A method for constructing non-elementary Cobb – Douglas production functions

Abstract

The actual scientific problem in machine learning is the development of new interpretable mathematical models, as well as methods and programs for their construction. Many well-known regression models have good interpretative properties, for example, linear and power models (Cobb – Douglas production functions). Previously, the author developed non-elementary linear regressions, the problem of constructing which was reduced to the mixed integer 0–1 linear programming problem.Based on non-elementary linear regressions, in this paper, for the first time, non-elementary Cobb-Douglas production functions are proposed, which include not only explanatory variables in degrees, but also all their possible pair combinations, transformed using binary operations min and max. The proposed models are linearized, which makes it possible to apply for their construction the mixed integer 0–1 linear programming problem formulated in the same way as for non-elementary linear regressions. As a result of its solution, the optimal structure model is automatically determined. The advantage of such a formulation is that the solution of the problem can be obtained faster than using enumeration procedures, and also that the signs of the estimates of the constructed model are guaranteed to be consistent with the meaningful meaning of the factors. At the same time, it is possible to control the requirements for the structure of the model using linear constraints on binary variables. In particular, the problem can be used to select the optimal structures for traditional elementary Cobb – Douglas production functions.The problem of modeling the gross regional product of the Tomsk region is solved. The following variables were chosen as explanatory variables: average per capita cash income of the population, investments in fixed capital, costs of innovative activities of organizations, average annual number of employees, cost of fixed assets, internal costs for research and development. The LPSolve package was chosen as the solver of the mixed integer 0–1 linear programming problem. As a result of solving this problem, the optimal structure of the non-elementary Cobb – Douglas production function was chosen, which contains all six explanatory variables in three regressors. The coefficient of determination for the constructed model turned out to be 0.997. All regression coefficients turned out to be significant according to Student's t-test, and their signs satisfy the meaningful meaning of the factors. An interpretation for the constructed model is given.