Okishio theorem

The Okishio theorem ( Japanese: Okishio no teiri置塩の定理) states: If a company increases its rate of profit by introducing a new technology that requires less labor but more material input, which then results in a higher profit rate for the economy as a whole, once the new technology has spread throughout the industry, provided that the real wages of workers, that is, the purchasing power of wages measured in certain goods, have not increased.

This theorem contradicts Karl Marx's law of the tendency of the rate of profit to fall , which assumes that after the new technology has spread to the entire industry, the overall economic rate of profit will be lower. The capitalists would be sitting in a trap of rationality , so to speak .

The model from Sraffa

The reasoning of Nobuo Okishio , a Japanese economist, is based on a Sraffa model. The national economy should consist of two departments I and II, whereby I the capital goods (means of production) and II the consumer goods for the workers - apart from the consumption of the capitalists ("classic" consumer function) - manufactures. The production coefficients indicate how much of the various inputs is necessary to produce a unit of a certain output. In this simple case there are only two outputs , the amount of capital goods and , the amount of consumer goods. ${\ displaystyle x_ {1}}$ ${\ displaystyle x_ {2}}$

The production coefficients:

${\ displaystyle a_ {11}}$ : Number of capital goods to manufacture one capital good.
${\ displaystyle a_ {21}}$ : Number of hours worked to produce an investment good.
${\ displaystyle a_ {12}}$ : Number of capital goods to manufacture a consumer good.
${\ displaystyle a_ {22}}$ : Number of hours worked to manufacture a consumer good.

The workers receive a certain wage at the wage rate per unit of work, which is expressed in consumer goods. ${\ displaystyle l}$

${\ displaystyle l \ cdot a_ {21}}$ : Number of consumer goods that are necessary to produce an investment good.
${\ displaystyle l \ cdot a_ {22}}$ : Number of consumer goods that are necessary to produce a consumer good.

Schematically, the economy can be represented as follows:

	Input ${\ displaystyle x_ {1}}$	Input ${\ displaystyle x_ {2}}$	output
Department I.	${\ displaystyle a_ {11} \ cdot x_ {1}}$	${\ displaystyle a_ {21} \ cdot l \ cdot x_ {1}}$	${\ displaystyle x_ {1}}$
Department II	${\ displaystyle a_ {12} \ cdot x_ {2}}$	${\ displaystyle a_ {22} \ cdot l \ cdot x_ {2}}$	${\ displaystyle x_ {2}}$

This can be represented in the following equations:

${\ displaystyle (a_ {11} \ cdot x_ {1} \ cdot p_ {1} + a_ {21} \ cdot l \ cdot x_ {1} \ cdot p_ {2}) \ cdot (1 + r) = x_ {1} \ cdot p_ {1}}$
${\ displaystyle (a_ {12} \ cdot x_ {2} \ cdot p_ {1} + a_ {22} \ cdot l \ cdot x_ {2} \ cdot p_ {2}) \ cdot (1 + r) = x_ {2} \ cdot p_ {2}}$

${\ displaystyle p_ {1}}$ : Price for the capital goods ${\ displaystyle x_ {1}}$
${\ displaystyle p_ {2}}$ : Price for the consumer good ${\ displaystyle x_ {2}}$
${\ displaystyle r}$ : General rate of profit. Due to the tendency described by Marx to equalize the profit rates between the branches (here departments) a uniform general profit rate develops in the national economy . ${\ displaystyle r}$

In Department I, the expenditure for constant capital , the expenditure for capital goods, is:

${\ displaystyle a_ {11} \ cdot x_ {1} \ cdot p_ {1}}$ and for the variable capital :
${\ displaystyle a_ {21} \ cdot l \ cdot x_ {1} \ cdot p_ {2}}$ .

In Division II, the expenditure for constant capital is :

${\ displaystyle a_ {12} \ cdot x_ {2} \ cdot p_ {1}}$ and for the variable capital :
${\ displaystyle a_ {22} \ cdot l \ cdot x_ {2} \ cdot p_ {2}}$ .

(You cannot simply add the constant and variable capitals of the two departments to a total economic figure, because you would have to know the size ratio between the two departments. To calculate this size ratio, see the informational calculations below.)

