Application of the Cobb-Douglas Production Model to Libraries

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Application of the Cobb-Douglas Production Model to Libraries

  • Robert M. Hayes
  • 2005


  • Production Functions in Economics
  • Applicable to Libraries?
  • Testing on Public Library Data
  • Optimization
  • Application to Academic Library Data

Production Functions - 1

  • In economics, a “production function" describes an empirical relationship between specified output and inputs. A production function can be used to represent output production for a single firm, for an industry, or for a nation. Just to illustrate, a production function of a wheat farm might have the form:
  • W=F(L,A,M,F,T,R)
  • That is, production of wheat in tons (W) depends on the use of labor measured in days (L), land in acres (A), machinery in dollars (M), fertilizer in tons (F), mean summer temperature in degrees (T), and rainfall in inches (R).

Production Functions - 2

  • In most applications of production functions, the input variables are simply labor (L) and capital (C). The output is usually measured by physical units produced or, perhaps, by their value.
  • Labor is typically measured in man-hours or number of full-time-equivalent (FTE) employees.
  • Capital is the variable that usually is most problematic. While data on output and labor are readily available, those for capital are not. It represents aggregations of diverse components, of different characteristics and vintage. Furthermore, only capital that is actually utilized should be treated as input, but it is difficult to determine the extent to which that is so.

Properties of Production Functions

  • It is generally assumed that a production function, F(L,C), satisfies the following properties:
    • F(L,0) = 0, F(0,C) = 0 (both factor inputs are required for output)
    • dF/dL > 0, dF/dC > 0 (an increase in either input increases output)
  • At a given set of inputs (L,C), the production function may show decreasing, constant, or increasing “returns to scale”:
    • If F(L, C) < F(L,C), there are decreasing returns to scale
    • If F(L, C) = F(L,C), there are constant returns to scale
    • If F(L, C) > F(L,C), there are increasing returns to scale
  • Constant returns to scale imply that the total income from output production equals the total costs from inputs:
  • pF(L,C) = wL + rC
  • (p the price per unit output, w and r costs of labor and capital).

The Cobb-Douglas Production Function

  • The simplest production function is the Cobb-Douglas model. It has the following form:
  • Q=aLbCc
  • where Q stands for output, L for labor, and C for capital. The parameters a, b, and c (the latter two being the exponents) are estimated from empirical data.
  • If b + c = 1, the Cobb-Douglas model shows constant returns to scale. If b + c > 1, it shows increasing returns to scale, and if b + c < 1, diminishing returns to scale.

Alternative forms

  • Equivalent is a linear function of the logarithms of the three variables:
  • log(Q) = log(a) + b log(L) + c log(C)
  • If b + c = 1, another equivalent form exhibits an underlying heuristic for the Cobb­Douglas model:
  • log(Q/L) = log(a) + (1 - b) log(C/L).
  • which says that the "production per employee" (Q/L) is a function of the capital investment per employee (C/L).

Market and Production

  • It should be recognized that allocation decisions must be concerned not only with productivity but with response to market demand. The manager must decide both how much should be invested in total, as determined by the market, as well as how resources should be divided between capital and labor but. If the relationship is homogeneous, the two decisions may be treated as independent, but if it is not homogeneous an optimum allocation from the standpoint of productivity could be inconsistent with the optimum from the standpoint of market.

Allocation of Resources in Libraries

  • Are production functions applicable to libraries?
  • Balance between Capital and Labor in Libraries
  • Output as represented by demand for services
  • Capital as represented by the Collection
  • Labor as represented by Services Staffing
  • Other determinants of demand for services

Are production functions applicable? - 1

  • The first issue that needs to be considered is whether it really make sense to discuss the relationship between "productivity" and the allocation of resources in the library or, indeed, in any service industry?
  • The answer is not obvious. In manufacturing, labor uses capital resources to produce a tangible product. The allocation of resources intuitively may be regarded as causal in its effect on production.
  • The market is in that respect separable from production, and one can determine optimal conditions for production.

Are production functions applicable? - 2

  • For the library, as for most service industries, however, the relationship between output and the allocation of resources is not at all clearly causal. In service industries, "production" is in the delivery of services in response to demand for them. While increased staff or capital resources may be needed to serve increased demand, it is not clear that they will generate increased demand. As a result, while models like the Cobb-Douglas may evidently apply to manufacturing industries, it needs to be demonstrated that they apply to service industries.

