Exposition and refutation of the transformation problem of Bortkiewicz, dating from 1922 (Under the Banner of Marxism, no.7-8, p.145-153).

To illustrate the transformation of values of commodities into prices of production, Marx gives the following 2 illustrative Tables: Table 1. Determination of value of commodities. Table 2. Determination of price of production of commodities.1

Each of these Tables operates with five groups of capitals, in which each has the same organic composition. For capitals of I this latter (c: v) is 4: 1, for capitals of II 7: 3 etc.; the absolute magnitude of capitals is not taken into account. The first table gives the value of products of production of enterprises with the same organic composition of capital under the assumption, that the rate of surplus value in all cases = 100% and that the used up part of constant capital (the transferred to the commodity value entering into it of the circulating part of constant capital and related [/сношенной] part of fixed capital) is different for I, II, III, IV, V. Under such conditions it is not difficult to calculate, for example, the value of the product of the production of enterprises, united in group III. The formula ac + v + m give us, in this case 52 + 40 + 40 = 132. In the same way we get also the rest of the results, signified in our Table under the heading (W).

If we calculate the rate of profit for I-V individually, then in each case, we obtain, as can be seen from Table 1, different results, since c + v = 100, but m (equal to v) depends on the organic composition of the corresponding capital. But a capitalist society does not condone this kind of status quo. Competition and the migration of capitals do their thing, and as a result along the whole of society there is a tendency to establish a general rate of profit, independent of the organic composition of this or the other capital. According to Marx, the general rate of profit is determined, as is known, by the expression in percentage of the ratio of the total surplus value to the total social capital. Consequently, if the organic composition of the latter is given and furthermore the rate of exploitation, then by the same token, also is given the general rate of profit. The organic composition of all five groups of capitals, taken together and representing themselves the total social capital, is defined by the ratio c = 390 to v = 110; but once the rate of surplus value = 100%, then m = v = 110, and if so, then the searched for average rate of profit is

110/(390+110) = 22%.

"So far as profits are concerned, - Marx notes, - the various capitalists are just so many stockholders in a stock company in which the shares of profit are uniformly divided per 100, so that profits differ in the case of the individual capitalists only in accordance with the amount of capital invested by each in the aggregate enterprise, i. e., according to his investment in social production as a whole, according to the number of his shares."2

Table 2 illustrates how the establishment of the general rate of profit leads to the formation of prices of production, differing from values. Here, in the calculation of the prices of production, e.g., the product of III, to the cost of production ac + v = 52 + 40 = 92 is added already not the entire surplus value of 40 as in Table 1, but only 22. The result is 114, i.e., 18 less: surplus value is like a common treasury of the whole class of capitalists and already then in the form of profit gets distributed among individual capitalists in proportion to their capital. But as the sole source of profit is surplus value, the first constructed by Marx theory of the average rate of profit must inevitably lead to the fact that if we add up separately all the without exclusion values of commodities and their prices of production, then the deviation of the first from the second, taking place for each commodity separately, gets mutually cancelled and as a result gives an equal sums.3 So, in our Table + 2 - 8 - 18 + 7 + 17 = 0,

Wi + Wii + Wiii + Wiv + Wv = 90 + 111 + 132 + 70 + 20 = 422.

Pi + Pii + Piii + Piv + Pv = 92 + 102 + 114 + 77 + 37 = 422.

This is the viewpoint of Marx's economic theory. But against Marx came out the well known Mr. v. Bortkiewicz, on the heels of whom followed Mr. Otto Kühne.4 Having undertaken mathematical criticism of Marx, they first encountered the "internal contradiction" of the above schemes. "There is nothing easier - writes Bortkiewicz - than to show that the procedure employed by Marx for the transformation of values into prices is erroneous" (meant are prices of production).5 In fact, - he and his friend opponent say - let's suppose that Table 1 refers to simple reproduction. And in simple reproduction the value of commodities, manufactured by spheres producing means of consumption for workers, should be equal to the aggregate variable capital; the value of commodities from manufacturing spheres producing means of consumption for the capitalists, should coincide with the aggregate of their total profit; the value of commodities, made means of production, should be equal to the used up part of constant capital of the whole society. All this we know from volume II of "Capital."6 Let's suppose further that the "spheres of production" I and V manufacture articles of consumption of the capitalist class. This, according to our authors, is perfectly assumable, as the value of the commodities produced by these spheres 90 + 20 = 110, i.e., the aggregate surplus values obtained by the capitalist class. Let's suposse by equal fashion the rightful assumption, that III and IV manufacture the means of production, because the value of the commodities manufactured by them (132 + 70) coincides exactly with the used up part of the social constant capital (202). Sphere II clearly produces the means of consumption for the working class, because the value of commodities manufactured by it 110 is equal of the sum of wages. In this way, in Table 1 all three conditions are met, under which is possible implementation in simple reproduction.