The following assumptions are now made:

${\ displaystyle p_ {2} = 1}$ : The consumer good should serve as a numéraire, so the price of the consumer good is 1. ${\ displaystyle x_ {2}}$ ${\ displaystyle p_ {2}}$
The real wage is . ${\ displaystyle l = 2 \ cdot p_ {2} = 2}$
The system of equations is normalized by setting the outputs and each equal to 1. ${\ displaystyle x_ {1}}$ ${\ displaystyle x_ {2}}$

In Marxism it is usually assumed that wages are equal to the value of labor, that is, just enough for workers to maintain their labor. It is assumed here that workers need two consumer goods per hour of work in order to maintain their labor.

At Sraffa, one technique is defined by the size of the production coefficient. For example, a technique is given with:

${\ displaystyle a_ {11} = 0 {,} 8}$ : Number of capital goods to manufacture one capital good.
${\ displaystyle a_ {21} = 0 {,} 1}$ : Number of hours worked to produce an investment good.
${\ displaystyle a_ {12} = 0 {,} 4}$ : Number of capital goods to manufacture a consumer good.
${\ displaystyle a_ {22} = 0 {,} 1}$ : Number of hours worked to manufacture a consumer good.

Then the following equilibrium system of equations is obtained, with the values still missing for the price and the rate of profit already calculated and inserted: ${\ displaystyle p_ {1} = 1 {,} 78}$ ${\ displaystyle r = 0 {,} 0961 = 9 {,} 61 \, \%}$

${\ displaystyle (0 {,} 8 \ cdot 1 \ cdot 1 {,} 78 + 0 {,} 1 \ cdot 2 \ cdot 1 \ cdot 1) \ cdot (1 + 0 {,} 0961) = 1 \ cdot 1 {,} 78}$
${\ displaystyle (0 {,} 4 \ cdot 1 \ cdot 1 {,} 78 + 0 {,} 1 \ cdot 2 \ cdot 1 \ cdot 1) \ cdot (1 + 0 {,} 0961) = 1 \ cdot 1}$

Introduction of technical progress

From Department I you can now single out a company that has the same production technology as the department as a whole. The following applies to this company:

${\ displaystyle (a_ {11} \ cdot x_ {1} \ cdot p_ {1} + a_ {21} \ cdot l \ cdot x_ {1} \ cdot p_ {2}) \ cdot (1 + r) = x_ {1} \ cdot p_ {1}}$

${\ displaystyle = (0 {,} 8 \ cdot 1 \ cdot 1 {,} 78 + 0 {,} 1 \ cdot 2 \ cdot 1 \ cdot 1) \ cdot (1 + 0 {,} 0961) = 1 \ cdot 1 {,} 78}$

This company now introduces technical progress by reducing the necessary inputs of labor as of on halved. This alone increases the technical composition of the capital , because now only half as much work, but just as much capital goods as before, is required to create a unit of the capital goods. In addition, it should be assumed that the labor savings go hand in hand with a greater consumption of capital goods, that is to say , increases. ${\ displaystyle a_ {21} = 0 {,} 1}$ ${\ displaystyle a_ {21} = 0 {,} 05}$ ${\ displaystyle a_ {11} = 0 {,} 8}$ ${\ displaystyle a_ {11} = 0 {,} 85}$

For this one company that has introduced technical progress, the following equation results at the present prices and the given wage rate - these quantities initially remain unchanged as long as only one company changes its technology:

${\ displaystyle = (0 {,} 85 \ cdot 1 \ cdot 1 {,} 78 + 0 {,} 05 \ cdot 2 \ cdot 1 \ cdot 1) \ cdot (1 + 0 {,} 1036) = 1 \ cdot 1 {,} 78}$

The company was able to increase its rate of profit from to . To this extent, this corresponds entirely to Marx's considerations, according to which it applies that a company only introduces a new technology if it increases its rate of profit. ${\ displaystyle r = 9 {,} 61 \, \%}$ ${\ displaystyle 10 {,} 36 \, \%}$

Karl Marx, Volume III of Capital, MEW 25, p. 275: "No capitalist applies a new mode of production, no matter how much more productive it is or no matter how much the rate of surplus value it increases, it voluntarily applies it as soon as it reduces the rate of profit."

