§3.1 Rate of surplus-value and rate of profit

In this section I use the following four main concepts and definitions. Circulating capital (k):

k_i = c_i + v_i [definition] (1)

where c and v are the parts invested in means of production and in labour-power (the wages sum of wl). For simplicity, we abstract throughout, like Marx, from fixed capital and from differences in turnover time.⁸ The subscript i refers to a specific sector of production (later, the subscript j will refer to any other sector).

The rate of surplus-value (s´):

s´_i = f(e*_i) [explanation – pro memory] (2a)

s_i = s´_iv_i [explanation; here reduced to definition] (2b)

(where e* is the ratio of ‘surplus labour’ and ‘necessary labour’).

Capital I posits surplus-value (s) and its increase as the driving force of capital. The focus of the middle part of Capital I is the explanation of surplus-value in terms of labour-time and of the productive forces of labour operating on the means of production.⁹ The rate of surplus-value (s´) is the concentration of these explanatory determinants. Here, as for Marx in the Kapital III manuscript, all this explanation is assumed. Thus we have s´ as a determinant for s. It remains to be seen (cf. the subsections below) how it is also a determinant for profit (r) and the rate of profit (r´):

r´_i = r_i/k_i [definition] (3)

The spectrum of wages is posited synchronically equal for all sectors (w_i = w), hence:

wl_i = v_i [approximation] (4)

For the sake of brevity, I posit throughout this section a number of assumptions without further argument. For our purposes there are three types: ‘simplifications’ serve to make a problem tractable; the same applies to ‘stage simplifications’ (these, however, are dropped in a later stage); ‘approximations’ {Annäherungen} set out the uninhibited result of forces (tendency laws).

The assessment of Marx’s general rate of profit transformation in the 1864–65 draft for Part Two of Das Kapital III depends very much on the view of how the categories presented in this part are connected to those of Capital I. We shall see (§4) that Marx is aware of the importance of the connection, and how he hesitates about the Part Two consequences (his self-interpretation!) for the, then, draft of Das Kapital I. Both in order to shortcut the discussion in the next section (§4), and because the categorical connections are not obvious, the current section sets out a brief formal presentation of the two main alternatives (§3.2–§3.5). Note that a formal presentation necessarily loses conceptual richness; on the other hand, once quantitative matters are involved – as is the case – it is a means of precision.

For the purposes of this section I reduce Marx’s complex method of many stages (levels of abstraction) to simply two stages: that of Capital I and of Part Two of Capital III. The movement to the later stage is one of increasing determination (concretion or completion).

In considering surplus-value and production, we can, of course, not get around exchange and prices. We assume that ‘market prices’ converge to what I provisionally call ‘supply prices’ (and what Marx will call “production prices” in Capital III). We want to explain production (x = pq) in terms of these convergence prices (p). In principle we can adopt two methods, and we can take these as possible interpretations of the two stages referred to. I call them ‘the method of concretion’ (§3.2–§3.3). and ‘the method of completion’ (§3.4–§3.5). For both methods, we shall be concerned throughout with synchronic matters only (some synchronic uniformity of sectors of production may go along with diachronic change).

§3.2 [CC-1] Method of Concretion at Volume I level: Abstract explanandum

Before we get to the explanation of the prices p, we first explain an abstract approximation of them, labelled ‘pi’ (π), whence we also have an abstract approximation of production: πq. Thus we have an abstractum or a theoretical construct. (This may be interpreted as a systematic-dialectical stage; or also as a theorisation of variables constructed in an (ideal) experimental constellation, so reaching ceteris paribus conditions.) In brief we posit:

π_iq_i = k_i + s´_iv_i [explanation of a construct/abstractum] (5)

Via the moment of s´_iv_i this incorporates the explanation represented by equations (2a and 2b). We proceed by adding two simplifying assumptions. First, that of synchronically uniform rates of surplus-value:

s´_i = s´ (for all i) [simplification or approximation] (6)

