# Cobb Douglas PPT

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<p>Application of the Cobb-Douglas Production Model to Libraries</p>
<p>Robert M. Hayes 2005</p>
<p>Overview Production Functions in Economics Applicable to Libraries? Testing on Public Library Data Optimization Application to Academic Library Data</p>
<p>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).</p>
<p>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.</p>
<p>Properties of Production Functions It is generally assumed that a production function, F(L,C), satisfies the following properties: </p>
<p>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)</p>
<p> At a given set of inputs (L,C), the production function may show decreasing, constant, or increasing returns to scale: </p>
<p>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</p>
<p> 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).</p>
<p>The Cobb-Douglas Production Function The simplest production function is the Cobb-Douglas model. It has the following form:</p>
<p>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.</p>
<p>Alternative forms Equivalent is a linear function of the logarithms of the three variables: log(Q) = log(a) + b log(L) + c log(C)</p>
<p> 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).</p>
<p>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.</p>
<p>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</p>
<p>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.</p>
<p>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.</p>
<p>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.</p>
<p>Capital Investment in the Library Capital Investment in Collection Capital Investment in Facilities Capital Investment in Technical Services</p>
<p>Service Costs in the Library Services staff Effect of Branch Library Operation Effect of Reference Services Effect of Departmentalization</p>
<p>Balance of Capital and Service Staff The Cobb-Douglas Model</p>
<p>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.</p>
<p>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</p>
<p>Circ is the circulation Srvst is the service staff Coll is the collection size</p>
<p>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.</p>
<p>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</p>
<p>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</p>
<p> These data present a qualitatively consistent picture, showing a high correlation between circulation per staff member and size of collection per service staff member.</p>
<p>Generalization to other States Illinois Public Libraries Ohio Public Libraries Missouri Public Libraries</p>
<p> 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</p>
<p> 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.</p>
<p>Resulting Estimates of Parameters The following table summarizes the results:</p>
<p>120 of 173 California 454 af 567 Illinois 230 of 251 Ohio 122 of 121 Missouri</p>
<p>log a .709 .633 .617 .670</p>
<p>1-b .654 .676 .691 .631</p>
<p>R .70 .79 .78 .64</p>
<p> 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.</p>
<p>Discussion of Variance Effect of Multi-collinearity Effect of Demographic Factors</p>
<p>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 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.</p>
<p>Effect of Multi-Collinearity - 2 To illustrate, consider the following two equations:</p>
<p>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.</p>
<p>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:</p>
<p>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.</p>
<p> 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.</p>
<p>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 marketoriented 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.</p>
<p>Effect of Demographic Factors - 2 Consider the following log linear form which combines Cobb-Douglas with some demographic factors: log(y0) = ailog(yi)i=0</p>
<p>6</p>
<p>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)</p>
<p>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: 21 Cobb-Douglas 2 Income 3 Education 4 Percent in school 5 Density of population 6 Distance ai .642 .101 1.856 .126 .076 .077 R .446 .119 .050 .030 .025 .005 beta .606 .067 .225 .236 .473 .241</p>
<p> These account for 67.5% of the variance (an R > .80).</p>
<p>Optimization Central Library Branch Libraries Division Between Central Library and Branches</p>
<p>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, letP=aX1bX2(1-b) - k(XlC1+X2C2 -TR).</p>
<p> 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)</p>
<p> This gives the following as the design equationsC1X1 = bTR C2 = (1 - b)TR</p>
<p>Optimization for Branch Libraries - 1 The management decisions where branches are involved, however, are more complex than is implied in the CobbDouglas model taken alone, since the effect of the inversedistance 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.</p>
<p>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 sy...</p>