920 4.867 0.0001 In S 0.285986 0.07618958 3.754 0.0003 In U 0.019001 0.02414557 0.787 0.4337 In D 0.030656 0.02543264 1.205 0.2318 In M -0.031324 0.01599503 -1.958 0.0538 In G -0.003305 0.01098661 -0.301 0.7643 In P 0.093980 0.04482646 2.097 0.0393 In R 0.024787 0.04541695 0.546 0.5868 In B 0.030396 0.04270257 0.712 0.4787 In L -0.042342 0.05273819 -0.803 0.4245 In C 0.005471 0.05031736 0.109 0.9137 R2 = 0.8142 Cases = 89 The regression results of this study also show that circulation is not a good measure of library output because it is not statistically significant. The Cost Function and Scale Economies in Academic Research Libraries 415 TABLE 2 Comparison between Studies on Scale Economies of Libraries Investigator Variable Coefficients Model Black, Inputs 1969, 1. Book stock 0.167 Log production function p. 595 2. Labor 0.833 Sum of the coefficients 1.0 Conclusion Constant returns to scale Goddard, Inputs Log production function 1973, 1. Book stock 0.486 p. 198 2. Labor 0.160 3. Materials 0.111 4. Capital 0.337 Sum of the coefficients 1.076* Conclusion Slight increasing returns to scale Feldstein, Outputs 1976 1. Circulation only. Total cost function Observed marginal cost Average cost function and average cost Conclusion The library system had diseconomies of scale, but some individual libraries had economies of scale. Cooper, Public Library Outputs 1979, 1. ILL borrowed 0.551 Log cost function p. 74 2. ILL lent -0.00058 3. Reference transactions -0.0062 4. Circulation 0.017 5. Volume added 0.467 Sum 1.028 (FY1975/76) Conclusion Constant return to scale 1983, Two-Year Public College Libraries ln cost function p. 216 Outputs 1. Volumes added 0.4218 2. Reference transactions 0.0921 3. Circulation 0.2705 4. Hours opened 0.5335 5. Interlibrary loan lending 0.0124 6. Interlibrary loan borrowing 0.0082 Sum 1.3 Conclusion Diseconomies of scale Two-Year Private College Libraries Outputs 1. Volumes added 0.3301 2. Reference transactions 0.0877 3. Circulation 0.1271 416 College & Research Libraries September 2002 4. Hours opened 0.1817 5. Interlibrary loan lending 0.0691 6. Interlibrary loan borrowing 0.0061 Sum 0.80 Conclusion Economies of scale Four-Year Public College and University Libraries Outputs 1. Volumes added 0.5554 2. Reference transactions 0.0957 3. Circulation 0.1374 4. Hours opened 0.3426 5. Interlibrary loan lending 0.0635 6. Interlibrary loan borrowing 0.0410 Sum 1.2 Conclusion Diseconomies of scale Four-Year Private Collection and University Libraries Outputs 1. Volumes added 0.5195 2. Reference transactions 0.1171 3. Circulation 0.2121 4. Hours opened 0.1706 5. Interlibrary loan lending 0.0708 6. Interlibrary loan borrowing 0.0359 Sum 1.1 Conclusion Diseconomies of scale Kantor, Outputs 1981, 1. In-house materials use 0.11 ln cost function Part II, 2. Circulation 0.32 p. 149 3. Reference queries received 0.32 Sum 0.75 Conclusion Economies of scale for scientific and technical libraries DeBoer, Inputs Translog cost function 1992, 1. Book stock p. 266 2. Supplies/service 3. Books Circulation Level 3,633 0.856 14,209 0.898 55,409 0.940 216,075 0.982 842,610 1.024 TABLE 2 (CONTINUED) Comparison Between Studies on Scale Economies of Libraries Investigator Variable Coefficients Model The Cost Function and Scale Economies in Academic Research Libraries 417 indicating that small economies of scale exist in academic research libraries. Comparing with Previous Research The studies on scale economies of librar- ies have reached different conclusions: diseconomies of scale, economies of scale, and constant returns to scale. The mixed findings should not be surprising for a number of reasons. First, studies were conducted on a wide range and diverse groups of libraries, including public li- braries, scientific and technical libraries, two- and four-year academic libraries, private and public college libraries, and academic research libraries. Second, the data were gathered at different levels. Some studies focused on libraries within one state, others gathered data from a re- gion or a number of states, and still oth- ers used the national data. Third, econo- metric models used in the studies vary from study to study. Some used the pro- duction function, others used the translog cost function, and still others used gen- eral cost functions. Fourth, the variables used in the models vary from study to study. Some used a single-output vari- able, and a few used multiple-output vari- ables. The production coefficient, which TABLE 2 (CONTINUED) Comparison Between Studies on Scale Economies of Libraries Investigator Variable Coefficients Model Conclusion Economies of scale for smaller-sized public libraries, but constant returns to scale for larger-sized public libraries Hammond, Outputs Translog cost function 1999, 1. Books 0.4682 p. 287 2. Audio/Visual Materials 0.0773 3. Inquiries 0.1303 Conclusion Increasing returns to scale Diseconomies of scope for the average British public library Liu, Outputs ln cost function 2002 Library Collections Outputs Volumes held 0.515291 Serials 0.285986 Audio 0.019001 Video 0.030656 Maps -0.031324 Graphs -0.003305 Library Service Outputs Group presentations 0.093980 Reference transactions 0.024787 Interlibrary loans borrowed 0.030396 Interlibrary loans lent -0.042342 Circulations 0.005471 Sum 0.928597 Conclusion Slight economies of scale for academic research libraries *The function coefficients do not add up to this sum, although the sum is close. 