Problem of conditioned optimization in the neoclassical theory of the firm

In a long period of time all production factors are variable, unlike the short period in which some of them have a fixed character. Although real production processes use many inputs, considering a production case with two variable inputs greatly simplifies the analysis of production over a long period of time. For the economic analysis of production that is characterized by variability of all production factors, an interesting decision is the one concerning the choice of a combination of two inputs, the usage of which will help achieve the maximum possible production or else the given volume of production will be achieved with the minimum amount of total costs. Starting from very simple algebraic functions (also known as Kob-Douglas production function) in this paper we will consider the problem of optimization, or the choice of using a combination of production inputs, so that with a given level of total costs, and given price of production inputs, we can achieve the maximum possible production (problem of conditioned maximization) and the choice of using a combination of production inputs at constant prices of inputs which leads to a given level of output with minimal production costs (problem of conditioned minimization)