The Cobb-Douglas Production Function - Admission/Application Essay Example

Production function & cost analysis Introductory outline In economic theory production function is used to depict the relationship between inputs and output while a cost function depicts the relationship between the costs of those inputs and the final output. Thus they are interrelated functions with emphasis on the differences between the short term and the long term. The production function would show increasing, decreasing or constant returns to scale as the time period progresses with inputs being altered. While there are simple production functions with minimal variability analysis, there are also those very complex ones such as the Cobb-Douglas Production Function. The firm might vary the inputs such as labor and capital in order to increase the output but nevertheless there is no certianity or rule which guarantees increasing returns to increasing inputs or scale of operations. Thus costs might increase, decrease or remain constant given the ceteris paribus principle operates. Production and cost functions can be shown both mathematically and graphically. Analysis Mathematically and symbolically the production function is generally depicted by the following equation: Q = f(X1, X2, X3,..., Xn) or Q = f(L,K), Where, Q stands for quantity; f for function of; X1 etc. for different inputs; and L and K for labor and capital respectively. The production function shows how a certain amount of inputs would lead to a certain amount of output while the corresponding cost function would show how the costs of those inputs would relate to the output. In other words the latter function would depict different components of the total cost function such as average cost, marginal cost, fixed cost, variable cost, semi-variable cost and so on (Jehle and Reny, 2000). In a simple hypothetical situation where the firm concerned produces a certain output of widgets during a certain period of time, the production function would show the different inputs of labor, capital and land needed to produce the planned quantity of widgets, 100,000 widgets per week. Thus the equation would show, 100,000 widgets = f (45L and 8K). This is a functional relationship between inputs and output. In other words it says quantity produced is a function of the variables. This relationship is what matters in a production function because all output decisions taken in conjunction with the firm’s current and predictable market share and sales, are in conformance with the firm’s own profit expectations. Therefore it’s obvious that the production function necessarily incorporates input variables after a series of prior calculations on marginal and average costs. The subsequent calculations are determined by the expected profit margins. The following diagram illustrates this relationship graphically. Figure 1: A typical production function A quadratic production function like this would simplify an otherwise complex relationship by depicting what possibilities are there for the firm to achieve output targets. For instance while every point below the function or the curve is attainable, none of the points above the function are attainable. This might be due to the fact that technology constraints exist within the production process of the firm. The vertical axis shows the output while the horizontal axis shows the quantity of variable inputs during a given time period. The top panel of the diagram illustrates how the firm goes from point “A” to point “B” and then to “C”. In fact when the point “C” on the top panel is reached the output is at its optimum. Thus optimum output level is reached by the firm when the optimum combination of variable inputs is achieved. Assuming that the firm would go on adding more and more inputs disregarding the diminishing returns to scale, the firm would incur losses. In other words the law of diminishing marginal returns begins to operate after a certain point has been reached on the production function (Shephard, 1981). The bottom panel of the diagram illustrates how the Marginal Physical Product (MPP) and the Average Physical Product (APP) are influenced by the varying levels of inputs. When output reaches from the point “A” to the point “B” on the top panel, APP rises but MPP falls. However MPP rises from the origin to the point “X” and this phenomenon reflects a production function related peculiarity. When MPP rises it rises much faster than APP and in the same vein when it falls it falls much faster than APP. In between it reaches the point “Y” on the bottom panel where it cuts through the APP curve and reaches the point “Z”. This in turn is a parallel to the law of diminishing marginal returns. Under the law of DMR, when the optimum output is attained by the firm its marginal output is equal to 0. In other words from the origin to the point “A” the firm attains increasing positive returns but after that the returns rise at a decreasing rate. The optimum output is shown by the physical distance between the point “C” on the top panel and the point “Z” on the bottom panel. Cost analysis Figure 2: Isoquants and cost function Costs are incurred in the process of purchasing variable inputs such as labor and capital and therefore the cost function of the firm acquires an additional significance. The fixed cost (FC) factor as against the variable cost (VC) plays now a significant role in this analysis. In fact the following two panels serve as a bridge from the production function to the short run cost function (Bairam, 1998). The second panel on production costs has been derived from the first panel’s information on isoquants. The top panel shows how the firm moves from the first labor (L1) isoquants to the second (L2) and so on with the fixed factor (K) at K*. Fixed costs don’t vary with output in the short run (Lau, 2000). It’s possible to have an isocost function too with isoquants but nevertheless this paper specifically concentrates on the cost function for analysis purpose only. So the subsequent derivation on the panel two is an illustration of the production cost function. As long as the production function operates according to this theoretical explanation in the short run and then moves into the long run in the next stage of the plant expansion coinciding with the removal of technology constraints, the firm would achieve returns to changing scale in the future without difficulty. However if snags were to develop the production function would have to be altered to accommodate such unforeseen variations. However it’s practically infeasible to do so. Now it’s possible to examine the various cost function related outcomes in the long run. The following two panels of the long run cost relations illustrate the fact that in the long term the Long Run Average Cost (LRAC) and the Long Run Marginal Cost (LRMC) decline up to a point shown by the tangency and then start rising again. Figure 3: Long run costs REFERENCES 1. Bairam, Erkin (Editor) Production and Cost Functions: Specification, Measurement and Applications. Surrey: Ashgate Publishing, 1998. 2. Jehle, Geoffrey A. & Reny, Philip J. Advanced Microeconomic Theory (2nd ed). New Jersey: Addison Wesley, 2000. 3. Lau, Lawrence J. (Editor) Econometrics: Econometrics and the Cost of Capital. Massachusetts: The MIT Press, 2000. 4. Shephard, Ronald William. Cost and Production Functions. London: Springer-Verlag,1981 Read More