A Fund Allocation Formula
Based on Demand, Cost,
and Supply
William H. Walters
This document is the final, published version of an article in Library
Quarterly, vol. 78, no. 3 (July 2008), pp. 303–314.
It is also available at http://www.jstor.org/stable/10.1086/588640
RESEARCH IN PRACTICE
A FUND ALLOCATION FORMULA BASED ON DEMAND,
COST, AND SUPPLY
William H. Walters1
Approximately 40 percent of academic libraries use formulas to allocate
book funds among departments or subject areas [1–3]. Consequently, dozens of published articles and reports have appeared on the topic of formulabased fund allocation. (See Walters [4] for a review of the literature.)
Unfortunately, every previous strategy for fund allocation requires a set of
weights or allocations that can be used as a starting point; for example,
fund allocation methods based on linear programming rely on the subjective assessment of each department’s importance or value to the university [5–8]. Likewise, regression-based methods require a set of current,
past, or hypothetical allocations from which a formula can be derived [4].
This essay presents a theoretically grounded method of developing a
fund allocation formula that does not rely on the initial estimation of
weights or allocations. This method is likely to be especially useful for
libraries that have never used fund allocation formulas—those that have
no prior basis for rating the importance of particular subject areas, departments, or variables.
Background
More than sixty variables have been identified as potentially important
determinants of departmental funding levels [4]. Although this wide array
of variables can be categorized in a number of ways [9–12], the most
conceptually useful classification is one of the simplest. Peter Sweetman
and Paul Wiedemann [13] group the variables into just three categories:
• Demand: the need for library materials in each department or subject
1. Dean of library services and associate professor of social sciences, Menlo College, Atherton,
CA 94027-4301. Telephone 650-543-3827; Fax 650-543-3833; E-mail [email protected].
[Library Quarterly, vol. 78, no. 3, pp. 303–314]
2008 by The University of Chicago. All rights reserved.
0024-2519/2008/7803-0004$10.00
303
304
THE LIBRARY QUARTERLY
area, as reflected in the size of the department as well as the extent of
its teaching and research activity
• Cost: the average cost of a monographic title in each subject area
• Supply: the number of new titles published annually in each subject area
The cost and supply categories can each be represented by a single variable.
Demand, however, is often represented by multiple variables representing
the various aspects of department size and activity—variables such as course
enrollment, the number of undergraduate majors, the number of faculty,
the number and level of degrees awarded, the number of courses offered,
and the extent to which those courses require library use.
The fourteen variables most often used in fund allocation formulas [4]
can all be placed into the three categories identified by Sweetman and
Wiedemann. Any formula that fully accounts for demand, cost, and supply
is therefore likely to represent all the factors generally regarded as important determinants of departmental funding levels.
Constructing a Demand Variable
The selection of appropriate demand variables can be challenging, especially when the library’s collection development policy and the institutional
climate provide no reason to favor one aspect of demand over another—
no reason to weight course enrollment more than research productivity
or to count the number of distinct course offerings more than the number
of students majoring in each field. If there are valid reasons for assigning
greater or lesser weights to particular components of demand, then other
methods will be more useful than the one described here. However, libraries that require a general indicator of demand may benefit from this
technique, which uses factor analysis to construct a single variable that
represents all the various components of demand.
Factor analysis was originally developed for use in psychometric research
but has since been adopted in a wide range of disciplines. (See [14], for
example.) Factor analysis is most often used to reveal the few principal
components or factors that underlie a larger set of variables. For instance,
it can be used to help researchers identify and understand the various
dimensions of intelligence represented by a battery of test questions. In
1969, William E. McGrath and associates [10] applied factor analysis to
library fund allocation, using it to identify the general factors underlying
a set of twenty-two departmental characteristics. They found that the set
of twenty-two variables could be adequately represented by just three factors: course-related demand, research-related demand, and the size of the
user population. McGrath and his team then determined which of the
twenty-two variables were most closely related to each of the three factors.
RESEARCH IN PRACTICE
305
(Course-related demand was most closely related to total library circulation
in each subject area; research-related demand was most closely related to
the number of works cited in the graduate theses accepted by each department over a five-year period; size was best represented by the total
number of undergraduate majors and graduate students registered with
each department.)2 In McGrath’s study, the three most representative variables were then used in a fund allocation formula to represent the entire
set of twenty-two variables for which data had been compiled. The same
three variables will not be appropriate for every institution, however, since
the results of each factor analysis are specific to the particular college or
university for which the analysis is conducted.