Marx expects, however, that when the new technology spreads throughout the department, the rate of profit will not only fall again for the “pioneer entrepreneur”, but that there will now be a lower general rate of profit for the whole economy. This is traditionally justified by the fact that only "living work" can create added value - but living work has been saved - and that the constant capital, the expenditure on capital goods, does not create any value, but merely transfers its value to the end products.

If one assumes that the new production technology is generally spreading in Division I and calculates the new equilibrium growth (and the new price ), one obtains: ${\ displaystyle p_ {2}}$

${\ displaystyle (0 {,} 85 \ cdot 1 \ cdot 1 {,} 77 + 0 {,} 05 \ cdot 2 \ cdot 1 \ cdot 1) \ cdot (1 + 0 {,} 1030) = 1 \ cdot 1 {,} 77}$
${\ displaystyle (0 {,} 4 \ cdot 1 \ cdot 1 {,} 77 + 0 {,} 1 \ cdot 2 \ cdot 1 \ cdot 1) \ cdot (1 + 0 {,} 1030) = 1 \ cdot 1}$

After generalization of new production technology in the example I Division and after reaching a new equilibrium growth rate of profit of the pioneering entrepreneur indeed has again become lower, but overall, the new remains general rate of profit with higher than the original from . ${\ displaystyle 10 {,} 30 \, \%}$ ${\ displaystyle 9 {,} 61 \, \%}$

Result

Nobuo Okishio gave this proof in general, that is, refuted the law of the tendency of the rate of profit to fall. However, in the above model there is only circulating capital , that is, means of production that go into the end products in the same period of production. Further studies have extended Okishio's results for the case of an economy with fixed capital , the means of production act longer than a production period. Labor-saving technical progress can therefore lead to increasing and not necessarily decreasing general profit rates.

Marxist replies

1) One reaction from Marxists was that the criticism was accepted. The law of the tendency of the rate of profit to fall is not a central component of Marx's theory. There are enough other reasons against capitalism that one does not have to rely on such a tendency to collapse .

A variant of this is that the law is accepted as an explanation for the regularly recurring economic fluctuations (crises), but not as a long-term trend.

Michael Heinrich opposes a breakdown crisis as well as an understanding of the crisis as a cyclical balancing movement and concludes: “However, since the conditions for the production of profit repeatedly collide with the elementary vital interests of the majority of the population, the Ask about the legitimacy of this social system and the possibility of a social alternative. ”Or:“ Whether the capitalist-accounting expression of capitalist exploitation rises or falls does not change the fundamentally narrow-minded character of the capitalist mode of production. "

2) A second reaction consisted in looking for ways to refute the Okishio argument within the Sraffa model, for example by introducing the law of the constant wage quota - albeit unknown to Marx - be it that the unions pursue productivity- oriented wage policy be it that the companies in the competitive struggle pass on part of the increased productivity in the form of lower prices and thus increase the purchasing power of wages, real wages . With a constant wage share, the rationality trap for companies is as follows: The individual company that introduces technical progress initially makes an extra profit. However, if technical progress is generalized, real wages adjust in such a way that the original, higher wage share is restored. However, the companies are left with the increased costs of the increased use of capital goods. The rate of profit has fallen.

The objection to this argument is that the assumption of a constant wage share is problematic. In any case, the rate of profit could be stabilized if the wage share were adjusted downwards. In the numerical example, the increase in the rate of profit is associated with a decrease in the wage share from 58.6% to 41.9%, see the detailed calculation below.

3) Finally, the third reaction is to reject the Sraffa model as a framework, in particular also the comparative static method . In capitalism one does not wait until a new equilibrium has been established, but the introduction of new technical methods is an ongoing process. The law of the tendency for the rate of profit to fall takes effect when an ever larger proportion of production is invested in each job and not in additional jobs. Such a continuous development cannot be captured by the comparative statics of the Sraffa models.

