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Application of the Cobb-Douglas Production Model to Libraries
Robert M. Hayes 2005
Overview 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=aLbCcwhere 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 CobbDouglas 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
Circ is the circulation Srvst is the service staff Coll is the collection size
Application to Public Libraries To see whether the Cobb-Douglas production model is applicable to public libraries, detailed data (1976) for several statesCalifornia, Illinois, Ohio, Missouri, Wisconsinwere 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)78 largest libraries 76 (not including LAPL or LA County) 35 of 78 with incomes less than $1,000,000 All 173 libraries 120 with incomes less than$1,000,000 log a .804 .806 .670 .770 .700 1-b .590 .590 .770 .592 .654 R .68 .68 .80 .67 .70
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:State California Illinois Missouri Ohio Number of Libraries 173 567 129 251 Average Collection 253,000 40,000 100,000 151,000 Average Budget $1,107,000 $143,000 $192,000 $254,000
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:
120 of 173 California 454 af 567 Illinois 230 of 251 Ohio 122 of 121 Missouri
log a .709 .633 .617 .670
1-b .654 .676 .691 .631
R .70 .79 .78 .64
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 interrelatedmulti-collinear. Both staff and collection are highly correlated with each other and with circulation. It