418 College & Research Libraries September 2002 measures scale economies, is very sensi- tive to the number of variables used and which ones are used. Table 2 shows such a diversity of studies on scale economies of libraries in terms of model, output and input variables, and findings. The findings of this study show that volumes held and total number of serials are significant output variables. The co- efficients of these two variables are sta- tistically significant at a very high level. A correlation analysis (not presented here) indicated that each of them is highly correlated with the dependent variable. The correlation between volumes held and total cost was found to be .89, and the correlation between serials and the total cost is .82, confirming that they are good indicators of outputs of academic research libraries. Group presentations that reflect the total number of library workshops and seminars offered also is a significant output. The data related to group presentations were only recently collected by the ARL. This study was not concerned with in- put substitution elasticities under the as- sumption that volumes held, serials, and group presentations, three major output variables, are not likely to be substitutes. Academic research libraries do not buy more books using serials expenditures simply because books are cheaper. Jour- nals provide up-to-date research findings. They are critical for research and teach- ing and cannot be replaced by books. It is also unlikely that libraries would reduce the number of library instructors teach- ing library workshops and use the sav- ings to purchase journals due to higher labor costs. Library instruction is indis- pensable training and education for stu- dents. It provides students with informa- tion competency they need to effectively access, retrieve, evaluate, and use infor- mation. The regression results of this study also show that circulation is not a good measure of library output because it is not statistically significant. The correlation analysis also revealed that the correlation between circulation and total cost is only .50, lower than volumes held (.89), seri- als (.82), group presentations (.54), and audios (.57). As it has been argued at the beginning of this article, given the nature of academic research libraries, a great deal of library materials, such as serials, refer- ence collections, and materials in carrels, are used in-house. Circulation records do not reflect such usage. The results in a separate regression that used volumes added as an output vari- able instead of volumes held with other variables unchanged showed that the R square decreased from .81 to .79 and the sum of the coefficients decreased from .93 to .86. As expected, using volumes added for the purpose of this study could have overestimated economies of scale because its use does not take into account staff and operating costs involved in maintaining existing collections. Summary and Conclusion This study examined scale economies of academic research libraries and reviewed the research literature on economies of scale in various libraries. It argued that academic research libraries are informa- tion provision organizations providing multiproducts and multiservices. The to- tal cost function was developed, and the natural log linear model was proved to be the best fit of the data. A wide range of collections and service outputs was incor- porated into the cost function to reflect the information provision function. Li- brary outputs were measured by library collections, including volumes held, se- rials, audio and video materials, maps and graphs, and library services, includ- ing workshops and seminars in the form of group presentations, reference trans- actions, circulation service, and ILL ser- vices. Three major output variables—vol- umes held, serials, and group presentations—stand out in terms of sta- tistical significance. It was found that slight economies of scale exist in academic research libraries. Previous research has made an impor- tant contribution to our understanding of scale economies of libraries and de- The Cost Function and Scale Economies in Academic Research Libraries 419 group presentations measured an impor- tant service activity of academic research libraries. More work needs to be done to collect data on capital, labor, and costs at the department or division level so that cross-section department or division production functions can be used to mea- sure the multiproduct activities of aca- demic research libraries. It is hoped that this investigation of scale economies in academic research li- braries will provide new insights into the existing literature in terms of understand- ing scale economies for libraries and out- put variables used in the total cost func- tion. Because scale economies are very sensitive to the number and nature of output variables used in the regression