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April 1991

FEE ASSESSMENT IN A MEDICAL LIBRARY:

AN APPLICATION OF THE CLUB PRINCIPLE

John J. Rousseau Senior Research Scientist Information Services Research

MRCA Information Services 3001 North Lamar Boulevard

AUSfill, Texas 78705 U.S.A.

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The Arabian Journal for Science and Engineering, Volul11e 16, NUl11her 2B. 283

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284

J. J. Rousseau

ABSTRACT

This paper explores the implications of a fee assessment policy for a library system where the annual joint fixed support costs of library operation are shared among the different institutional members of the system according to a club principle. Linear programming and game theoretic techniques are used to address questions of the optimal size of the library system and optimal pattern of membership, and to characterize the set of fair and efficient institutional fees. Two alternative rules for selecting particular fee schedules based on the Shapley value and nucleolus solution concepts in game theory are also presented.

The present development extends in two directions an earlier study in cost recovery conducted at the Houston Academy of Medicine-Texas Medical Center Lib- rary, providing additional insights and an improved basis for policy analysis. For purposes of illustration and comparison the model is applied to this library to evaluate its assessment policy using data from that study, which reflect operating conditions in 1976. Our results represent a superior assessment policy as measured by our criteria of fairness and efficiency, and offer support for the use of the club principle not only in libraries, but for transactions pricing and cost recovery in the general context of information services.

The Arabian Journal for Science and Engineering, Volume 16, Number 2B. April 1991

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J. J. RousJeau

FEE ASSESSMENT IN A MEDICAL LIBRARY: AN APPLICATION OF THE CLUB PRINCIPLE

1. INTRODUCTION

A service institution such as a library provides a variety of related and interrelated services to a heterogeneous mix of users. Moreover, there are many so-called support functions, shared across services, which are necessary to the overall operation of the library. Together these direct services and support functions give rise to the costs that must be recovered somehow in maintaining the operation of the library. However, relating, or more appropriately al- locating, these costs to the different services and support functions that generate them is a very difficult administrative problem. The library's administration must deal with this issue either in justifying its budget, particularly when subscription to the library system is required, or in monitoring its performance and responding to changing patterns of usage and expanding needs for resources.

In a joint study with library administrative personnel, Bres et al. [1, 2], looked at the problem of cost recovery at the Houston Academy of Medicine- Texas Medical Center (HAM-TMC) Library. The library's annual operating budget is recovered through assessments to the different institutional or departmental members of the library system. In the past, assessments were determined largely through negotiations among the various players, and a more objective means of determining annual fees was considered necessary. Although their procedures provided (at least) a first-cut at the cost allocation process, several important issues such as equity of the assessments and the optimal size of library facilities were not addressed in that study.

Clearly, and as recognized by Bres et al. [1, 2], fees assessed to the institutions should reflect so far as possible the way in which costs are actu<;llly generated, directly incorporating both individual user be- havior and variation among institutions in informational requirements. In addition to recovering total cost, however, consideration must also be paid to the

"fairness" and "affordability" of any resulting fee structure since they are important factors in maintaining the institutions' continued membership in the library system. Many of the costs involved are common or joint in nature and this causes significant dif- ficulties in any such assessment process.

In the general setting, while there may be agree- ment that separable user costs should be incorpo- rated in user charges, there is disagreement on how common capacity costs should be recovered. Among the various suggestions is that common costs should be shared among users according to a "club principle" [3, 4]. Littlechild [5] formalized the common cost problem within the context of multi-product public enterprises and used the techniques of linear programming and game theory to derive the implications of the club principle for efficient and equitable pricing and investment policies. Littlechild and Thompson [6] subsequently applied this analysis to the determination of aircraft landing fees.

The purpose of the present paper is to show how these club principle ideas may be adapted to cost recovery and fee assessment in a library context, thereby extending the initial work of Bres et al. [1,2]

by incorporating notions of equity and optimal capacity, and to illustrate the application by calculations based on the HAM-TMC Library. For purposes of comparison it will prove convenient to employ the data of Bres et al. which reflect operating conditions of the library in 1976. More precisely, we shall at- tempt to ascertain whether the HAM -TMC library system was operating at its "optimal" size; whether the institutional assessments were both "fair" and

"efficient"; and how those assessments compare to ones derived from solution concepts in game theory which incorporate further notions of equity or stability and which can be used to select particular fee schedules from the optimal (i.e., fair and efficient) set.

Although the present paper is set in a library context. it is apparent that many of the issues involved are common to transactions pricing and cost recovery for information services in general. Thus we hope that this paper will also indicate how the club principle may be a useful approach to such problems in the wider context.

We begin with a brief discussion of library operating costs and examine the case for a pricing policy based on the club principle. This is followed by our mathematical model, application. and presentation of results. A concluding section summarizes our procedures and findings and discusses some qualifica- tions and extensions.

April 1991 The Arabian Journal for Science and Engineering, Volume 16, Number 2B. 285

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286

1. 1. Rousseau

THE CLUB PRINCIPLE

A reasonable division of the annual operating costs of a library distinguishes those attributable to each specific direct service provided (called direct costs) and those essential to general library operations as an integrated facility (called support costs).

*

Direct costs are variable in nature, responding to the usage experience of each service; whereas support costs, which may be relatively substantial in some libraries, are characteristically more complex to analyze.