McGrath’s technique is a statistically rigorous method of identifying and
representing the various components of demand. However, some libraries
may not have the resources to compile large data sets or the expertise to
take full advantage of the factor analysis results. Fortunately, factor analysis
can also be used to construct a single variable representing all the various
aspects of demand. The goal, in this case, is not to reveal the principal components underlying a large set of variables but to construct a general indicator
of demand using simpler methods and a smaller number of variables.
The procedure begins with the compilation of data for those variables
felt to be important components of demand. (Cost and supply variables
should not be included at this stage.) The choice of variables is always
subjective and dependent on local conditions, although a typical liberal
arts college might want to include variables such as
e Total enrollment in courses offered by the department or program;
f Number of regular faculty positions plus one-fourth the number of
adjunct instructors and other part-time academic staff not on the
faculty list;
m Number of undergraduate majors and graduate students in the
department;
o Number of distinct courses offered by the department or program;
and
s Number of senior projects and master’s theses submitted.
While some other methods of fund allocation require the identification
and exclusion of variables that are closely related, no such exclusion is
necessary when factor analysis is used. Each value should be expressed as
a percentage of the total for all departments combined. Example data for
St. Lawrence University, a liberal arts college in upstate New York, are
presented in table 1. The example data are for 2004–5, except for variable
s, which covers a three-year period.
To perform the factor analysis, first type, paste, or import the depart2. These three variables were described incorrectly in an earlier paper [4].
306
THE LIBRARY QUARTERLY
TABLE 1
Example Data for St. Lawrence University
e
(Enrollment)
African studies
Anthropology
Asian studies
Biology
Canadian studies
Chemistry
Economics
Education
English
Environmental studies
Fine arts
French
Gender studies
Geology
German
Global studies
Government
History of science
History
Italian
Japanese
Latin American studies
Mathematics
Music
Philosophy
Physics
Psychology
Religious studies
Russian
Sociology
Spanish
Speech and theater
Sports and athletics
Column sum
f
(Faculty)
m
(Majors)
o
(Courses)
s
(Projects)
.57
2.78
.00
4.03
.51
2.76
6.99
7.82
8.49
2.88
4.37
1.42
1.10
1.61
.51
2.30
6.92
.00
5.31
.35
.23
.42
8.83
1.97
2.52
1.54
8.71
2.70
.00
4.59
1.79
3.02
2.96
.00
2.15
.00
5.91
1.61
3.23
4.84
6.99
11.29
2.69
3.76
2.15
.54
2.69
1.07
2.69
5.38
.00
5.38
.54
.54
.00
6.45
1.61
2.15
2.69
6.45
2.15
.00
4.84
2.15
3.76
4.30
.00
1.08
.00
8.94
.10
1.47
9.33
4.72
11.59
3.93
4.22
.10
.00
2.36
.20
1.57
9.72
.00
7.37
.00
.00
.00
6.97
.79
.49
1.08
14.34
1.18
.00
4.72
1.47
2.26
.00
1.03
2.63
2.75
3.89
1.14
2.18
3.32
10.08
7.10
4.12
4.70
1.14
3.09
2.52
.92
6.41
3.78
.00
6.19
.23
.23
2.86
4.93
3.44
2.75
1.95
3.09
1.95
.00
3.67
1.26
4.24
2.41
.00
1.75
.58
16.86
.00
1.74
5.81
.58
8.72
1.74
.58
.58
.00
3.49
.00
1.75
15.12
.00
1.75
.00
.00
.00
11.63
.58
1.74
.58
15.12
.58
.00
6.40
1.74
.58
.00
100.00
100.00
100.00
100.00
100.00
Note.—Each value is expressed as a percentage of the total for all departments combined.
mental data into SPSS, Minitab, SAS, Stata, or another statistical package.
Use the format shown in table 1 but omit the final row (column sum).
The exact procedure for conducting the analysis varies with the software
chosen, although SPSS, Minitab, SAS, and Stata all have factor analysis
capabilities.
• In SPSS, select Analyze—Data Reduction—Factor. Move all the numeric
variables to the box marked Variables. There is no selection variable.