4) In a fourth reaction it is doubted that Okishio refuted the law of the tendency of the rate of profit to fall by referring to the cost criterion of the choice of technology and justified an increase in the Marxian rate of profit. Okishio understands the rate of profit to be the quotient of the difference between the price of goods and the cost price of the goods to the cost price. If the new technology is used under the premise, also met by Marx, that the consumption of constant capital increases less than the consumption of variable capital decreases, i.e. the production costs decrease, the rate of profit thus defined must inevitably increase. It is true that the ratio of surplus value to the cost price of goods can be called the rate of profit, writes Klaus Müller . Only that is not the rate of profit whose development tendency was founded by Marx. Marx and Engels did not respect the value of the consumed capital - the cost price - but to the pre-imposed total capital. The "cost price profit rate" could rise while at the same time the Marxian capital advance profit rate falls. "The ratio of surplus value to cost price is only then identical with the" real profit rate "if the capital advance and capital consumption are equal or, which amounts to the same thing if capital Turns exactly once a year The claim that technical innovations would increase Marx's rate of profit is therefore wrong.

Alfred Müller and Stephan Krüger

In the case of Alfred Müller and Stephan Krüger, the prerequisite for the fall in the rate of profit (as well as cyclical fluctuations) is that production takes place industrially with the help of machinery . The capitalist companies seek to increase their profits independently of one another on their own responsibility, there is no central planning authority of the economy. Okishio, on the other hand, abstracts from fixed capital in his model , there is only circulating capital . The admissibility of this abstraction is controversial.

According to Alfred Müller, the Okishio theorem does not take into account the special features of capitalist society. “If there were a macroeconomic coordinating body and capitalist private ownership of the means of production were abolished, if the means of control were mastered and the information problem was solved, supply and demand would be balanced, and the economy would move on a steady path of equilibrium. The difficulties arise as soon as the capital ratio comes in and the production process becomes only a means for the valorization process of capital. ”The latter happens through the individual capitalists without macroeconomic coordination.

Similar to Stephan Krüger, here the companies make "autonomous" investments, that is, independent of the current profit or cost situation, depending on the situation. In the crisis, investments in rationalization are made, which increase both labor productivity and the technical composition of capital in order to get out of the crisis. The momentary advantage for the individual enterprise is in the long run contrasted with the fall in the general rate of profit because of the increasing composition of capital.

For information: the material side

The dual system of equations

For the considerations so far, it was sufficient to only consider the size of the money. Should further studies be carried out in order to calculate the quantities constant capital c, variable capital v and surplus value (or profit) m for the economy as a whole, or relationships between these quantities such as the rate of surplus value m / v or the value composition of capital c / v, then the size ratio between departments I (capital goods) and II (consumer goods) must be determined. This is done by showing the steady growth from the material side.

In the previous equations it was calculated how, under certain technical conditions, which were represented by the production coefficients, and under a certain given real wage, which was represented as the amount of consumer goods per hour worked, a uniform rate of profit r emerges, whereby a price still had to be fixed . Here the price for the consumer good was set equal to 1 (Numéraire) and the price for the capital good was calculated. Expressed in monetary terms, the conditions for steady growth were thus determined. ${\ displaystyle x_ {2}}$ ${\ displaystyle p_ {2}}$ ${\ displaystyle x_ {2}}$ ${\ displaystyle x_ {1}}$

The general equation

In order that steady growth is also possible in terms of material, the following must apply:

${\ displaystyle (a_ {11} \ cdot x_ {1} + K \ cdot a_ {12} \ cdot x_ {2}) \ cdot (1 + g) = x_ {1}}$
${\ displaystyle (a_ {21} \ cdot l \ cdot x_ {1} + K \ cdot a_ {22} \ cdot l \ cdot x_ {2}) \ cdot (1 + g) = K \ cdot x_ {2} }$

An additional size K must now be calculated, which reflects the size ratio between departments I and II, where weight 1 is assigned to department I and weight K to department II.

If in such growth models it is assumed that the total profits in the next period will be used to produce more on the same technical basis in accordance with the rate of profit r, then the result is that the growth rate from a material point of view (g) is equal to the rate of profit r.