A large portion of support costs are related to physical inputs, such as books and journals purchased or plant (space) and equipment leased. Also to be considered are those personnel and their activities that contribute to library operations overall, but at levels not determined by the quantity of direct library usages. One aspect of support costs, then, arises from the necessity to establish and maintain a basic facility with the requisite capacity to provide a predeter- mined set of services for its users. In a sense, these costs are relatively fixed with respect to usage once having defined the type of services to be offered.

However, qualitative considerations of servi,ces may require changes in supportive facilities and activities as usage varies. Thus, some portion of support costs is related to direct service usage.

The total operating costs of a library might, therefore, be described as having the following three components: direct costs for each service provided; variable support costs related to usage of services; and fixed support costs related to certain characteristics and functions of the institutions using the library.

A reasonable and equitable charging mechanism, therefore, might recover the direct costs and variable portion of support costs through usage based fees, and the fixed support costs through institutionally based (or "membership") fees. Consequently, an institutional member of a library system would pay a fee which is partially based on the costs generated through its members' use of services and partially based on the costs of maintaining that type and quality of library that the institution would need to have available whenever its members wished to use it.

Following Littlechild and Thompson [6], we define the optimal extent of library operation and pattern of institutional membership as those which maximize the net present value of total benefits. We call a fee

"'The following discussion is partially based on Bres, ef al. [1. 2].

assessment policy efficient if it satisfies the following three conditions:

(1) It should give the library authority the incentive to operate the library to its optimal extent and accept the optimal pattern of membership;

moreover, the authority should just break even.

(2) There should be no cross subsidization whereby any group of institutions pays more than the costs of accommodating that group alone.

Hence, there should be no incentive to operate another library to cater for any group of institutions.

(3) The fee assessment scheme should give all institutions which can cover their user cost of library services the incentive to use the library by setting institutional fees in such cases within

"ability-to-pay" .

If one set of institutional assessments is efficient, then in general many efficient sets of assessments will exist. As a first step in choosing among them we consider some basic notions of equity. We call a fee assessment policy fair if (a) institutions with less costly demands pay no more than institutions with more costly demands and (b) the amount by which the charge to an institution with more costly demands exceeds that to an institution with less costly demands does not exceed the difference in costs of providing for the two types.

Although we will not employ them here, two other criteria may be suggested: (c) institutions whose value of service is higher may be charged more and (d) institutions whose membership provides benefits to other institutions may be charged less.

Even with these efficiency and fairness criteria, we can expect that additional restrictions will be needed to narrow down the set of fee schedules obtained.

For this we may employ some game theoretic notions of solutions, each with its own assumptions and ra- tional, to select a specific fair and efficient allocation of costs; in particular, we examine the Shapley value and the nucleolus.

THE MODEL

The model presented here is an application of Littlechild's [5] general development. We deal with the case of a single library, and consider a single time period of one year rather than a multiperiod model.

Initially, our assumption is that the library authority seeks to maximize net social benefit as defined

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below, and that there are no externalities associated with membership. In our practical application, however, we wlll examine rules for determining institutional assessments which depend upon the relatively objective and ascertainable element of cost, but not upon benefits. We exclude the possibility of collusion among institutions to influence charges, although the game theoretic solutions may be interpreted as reflecting coalitional bargaining power. In a concluding section we examine the possibilities of relaxing some of these assumptions.

Let N

=

{I, 2, ... , n} denote a set of potential institutional members of a library system over the (one year) planning period, with the typical member denoted by subscript j, and the typical subset denoted by 5. Further, let there be p different services, denoted by subscript k, available to q different user types, denoted by subscript i, who are individual members of the institutions within the library system.

Let

b_i gross benefit to institution j and its members if that institution joins the library system,

Cklj usage cost of service k by all users of type i

who are members of institution j,

G(5) annual joint fixed support costs of a library designed to accommodate subset 5 of institutions.

The net social benefit of operating a library for subset 5 of institutions, denoted Jt(5), is defined by

(1)

Suppose there are m different types of institution, denoted by subscript i, with n_iinstitutions of type i and Ni the set of institutions type i. Then

III III

[ n_i

=

nand

U

^Ni

=

^N

i = ^I ⁱ= ^I

Let gi denote the annual fixed support costs of a library designed to accommodate institution type i.

In a later section we shall argue that the institution types can be ranked by "size" and that the cost of accommodating any subset of institutions can be approximated by the cost of accommodating the largest institution type in that subset. Hence

G(5)

=

max{gi: 5 n Ni -::/= 0} (2)

1.1. Rousseau

Let x denote a 211-dimensional decision vector whose typical component Xs takes the value unity if a library is operated to accommodate precisely the subset 5 of institutions, and takes the value zero otherwise.

Consider the following pair of dual linear programs:

Max [ Jt(5)xs

s~ N

[ Xs :::; 1, j

=

^{1, ... ,}ⁿ

S E i

Xs 2: 0, 5 c;: N

II

Min[u;

i= I

[Ui

2: Jt(5), 5 eN'

j E 51

Uj2:0,j= 1, ... ,n.

The primal problem I chooses that extent of library operation and pattern of institutional membership which maximizes total net social benefit, with each potential institutional member allotted to at most one library. The dual problem II determines the minimum total payoff consistent with the payoff to any set of institutions being not less than the net benefit they could obtain by forming a club.

