Purpose. Development of a procedure for determining the optimal capitallabor ratio within the framework of twofactor pro duction functions, as well as an analysis of the conditions for its implementation in practice. Methodology. In the process of the conducted mathematical analysis, the search for the extrema of the most popular in eco nomic studies, of twofactor production functions, namely the CobbDouglas production function, CESfunction, linear func tion, Allen's function in terms of capital assets, was performed. The limit rate of the technological substitution of factors in condi tions of optimal capitallabor ratio was found, which proved to be unique in all investigated production functions. The indicated relationship was verified on the basis of microeconomic theory, in particular, using the equilibrium principle. Findings. The hypothesis of the single marginal rate of technological substitution in conditions of optimal capitallabor ratio is verified. On the basis of the microeconomic theory, the procedure for determining the optimal capitallabor ratio in the framework of any twofactor production functions is substantiated due to the marginal rate of technological substitution. The proposed pro cedure consists in determining the marginal rate of technological substitution as the ratio of the marginal products of the produc tion factors and equating it to one unit. The obtained ratio makes it possible to deduce the formula of optimal capitallabor ratio, provided that the indicators of the dynamics of realized products, main assets and labor remuneration in the enterprise are ade quately described by the corresponding production function with nonzero substitution. Originality. Development of a procedure for optimization of capitallabor ratio within the framework of substitutional two factor production functions based on the formulas of their marginal rate of technological substitution. Practical value. Theoretical conclusions and proposals were tested on the example of a Ukrainian enterprise in the field of mechanical engineering. The results of the research can be used to manage capitallabor ratio by determining its optimal volume in production, which opens the possibility of rational use of basic productive assets and labor at domestic enterprises which are characterized by relative overequipment with high moral and physical equipment decay.
The most popular two-factor production functions used in the process of modeling sustainable economic development are examined. Economic and mathematical characteristics of Cobb-Douglas production functions, CES-function, linear function, Leontief and Allen functions are considered, in particular, type of dependence of labour productivity in relation to capital-labour ratio of commodity production system within mentioned production functions. Their most important economic and mathematical characteristics are presented: factors average and marginal return, demand for production resources, factor substitution, factors marginal rate of technical substitution, output elasticity by factors, elasticity of factors technical substitution, optimal capital-labour ratio according to the criterion of maximum output. Comparative analysis is given to Cobb-Douglas and CES-functions, which are two production functions mostly required in practice.
Analysis of the advantages and disadvantages of indicators of the internal rate of return and the modified internal rate of return as criteria for the effectiveness of an investment project. The mathematical and graphic apparatus for studying the functions and dependencies between the economic characteristics of future financial and production activities of entrepreneurial activity was used in the process of the analysis of the most important indicators of the effectiveness of the investment project. The characteristics of the criterion of the internal rate of return are studied on the basis of the properties of the function which describes the dependence of the net present value of the project on the value of the discount rate. The main advantages of the modified internal rate of return in comparison with its unmodified analogue are revealed. The inequalities between the indicated indicators are mathematically proved, their influence on the stability (safety) characteristics of the investment project is analyzed, and the interrelation of the criterion of the modified internal rate of return with the value of the profitability index is shown. The scientific novelty consists in the mathematical proof of the interrelation between the indicators of the internal rate of return and the modified internal rate of return, as well as the interrelation between the criterion of the modified internal rate of return and the profitability index of the investment project. Theoretical conclusions and suggestions can be used in the investment analysis of future financial and production projects in the domestic economy, which opens up the possibilities of rational use of resources in entrepreneurial activity at all levels of business process management.
The article examines the situation of appearance of a non-optimal capital-labor ratio on the basis of a comparison of the relative speed of the dynamics of indicators of labor productivity, capital productivity and capital-labor ratio in machine building industry of Ukraine in recent years. Mathematical determination of the optimal capital-labor ratio is substantiated on the basis of production functions taking into account dynamics of the most important indicators of economic activity, presented in value terms. Methodological and applied aspects of the use of the equimarginal principle from microeconomics are discussed to determine the optimal capital-labor ratio within the limits of substitutional production functions. It is proved that at the point of optimal capital-labor ratio, the marginal rate of replacement of production factors’ substitution is equal to one. The resulting conclusion is used as a basis for development of a procedure for finding optimum capital-labor ratio using econometric models, which adequately describe the relationship of time series of product sales, basic productive assets and labor costs based on substitutional production functions. The use of the proposed procedure for determining the optimal capital-labor ratio is carried out on the example of the Cobb-Douglas-Tinbergen production function, the dynamised CES-function and the linear function. The methodological recommendations on calculation of unknown parameters are presented for these functions, as well as the formulas of optimal capital-labor ratio with indicated extreme values of products sold and the total costs for basic production assets and labor payment. The obtained theoretical results are tested according to the data of Ukraine’s machine building. The hypothesis about non-optimal capital-labor ratio in 2007-2015 is confirmed in terms of volumes of sales of the industry production. It turns out that for the analyzed period of time, the basic production assets of machine building were relatively abundant compared with the payment of labor. In 2016-2017, a positive trend begins in dynamics of the capital-labor ratio in the industry to a certain reduction and a gradual approach to the optimal value.
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