The monograph develops the theory of production functions and their systematic typology. It looks at the relationship between inputs and outputs as a universal relationship that is used not only in economics but also in other disciplines. In addition to the static production function, special attention is paid to the dynamization of individual quantities and the issue of expressing the effect of changes in these quantities on the change in production. It is explained why in the aggregate production function expressed through aggregate factor input and aggregate factor productivity it is necessary to use a multiplicative relationship, why the multiplicative link is also suitable in terms of total input factor and why the share of weights in labor and capital should be the same. The use of the production function is demonstrated on the development of the economies of the USA, China and India and on the ten largest economies of the world in terms of absolute GDP, on cryptocurrencies and on the so-called farming role.In addition to a comprehensive overview of production functions, the monograph also enriches new ideas that arose during long-term computational and analytical activities of economic and business. Particularly innovative is the generalization of the production function to any system with variable inputs and outputs. The production function can thus be recognized in many identities. The original intention of the research was to examine the intensity of economic development, but it turned out that it is closely related to production functions. The impetus for this research comes from Prof. Ing. František Brabec, DrSc. a genius mathematician, designer, economist and manager, former general director of Škoda in Pilsen and later rector of ČVÚT.The presented typology of production functions is not limited to one area of economics, but goes beyond it. The monograph respects the definition of the static production function as the maximum amount of production that can be produced with a given number of production factors. On this function, which can be effectively displayed using polynomial functions of different orders,significant points can be systematically defined, ie the inflection point, the point of maximum efficiency, the point of maximum profit and the point of maximum production. The purpose is to optimize the number of inserted production factors. The text is preferred the point with the greatest effectiveness. If this quantity does not correspond, for example, to demand, it is possible to choose another technology, which will be reflected in a shift in the static production function. At the same time, the important points of these functions describe the trajectory, which has the nature of a dynamic production function. For a dynamic production function, the crucial question is how the change in individual factors contributes to the overall change in output. If the production function is expressed through inputs and their efficiency, dynamic parameters of extensibility and intensity can be defined, which exactly express the effect of changes in inputs and the effect of changes in efficiency on changes in outputs for all possible situations. Special attention is paid to the aggregate production function. It explains why it should be expressed as the product of the aggregate input factor (TIF) and aggregate factor productivity (TFP), or why the term TIF should be expressed as a weighted product of labor and capital, in which the value of labor and capital weights could be and identical. The monograph here surpasses the traditional additive view of the multi-factor production function by proposing a multiplicative link, which also allows the derivation of growth accounting, but with a new interpretation of weights and (1-), which do not need to be calculated for each subject and each year.The time production function is used to forecast the GDP development of the US, China and India economies until 2030 and 2050, respectively. It is also predicted an increase in the absolute GDP of Indonesia, a stable position of Russia and the loss of the elite position of Japan and Germany.The monograph also deals with the hitherto unresolved question of whether, even in economics, it is also necessary in certain circumstances to take into account a phenomenon called quantization in physics. It turns out that quantization is a common thing in economics, which is documented on specific forms of production functions that respect quantization in economics.The monograph also deals with the relationship between the efficiency of an individual given the use of a certain point on a specific static production function and common efficiency, ie all actors together. These examples assume limited resources. The sum of the outputs of all actors depends on how the actors share these limited resources. It can be expected that there will be at least one method of distribution that will bring the highest sum of outputs (products, crops) of all actors. This result, however, also depends on the shape of the production functions. This is investigated using EDM, i.e.elementary distribution models. EDM for polynomial production functions of the 2nd to 5th order are not yet published in summary. Of the new findings, they are the most interesting. When using two polynomial production functions, the EDM boundary becomes linear if the inflection point is used for both production functions. If we are above the inflection point, the EDM is properly concave. It turned out that the "bending" of the production function in the region of the inflection point can be modeled using a quantity of the order of the respective polynomial. The higher the order of the polynomial, the higher the deflection can be achieved. This proved to be a very important finding in modeling specific production functions. This effect cannot be achieved by combining other parameters.
Decentralized Decisions and Technological Progress: A Formal Proof