Capital and Labor in Libraries

  • But, recognizing that complexity, the question of how resources are to be allocated between capital and labor is a crucial management decision. It may be based on a view of causality between production and resource components; it may be based on the need to respond to demand for services. But, in either event, it should represent the optimum allocation within the constraints of the relationships among the variables involved.

Capital Investment in the Library

  • Capital Investment in Collection
  • Capital Investment in Facilities
  • Capital Investment in Technical Services

Service Costs in the Library

  • Services staff
  • Effect of Branch Library Operation
  • Effect of Reference Services
  • Effect of Departmentalization

Balance of Capital and Service Staff

  • The Cobb-Douglas Model
    • The Capital component of the Cobb-Douglas model will be measured by the size of the Collection of the library. It is assumed that costs in acquisition of it, in facilities to house it, and in the technical services staff for building the collection are proportional to the size of the collection.
    • The Labor component will be measured by the services staff, which will be calculated as the total staff minus the technical services staff
    • The Production will be measured by the circulation, as a surrogate for all of the uses of the library and its services.

Estimation of the Parameters

  • The log-linear form of the Cobb-Douglas model will be used to estimate the parameters:
  • log(Circ/Srvst) = a + (1 – b) log(Coll/Srvst)
  • where

Application to Public Libraries

  • To see whether the Cobb-Douglas production model is applicable to public libraries, detailed data (1976) for several states—California, Illinois, Ohio, Missouri, Wisconsin—were used to determine the relevant parameters.
  • Data for a portion of the California libraries (the 78 serving the largest populations) provided the primary basis for exploration of the Cobb-Douglas model, while those for the rest of the California libraries and for the other states and national libraries served as the means for testing and evaluating the results.

Testing on Data for California Libraries

  • 78 largest libraries
  • 76 largest, not including LAPL or LA County
  • All 173 libraries
  • 35 of 78 largest with budgets less than $1,000,000
  • 120 of all 173 with budgets less than$1,000,000

The Results for California Libraries

  • log (Circ/Srvst) = a + (1-b) log(Coll/Srvst)
  • These data present a qualitatively consistent picture, showing a high correlation between circulation per staff member and size of collection per service staff member.

Generalization to other States

  • Illinois Public Libraries
  • Ohio Public Libraries
  • Missouri Public Libraries

The overall size of libraries in each of these states is relatively smaller than those in California:

  • The overall size of libraries in each of these states is relatively smaller than those in California:
  • In order to make comparison more meaningful, the estimation of the parameters was limited to libraries with income of less than $1,000,000 in California and each of the other states.

Resulting Estimates of Parameters

  • The following table summarizes the results:
  • In summary, the Cobb-Douglas equation appears consistently to describe the behavior of libraries of a size determined by budget of less than $1 million, across a set of four states (California, Illinois, Ohio, and Missouri). In each case, there is a relatively high correlation. There is close agreement among the values for the parameters for the four regressions.

Discussion of Variance

  • Effect of Multi-collinearity
  • Effect of Demographic Factors

Effect of Multi-Collinearity - 1

  • The use of regression equations is an easy way to deal with the kind of analyses involved in evaluating the Cobb-Douglas equation. However, although easy, it is a way fraught with pitfalls. In particular, the variables involved are closely interrelated—multi-collinear. Both staff and collection are highly correlated with each other and with circulation. It is therefore easy to investigate equations that will almost automatically result in high correlation, but will simply reflect the self-evident correlations.
  • In particular, different forms of the Cobb-Douglas equation, though arithmetically equivalent, can exhibit radically different correlations.

Effect of Multi-Collinearity - 2

  • To illustrate, consider the following two equations:
  • log CIRC = log(a) + blog(SRVST) + (1 - b)log(COLL)
  • log CIRC/SRVST = log(a) + (1 b)log(COLL/SRVST)
  • The correlations for these two equations, for the largest 78 California libraries and for the same values of a and (1 - b), (viz., log(a) = .804 and 1 - b = .590) are, respectively, R = .96 and R = .68. The reason is simple: The first equation is controlled by the close relationship between circulation, on the one hand, and service staff and collection size on the other; the second equation depends upon the less clear-cut relation between the ratios.

