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Unformatted text preview: Technical Efficiency R. Pope Definition 1: A production function represents the technically efficient combinations of inputs and outputs when prices are only known to be positive. Definition 2: Technically efficient input and output combinations are those for which output is maximized for a particular combination of inputs. Definition 3: Technical efficient input and output combinations for which inputs can be minimized for a given output. The above pictures are for a single input and a single output. Modern production theory has many inputs and outputs. However, at the intermediate theory level a production function with many inputs (often two) and a single output is handy and still widely in use empirically. The two input production function is often written ( , ) q f k l = (1) where q is output and k is capital while l is labor. This is a concept that economists, business analysts, and engineers use. Unlike utility, it is cardinal. Marginal Physical Product or Marginal Product in Short Definition 4: The marginal physical product is the change in output over the change in an input. Thus, ; k l q q MP MP k l = = . out- put input Inefficient Efficient Example: A Cobb-Douglas Production function is of the form , , q Ak l A = > . A, , are just numbers. Taking partial derives gives the marginal product functions: 1 k MP Ak l - = (2) 1 l MP Ak l - = (3) Note that the marginal product of labor depends on the quantity of capital and the marginal product of capital depends on the quantity of labor. As with utility if one multiplies (2) by / k k , it can be written as: k q MP k = and similarly for l . Because marginal products are presumed positive in the region of economic concern, this requires in the Cobb-Douglas that , > . Concept 2: Diminishing Marginal Productivity Definition: Marginal product is diminishing if marginal product falls as more of the input is applied, e.g., 2 2 2 2 0, k l MP MP q q k k l l = < = < . Question: For the Cobb-Douglas above, what restrictions on parameters imply that marginal products diminish? 2 2 2 ( 1) q Ak l k - =- (4) 2 2 2 ( 1) q Ak l l - =- . (5) Hence, positive marginal products and diminishing marginal products in a Cobb-Douglas requires the exponents to be positive and less than 1. Input Elasticities It is sometimes helpful to use the concept of output elasticities defined as: , , ln( ) ln( ) ; ln( ) ln( ) q k q l f k q f l q k f k l f l = = = = . It measures the percentage response in output for a 1 (small) percentage input in the input holding all other inputs fixed....
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- Winter '08