RESEARCH IN PRACTICE
307
The default settings (principal components extraction without rotation)
will not need to be changed.
• In Minitab, select Stat—Multivariate—Factor Analysis. Move all the numeric variables to the box marked Variables. Enter “1” as the number
of factors to extract, “principal components extraction” as the method
of extraction, and “none” as the type of rotation.
The statistical method used here is often known as principal components
analysis rather than factor analysis, although the distinction between the
two is not important in this context. For further information on factor
extraction and rotation methods, see the guides by Dennis Child, Jae-On
Kim and Charles W. Mueller, and Paul Kline [15–18].
The only results needed for this purpose are the communality estimates
reported near the beginning of the factor analysis output. In SPSS, these
appear in the Extraction column of the first table (Communalities). In
Minitab, they appear in the Communality column of the first table (Unrotated Factor Loadings and Communalities). Each communality estimate
(sometimes designated h2) represents shared or common variance—the
extent to which a particular variable can be expressed as a linear combination of the others. For example, variable e has a communality of 0.93,
indicating that for this particular set of academic departments, there is a
93 percent “overlap” between e and the rest of the demand variables ( f,
m, o, and s combined). Conversely, 7 percent of the interdepartmental
variation in variable e is unique to that variable and cannot be found within
the other variables in the set.
Variables with higher communalities (approaching 1.00) are more closely
related to the general construct represented by the entire set of variables.
That is, variables with higher communalities are more closely related to the
overall demand for library resources. We can therefore construct a variable
representing overall demand simply by taking a weighted average of the five
component variables (e, f, m, o, and s). Table 2 shows these calculations using
the example data. The five communality estimates were taken from the SPSS
output. Each value in columns e, f, m, o, and s is the corresponding value
from table 1 multiplied by the appropriate communality estimate. For example, the value of 0.53 reported in column e for African studies is 0.57
(from table 1) # 0.93 (the communality estimate for variable e).
The second-to-last column (row sum) is the sum of the values in columns
e, f, m, o, and s. Each row sum is therefore a weighted average of the
component variables. All five component variables contribute to each row
sum in proportion to their communalities. That is, a component variable
with a communality of 0.90 counts for twice as much as a variable with a
communality of 0.45. In this example, variable f (number of faculty) counts
for more than variable o (number of courses offered) because variable f is
TABLE 2
Calculation of the Demand Variable
Communality estimate
African studies
Anthropology
Asian studies
Biology
Canadian studies
Chemistry
Economics
Education
English
Environmental studies
Fine arts
French
Gender studies
Geology
German
Global studies
Government
History of science
History
Italian
Japanese
Latin American studies
Mathematics
Music
Philosophy
Physics
Psychology
Religious studies
Russian
Sociology
Spanish
Speech and theater
Sports and athletics
Column sum
e
f
m
o
s
Row
Sum
.93
.53
2.59
.00
3.75
.47
2.57
6.50
7.27
7.90
2.68
4.06
1.32
1.02
1.50
.47
2.14
6.44
.00
4.94
.33
.21
.39
8.21
1.83
2.34
1.43
8.10
2.51
.00
4.27
1.66
2.81
2.75
.90
.00
1.94
.00
5.32
1.45
2.91
4.36
6.29
10.16
2.42
3.38
1.94
.49
2.42
.96
2.42
4.84
.00
4.84
.49
.49
.00
5.81
1.45
1.94
2.42
5.81
1.94
.00
4.36
1.94
3.38
3.87
.89
.00
.96
.00
7.96
.09
1.31
8.30
4.20
10.32
3.50
3.76
.09
.00
2.10
.18
1.40
8.65
.00
6.56
.00
.00
.00
6.20
.70
.44
.96
12.76
1.05
.00
4.20
1.31
2.01
.00
.53
.55
1.39
1.46
2.06
.60
1.16
1.76
5.34
3.76
2.18
2.49
.60
1.64
1.34
.49
3.40
2.00
.00
3.28
.12
.12
1.52
2.61
1.82
1.46
1.03
1.64
1.03
.00
1.95
.67
2.25
1.28
.64
.00
1.12
.37
10.79
.00
1.11
3.72
.37
5.58
1.11
.37
.37
.00
2.23
.00
1.12
9.68
.00
1.12
.00
.00
.00
7.44
.37
1.11
.37
9.68
.37
.00
4.10
1.11
.37
.00
1.08
8.00
1.83
29.88
2.62
9.05
24.64
23.48
37.72
11.89
14.07
4.32
3.15
9.59
2.10
10.47
31.61
.00
20.74
.93
.82
1.91
30.28
6.18
7.29
6.22
37.98
6.90
.00
18.87
6.69
10.82
7.90
.28
2.06
.47
7.68
.67
2.33
6.33
6.04
9.70
3.06
3.62
1.11
.81
2.46
.54
2.69
8.13
.00
5.33
.24
.21
.49
7.78
1.59
1.87
1.60
9.76
1.77
.00
4.85
1.72
2.78
2.03
389.00
100.00
Demand
Note.—Each value in columns e (enrollment), f (faculty), m (majors), o (courses), and s (projects) is the corresponding
value from table 1 multiplied by the appropriate communality estimate.