Effect of Multi-Collinearity - 3

  • If this problem were treated as a multiple regression problem, in which an effort were made to represent log(CIRC) as a function of the two independent variables log(SRVST) and log(COLL), several technical problems would arise: 
    • 1. The determinant of the matrix of inter-variable correlations would be near zero, making it difficult to calculate the regression coefficients;
    • 2. As a result, the computation would provide imprecise, highly variable estimates of those coefficients; and
    • 3. There would be large sampling variances.
  • The use of the ratios, CIRC/SRVST and COLL/SRVST, significantly reduces the impact of the multi-collinearity. It permits one to obtain consistent estimates and to avoid the technical problems of multi-collinearity.

Effect of Demographic Factors - 1

  • The second, and more important, issue involved in evaluating the correlations found for the Cobb-Douglas equation arises because the library is not a market-oriented organization. The management decision with respect to allocation of resources between capital (i.e., collection) and service staff therefore is likely to account for only a portion of the variance among libraries, with at least part of the remaining variance being determined by demographic factors. 
  • The analysis presented of the California data considers only those issues affected by library management decisions and accounts for only 50% of the variance among California libraries. It is, therefore, worthwhile to assess the effect of some demographic factors.

Effect of Demographic Factors - 2

  • Consider the following log linear form which combines Cobb-Douglas with some demographic factors:
  • 6
  • log(y0) = ailog(yi)
  • i=0
  •   y0 = circ/popl,
  • y1 = a(srvst)b(coll)1-b/popl,
  • y2 = (average income),
  • y3 = (average years of education)
  • y4 = (number in school)/popl,
  • y5 = (area/popl)
  • y6 = (average distance)

Effect of Demographic Factors - 3

  • The first, y0, is the same dependent variable used before; the second, y1, is the Cobb-Douglas formula for circulation, divided by population; the remaining are the typical demographic variables.
  • The regression for California libraries on this equation, combining Cobb-Douglas and demographic factors, was as follows:
  •  These account for 67.5% of the variance (an R > .80).


  • Central Library
  • Branch Libraries
  • Division Between Central Library and Branches

Optimization for a Central Library

  • For a central library, the management decision is to maximize X0 = aX1bX2(1-b) , subject to X1C1 + X2C2 = TR
  •   where TR is the budget available for the central library.
  • Using a Lagrangian multiplier, let
  • P=aX1bX2(1-b) - k(XlC1+X2C2 -TR).
  • To maximize P, take partial derivatives:
  • dP/dX1 = abX1(b-1) X2(1-b) – kC1 = 0
  • dP/dX2 = a(l - b)Xl b X2(1-b) – kC2 = 0
  •   dP/dk = X1C1 + X2C2 - TR = 0
  • Taking the ratio of the first iwo equations,
  • C1/C2 = (b/(1-b))(X2/X1)
  • This gives the following as the design equations 
  • C1X1 = bTR
  • C2 = (1 - b)TR

Optimization for Branch Libraries - 1

  • The management decisions where branches are involved, however, are more complex than is implied in the Cobb-Douglas model taken alone, since the effect of the inverse-distance law on utilization of a library makes the number of branches crucial. Assuming that the inverse distance law is applicable, take XB= (B/B0)X0
  • That is, given the actual circulation, X0, for a given number of branches, B0, the circulation for another number of branches, B, would be proportional to B/B0.
  • However, a change in the number of branches would also change the number of service staff needed, resulting in a different distribution of resources between service staff and collection, and lead to changes in circulation as represented in the Cobb-Douglas model.