In principle, it might be optimal to operate more than one library, or even part-time libraries, but Littlechild [5] has shown that if the net benefit function Jt is convex, then it will be optimal to operate precisely one library. In the present example this translates to the joint fixed support cost function G(5) being concave reflecting economies of scale, that is:

G(5 u 1):::; G(5)

+

G(1) - G(5 n 1) for all 5, Tc;; N.

However, this single club need not be a unique optimum unless G is strictly concave. In addition, if there is a positive net benefit for each single member club then the club consisting of all potential members N is an optimum.

EFFICIENT AND FAIR FIXED CHARGES Since our primary concern is with joint fixed costs, we will proceed as if usage costs are charged for as incurred. That is, for each direct service provided, direct costs plus that portion of variable support costs attributed to the service are recovered through a charge (possibly dependent upon user type) per unit use of the service by individuals. (See, for example, Bres et ai. [1], Part II).

April 1991 The Arabian lournal for Science and Engineering, Voilime 16, Number 2B.

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288

1. 1. Rousseau

Then the fee assessed to institution j may be considered as having two components: one equal to the

p q

usage cost L L Cklj of all services by users of all

k = 1 1 = 1

types who are members of institution j and the other a fixed charge or membership fee, Yj' representing a contribution to fixed costs. The net payoff to institution j is the gross benefit less the institutional assessment, that is,

Uj

=

bj -

(t t Cklj + Yj ) ,j = 1, ... , n

k = 1 1 = 1

(3)

Substituting from (1) and (3) into the dual problem II, cancelling, and rearranging yield the equivalent problem

II'

LYj ~ G( S), SeN

jES

p q

L LCklj+Yj~bj,j=l, ... ,n

k = 1 1= 1

The equivalent dual problem II' determines that schedule of fixed charges which maximizes total contribution. By complementary slackness between I and II' we have that total contribution will just equal total fixed support costs for whichever library is actu- ally operated, which is condition (1) of an efficient fee assessment policy. The first set of constraints in II' requires that no subset of institutions contributes more than the cost of operating a library for that subset alone. That is, the fixed charges must avoid cross-subsidization, which is efficiency condition (2).

The second set of constraints requires that no institution is charged more than its associated net benefit (i.e., gross benefit less user cost). That is, for those institutions which can cover their usage cost of library services, fixed charges must be set within ability to pay, which is efficiency condition (3). Consequently, the set of optimal solutions to problem II' is precisely the set of efficient fixed support charges Yj; the efficient institutional fees are obtained by adding user

p q

cost

L L

Ck1r Since problem I is feasible and

k = 1 1= 1

bounded, we are assured that an efficient set of in-

stitutional fees exists, but it will generally not be unique.

The two fairness criteria (a) and (b) given earlier may be formally represented by

< <

+

gi - gi - 1 . - 1

Yi-l-Yi-Yi-l - '-- - , l - , ... ,m.

ni

These conditions are independent of benefits.

GAME THEORETIC METHODS FOR FEE ASSESSMENT

(4a, b)

Let nOeS) be the maximum net benefit obtainable by the subset S of institutions or any smaller set, i. e., nOeS)

=

Maxn(T). Formally, n° is the monotonic

Tr;;;S

cover of n. We may consider the set of institutions N as a set of players in a game and n° as a characteristic function. Littlechild [5] has shown that the set of optimal solutions to the dual problem II comprises precisely the core of the game (N, nO). Thus, the set of payoffs which cannot be improved upon by any coalition of institutions is precisely the set of payoffs which corresponds to efficient institutional assessments.

This allows us to determine the efficient assessments by going straight to the core of the appropriate game, by-passing the linear programming formula- tion. We can then look to other game theoretic solution concepts as a means of selecting a particular fee assessment schedule from the efficient set which is generally not a unique vector.

We may also develop assessment rules based on the cost function alone with no regard to benefits, subsequently checking that resulting assessments satisfy ability-to-pay. Thus, we will consider the library cost game (N, -G) in which the characteristic function is the negative of the joint fixed support cost function. Two solution concepts are explored. The Shapley value [7] is a unique vector of payoffs which satisfies certain general and reasonable critera, and gives to each player a weighted average of his contribution to each coalition, namely

(n - s)!(s - I)!

Yj

= L - _ . - .. - - , - ---

SeN n.

S;:) j

[ G (S) - G (S - {j})], j

=

1, ... , n (5) where s

=

lSi.

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In our example there is a remarkable simplification in the formula for the Shapley value, as demonstrated by Littlechild and Owen [8].

The nucleolus of Schmeidler [9] is that unique vector of payoffs which maximizes the minimum benefit (relative to the characteristic function) lexicographi- cally over coalitions, and in that sense minimizes the maximum complaint that could be raised. In general, the nucleolus cannot be written down in explicit form, but may be calculated by a sequence of linear programs which in the present example take a particularly simple form. Since the present game is convex, both the Shapley value and the nucleolus lie in the core, and hence the institutional assessments are efficient if they also are within ability-to-pay. These assessments will also be fair if they satisfy conditions (4a, b).

AN APPLICATION TO THE HAM·TMC LIBRARY

The Houston Academy of Medicine - Texas Med- ical Center (HAM-TMC) Library is an independent medical library that jointly serves and is supported by 22 academic, research and clinical institutions. Serv- ing over 10,000 individuals, the library ranked second in 1976 in total expenditures of all medical libraries in the U.S. and Canada, with an annual budget in excess of $ 1.27 million which is recovered through fees assessed to the supporting institutions.