RESEARCH IN PRACTICE
309
more closely related to the underlying concept represented by the set of
five variables.
For each academic department, the last column (Demand) is simply the
row sum divided by 389.00 (the combined total of all the row sums), then
multiplied by 100 (to express each value in percentage terms). This final
calculation produces a variable representing each department’s share of
the total institutional demand for library resources.
Accounting for Cost and Supply
Because factor analysis is a flexible technique, it can be used to construct
either a single demand variable or several variables representing the various
aspects of demand. (The analysis by McGrath and associates [10] is a good
example of the latter approach.) The use of a single variable to represent
demand has two advantages in this context: (1) it avoids the need to subjectively estimate which components of demand are most appropriate for
inclusion in the fund allocation formula, and (2) it maintains consistency
with the demand-cost-supply framework established by Sweetman and Wiedemann [13].
Some libraries might want to allocate book funds strictly in accordance
with demand. Anthropology’s demand score of 2.06 suggests that 2.06
percent of the allocated budget should be made available for resources
recommended by the librarians and faculty responsible for that subject
area. A strictly demand-driven approach is appropriate if we accept the
assumption that departments with equal demand for library materials
should be entrusted with equal amounts of money.
However, many libraries prefer to account for the fact that the cost of
books and other monographic items varies by subject area. Books in the
fields of art or chemistry, for example, are more expensive than those
in education or English literature. Within Sweetman and Wiedemann’s
demand-cost-supply framework, we can account for cost simply by multiplying demand by price per title, then dividing the resulting values by a
constant (the sum of the values) to ensure that the total equals 100 percent
(see table 3). This approach ensures that departments with equal demand
will be able to purchase an equal number of monographic items. Because
nearly 60 percent of all published allocation formulas include a cost variable [4], we can conclude that most libraries view equity this way, on the
basis of items purchased rather than funds spent. Average price data for
particular subject areas can be obtained from a variety of sources, including
published guides, approval-plan data, and past years’ acquisitions records
[19–20].
310
African studies
Anthropology
Asian studies
Biology
Canadian studies
Chemistry
Economics
Education
English
Environmental studies
Fine arts
French
Gender studies
Geology
German
Global studies
Government
History of science
History
Italian
.28
2.06
.47
7.68
.67
2.33
6.33
6.04
9.70
3.06
3.62
1.11
.81
2.46
.54
2.69
8.13
.00
5.33
.24
54.02
50.18
49.72
50.43
39.80
68.64
56.91
34.86
50.52
47.35
72.02
43.82
39.60
76.43
38.02
62.69
52.90
50.86
45.63
42.98
15.13
103.37
23.37
387.30
26.67
159.93
360.24
210.55
490.04
144.89
260.71
48.64
32.08
188.02
20.53
168.64
430.08
.00
243.21
10.32
.29
1.96
.44
7.34
.51
3.03
6.82
3.99
9.28
2.74
4.94
.92
.61
3.56
.39
3.19
8.15
.00
4.61
.20
232
678
1,004
1,122
244
248
1,082
946
2,432
886
1,028
130
662
160
164
694
1,528
280
2,388
50
3,509
70,085
23,462
434,553
6,507
39,663
389,780
199,184
1,191,787
128,373
268,012
6,323
21,234
30,083
3,367
117,033
657,158
0
580,780
516
.06
1.22
.41
7.57
.11
.69
6.79
3.47
20.77
2.24
4.67
.11
.37
.52
.06
2.04
11.45
.00
10.12
.01
Allocation Based
Allocation Based
Demand Allocation Based on
Demand # on Demand, Cost,
on Demand*
Cost† # Cost Demand and Cost Supply‡ Cost # Supply
and Supply
TABLE 3
Calculation of Allocations Based on Demand, Cost, and Supply
311
100.00
.21
.49
7.78
1.59
1.87
1.60
9.76
1.77
.00
4.85
1.72
2.78
2.03
* From table 2.