1) mi+ii+iii+iv+v = Wi+v (means of cons. capitalists)

2) vi+ii+iii+iv+v = Wii (means of cons. workers) }(alpha)

3) aci+ii+iii+iv+v = Wiii +iv (means of cons. production)

Let's note that the same table, taken according to Marx ("Capital," vol. III, part1, Moscow, 1907, p. 134.), gives not such results, since Marx takes acii = aciii = 51. Therefore his table gives, for example, the inequality

vi+ii+iii+iv+v < Wii or 110 < 111.

But as the change Bortkiewicz made, which as already noted (footnote 1) has nothing unjustified in itself - the magnitudes of used up part of constant capital are chosen quite arbitrarily also in Marx - we also continue to follow the exposition of his opponents, who precisely for objection also required to make in the schemes of Marx a change of numerical data.

But to what results do we come, if we put the system of equations (alpha), characterizing simple reproduction to Table 2, expressing the corresponding commodities in prices of production? It appears that

mi+ii+iii+iv+v < Pi+v, or 110 < 129

vi+ii+iii+iv+v > Pii, or 110 > 102

aci+ii+iii+iv+v > Piii+iv, or 202 > 191.

"We believe therefor, - writes Kühne, - that these results prove that the transformation of values in prices (of production), as Marx gives it, should decisively be rejected, precisely because it leads to unacceptable absurdities (unerträgliche Inkonzinnitäten)."

These are the conclusions of Bortkiewicz and Kühne.

With a more or less careful analysis it is not difficult to see, that these conclusions, so unfavorable for Marx's economic theory, are entirely based on a "little" falsification. The fact is that Marx here understands by "production spheres" I, II, III etc. not certain industrial branches, producing certain use-values, but groups of enterprises with the same organic composition of capital: this latter also is in the given case for Marx the fundamentum divisionis. And if this is so, then it is entirely obvious that every of our "spheres" can produce at the same time means of consumption of workers, and means of consumption of capitalists, and even elements of constant capital. When the Mr. critics fix in advance for certain "spheres" of production of commodities certain use-values, then such a priori assumption is justified alone really just by the artificially selected by them numerical data.

The only rightful question, that here could have been presented, reads: which of the "spheres" I - V or which parts of these "spheres" should engage in the production of consumer commodities of capitalists, which should produce the means of consumption of workers and which the means of production, in order for conditions of simple reproduction to be met both in the 1st, and in the 2nd Tables? The answer to this question (and herewith not the only) can be obtained using a system of two equations with two unknowns. Let's assume that on the production of means of consumption of capitalists is spent a part of the Ist and a part of the IInd spheres. Let the corresponding part of I be x1, and the corresponding part of II - y1. In this case the value of means of consumption of capitalists, expended by the searched for parts of the Ist and the IInd sphere, on the basis of Table 1, constitutes 90x1 + 110y1; the price of production of these means of consumption, on the basis of Table 2 is determined through 92x1 + 102y1. And both these sums [see the first equation system (alpha)] must be equal to the total surplus value, or what is the same, total profit of the class of capitalists, ie, 110. We obtain thus a system of two equations with two unknowns

90x1 + 110y1 = 110

92x1 + 102y1 = 110

Solution of these equations gives x1 = 44/47 and y1 = 11/47. In other words, in the production of means of consumption of the capitalist class must expend 44/47 of sphere I and 11/47 of sphere II. Suppose further that on production of means of consumption for the workers is expended a part of sphere III, defined by x2, and a part of sphere IV, defined by y2.

Then we, as in the first case, we will have the system of equations:

132x2 + 70y2 = 110

114x2 + 77y2 = 110

Solving these equations, we obtain x2 = 55/156 and y2 = 495/546; that is, 55/156 of sphere III and 495/546 of sphere IV is employed for the manufacture of consumer goods of workers. It is easy to see that the production of elements of constant capital employs 3/47 of sphere I, 36/47 of II, 101/156 of III, 51/546 of IV and the whole of V.

