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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