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  • 村上 文男
    繊維工学
    1967年 20 巻 2 号 P111-P117
    発行日: 1967/02/20
    公開日: 2010/09/27
    ジャーナル フリー
  • 椿原 三郎
    数学教育学会誌
    1968年 9 巻 2 号 3-18
    発行日: 1968年
    公開日: 2020/09/11
    ジャーナル フリー
  • 日本数学教育会誌
    1958年 40 巻 9 号 152-
    発行日: 1958年
    公開日: 2021/10/01
    ジャーナル フリー
  • 関 恒義
    季刊 理論経済学
    1956年 7 巻 1-2 号 243-248_1
    発行日: 1956/11/26
    公開日: 2008/02/28
    ジャーナル フリー
  • 李 幇喜, 藤森 頼明
    季刊経済理論
    2011年 48 巻 3 号 56-68
    発行日: 2011/10/20
    公開日: 2017/04/25
    ジャーナル フリー
    In this paper, we inherited Hua LooKeng's problem consciousness, and we investigate the stability of price equilibria of Marx-Sraffa joint-production system, where equilibrium is defined in terms of equalities with rectangular matrices, from the angle of eigensystems. We transform the problem of searching equilibiria of Marx-Sraffa joint-production systems into eigenvalue problems by means of Moore-Penrose inverses. First, we give an overview of Sraffa-Okishio-Nakatani (SON) reduction from the angle of nonsingular transformation. SON dealt with a limitted joint production system in which only aged fixed capital is produced jointly. In SON, elimination of terms related to aged fixed capital leads to the determination of profit rates and prices of brandnew commodities. It is shown that this elimination in SON gives a decomposition of coefficient matrix into three parts; the first one concerns only brandnew commodities, the second aged fixed capital to brandnew fixed capital, and the third part zeros. Second, it is shown that the second part of the above decomposition is exactly the companion matrices in renewal dynamics of fixed capital discussed by Yamada-Yamada. Thirdly, by assuming the rank condition on input and output matrices, we demonstrate that price equilibria of general joint production system of the Marx-Sraffa style can be transformed into an equivalent eigenvalue problem by means of MPinverses. We give a numerical example after the theory. Since the resulting square matrices of the eigensystem are no longer necessarily nonnegative, the dominant eigenvalue is not the one corresponding to equilibria. In this sense, one cannot assert that price equilibrium is stable. In SON, price equilibrium is dynamically stable, because the relative prices of aged fixed capital are determined separated from the determination of profit rates. The rate of depreciation which depends on profit rates is applied therein. In this sense, the accounting system of amortisation is an stabilizing effect in the economy.
  • 藤森 頼明
    季刊経済理論
    2009年 46 巻 3 号 6-20
    発行日: 2009/10/20
    公開日: 2017/04/25
    ジャーナル フリー
    The paper reviews mathematical development of Marx's economics before, by, and after Okishio. Marx's economics is based on dialectics, such as quality and quantity etc., and dialectic feature is not peculiar to Marx's economics, as seen from comparison with physics. Marx's economics has mathematical nature in many fields. The value form theory is established on the distinction between assignment and comparison, for instance. Values and production prices of commodities, the fundamental Marxian Theorem (FMT), the transformation problem, depression theory, socialist planning and relationship among men, all those are mathematical topics. Propositions within linear framework are established mainly by Okishio and his followers. Marx's theory in terms of the Leontief model turned out to be fully dependent on Perron-Frobenius (PF) theorem on nonnegative matrices. Okishio's theorem on technical changes is a corollary of PF theorem. Theory of fixed capital is one of greatest merits and contributions of economic theory of Marx and Sraffa. Fixed capital is a materialistic basis of capitalist production, so that movement of capital depends on fixed capital. It has been disclosed that the renewal process of fixed capital is described as a Markov process, that is accompanied with damped oscillation and initial spikes called Ruchti-Lohmann effect. The cycle of resonance is, nonetheless, as long as twice of durability of fixed capital. Some may argue that Marx's economics lacks micro foundation, but it is not a big point at all. Mathematical theory of Marx's economics follows Marx's fundamental idea that competition in the market is a phenomenal form of the internal dynamics of capitalist production system that governs the process. The equilibrium value itself is, if it exists, determined by a system of simultaneous equations that depict rather structural aspects of economies. Marx's economics is of computational nature, and, in this sense, highly critical against neoclassical economics, that suffers from computational difficulties. Criticism against neoclassical economics is one of important tasks for Marxian economics. It is pointed out that neoclassical macro dynamics should suffer from various formal difficulties, and neoclassical theorem based on saddle path fails in explaining cyclical depression, like 2008 world depression. Future development of Marx's economics should also include topics related to revival and renewed models of socialist planned economies, and nonlinearity or complexity of economic phenomena. As for socialist planning, there is a some stock of discussions before 1990. Fundamental problems are still worth pursuing. Meanwhile, Mandelbrot and economic physics present, from large scale data analyses, serious counter examples against neoclassical economics. This type of approach and method should be digested by Marxian economics. A basic stance of Marx's economics is that relationship among men appears in the relationships among commodities, and that the latter is a projection of the former. Hence Marx's economics is of steric nature. On the contrary, most of bourgeois economics does not analyse relationship of men, so that it is planar, and gives us no deeper insight of our economy.
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