Checking our solution shows clearly that both Table 1 and 2 fully satisfy the conditions of simple reproduction:

44/47 . 90 + 11/47 .110 = 44/47 . 92 + 11/47 . 102 = 110

55/156. 132 + 495/546 . 70 = 55/47 . 114 + 549/546 . 77 = 110

3/47 . 90 + 36/47 . 110 + 101/156 . 132 + 51/456 . 70 + 20 = 202

3/47 . 92 + 36/47 . 102 + 101/156 . 114 + 51/456 . 77 + 37 = 202.

Splitting I in proportion 44:3, II in proportion 11:36, III in proportion 55:101 and IV in proportion 495:51, we instead of a sphere of production uniting only just the same organic composition of capital, obtain a branch of production manufacturing certain groups of use-values. It is entirely clear that by making this split of the relevant "spheres," we neither violate the organic composition of capital (c:v), nor the rate of surplus value (m/v), nor the rate of profit (m/c+v), because in all cases, we multiply the numerators and denominators of these fractional expressions by one and the same number. The same applies also to the used up part of constant capital (a). Making these estimates up to 0,001, we instead of Tables 1 and 2 get Tables 3 and 4:

I A and II A manufacture means of consumption of the capitalists,

III A and IV A manufacture means of consumption for the workers,

I B, II B, III B, IV B and V manufacture elements of constant capital.

The ratios of system (alpha) are met:

mi+ii+iii+iv+v = WiA+iiA, or 110 = 84,235 + 25,745

vi+ii+iii+iv+v = WiiiA+ivA, or 110 = 46,539 + 63,461

aci+ii+iii+iv+v = WiB+iiB+iiiB+ivB+v, or 202 = 5,745 + 84,255 + 85,481 + 6,539 + 20, 000.

In the same way we get the transformed Table of prices of production. (see Table 4).

The application to this Table of criteria of simple reproduction leads us to the same results, as also in the analysis of Table 3. In fact:

m'i+ii+iii+iv+v = PiA+iiA, or 110 = 86,128 + 23, 872

vi+ii+iii+iv+v = PiiiA+ivA, or 110 = 40,132 + 69,868

aci+ii+iii+iv+v = PiB+iiB+iiiB+ivB+v or 202 = 5, 872+ 78,128 + 73,808 + 7,192 + 37,000.

And if this is so, then between the schemes of volume III of "Capital" (calculation of prices of production) and the schemes of volume II (in conditions of simple reproduction) there is absolutely no contradiction.

original title: Об одном „противоречии“ в экономической системе К. Маркса. - Шолом Дволайцкий.

source: Под знаменем марксизма (first volume online).

- 1. 1) Marx instead of 50 and 52 took in both cases 51 (ac, the used up constant capital, of sphere II and III). We take the table with the changes made by prof. L. Bortkiewicz, as they don't change the heart of the matter. See L. v. Bortkiewicz: "Wertrechnung und Preisrechnung im Marxschen System," 2. Artikel. Archiv fur Sozialwissenschaft und Sozialpolitik," Neue Folge, XXV Band, 1. Heft, Tübingen 1907, p. 14.
- 2. K.Marx, "Capital," vol.III, part 1, Moscow 1907, p.134. https://www.marxists.org/archive/marx/works/1894-c3/ch09.htm
- 3. I leave out a lengthy footnote in the original text with the mathematical formulation - translator's remark
- 4. Dr. Otto Kühne, Untersuchungen über die Wert- und Preisrechnung des Marx'schen Systems. Eine dogmen-kritische Auseinandersetzung mit L. v. Bortkiewicz, Greifswald 1922, p.19 et seq.
- 5. Bortkiewitcz, cit. work, p. 15.
- 6. K. Marx, "Capital," vol.II, Moscow 1920.

## Comments

I made some corrections in my translation.

For information about the author of this article: http://libcom.org/forums/theory/forgotten-great-theoreticians-02042010#c...

I think this article is the earliest Marxist response to Bortkiewicz.

Thank you.

Photobucket no longer allows free third party hosting, so here's the image of tables 3 and 4 (via cubeupload):

I added them to the body of the text. Are they the correct ones? And shouldn't tables 1 and 2 be in the text rather than just the intro? If you let me know where tables 1 and 2 should appear in the text, I can add them.

Excellent. The discussion of the first couple of tables begins immediately, so it's fine to leave them in the intro.

btw, I think the point of this text was made also in a footnote of a contribution (just remember the author's name sounded Spanish) in Ricardo, Marx, Sraffa. The Langston Memorial Volume (1985).