The 22 institutions can be categorized under six general headings.

Medical Schools:

Baylor

University of Texas (UTMS) Medical Associations:

Houston Academy of Medicine (HAM) Nursing Schools:

Texas Women's University (TWU) U.T. System School of Nursing (UTSSN)'

Prairie View A&M University School of Nursing (PVAMUSN)

Graduate Medical and Health Related Schools:

U.T. Graduate School of Biomedical Sciences (GSBS)

U.T. School of Public Health (SPH) U.T. School of Allied Health (SAH) U.T. Dental Branch (DB)

1. J. Rouss(}aU

Specialized Research Institutes:

U.T. System Cancer Center M.D. Anderson Hospi- tal and Tumor Institute (UTMDA)

Texas Institute of Rehabilitation and Research (TIRR)

Texas Research Institute of Mental Sciences (TRIMS)

Speech and Hearing Institute (SHI) Institute of Religion (IR)

Teaching and Research Hospitals:

Methodist Hospital

St. Luke's Episcopal/Texas Children's Hospital (SL/

TCH)

Harris County Hospital District (HCHD) Hermann Hospital

St. Joseph Hospital Memorial Hospital

Veteran's Administration Hospital (V A)

The categories are necessarily broad with, in some cases, substantial variation in size and emphasis of institutions under a particular heading. In addition, the nature of the institutions is such that more than one classification is possible. For example, UTMDA might be classified as a graduate medical school, a research hospital, or a specialized research institute;

SHI could also be considered as a graduate college;

TIRR, TRIMS, and IR are small in comparison with UTMDA and perhaps more closely resemble research hospitals than specialized research institutes;

even the hospitals themselves vary as to their emph- ases on teaching and research.

In addition, Baylor Medical School includes smaller departments with programs in Allied Health and Biological Sciences, but data are not available on them in any disaggregated fashion, whereas they are available under the U.T. System. In the interests of comparability, therefore, we absorb SAH and GSBS into the U.T. Medical School, so that Baylor and UTMS may be considered to conduct essentially the same programs.

USAGE COSTS

Our primary focus in this paper is on determining the fixed charge components of institutional assessments. For completeness, however, we include the following brief discussion on usage costs. Eight direct services were considered:

1. Self service - use of material in building, study space

April 1991 The Arabian Journal for Science and Engineering, Volume 16, Number 2B. 289

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1. 1. Rousseau

2. Circulation

3. Interlibrary loans - within TMC

4. Reference questions under ten minutes - includes telephone holds, paging, holds and rushes, etc.

5. Reference questions over ten minutes 6. Interlibrary borrowing

7. Manual bibliography searches

8. Computerized searches - MEDLINE, SDILINE, SOC services, etc.

Using estimates by library personnel for direct costs and variable support costs for each service pro

vided, a goal programming model was developed to determine unit charges for the services. ([2], Part II).

The usage cost for each service for each institution was then obtained using projections (based on survey techniques) of usage. data. Unit charges and usage costs are reproduced here in Table 1. For later pur

poses of comparison we derive the schedule of fixed charges implied by actual institutional assessments in 1976 by subtracting the usage costs of Table 1 from those assessments. These implied fixed charges are given in Table 3, below.

JOINT FIXED SUPPORT COSTS

The fixed portion of the annual support costs of operating a library typically include costs associated with library acquisitions (such as purchases of books and serials); costs of different types of personnel; cer

tain administrative costs (such as legal and profes

sional fees, supplies, etc); utilities and maintenance;

and expenditures on various types of equipment.

Such non-usage related costs must be incurred to ac

commodate the demands of all the member institu

tions as a group. However, distinct differences in in

formational requirements and quality and type of facilities demanded may exist among the member in

stitutions. Such differences are determined by the various (non-usage related) characteristics peculiar to different institutions, such as the set of academic pro

grams offered, accreditation standards, and potential user population. However, there is little or no quan

titative evidence that systematically relates the level of support costs back to the institutional characteris

tics which generate these costs.

Therefore we make the following observation and assumption. The relevant costs are those which must be incurred in order to maintain a certain quality and type of library as dictated 6y the member institutions' diverse requirements, irrespective of library usage levels. The initial set-up costs of the library are the fixed joint costs of constructing and equipping this

quality and type of facility in the first place, and are determined by this same set of (non-usage related) institutional characteristics. On this basis we asssume (as a first approximation) that, for any group of in

stitutions, the fixed portion of annual support costs of operating a library to accomodate their joint re

quirements would be proportional to the fixed joint set-up costs of that library; and that the constant of proportionality would be the same for all groups of institutions. In this way the relative ranking of institu

tions would remain the same for either initial set-up costs or fixed annual support costs.

Bres et al. [1, 2] have constructed a general expres

sion that defines, for any group of institutions, the initial costs of building and equipping a library to meet that group's requirements. The expression is derived from such items as purchases of books and journals; purchases of equipment, furniture, and fur

nishings; accreditation standards; and building con

struction costs. The cost for any group of institutions is ultimately determined both directly and indirectly, by two factors: the size and type of collections of books and journals required to support that group's programs, and the amount of seating and study capacity which that group needs to have available at all times. These two factors reflect the essential (non

usage related) differences between institutions, namely, differences in the amount, quality, and type of their informational requirements and the facilities demanded.