The concrete numerical examples

The first numerical example with the rate of profit results in: ${\ displaystyle r = 9 {,} 61 \, \%}$

${\ displaystyle (0 {,} 8 \ cdot 1 + 0 {,} 2808 \ cdot 0 {,} 4 \ cdot 1) \ cdot (1 + 0 {,} 0961) = 1}$
${\ displaystyle (0 {,} 1 \ cdot 2 \ cdot 1 + 0 {,} 2808 \ cdot 0 {,} 1 \ cdot 2 \ cdot 1) \ cdot (1 + 0 {,} 0961) = 0 {, } 2808 \ cdot 1}$

The weight of Division II is . For the second numerical example with the rate of profit we get: ${\ displaystyle K = 0 {,} 2808}$ ${\ displaystyle r = 10 {,} 30 \, \%}$

${\ displaystyle (0 {,} 85 \ cdot 1 + 0 {,} 14154 \ cdot 0 {,} 4 \ cdot 1) \ cdot (1 + 0 {,} 1030) = 1}$
${\ displaystyle (0 {,} 1 \ cdot 2 \ cdot 1 + 0 {,} 14154 \ cdot 0 {,} 05 \ cdot 2 \ cdot 1) \ cdot (1 + 0 {,} 1030) = 0 {, } 14154 \ cdot 1}$

Division II weight is now . The respective growth rates are equal to the profit rates . ${\ displaystyle K = 0 {,} 14154}$ ${\ displaystyle g}$ ${\ displaystyle r}$

On the left-hand side of the equations, the inputs are in the first equation in each case and the quantity that is required as input is in the second equation on the left . On the right-hand side of the equations, the first equation shows a good from and the second equation K goods from . ${\ displaystyle x_ {1}}$ ${\ displaystyle x_ {2}}$ ${\ displaystyle x_ {1}}$ ${\ displaystyle x_ {2}}$

If one evaluates the input an with the price , then one obtains the constant capital c. If you evaluate the input with the set price , you get the variable capital v. If one evaluates the output of a good and K goods of with the respective prices and , one obtains the total turnover of the economy c + v + m. ${\ displaystyle x_ {1}}$ ${\ displaystyle p_ {1}}$ ${\ displaystyle x_ {2}}$ ${\ displaystyle p_ {2} = 1}$ ${\ displaystyle x_ {1}}$ ${\ displaystyle x_ {2}}$ ${\ displaystyle p_ {1}}$ ${\ displaystyle p_ {2}}$

If you subtract the amount for constant and variable capital (c + v) from this turnover, you get profit m as the remainder.

The composition of the value of capital c / v, the rate of surplus value m / v and the “ wage share ” v / (m + v) can then be calculated.

For the wage share results in the first case and in the second case . This corresponds to added value rates of 0.706 and 1.389 respectively. The value composition of the capital c / v results in 6.34 in the first case and 12.49 in the second. According to the formula ${\ displaystyle 58 {,} 6 \, \%}$ ${\ displaystyle 41 {,} 9 \, \%}$

            Profitrate  $p={\frac {\frac {m}{v}}{{\frac {c}{v}}+1}}$

The two profit rates and are calculated again . ${\ displaystyle 9 {,} 61 \, \%}$ ${\ displaystyle 10 {,} 30 \, \%}$

Comparative static method

In this example, it should be noted that the comparative static method is used, that is, different economies that are on an equally balanced growth path are compared with one another. If one moves from an equilibrium consideration to imbalance models, other results can certainly arise. If the capitalists increase the technical composition of capital, because this increases the rate of profit for them, this can become an ongoing process, even under the pressure of competition, which means that the economy no longer has the time to turn into a new, equilibrium growth path , but continued labor-saving investments in rationalization lead to a tendency to stagnate. The law of the tendency of the rate of profit to fall is nowadays understood as an imbalance model, also under the impression of the Okoshio criticism.

