Economists usually obtain empirical production functions from statistical analysis of industry data. Thus the production firm represents the representative firm in the industry. If the exponent is less than or equal to 1, the production function is concave. The cost function is the minimum cost of producing the output over its range. The cost function is obtained by solving a cost minimization problem which also provides the conditional demands for inputs to produce each level of output.
Cost minimization is an important decision problem in industry.
A corporation also wants to known how to minimize the cost of producing a sales forecast. Over the year a public utility manage wants to know how to minimize the cost of producing power for all the various levels of demand. First let us consider the one input variable case. This problem is very simple to solve because for each amount of output there is only one amount of input to produce it. This production function is usually used for industry wide studies.