Q. Implications for the shape of cost function?

A cost function is also a mathematical relationship, one which relates the expenses an organisation incurs on the quantity of output it generates and to the unit prices it pays. Arithmetically, let E denote the expense an organisation incurs in production of output quantity Y when it pays unit prices (p1... pn) for the inputs it uses. Then cost function  C(y, p1, ...,  pn) describes the minimum expenditure essential 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), though the degrees to which minimum cost increases with an increase in the quantity of output produced or in any input price relies on the aspects describing the structure of production technology. For illustration, scale economies enable output to expand faster than input usage. Or we can say, proportionate increase in output is larger than proportionate increase in inputs. Such a situation is also referred as elasticity of production in relation to inputs being greater than one scale economies so create an incentive for large-scale production and by analogous reasoning scale diseconomies create a technological deterrent to large-scale production. For another instance, if a pair of inputs is a close substitute and unit price of one of the inputs increases, resulting increase in cost is less than if two inputs were poor complements orsubstitutes. Lastly, 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 as well as allocative inefficiency are expensive.

As these illustrations suggest, under fairly general conditions shape of the cost function is a mirror image of shape of the production function. So the cost function and production function normally afford equivalent information concerning the structure of production technology. This equivalence relationship between cost functions and production functions is called 'duality' and it states that one of the two functions has certain aspects if and only if, the other has certain aspects. Such a duality relationship has some significant implications. Since production function and cost function are based on different data, duality allows us to use either function as the basis of an economic analysis of production, without fear of attaining conflicting inferences. Theoretical properties of associated input demand and output supply equations may be inferred from either theoretical properties of the production function or more easily for those of the dual cost function.