Optimization for Branch Libraries - 2

  •  Assume the following form for the Cobb-Douglas model: XB = (B/B0) a(Bm)b (X2)(1-b)
  • where m is the minimum staffing required per branch.
  • We want to choose B and X2 so as to maximize XB, subject to the boundary condition that the total resources available for the branch library system are fixed:
  • BmC1 + X2C2 = TB  
  • Using a Langranian multiplier, let
  • P = (B/B0)a(Bm)b (X2)(1-b) - y (BmC1 + X2C2 - TB)
  • = (B(1+b)/B0)a(m)b (X2)(1-b) - y (BmC1 + X2C2 - TB) 

Optimization for Branch Libraries - 2

  • Taking partial derivatives with respect to B, X2, and y:
  • dP/dB = (1 + b)(Bb/B0) a(m)b (X2)(1-b) - y (mC1)
  • dP/dX2 = (1 – b)(B(1+b)/B0)a(m)b (X2)-b - y (C2)
  • dP/dy = BmC1 + X2C2 - TB 
  • Taking ratios of the first two equations,
  • mC1/C2 = ((1 + b)/(1 – b))(X2/B)  
  • From the third equation, 
  • X2C2 = (1 – b)TB/2, BmC1 = (1 + b)TB/2

Optimization for Branch Libraries - 2

  • Taking partial derivatives with respect to B, X2, and y:
  •   dP/dB = (1 + b)(Bb/B0) a(m)b (X2)(1-b) - y (mC1)
  • dP/dX2 = (1 – b)(B(1+b)/B0)a(m)b (X2)-b - y (C2)
  • dP/dy = BmC1 + X2C2 - TB
  • Taking ratios of the first two equations,  
  • mC1/C2 = ((1 + b)/(1 – b))(X2/B)
  • From the third equation above, 
  • X2C2 = (1 – b)TB/2, BmC1 = (1 + b)TB/2

Central Library and Branches

  • The design equations for a total system are therefore:
  • (1 – b)TB/2 + (1 – b)TR = CC2
  • (1 + b)TB/2 + bTR = (S – 0.05C)C1
  • TB + TR = T
  • X0 = (B/B0)(aTB/2)((1 + b)b(1 – b)(1–b) + aTRb b (1 – b) (1–b)
  • where TB and TR are the allocations of the total budget to the central library and the branch library; S is the total staff; and C is the collection size. In these equations, the service staff is represented by (S - 0.05), the total staff reduced by those committed to capital resource maintenance.

Application to Academic Libraries

  • Turning now to the application of the Cobb-Douglas model to academic libraries, two contexts will be considered:
    • Library productivity
    • Institutional productivity
  • For each, data were acquired from ARL statistics and, for institutional productivity, from Social Science Citation Index.

The following variables were acquired for each ARL library and institution:

  • The following variables were acquired for each ARL library and institution:
    • For x0, (1) Number of interlibrary loans
  • (2) Number of Ph.D.s graduated
  • (3) Number of faculty
  • (4) Number of publications
    • For x1, (1) Number of Reader Services staff
  • (2) Number of faculty
    • For x2, (1) Size of library collection
  • (2) Index of library rank (from ARL)
  • (3) Index of library size (from ARL)

Statistics were acquired for the ARL libraries and institutions for academic year 1973/74 and for the total number of publications attributable to each institution for the period 1971/­1981. Data of a similar nature were obtained for each of the past 17 years; for each of them, the analyses are quite consistent in the overall patterns.

  • Statistics were acquired for the ARL libraries and institutions for academic year 1973/74 and for the total number of publications attributable to each institution for the period 1971/­1981. Data of a similar nature were obtained for each of the past 17 years; for each of them, the analyses are quite consistent in the overall patterns.
  • Ordinary least squares regression analyses are applied to these data for the several ARL libraries. In addition to these regression analyses, some effort was made to identify major differences among the libraries.

Library Productivity

  • Measures of Production
  • Measures of Labor
  • Measures of Capital Investment
  • Results

Measures of Production- 1

  • The characterizing function of the academic research library is support to research of faculty and students, primarily doctoral students. However, this is a very difficult function to measure. Unfortunately, statistics for "circulation" were not reported in ARL statistics until 1995, but even so it is a matter of some debate concerning whether they are an adequate measure of research use.
  • On the one hand, the claim has been made that “circulation is a reasonably reliable index of all use, including the unrecorded, consultative, or browsing use within the library". On the other hand, other analyses that in-house use is significantly different.

Measures of Production- 2

  • The Faculty. If we regard the faculty as the primary research users, might not the number of them be a measure of the amount of research use made of the collection? Underlying that view are a number of assumptions (e.g., the average use of the collection by a faculty member is not a function of the library as such and will be uniform from institution to institution, even if not from faculty member to faculty member).
  • Ph.D. Graduates. In the same vein, the Ph.D. students also are heavy research users of a collection. With the same kinds of assumptions that apply to faculty, might not the number of Ph.D. graduates be a measure of the research use?