The HAM-TMC Library collection consists of ap

proximately 133 000 books and serials, encompassing all fields of medicine, nursing and some special areas of research, different amounts of which are required by the different institutions to support their pro

grams. We proceed on the basis that all institutions require a core collection consisting minimally of a certain amount of literature on general medicine and nursing. Differences in informational requirements between institutions are then reflected in the extent of this collection and the amount of specific research items needed.

Six institutions (DB, SPH, UTMDA, PVAMUSN, TRIMS, and VA) already have their own collections consisting either of specialized materials relevant to their particular fields or smaller quantities of much the same type of literature as that at the HAM-TMC Library. These institutions essentially use the HAM

TMC collections as supplementary literature for back-up purposes only, although to varying degrees.

290 The Arabian Journal for Science and Engineering, Voillme 16, Number 2B. April 1991

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). "1:::l ~ ... 'Cl 'Cl ... Table 1. Unit Charges and Usage Costs for Library Services (dollars) S.S. CIRC ILL REF < 10 REf> 10 ILB MAN.BIB COMP.S TOTAL UNIT CHARGE 0.30 l.70 2.19 2.50 5.00 7.00 59.36 12.33 INSTITUTIONS BAYLOR 37468.64 87254.13 0.00 11867.50 736.43 2988.68 712.30 12588.10 153615.77 HAM 2480.41 5776.17 0.00 5285.00 407.20 231. 71 2552.40 5252.23 21985.11 TWU 22750.33 52979.36 0.00 7782.50 519.83 278.72 0.00 1997.33 86308.08 UTHSC (28388.06) (64716.88) (1 079.35) (9607.50) (706.11) (1 527.92) (1543.31) (12057.94) (119627.06) MS 13 607.86 30299.40 0.00 4285.00 424.53 648.11 1424.59 8568.78 59258.28 GSBS 4157.73 9680.18 0.00 535.00 30.32 560.80 0.00 554.81 15518.85 SAH 1010.28 2352.94 0.00 167.50 0.00 0.00 0.00 0.00 3530.72 ;i SHI 760.93 1 771. 92 0.00 240.00 21.66 80.59 0.00 86.30 2961.41 ~ ). UTSSN 4946.17 11518.36 0.00 2335.00 177.61 23.51 0.00 172 .61 19173.25 ... s:';) 0s:; . ::::

DB SPH

1618.13 2286.96 3768.10 5325.97 579.16 500.18 570.00 1475.00 0.00 51.98 40.30 174.62 0.00 118.72 357.55 2317.89

6933.23 12251.32 ... ~ UTMDA 2216.40 5 161.18 4501.66 1037.50 30.32 0.00 0.00 0.00 12947.06 ~ PVAMUSN 1207.32 2811.64 0.00 300.00 8.66 0.00 0.00 24.66 4352.27 ~ Ci' TIRR 337.86 786.58 4975.52 355.00 8.66 0.00 0.00 702 .76 7 166.38 ... v; TRIMS 417.99 973.45 3869.85 240.00 43.32 0.00 0.00 789.07 6333.68 r, ~. :::: I RELIGION 436.22 1015.93 0.00 132.50 0.00 33.58 0.00 234.25 1 852.49 r, ~ HCHD 850.02 1979.19 0.00 185.00 8.66 0.00 0.00 234.25 3257.13 s:';) :::: ~ HERMANN 1630.38 3796.98 236.93 342.50 0.00 23 .51 0.00 382.20 6412.51 ~ oti s· METHODIST SLfTCH

1658.19 1488.36 3861.54 3465.70 0.00 1026.69 595.00 565.00 30.32 82.31 23.51 23.51

0.00 0.00

320.56 739.75 6489.12 739l.33

~ ~ MEMORIAL 113.02 263.33 0.00 25.00 0.00 6.72 0.00 0.00 408.06 ~. ~ ST.l0SEPH 129.79 302.40 0.00 72.50 8.66 0.00 0.00 0.00 513.32 VA 448.78 1044.81 0.00 265.00 30.32 6.72 0.00 0.00 1 795.63

2:

§ 1'i) 102021.75 38657.50 2620.83 ~ TOTALS 236189.25 15690.00 5144.56 4808.00 35323.11 440455.00 ... ?~ ~ :::0 § 0

\:) :;:: c;" ~ c;" tv ~

(';) l::l :;:: ~ """" ."NOI.L V)/7dclY !JNUFJ::lNI!JN3 (JN t/ ~{~/().:JlJl_ :NOll. \12 I J1'I.Ld () :3l\'.:JI~/.L

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1. 1. Rousseau

It will prove convenient to consider the composite

"institution" comprised of UTMS, GSBS, SAH, SHI, and UTSSN. This grouping of institutions pre

sents no real administrative problems since their fees traditionally come from the same source. If neces

sary, subdivision of the composite fee may be deter

mined along lines similar to those presented here.

The advantage of this grouping is that the entire HAM-TMC collection is now required by this com

posite institution and every other institution requires only some proportion of this total in its core collec

tion. In a similar manner we can argue that this com

posite institution dictates the extent to which capacity is provided and every other institution requires only some fraction of this capacity.