Individual evidence

↑ No statement is made about the division of the total work volume in hours among the individual workers. One could assume, for example, that a worker works around eight hours a day with a correspondingly high daily wage. The number of workers then changes accordingly with the total work volume measured in hours.
↑ Bertram Schefold (1979): "Fixed capital as a co-product and the analysis of accumulation in different forms of technical progress" in: Contributions to Marxian theory 13, Frankfurt am Main. See p. 284ff there.
↑ ^a ^b Michael Heinrich: Critique of the political economy . Butterfly Verlag, Stuttgart 2004, ISBN 3-89657-588-0 , p. 148 ff .
↑ See Helmut Dunkhase: On the profit rate discussion in the Marxist papers. Marxist Blätter November 2010. On the profit rate discussion in the Marxist Blätter ∗ Helmut Dunkhase
↑ Michael Heinrich: The science of value . Westphalian steam boat, Münster 2003, ISBN 3-89691-454-5 , p. 369 f .
^ Price, value and profit - a continuous, general, treatment. By: Freeman, Alan, April 1996, Online at http://mpra.ub.uni-muenchen.de/1290/MPRA Paper No. 1290 (of November 7, 2007 / 01:40; PDF; 685 kB)
^ Karl Marx: The capital . In: Marx-Engels works . tape 25 . Dietz-Verlag, Berlin 1972, p. 237 .
↑ Klaus Müller: Trendy fall or rise? On the complexity of economic phenomena using the example of the general average profit rate . Marx-Engels-Jahrbuch 2009. Akademie-Verlag, Berlin 2010, ISBN 978-3-05-004677-8 , pp. 47-75 .
↑ Stephan Krüger, p. 414f.
↑ Alfred Müller, p. 103f.
↑ Alfred Müller: Die Marxsche Konjunkturtheorie - An over-accumulation-theoretical interpretation. PapyRossa Cologne, 2009 (dissertation 1981), p. 160.
^ Stephan Krüger: General theory of capital accumulation - business cycle and long-term development tendencies. Critique of Political Economy and Analysis of Capitalism, Volume 1 . Hamburg 2010, ISBN 978-3-89965-376-2 .

literature

Duncan K. Foley: Understanding Capital: Marx's Economic Theory. Harvard University Press 1986, ISBN 0-674-92088-0 .
Alan Freeman: Price, value and profit - a continuous, general, treatment. In: Alan Freeman, Guglielmo Carchedi (Ed.): Marx and non-equilibrium economics. Elgar, Cheltenham, Brookfield, US 1996. (a mathematical argument against Okishio's theorem, with typos in the formulas) online .
Michael Heinrich : The science of value. Westphalian steam boat, Münster 2003, ISBN 3-89691-454-5 .
Nobuo Okishio: Technical Changes and Rate of Profit. 1961, German in: HG Nutzinger , E. Wolfstetter (Ed.): The Marx theory and its criticism, 2 volumes. Frankfurt am Main 1974.
Piero Sraffa: Goods production using goods. Afterwords by Bertram Schefold. First published in 1960. Suhrkamp, Frankfurt am Main 1976.

[1] No statement is made about the division of the total work volume in hours among the individual workers. One could assume, for example, that a worker works around eight hours a day with a correspondingly high daily wage. The number of workers then changes accordingly with the total work volume measured in hours.

[2] Bertram Schefold (1979): "Fixed capital as a co-product and the analysis of accumulation in different forms of technical progress" in: Contributions to Marxian theory 13, Frankfurt am Main. See p. 284ff there.

[MH-3] Michael Heinrich: Critique of the political economy . Butterfly Verlag, Stuttgart 2004, ISBN 3-89657-588-0 , p. 148 ff .

[4] See Helmut Dunkhase: On the profit rate discussion in the Marxist papers. Marxist Blätter November 2010. On the profit rate discussion in the Marxist Blätter ∗ Helmut Dunkhase

[5] Michael Heinrich: The science of value . Westphalian steam boat, Münster 2003, ISBN 3-89691-454-5 , p. 369 f .

[6] Price, value and profit - a continuous, general, treatment. By: Freeman, Alan, April 1996, Online at http://mpra.ub.uni-muenchen.de/1290/MPRA Paper No. 1290 (of November 7, 2007 / 01:40; PDF; 685 kB)

[7] Karl Marx: The capital . In: Marx-Engels works . tape 25 . Dietz-Verlag, Berlin 1972, p. 237 .

[8] Klaus Müller: Trendy fall or rise? On the complexity of economic phenomena using the example of the general average profit rate . Marx-Engels-Jahrbuch 2009. Akademie-Verlag, Berlin 2010, ISBN 978-3-05-004677-8 , pp. 47-75 .

[9] Stephan Krüger, p. 414f.

[10] Alfred Müller, p. 103f.

[AM-11] Alfred Müller: Die Marxsche Konjunkturtheorie - An over-accumulation-theoretical interpretation. PapyRossa Cologne, 2009 (dissertation 1981), p. 160.

[12] Stephan Krüger: General theory of capital accumulation - business cycle and long-term development tendencies. Critique of Political Economy and Analysis of Capitalism, Volume 1 . Hamburg 2010, ISBN 978-3-89965-376-2 .