†
Estimated price per title.
‡
Estimated number of relevant titles published.
Column sum
Japanese
Latin American studies
Mathematics
Music
Philosophy
Physics
Psychology
Religious studies
Russian
Sociology
Spanish
Speech and theater
Sports and athletics
47.64
58.61
55.93
48.10
49.32
50.60
57.83
34.36
49.13
49.15
75.97
50.19
48.53
5,279.56
10.00
28.72
435.14
76.48
92.23
80.96
564.42
60.82
.00
238.38
130.67
139.53
98.52
100.00
.19
.54
8.24
1.45
1.75
1.53
10.69
1.15
.00
4.52
2.47
2.64
1.87
54
326
1,082
458
688
514
546
1,076
132
1,926
124
576
162
5,737,483
540
9,362
470,817
35,027
63,453
41,613
308,174
65,439
0
459,115
16,203
80,368
15,960
100.00
.01
.16
8.21
.61
1.11
.73
5.37
1.14
.00
8.00
.28
1.40
.28
312
THE LIBRARY QUARTERLY
By introducing a third variable representing supply, we can ensure that
departments with equal demand will be able to purchase an equal proportion of all the relevant titles published each year. Approximately 20
percent of published allocation formulas include a supply variable [4]. To
account for all three variables—demand, cost, and supply—simply multiply
the three together, then divide each resulting value by the total for all departments combined. Cost is usually defined as the average price per title,
while supply is the number of titles published, cataloged, or reviewed each
year. Supply data can be found in American Book Publishing Record, Books in
Print, the Bowker Annual, Choice, Publishers Weekly, or WorldCat, or compiled
from vendors’ approval plan records. Because many libraries will find it
helpful to exclude certain kinds of titles—juvenile fiction or mass market
paperbacks, for example—Choice and other databases that allow users to
limit their searches by genre, language, or place of publication can be
especially good sources of supply data. The exact specification of the supply
variable will depend on the availability of data as well as local factors such
as the type of library, its user population, and its mission.
Table 3 shows how to calculate departmental allocations using data for
demand, cost, and supply. (Cost, in this example, is the estimated price
per title, while demand is the estimated number of relevant titles published each year. Both the cost and supply variables refer to titles covered
by St. Lawrence University’s book and slip approval plans.)
As the table shows, departmental allocations that incorporate both demand and cost are very similar to those based solely on demand. (The
correlation between the two sets of allocations is 0.98.) For example, anthropology receives 2.06 percent of the allocated budget if we use demand
as the sole determinant of funding levels. When cost is considered along
with demand, anthropology’s share declines by just a tiny amount, to 1.96
percent. This reflects the fact that books cost about the same amount in
most subject areas. The exceptions are fields such as fine arts and chemistry,
where the inclusion of a book-cost variable significantly increases their
departmental allocations.
The use of all three variables—demand, cost, and supply—produces a
greater change in the resulting allocations. (See table 3.) The correlation
between the demand-based allocations and the demand-cost-supply allocations is 0.85—still a close relationship, but with substantial variations for
quite a few subject areas. Specialized subjects, such as African studies, get
far lower allocations when supply is included, since relatively few books are
published in those fields. In contrast, English and history more than double
their allocations due to the number of books published in such broad subject
areas.
Allocation formulas that account for supply are likely to be sensitive to
changes in data quality and measurement. For instance, a supply variable
RESEARCH IN PRACTICE
313
based on the total number of titles published worldwide will lead to different results than one based on the number of English-language academic
titles published in North America or the number reviewed in Choice. Although the method described here does not rely on the advance specification of weights for particular subjects or variables, it does require the
careful selection of variables that are appropriate to local circumstances.
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