In this way, since we are referring to the same general type of literature and type of capacity, we can now approximate the requirements of any group of institutions by the requirements of the "largest" indi

vidual member of that group. Although this one hundred percent overlap of institutional require

ments may not be a strictly accurate description for a few of the coalitions, it certainly is the case for the overwhelming majority of the one-quarter million

plus involved, and it is expected that any error intro

duced will be relatively slight. Moreover, this approx

imation considerably simplifies calculation of the characteristic function values and the solution proce

dures for our game. We note that precise figures are not essential here; rather, we are concerned with methodologies for allocation of cost.

On this basis we develop an index based on library personnel estimates for each of the collection re

quirements and capacity factors in terms of which the different institutional types can be ranked. Library records show that historically approximately sixty percent of annual expenditures are directly or indi

rectly related to collections of books and journals, so that an overall index can then be obtained by standar

dizing the indices and combining them in the propor

tion 3:2.

The overall index then gives a reasonable estimate of the proportions in which initial set-up costs (and hence , by assumption, annual fixed suppor~ costs in

cluding utilities and maintenance) would be gener

ated by different institutional types.

These index numbers (IJ, standardized to unity for the "largest" institutional type (our composite institu

tion which we call UT), are given in Table 2, where the institutional types have been labeled in increasing order of "size".

Table 2. Institutional Fixed Support Costs

Institution Institution Number Fixed Support Fixed Support

Type Institutions Cost Index Cost

ni Ii gi

VA 0.0111 $ 9257

2 TRIMS 0.0323 26937

3 PVAMUSN 1 0.0351 29272

4 TIRR 8 0.0429 35777

IR 0.0429 35777

HCHD 0.0429 35777

HERMANN 0.0429 35777

METHODIST 0.0429 35777

SLrrCH 0.0429 35777

MEMORIAL 0.0429 35777

ST. JOSEPH 0.0429 35777

5 SPH 0.0509 42449

6 DB 0.0547 45 618

7 UTMDA 0.1370 114254

8 TWU 0.1554 129 599

9 HAM 1 0.9062 239 100

10 BAYLOR 0.9062 755746

11 UT 1 1.0000 833 973

292

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Total annual fixed support costs are obtained by subtracting total usage costs from the annual budget, and amounted to $ 833 973 in 1976. Annual fixed support costs for different institutional types are related by the equations.

(6) gll

=

833973

which may be solved for the parameters gi given in Table 2.

BENEFITS FROM INSTITUTIONAL MEMBERSHIP

A complete application of the club principle requires reasonable estimates of the annual benefits derived by institutions from membership in the library system. Such estimates are difficult to obtain and no totally suitable proxy measure is available.

A consumer surplus approach might be considered but there is little or no data on past "prices", "quantities", and "elasticities" from which to derive a (linear) demand curve for each institution.

Moreover, it is difficult to determine a suitable measure for "quantity of library services" since this means different things to different institutions due to their varying usage patterns and extent of informational requirements. The annualized cost of constructing, equipping and operating a library to meet each individual institution's requirements may be an appropriate measure of benefit but it is impossible to derive from the present data.

Instead, we develop assessment rules based on the cost function alone with no regard to benefits, subsequently making a (partial) check that resulting assessments satisfy ability-to-pay by comparing them with actual payments made by institutions in 1976, which are probably underestimates of benefits received. We note here that this is equivalent to using the sum of usage costs and (approximated) j<?int fixed support costs for each institution as a lower bound on benefit in the more general procedure. We shall now consider the three issues raised at the end of the introduction.

OPTIMAL SIZE OF LIBRARY SYSTEM

Without an appropriate measure of benefit it has not been possible to determine whether the existing size of library operations is optimal, that is, whether the library system should be expanded to accommo-

1. 1. Rousseau

date other health-related institutions, remain as is, or be contracted in scope. Moreover, even with benefit figures there seems to be no shortcut way to determine optimal size, circumventing linear programming problem I.

Since usage patterns vary across services and even across institutions of the same type, it is not to be expected that usage costs (for institutions or for any given service} will follow the same hierarchy as for joint fixed support costs. As a result it is not possible to resort to "average" benefit and usage cost figures for each type of institution which would have greatly simplified matters since the optimal policy would then have been to accommodate all institutions of all types up to but not exceeding some critical size. The primal problem I would then have reduced to a comparison of the m + 1 values of

n(;S

N;) =;ton;(b; - c;) - gk

over

^k⁼0,1, ... , m wht;re bi and ^Ci denote average benefit and usage cost, respectively, for an institution of type i.

However, since our joint fixed support cost function is concave we do know that it will be optimal to operate precisely one library, although this need not be a unique optimum unless strict concavity holds. In addition, we know that if each single member library system has a positive net benefit, than the system consisting of all potential members N is an optimum.

The 1976 INSTITUTIONAL ASSESSMENTS In this section we may confine our attention to the fixed charge components of the institutional assessments. The actual 1976 assessments and the contributions to joint fixed support costs implied therein are given in Table 3. The implied fixed charges are obtained by subtracting the appropriate usage costs of Table 1 from the total 1976 assessment for each institution.

