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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Okishio's theorem ： ウィキペディア英語版 Okishio's theorem Okishio's theorem is a theorem formulated by Japanese economist Nobuo Okishio. It has had a major impact on debates about Marx's theory of value. Intuitively, it can be understood as saying that if one capitalist raises his profits by introducing a new technique that cuts his costs, the collective or general rate of profit in society – for all capitalists – goes up. Okishio () establishes this theorem under the assumption that the real wage – the price of the commodity basket which workers consume – remains constant. Thus, the theorem isolates the effect of 'pure' innovation from any consequent changes in the wage. For this reason the theorem, first proposed in 1961, excited great interest and controversy because, according to Okishio, it contradicts Marx's law of the tendency of the rate of profit to fall. Marx had claimed that the new general rate of profit, after a new technique has spread throughout the branch where it has been introduced, would be lower than before. In modern words, the capitalists would be caught in a rationality trap or prisoner's dilemma: that which is rational from the point of view of a single capitalist, turns out to be irrational for the system as a whole, for the collective of all capitalists. This result was widely understood, including by Marx himself, as establishing that capitalism contained inherent limits to its own success. Okishio's theorem was therefore received in the West as establishing that Marx's proof of this fundamental result was inconsistent. More precisely, the theorem says that the general rate of profit in the economy as a whole will be higher if a new technique of production is introduced in which, at the prices prevailing at the time that the change is introduced, the unit cost of output in one industry is less than the pre-change unit cost. The theorem, as Okishio (1961:88) points out, does not apply to non-basic branches of industry. The proof of the theorem may be most easily understood as an application of the Perron–Frobenius theorem. This latter theorem comes from a branch of linear algebra known as the theory of nonnegative matrices. A good source text for the basic theory is Seneta (1973). The statement of Okishio's theorem, and the controversies surrounding it, may however be understood intuitively without reference to, or in-depth knowledge of, the Perron–Frobenius theorem or the general theory of nonnegative matrices. ==The Sraffa model== The argument of Nobuo Okishio, a Japanese economist, is based on a Sraffa-model. The economy consists of two departments I and II, where I is the investments goods department (means of production) and II is the consumption goods department, where the consumption goods for workers are produced. The coefficients of production tell, how much of the several inputs is necessary to produce one unit of output of a given commodity ("production of commodities by means of commodities"). In the model below two outputs exist $x\_1$, the quantity of investment goods, and $x\_2$, the quantity of consumption goods. The coefficients of production are defined as: *$a\_$: quantity of investment goods necessary to produce one unit of investment goods. *$a\_$: quantity of hours of labour necessary to produce one unit of investment goods. *$a\_$: quantity of investment goods necessary to produce one unit of consumption goods. *$a\_$: quantity of hours of labour necessary to produce one unit of consumption goods. The worker receives a wage at a certain wage rate w (per unit of labour), which is defined by a certain quantity of consumption goods. Thus: *$w\; \backslash cdot\; a\_$: quantity of consumption goods necessary to produce one unit of investment goods. *$w\; \backslash cdot\; a\_$: quantity of consumption goods necessary to produce one unit of consumption goods. This table describes the economy: w x_1 |$x\_1$ |- |Department II |$a\_\; x\_2$ |$a\_\; w\; x\_2$ |$x\_2$ |} This is equivalent to the following equations: *$(a\_\; x\_1\; p\_1\; +\; a\_\; w\; x\_1\; p\_2)\; (1+r)\; =\; x\_1\; p\_1$ *$(a\_\; x\_2\; p\_1\; +\; a\_\; w\; x\_2\; p\_2)\; (1+r)\; =\; x\_2\; p\_2$ *$p\_1$: price of investment good $x\_1$ *$p\_2$: price of consumption good $x\_2$ *$r$: General rate of profit. Due to the tendency, described by Marx, of rates of profits to equalise between branches (here departments) a general rate of profit for the economy as a whole will be created. In department I expenses for investment goods or for ''constant capital'' are: *$a\_\; x\_1\; p\_1$ and for ''variable capital'': *$a\_\; w\; x\_1\; p\_2$. In Department II expenses for ''constant capital'' are: : $a\_\; x\_2p\_1$ and for ''variable capital'': : $a\_\; w\; x\_2\; p\_2.$ ''(The constant and variable capital of the economy as a whole is a weighted sum of these capitals of the two departments. See below for the relative magnitudes of the two departments which serve as weights for summing up constant and variable capitals.)'' Now the following assumptions are made: * $p\_2\; =\; 1$: The consumption good $x\_2$ is to be the Numéraire, the price of the consumption good $p\_2$ is therefore set equal to 1. *The real wage is assumed to be $w\; =\; 2\; p\_2\; =\; 2.$ *Finally, the system of equations is normalised by setting the outputs $x\_1$ und $x\_2$ equal to 1, respectively. Okishio, following some Marxist tradition, assumes a constant real wage rate equal to the value of labour power, that is the wage rate must allow to buy a basket of consumption goods necessary for workers to reproduce their labour power. So, in this example it is assumed that workers get two pieces of consumption goods per hour of labour in order to reproduce their labour power. A technique of production is defined according to Sraffa by its coefficients of production. For a technique, for example, might be numerically specified by the following coefficients of production: *$a\_=0.8$: quantity of investment goods necessary to produce one unit of investment goods. *$a\_=0.1$: quantity of working hours necessary to produce one unit of investment goods. *$a\_=0.4$: quantity of investment goods necessary to produce one unit of consumption goods. *$a\_=0.1$: quantity of working hours necessary to produce one unit of consumption goods. From this an equilibrium growth path can be computed. The price for the investment goods is computed as (not shown here): $p\_1=\; 1.78$, and the profit rate is: $r\; =\; 0.0961\; =\; 9.61\; \backslash \%$. The equilibrium system of equations then is: *$(0.8\; \backslash cdot\; 1\; \backslash cdot\; 1.78\; +\; 0.1\; \backslash cdot\; 2\; \backslash cdot\; 1\; \backslash cdot\; 1)\; \backslash cdot\; (1+0.0961)\; =\; 1\; \backslash cdot\; 1.78$ *$(0.4\; \backslash cdot\; 1\; \backslash cdot\; 1.78\; +\; 0.1\; \backslash cdot\; 2\; \backslash cdot\; 1\; \backslash cdot\; 1)\; \backslash cdot\; (1+0.0961)\; =\; 1\; \backslash cdot\; 1$ 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Okishio's theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.