Q. What do you mean by Cost Function?
Cost function is a derived function. It's derived from the production function that describes the efficient method of production at any given time. Production function specifies the technical relationships between inputs and level of output. So cost will vary with the changes in level of output, nature of production function, or factor prices. So symbolically we may write the cost function as
C = ƒ(X, T, Pf)
Where, C = Total cost, X = Output, T = Technology, Pf = Prices of factors.
Total cost is evidently, an increasing function of output, C = ƒ (X), ceterius paribus. Clause 'ceteris paribus' implies that 'all other factors that determine costs are constant'. If these factors change, they would affect the cost. Technology is itself determined by physical quantities of the factor inputs, quality of the factor inputs, and efficiency of the entrepreneur, both in organising the physical side of production and in making the correct economic choice of techniques. So, any change in these determinants will shift the production function and therefore will shift the cost curve. For example the introduction of a better method of organising production or application of an educational programme to the existing labour would shift the production function upwards and therefore will shift down the cost curve. In the same way, the improvement of raw material, or improvement in the use of the same raw materials will result in a downward shift of the cost function.
Because no output is possible without an input, ceteris paribus, an increase in factor prices will lead to an increase in the cost. Factor prices relies on the demand and supply of factors in the economy.
Of all the determinants of cost, cost-output relationship is considered as the most significant one. So in economic analysis cost function is analysed with respect to output. This is since the cost-output relationship is subject to faster and more frequent changes. Relationship between output and cost is analysed with respect to long-run and short-run.