The implied charges are (by definition) within ability-to-pay and recoup total fixed support costs, but they are not quite efficient since they result in some cross-subsidization among institutions. For example, the six smallest institution types are collectively charged close to $ 70 000 but the cost of providing for that group alone is less than $ 46 000; likewise, the first five institution types are collectively overcharged by almost $ 5000. The group consisting of types 4, 5, and 6 and the group consisting of types 4 through 8 are both overcharged by $17 000. The subsets of institutions consisting of all but the three largest types

April 1991 The Arabian Journal for Science and Engineering, Voillme 16, Number 2B. 293

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1.1. Rousseau

Table 3. Contributions to Joint Fixed Support Costs

Institution Institution Total Implied Fee Shapley Nucleolus

Type 1976 in 1976 Value

Assessment Assessmentⁱ

1 VA $ 2ROO $ 1004.37 $ 514.2R $ 29Rl.42

2 TRIMS R260 1926.32 1554.28 2981.42

3 PVAMUSN R284 3931.73 1700.22 29R1.42

4 TIRR 10 129 2962.62 2133.89 29Rl.42

IR 6751 4R9R.51 2133.89 29R1.42

HCHD 6442 3184 .87 2133.R9 2981.42

HERMANN R121 1708.49 2133.89 2981.42

METHODIST 11 687 5197.8R 2133.89 2981.42

SUTCH 1[406 4014.67 2133 .89 29Rl.42

MEMORIAL 3600 3191. 94 2133.89 2981.42

ST. JOSEPH 3606 3092.6R 2133.89 29R1.42

5 SPH 24360 12 108.68 30R7 .03 4274.14

6 DB 29 198 22264.77 3615.20 4274.14

7 UTMDA 36042 23 094.94 17 342.40 29 418.38

R TWU 147 267 60 958.92 21 178.65 29 418.38

9 HAM 118 261 96275.R9 57 678.98 69 459.69

10 BAYLOR 425 37R 27[ 762.23 316 001. 98 293 052.85

11 UT 4 [2 837 312 394.49 394228.9R 371 279.85

I Obtained by subtracting the usage costs of Table 2 from actual 1970 assessments.

and all but the two largest types are overcharged by

$24 000 and $ 10 000, respectively.

Something of a pattern appears to be present in the implied fee structure. Although each individual in

stitution is charged less than its stand alone cost, the

benefits from the economies of scale inherent in their cooperation seem to be disproportionately in favor

of the very large institutions. The four smallest types

are collectively charged only slightly less than the cost to accommodate that group alone. Coalitions of the middle sized institutions are either overcharged or

paying essentially "breakeven" fees. The two largest

institutions, on the other hand, enjoy significant sav

ings from their membership.

The implied charges also do not satisfy our twin conditions of fairness, (4a , b). TRIMS (type 2) is charged more than Hermann (type 4). PVAMUSN (type 3) is charged more than five institutions of the next largest type. Even among institutions of the

same type 4 there is variation in the charges.

Moreover, the difference between the charges for types 6 and 5 is $ 10 000 and yet the cost difference is only $3200; types 8 and 7 have a charge differential of $ 38 000 but a cost difference of only $ 15 000. By

way of contrast the largest institutions have charge

differentials significantly less than their correspond

ing cost differences, which again indicates a bias in

the implied fee structure against the middle sized in

stitutions.

GAME THEORETIC FEE ASSESSMENTS We here discuss the characteristics of two

schedules of fixed charges derived from the Shapley

value and nucleolus notions of solution in game theory.

As demonstrated by Littlechild and Owen [8], there is a remarkable simplification in the formula (5) for the Shapley value to

gl-g l - l .

YI = YI- 1+ ^11/ ^'¹⁼1, ... , m. (7)

k =

Ink

I

Intuitively, the annual fixed support costs of operat

ing the library required by the smallest institution type is divided equally among all institutions of all types. The incremental cost of accommodating the next smallest institution type is divided equally among all institutions of all types except the smallest.

294

The Arabian Journal for Science and Engineering, Volume 16, Number 2B. April 1991

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We continue in this fashion until the incremental cost of accommodating the largest type (over the next to largest type) is divided equally only among institutions of the largest type.

Littlechild [10] has also derived a simple expression for the (negative) nucleolus of a characteristic function cost game exhibiting the present special structure.

i

Letting Mi

= L

^ni'the (negative) nucleolus is given

by i= I

Yi

=

rt: for it: _ 1< i:::; it: and k

=

1, ... , k', (R) where rt: and it: are defined inductively by

(9)

and it: denotes the (largest) value of i for which the above expression attains its minimum, beginning with ro

=

^io

=

^go

=

^{0 and}continuing for k

=

^1,^...^,^k'

where i_k'

=

m. The Shapley value and nucleolus based contributions to joint fixed support costs are given in Table 3. In contrast to the implied fixed charges of the previous section, both the Shapley value and nucleolus based fees are fair in both senses of (4a, b). They also recoup total fixed costs and avoid cross-subsidization, and hence they are also efficient if they satisfy ability-to-pay.

Using the implied fee structure as a (conservative) lower boundary on ability-to-pay, the Shapley value based contributions do satisfy this condition for all institutions except the two largest (Baylor and UT) and one of the smaller types (Hermann), whereas the nucleolus satisfies this condition for eleven of the eighteen institutions but with no clear pattern established. We note that the nucleolus allocation is con- sistently higher than that of the Shapley value except for the two largest institutions. Clearly, the nucleolus slightly favors the largest institution types at the ex- pense of all others whereas the Shapley value has the opposite effect, but in both cases the bias against the middle sized institutions is corrected.

