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 goes up for all capitalists. In 1961, Okishio established this theorem under the assumption that the real wage 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.
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 , the quantity of investment goods, and , the quantity of consumption goods.
The coefficients of production are defined as:
: quantity of investment goods necessary to produce one unit of investment goods. : quantity of hours of labour necessary to produce one unit of investment goods. : quantity of investment goods necessary to produce one unit of consumption goods. : 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:
: quantity of consumption goods necessary to produce one unit of investment goods. : quantity of consumption goods necessary to produce one unit of consumption goods.
This table describes the economy:
Investment goods usedConsumption goods usedOutputDepartment IDepartment II
This is equivalent to the following equations:
: price of investment good : price of consumption good : 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: and for variable capital: .
In Department II expenses for constant capital are:
and for variable capital:
(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:
: The consumption good is to be the numéraire, the price of the consumption good is therefore set equal to 1. The real wage is assumed to be Finally, the system of equations is normalised by setting the outputs and 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:
: quantity of investment goods necessary to produce one unit of investment goods. : quantity of working hours necessary to produce one unit of investment goods. : quantity of investment goods necessary to produce one unit of consumption goods. : 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): , and the profit rate is: . The equilibrium system of equations then is:
Introduction of technical progress
A single firm of department I is supposed to use the same technique of production as the department as a whole. So, the technique of production of this firm is described by the following:
Now this firm introduces technical progress by introducing a technique, in which less working hours are needed to produce one unit of output, the respective production coefficient is reduced, say, by half from to . This already increases the technical composition of capital, because to produce one unit of output (investment goods) only half as much of working hours are needed, while as much as before of investment goods are needed. In addition to this, it is assumed that the labour saving technique goes hand in hand with a higher productive consumption of investment goods, so that the respective production coefficient is increased from, say, to .
This firm, after having adopted the new technique of production is now described by the following equation, keeping in mind that at first prices and the wage rate remain the same as long as only this one firm has changed its technique of production:
So this firm has increased its rate of profit from to . This accords with Marx's argument that firms introduce new techniques only if this raises the rate of profit.Volume III of Capital, chapter 15: "No capitalist ever voluntarily introduces a new method of production, no matter how much more productive it may be, and how much it may increase the rate of surplus-value, so long as it reduces the rate of profit."
Marx expected, however, that if the new technique will have spread through the whole branch, that if it has been adopted by the other firms of the branch, the new equilibrium rate of profit not only for the pioneering firm will be again somewhat lower, but for the branch and the economy as a whole. The traditional reasoning is that only "living labour" can produce value, whereas constant capital, the expenses for investment goods, do not create value. The value of constant capital is only transferred to the final products. Because the new technique is labour-saving on the one hand, outlays for investment goods have been increased on the other, the rate of profit must finally be lower.
Let us assume, the new technique spreads through all of department I. Computing the new equilibrium rate of growth and the new price gives under the assumption that a new general rate of profit is established:
If the new technique is generally adopted inside department I, the new equilibrium general rate of profit is somewhat lower than the profit rate, the pioneering firm had at the beginning (), but it is still higher than the old prevailing general rate of profit: larger than .
Result
Nobuo Okishio proved this generally, which can be interpreted as a refutation of Marx's law of the tendency of the rate of profit to fall. This proof has also been confirmed if the model is extended to include not only circulating capital but also fixed capital. Mechanisation, defined as increased inputs of machinery per unit of output combined with the same or reduced amount of labour-input, necessarily lowers the maximum rate of profit.Anwar Shaikh (1978): Political economy and capitalism: notes on Dobb's theory of crisis. Cambridge Journal of Economics, 1978, 2, 233-251. In this article is a reference to Bertram Schefold (1976): Different forms of technical progress. Economic Journal
Marxist responses
Some Marxists simply dropped the law of the tendency of the rate of profit to fall, claiming that there are enough other reasons to criticise capitalism, that the tendency for crises can be established without the law, so that it is not an essential feature of Marx's economic theory. Others would say that the law helps to explain the recurrent cycle of crises, but cannot be used as a tool to explain the long term developments of the capitalist economy.
Others argued that Marx's law holds if one assumes a constant ‘’wage share’’ instead of a constant real wage ‘’rate’’. Then, the prisoner's dilemma works like this: The first firm to introduce technical progress by increasing its outlay for constant capital achieves an extra profit. But as soon as this new technique has spread through the branch and all firms have increased their outlays for constant capital also, workers adjust wages in proportion to the higher productivity of labour. The outlays for constant capital having increased, wages having been increased now also, this means that for all firms the rate of profit is lower.
However, Marx did not know the law of a constant wage share. Mathematically the rate of profit could always be stabilised by decreasing the wage share. In our example, for instance, the rise of the rate of profit goes hand in hand with a decrease of the wage share from to , see computations below. However, a reduction in the wage share is not possible in neoclassical models due to the assumption that wages equal the marginal product of labour.
A third response is to reject the whole framework of the Sraffa-models, especially the comparative static method.Andrew Kliman: The Okishio Theorem: An Obituary: AN OBITUARY Kliman thus argues that the model supposes that input and output prices are the same and that only one technological shift happens, rather than a series of successive innovations. "Even a cursory reading of the law immediately discloses that it refers to the impact of a continuing series of technical changes), and that these technical changes are held to lead to a continuing series of reductions in commodities' prices (given a constant monetary expression of value)", he writes. In a capitalist economy entrepreneurs do not wait until the economy has reached a new equilibrium path but the introduction of new production techniques is an ongoing process. Marx's law could be valid if an ever-larger portion of production is invested per working place instead of in new additional working places. Such an ongoing process cannot be described by the comparative static method of the Sraffa models.