model, it is very important for research- ers to carefully select output variables. Good output indicators should reflect the true costs of libraries’ outputs. veloped various methodologies that can be used in the later studies. But previ- ous research also has some limitations. One of the obvious limitations is apply- ing the Cobb-Douglas production func- tion, which is normally used to measure a single output, to libraries that provide a wide range of outputs and services. This problem could be overcome by ap- plying the production function to cross- section data at the departmental or divi- sion level. But this did not happen prob- ably because of the lack of understand- ing of library operations by some re- searchers and the lack of consistent and reliable data related to library outputs, which frustrated researchers. The ARL has been collecting data for academic re- search libraries for many years. The ARL statistics have increasingly reflected the multiproduction nature of academic re- search libraries. For example, data on Notes 1. American Library Association. ALA Library Fact Sheet Number 1 (Chicago: American Li- brary Association, 2001). Available online from http://www.ala.org/library/fact1.html. Retrieved 29 March 2002. 2. Edwin Mansfield, Microeconomics: Theory and Applications, 9th ed. (New York: W.W. Norton, 1997), 191. 3. Paul W. Bauer, et al., “Optimal Use of Scale Economies in the Federal Reserve’s Currency Infrastructure,” Federal Reserve Bank of Cleveland Economic Review 36, no.3 (3rd quarter 2000): 13–27. 4. Elchanan Cohn, Sherrie L. W. Rhine, and Maria C. Santos, “Institutions of Higher Educa- tion as Multi-product Firms: Economies of Scale and Scope,” Review of Economics and Statistics 71, no.2 (May 1989): 284–90; Rajindar K. Koshal and Manjulika Koshal, “Do Liberal Arts Colleges Exhibit Economies of Scale and Scope?” Education Economics 8, no.3 (Dec. 2000): 209–11; Kalyan Chakraborty, Basudeb Biswas, and Cris W. Lewis, “Economies of Scale in Public Education: An Econometric Analysis,” Contemporary Economic Policy 8, no.2 (Apr. 2000): 238–48. 5. William J. Baumol, et al., “The Costs of Library and Informational Services,” in Libraries at Large, eds. D. M. Knight and E. S. Hourse (New York: R.R. Bowker, 1969), 168–227; Larry DeBoer, “Economies of Scale and Input Substitution in Public Libraries,” Journal of Urban Economics 32, no. 2 (Sept. 1992): 257–68. 6. Lewis Guodo Liu, “The Contribution of Library Collections to Prestige of Academic Pro- grams of Universities: A Quantitative Analysis,” Library Collections, Acquisitions, and Technical Services 25, no.1 (spring 2001): 49–65. 7. Baumol, et al., “The Costs of Library and Information Services.” 8. Stanley W. Black, “Library Economics,” in Libraries at Large, eds. D. M. Knight and E. S. Hourse (New York: R.R. Bowker, 1969), 590–98. 9. Baumol et al., “The Costs of Library and Information Services.” 10. William J. Baumol and Matityahu Marcus, Economics of Academic Libraries (Washington, D.C.: American Council on Education, 1973). 11. Haynes C. Goddard, “Analysis of Social Production Functions: The Public Libraries,” Public Finance Quarterly 1, no.2 (Apr. 1973): 191–204. 12. Ibid., 224. 13. Robert M. Hayes, “The Management of Library Resources: The Balance between Capital and Staff in Providing Services,” Library Research 1 (1979): 119–42. 420 College & Research Libraries September 2002 14. Larry DeBoer, “Economies of Scale and Input Substitution in Public Libraries,” Journal of Urban Economics 32, no. 2 (Sept. 1992): 257–68. 15. Christopher J. Hammond, “The Technology of Library Service Provision: A Cost Function Analysis of Public Library Systems in the United Kingdom,” Information Economics and Policy 11, no. 3 (Sept. 1999): 271–95. 16. Kathleen Foley Feldstein, “The Economics of Public Libraries” (Ph.D. diss., Massachu- setts Institute of Technology, 1976). 17. Michael D. Cooper, “The Economies of Library Size: A Preliminary Inquiry,” Library Trends 28, no.1 (summer 1979): 63–78; ———, “Economies of Scale in Academic Libraries,” Library and Information Science Research 5 (summer 1983): 207–19. 18. Paul Kantor, “Levels of Output Related to Cost of Operation of Scientific and Technical Libraries: Part I. Techniques and Cumulative Statistics,” Library Research 3 (1981): 1–28; —–, “Levels of Output Related to Cost of Operation of Scientific and Technical Libraries: Part II. A Capacity Model of the Average Cost Formula,” Library Research 3 (1981): 141–54. 19. Feldstein, “The Economies of Public Libraries.” 20. Cooper, “The Economies of Library Size,” 66. 21. ARL Statistics (Washington, D.C.: Association of Research Libraries, 2001). Available online from http://www.arl.org/stats/arlstat/index.html. Retrieved 10 December 2001. 22. Cooper, “The Economies of Library Size,” and “Economies of Scale in Academic Librar- ies”. 23. Stephen R. Lawrence, Lynn Silipigni Connaway, and Keith H. Brigham, “Life Cycle Costs of Library Collections: Creation of Effective Performance and Cost Metrics for Library Resources,” College & Research Libraries 62, no. 6 (Nov. 2001): 541–44. 24. Ibid., 543. 25. Hammond, “The Technology of Library Service Provision,” 274.