Finally, we note that for a given optimal size of library the nucleolus fee structure is independent of benefits [11] ~ this is not the case, however, for the Shapley value.

1. 1. Rousseau

SUMMARY AND CONCLUDING REMARKS This paper has been concerned with the problem of recovering the annual operating costs of a library through assessments to the various diverse institutions (members of the library system) that jointly support and are served by the library. To maintain cooperative support and stability, the fees assessed to the institutions must be both equitable and efficient. We have argued that a reasonable and equitable charging mechanism, therefore, might recover the direct costs of services provided and the variable portion of support costs through usage based fees, and the joint fixed support costs through institutionally based (or "membership") fees based on the club principle of Littlechild [5]. Linear programming and game theoretic techniques were used to address such issues as optimality, efficiency, and fairness; and the Shapley value and nucleolus notions of solution were discussed as particular rules for cost allocation.

This paper extends the work of Bres et al. [1,2] in two directions, incorporating notions of fairness as well as efficiency and attempting to address questions of the optimal size of library facilities. For conveni- ence and purposes of comparison the model was applied to the Houston Academy of Medicine - Texas Medical Center Library, employing the data of Bres et al. which reflected operating conditions in 1976. We established that a single library was optimal, but with the present data we were unable to determine the optimal size of the library system. The fixed charge components implied by actual 1976 assessments did not meet the fairness and efficiency criteria we proposed, and appeared to be biased against the middle sized institutions. By way of contrast, both our Shapley value and nucleolus alloca- tions were fair and also efficient provided they satis- fied ability-to-pay; this latter criterion was close to being met for all institutions even though a conservative measure of ability-to-pay was employed. In addition, both fee schedules corrected the aforemen- tioned bias.

The calculations of costs used in the present application were exceedingly rough and ready and some of our assumptions represent convenient simplification, but they suffice to illustrate the principles involved. The results of the analysis offer encourage- ment for the procedures used, leading to greater insights than previously available, but a more serious application with current data would be necessary be- fore any specific policy recommendations could be provided. In principle the model could be generalized

April 1991 The Arabian Journal for Science alld Engineering, Volume 16. Number 2B. 295

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1. 1. Rousseau

to incorporate a more complex joint fixed support cost function, multiple time periods, and explicit measures of benefit; of course, the attendant compu

tations would become more difficult, especially in terms of calculating various game theoretic notions of solution. Alternative notions of solution could also be explored. In other work already underway the mollifier representations and disruption solutions of Charnes, Rousseau , and Seiford [12] are being explored as distributive devices.

An important issue that arises in a multiperiod model is the possible fluctuation in the number of members due to addition and/or withdrawals from the library system. Complications from a fairness perspective immediately arise since new members will instantly benefit from a system already built up and paid for by other, particularly the largest, institu

tional types. On the other hand, withdrawals from the system leave the remaining members with a smal

ler base over which to distribute the financial load with no increase in benefit received. Such extensions, however, are beyond the scope of the present paper.

REFERENCES

[1] E. Bres, A. Charnes, D. Cole Eckels, S. Hitt, R. Ly

ders, J. J. Rousseau, K. Russell, and M. Schoeman,

"Costs and Their Assessment to Users of a Medical Library, Parts 1-4", CCS Reports 301-304 Center for Cybernetic Studies, The University of Texas at Au

stin, 1977.

[2] E. Bres, "Costs and Their Assessment to Users of a Medical Library: A Game Theoretic Method for Al

locating Joint Fixed Costs", in Applied Game Theory,

ed. S. J. Brams, A. Schotter and G. Schwodiauer, Wurzburg-Wien; Physica-Verlag, 1979, p. 334.

[3] R. H . Coase, "The Marginal Cost Controversy", Economica, 13(3) (1946), p. 169.

[4] J. Wiseman, "The Theory of Public Utility Pricing An Empty Box", Oxford Economic Papers, 9 (1957), p.56.

[5] S. C. Littlechild, "Common Costs, Fixed Charges, Clubs and Games", Review of Economic Studies, 42(1) (1975), p. 117.

[6] S. C. Littlechild and G. F. Thomson, "Aircraft Land

ing Fees: A Game Theory Approach", The Bell Jour

nal of Economics, 8(1) (1977), p. 186.

[7] L. S. Shapley, "The Value of an N-person Game", ed.

H. W. Kuhn and A . W. Tucker, in Contributions to the Theory of Games, Vol. 11. Princeton University Press, 1953, p. 307.

[8] S. C. Littlechild and G. Owen, "A Simple Expression for the Shapley Value in a Special Case", Management Science, 20(3) (1973), p. 370.

[9] D. Schmeidler, "The Nucleolus of a Characteristic

Function Game" , SIAM Journal of Applied Mathema

tics , 17(6) (1969), p. 1163.

[10] S. C. Littlechild, "A Simple Expression for the Nuc

leolus in a Special Case", International Journal of Game Theory, 3(1) (1974), p. 21.

[11] S. C. Littlechild and G. Owen, "A Further Note on the Nucleolus of the 'Airport Game"', International Journal of Game Theory, 5(3) (1977), p. 91.

[12] A. Charnes, J. J . Rousseau, and L. Seiford, "Comple

ments, Mollifiers and the Propensity to Disrupt", In

ternational Journal of Game Theory , 7(1) (1978), p.

37.

Paper Received 21 August 1989; Revised 4 June 1990.

296