von Neumann Output Ratio in Marx's Social Reproduction with the Two Major Categories—Obtaining von Neumann Output Ratio through Preferential Growth of the Means of Production

von Neumann Output Ratio is a widely influential concept and term in mathematical economics. von Neumann Output Ratio of Marx's reproduction with two major categories is the ratio between the total output of the means of production category and the total output of the consumption category while the two major categories obtain the maximum equilibrium growth rate. It can most centrally formalize the social structure in the state of the highest equilibrium growth. It is of theoretical significance to establish the relationship between the quantity relation of Marx's theory of social capital reproduction and von Neumann Output Ratio, and to identify and clarify von Neumann Output Ratio of Marx's reproduction with two major categories. Based on Marx's theory of social reproduction, following the stand, viewpoint, basic model and analysis method of this scientific theory, this paper focuses on the unity of opposites between von Neumann Output Ratio of Marx's reproduction with two major categories and personal consumption of capitalists, and connects Marx's social reproduction formula and the equation system of output growth of the two major categories. By finding and determining the equilibrium growth transfer matrix of output column vector of the two major categories from this term to next term, the paper proved just the development rate of the output of the two major categories which maintains original output structure is the eigenvalue of the transfer matrix. Then, by using the constant capital growth rate of the whole society of social reproduction as an intermediary, it is determined that the equilibrium growth rate of output of the two major categories is a monotone increasing function of the ratio of output of category Ⅰ to output of category Ⅱ, so as to determine that "preferential growth of means of production" is the path to obtain von Neumann Output Ratio of Marx's reproduction. By use of the research result that the ratio between the two major categories in Marx's social reproduction can be convergent through the preferential growth of the means of production, von Neumann Output Ratio of Marx's reproduction is obtained, and then the maximum equilibrium growth rate of the output of the two major categories and the minimum personal consumption of capitalists are also determined. von Neumann Output Ratio of Marx's reproduction is formalized in terms of several basic concepts and terms in Marx's theory of social reproduction. It was also verified that von Neumann Output Ratio of Marx's reproduction, the maximum equilibrium growth rate and the minimum personal consumption of capitalists satisfy the equation system of output growth of the two major categories.