ペロン-フロベニウス定理の経済学への応用 : 数学者たちの「マルクスの基本定理」(<特集>経済学の数理的方法と記述的方法)

抄録

Georg von Charasoff, a Russian mathematician, was one of the first researchers to recognise that the price of production is an eigenvector of the input matrix, and to determine the rate of profit using its eigenvalue. He anticipated, at this analytical level, most of the arguments that were to be proposed later in the course of the 'transformation problem', i.e. Fundamental Marxian Theorem (FMT), convergence theorem for Marxian transformation procedure and the theorem of rising rate of profit. Moreover, he developed, prior to W. Leontief, P. Sraffa and J. v. Neumann, such ideas as the power series of the Leontief inverse, the basic and non-basic products and the duality of the growth and profit rate in the balanced growth. Although Charasoff's name and his work had been forgotten in the research of economics, his ideas were rediscovered and further developed particularly by Kei Shibata, N. Okishio and M. Morishima without any reference to Charasoff's original contribution. Maurice Potron was a French mathematician whose largely unknown contributions to economic analysis should be acknowledged as pioneering achievements. First, Potron proved de facto Fundamental Marxian Theorem 48 years earlier than Morishima, Seton and Okishio by adapting the Perron-Frobenius theorem and he proved de facto FMT by considering heterogeneous labours 65 years earlier and even more generally than Bowles and Gintis. Second, in doing so, Potron was the first to apply the Perron-Frobenius theorem to economics; third, he laid the foundations of input-output table long before Leontief by calculating input coefficients for some products. And finally, Potron articulated and proved the socalled Hawkins-Simon condition. Furthermore, a comparison between Charasoff's and Potron's treatment of FMT provides us with some insights into the role of mathematics in economic thoughts. Both mathematicians devoted themselves to economic research, and at last they could reach comparable results concerning price determination and especially FMT by using the concepts of eigenvectors and eigenvalues. They had, however, quite different ideological backgrounds, and correspondingly they took quite different positions on normative issues in the face of the so-called social problems around the turn of the 19^<th> to the 20^<th> century where the impoverishment of the working class was increasingly serious. Starting from the FMT, the existence of profit and, equivalently, the execution of surplus labour was condemned by Charasoff from the Marxian or 'human' viewpoint of economizing of living labour. Based on the same FMT, Potron affirmed surplus labour and therefore profit as the condition that enabled and stabilised the compatibility of just price and just wage, i.e. those two principles of justice which were thought to incorporate the Roman Catholic position of reconciling class interests. Both mathematicians and their economic research exemplified a limit to mathematical reasoning in economics in that FMT just as mathematical propositions in general cannot justify any specific normative assertion.