Measure of Labor - 1

  • The costs involved in technical services (basically, in selecting, acquiring, processing, and cataloging of acquired materials) are regarded as part of the capital investment.
  • Doing this requires that the staff involved in technical services be estimated, since the published data do not clearly identify it. The basis for doing so was identical with that used for public libraries
  • In the following tabulation, the FTE estimates are based on acquisitions (ACQ), using a standard ratio of 1.5 for volumes per title Serial titles are assumed to account for about one-fourth of the volumes acquired based on the apparent ratio of serial volumes to total volumes and one volume per year per serial).

Measure of Labor - 2

Measure of Labor - 3

  • Those figures are all for acquisitions. In addition, continuing with the same pattern followed in the analysis of public libraries, the labor required to maintain the physical facilities in which the collection is housed has been estimated to be .012 FTE per 1000 volumes of collection. Thus, the technical services staff has been estimated as:
  • (1.26 x ACQ) + (.012 x COLL)
  • To obtain the estimates for Reader Services Staff, this estimate of technical processing staff is then subtracted from the data, that are available, on total staff.

Measures of Capital - 1

  • The underlying rationale for applying the Cobb-Douglas model to libraries is that "information" is at least analogous to if not identical with "capital investment". It seems evident that the primary capital resource of the library in fact is its collection, so the size of collection will be used as the measure for x2.
  • However, there are serious difficulties in measuring even so basic a variable as the size of the collection. Libraries differ in whether or not they include microforms, government documents, special collections, etc. They differ in the means by which they measure any of those.

Measures of Capital - 2

  • Despite that difficulty, the analysis took the number of volumes in the collection, as reported in ARL Statistics, as the primary value for this variable.
  • An alternative measure of library resources could be the Library Resources Index, which was derived from three component variables:
    • total volumes held
    • volumes added
    • current serials.

Measures of Capital - 3

  • A third alternative is the ARL Library Index. Through factor analysis, the ARL data variables were reduced from 22 down to 10. A weight is derived for each of the variables on the factor to which they relate.
  • These weights are then applied to the variables for a given library to derive the index value for that library.
  • The weights for the ten variables in that principal factor analysis for 1980/81 were as follows (the comparable values reported for 1979/80 are quite similar):

Measures of Capital - 4


  • Using Faculty as the measure of productivity
  • Using PhD graduates as the measure of productivity

ARL-2000 Data

  • Since 1995, ARL statistics have included circulation and reference use data, so it is now possible to apply the Cobb-Douglas model with them as output measures.
  • The results are the following two regression equations:
  • log(Circ/Srvst) = -0.095 + 0.887 log(Coll/Srvst), R = 0.52
  • log((Circ+Ref)/Srvst) = 5.92 + 0.478(Coll/Srvst), R = 0.62
  • The following charts show the scattergrams for the ARL libraries that reported circulation and reference data (eight did not).
  • While the correlation values of R = 0.52 and R = 0.62 are not large, they are substantial.

Scattergram for (Circ)/(Srvst)

Scattergram for (Circ + Ref)/(Srvst)

Institutional Productivity

  • Measures of Labor
  • Measures of Capital Investment
  • Measures of Production
  • The Results

Measure of Labor

  • The "producing labor" for the university is almost self-evident. It's the faculty, with all other university staff simply being supportive to them.

Measures of Capital Investment

  • The measurement of "capital investment" for the university turns out to be quite complex. In fact, there do not appear to be any generally available or published data that describe the total capital resources in any given university. However, the size of the collection of each university's library is published data. Furthermore, use of it as a surrogate for the total capital investment will provide a valuable test of the degree to which it is indeed analogous, as an information investment, to capital investments generally. We therefore first use the size of collection as our measure or surrogate for the university's capital investment.

Measures of Production

  • The university serves many functions: education, research, and public service. The measure of "production" for the first two, at least, appears to be straightforward. Number of students graduated should measure the function of education reasonably well; number of doctoral students or number of papers published should measure the second. Focusing attention on the research functions, though, it is that which will be measure and using the two measures appropriate to it.



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