The inputs L = 59.36, K = 42.05, and M = 40.64 are the least-cost combination of inputs that will produce q = 35 units of product at the input prices wL = 7, wK = 13, and wM = 6. With these inefficient input combinations:ġ. But, these combinations will be more costly at the given factor prices.
Other combinations of factor inputs will also produce 35 units of product, like L = 74.01, K = 37.44, and M = 36.19. Then to produce 35 units of product at minimum cost, it should use:
Suppose the firm can buy its factors at the prices:Ĭ(q) = wL * L + wK * K + wM * M = 7 * L + 13 * K + 6 * L Decreasing returns to scale: alpha + beta + gamma
If for given values of L,K, and M, the Hessian of the production function f is negative definite, then its isoquants at that point are concave to the origin. Production functions need to have certain properties, to ensure that we can solve the least-cost problem: Check any of the many textbooks. L^ alpha means "raise the value of L to the power of the value of alpha." The symbol "^" means "raise to the power," i.e. Where L = labour, K = capital, M = materials and supplies, and q = product. The three factor Cobb-Douglas production function is: If you have not already done so, look at how the parameters of a Cobb-Douglas production function can be estimated: Estimating a Cobb-Douglas production function. A profit maximizing firm will try to use a combination of inputs that will minimize its cost of producing a given level of output. Recall that a production function produces levels of output for combinations of inputs. Now I will show how cost functions look when they are obtained from a production function. In the oligopoly / government enterprise model, I derived each firm's short run cost function from a quadratic, long run, average cost function. Please show your support by joining Egwald Web Services as a Facebook Fan: It could also play a role in certain marketing courses, especially at the graduate level.Egwald Economics - Production Functions: Cobb-Douglas Production FunctionĮgwald's popular web pages are provided without cost to users. The information contained in this paper could act as a pedagogical aide in any microeconomics-based course or in a production management class. Also explored are some of the duality issues that potentially exist between certain production and cost functions. These would include, but are not limited to, items such as the returns to scale of the function, the separability of the function, the homogeneity of the function, the homotheticity of the function, the output elasticity of factors (inputs), and the degree of input substitutability that each function exhibits. Several important characteristics of the production function are also explained in this paper. This paper also covers some relatively newer production functions, such as the Arrow, Chenery, Minhas, and Solow (ACMS) functions, the transcendental logarithmic (translog), and other flexible forms of the production function. These consist of some well known functions, such as the Cobb-Douglas, Constant Elasticity of Substitution (CES), and Generalized and Leontief production functions. Several various production functions that have played an important historical role in economics are explained. The origin and development of this function over time is initially explored. This paper offers a brief overview of the historical significance and operational role of the production function in business and economics. The production function explains a basic technological relationship between scarce resources, or inputs, and output.