Duality between Production Function and Cost Function

Production functions and cost functions are the cornerstones of business and  managerial economics. A production function is a mathematical relationship that captures  the essential features of the technology by means of which an organisation metamorphoses  resources such as land, labour and capital into goods or services such as steel or cement. It is  the economist’s distillation of the salient information contained in the engineer’s blueprints.  Mathematically, let Y denote the quantity of a single output produced by the quantities of  inputs denoted (x1,…, xn). Then the production function f(x1,…,xn) describes how a given  output can be produced by an infinite combinations of inputs (x1,.., xn), given the technology  in use. Several important features of the structure of the technology are captured by the  shape of the production function. Relationships among inputs include the degree of  substitutability or complementarily among pairs of inputs, as well as the ability to aggregate  groups of inputs into a shorter list of input aggregates. Relationships between output and  the inputs include economies of scale and the technical efficiency with which inputs are  utilized  to produce a given output.

Each of these features has implications for the shape of the cost function, which is  intimately related to the production function. A cost function is also a mathematical  relationship, one that relates the expenses an organisation incurs on the quantity of output  it produces and to the unit prices it pays. Mathematically, let E denote the expense an  organisation incurs in the production of output quantity Y when it pays unit prices (p1,…, pn)  for the inputs it employs. Then the cost function C(y, p1, …, pn) describes the minimum  expenditure required to produce output quantity Y when input unit prices are (p1,…, pn),  given the technology in use and so E ‰¥C(y, p1,…,pn). A cost function is an increasing function  of (y, p1,…, pn), but the degrees to which minimum cost increases with an increase in the  quantity of output produced or in any input price depends on the features describing the  structure of production technology. For example, scale economies enable output to expand  faster than input usage. In other words, proportionate increase in output is larger than the  proportionate increase in inputs. Such a situation is also denoted as elasticity of production  in relation to inputs being grater than one scale economies thus create an incentive for  large-scale production and by analogous reasoning scale  dis-economies  create a  technological deterrent to large-scale production. For another example, if a pair of inputs is  a close substitute and the unit price of one of the inputs increases, the resulting increase in  cost is less than if the two inputs were poor substitutes or complements. Finally, if wastage  in the organisation causes actual output to fall short of maximum possible output or if inputs  are misallocated in light of their respective unit prices, then actual cost exceeds minimum  cost; both technical and allocative inefficiency are costly.

As these examples suggest, under fairly general conditions the shape of the cost  function is a mirror image of the shape of the production function. Thus, the cost function  and the production function generally afford equivalent information concerning the  structure of production technology. This equivalence relationship between production  functions and cost functions is known as ‘duality’ and it states that one of the two functions  has certain features if and only if, the other has certain features. Such a duality relationship  has a number of important implications. Since the production function and the cost function  are based on different data, duality enables us to employ either function as the basis of an  economic analysis of production, without fear of obtaining conflicting inferences. The  theoretical properties of associated output supply and input demand equations may be  inferred from either the theoretical properties of the production function or, more easily, for  those of the dual cost function.

Empirical analysis aimed at investigating the nature of scale economies, the degree  of input substitutability or complementarily, or the extent and nature of productive  inefficiency can be conducted using a production function or again more easily using a cost  function.

If the time period under consideration is sufficiently short, then the assumption of a  given technology is valid. The longer-term effects of technological progress or the  adaptation of existing superior technology can be introduced into the analysis. Technical  progress increases the maximum output that can be obtained from a given collection of  inputs and so in the presence of unchanging unit prices of the inputs technical progress  reduces the minimum cost that must be incurred to produce a given quantity of output. This  phenomenon is merely an extension to the time dimension of the duality relationship that  links production functions and cost functions. Of particular empirical interest are the  magnitude of technical progress and its cost-reducing effects and the possible  labor-saving  bias of technological progress and its employment effects that are transmitted from the  production function, to the cost function and then to the  labor  demand function.

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