A key question is whether this is indeed a simplifying assumption or rather an approximation of a systemic force (see §3.3, §4.1 and §5). Second, that of synchronically uniform compositions of capital:

(k/v)_i = k/v (for all i) [stage simplification] (7)

Dividing (5) through v_i, and making use of (6) and (7), we have:

π_iq_i/v_i = k/v + s/v (for all i) [implication] (8)

Because wl_i = v (eqn. 4), then substituting wl_i for v_i, and multiplying by w, we also have:

π_iq_i/l_i = k/l + s/l (for all i) [implication] (9)

In the way I have presented it here, it is the implication of the construct that its prices (π) are proportional to labour commanded (Smith) as well as to labour embodied (Ricardo).

Even if Marx introduces the (general) rate of profit only in Capital III, another implication is that we have uniform rates of profit since (cf. eqn. 3):

r´_i = s´v_i/k_i = s´/(k_i/v_i) = s´(k/v) [implication] (10)

(This is roughly how I interpreted Capital I until my study of the 1864–65 manuscript discussed in this essay.¹⁰ In §4 we will see that this interpretation makes Marx’s Das Kapital III manuscript transformation procedure methodologically mistaken.)

§3.3 [CC-2] Method of Concretion at Volume III level: concretion

We drop the construct π and proceed to ‘production prices’ (p). We assume that real market prices converge to these production prices (‘centres of gravitation’).

Variant (a)

Marx (at some point, see §4.2-§4.3) defines this constellation as follows.

• Analogous to the πq equation (5):

p_iq_i = k_i + s´_iv_i [explanation] (11)

Via the moment of s´_iv_i this again incorporates the explanation represented by equations (2a and 2b).

• Rates of profit converge to uniform rates:

r´_i = r´ (for all i) [approximation] (12)

• Non-uniform compositions of capital (thus 7 is dropped). Hence in general:

(k/v)_i ≠ (k/v)_j [empirical observation] (13)

• Uniform rates of surplus-value (thus 6 is maintained):

s´_i = s´ (for all i) [simplification or approximation] (6)=(14)

Because of the diverging k/v there is no labour-commanded or labour-embodied proportionality for the production prices p (we merely have p_iq_i/v_i = (k/v)_i + s´). Marx is aware and explicit about this. Rather more problematical is that the combination of the three restrictions (12), (13) and (14) is impossible.

r´_i = s´v_i/k_i = s´/(k/v)_i [implication] (15)

Thus r´_i ≠ r´_j. We can have uniform profit rates only with either (1) both of s´ and (k/v) uniform (the πq constellation) or (2) both of them non-uniform.

Variant (b): reconstruction of variant (a)

The obvious way to mend the incompatibility is to reconstructively drop the (simplifying) assumption of uniform rates of surplus-value (14).¹¹ So we have:

s´_i ≠ s´_j [empirical observation] (14*)

Whence we derive:

r´_i = r´ = (s´_iv_i)/k_i (for all i) [implication] (15*)

In this reconstruction the micro equality of profits and surplus-value (r = s) is maintained;¹² by implication we also have the macro R = S. Each at this level.

Variants (a) and (b)

Instead, Marx endeavours to get around the incompatibility via a macro-micro détour in which he transposes quantities between the two levels, and that in effect results in:

r´_i = r´ = [(s´v_i)+δ_i]/k_i [‘methodologically illegitimate détour’] (15**)

(with the aggregate sum of the δ_i amounting to zero).

However, within the constellation of the Concretion Method this is methodologically illegitimate (though not for the reasons usually stated). Since the two explananda πq and pq are non-identical, transposition of quantities from the one to the other level makes no sense. (Certainly, it does make sense to apply the qualitative conclusions from the earlier level to quantification at the lower level – that is, as long as any ceteris paribus conditions of the earlier level allow for it.)

§3.4 [CP-1] Method of Completion at Volume I level: partial explanation of ‘concretum’

There is also a very different interpretation of what happens at Capital I level. In this alternative we have no π construct but we rather start immediately with the convergence price p (and hence pq), thus prices of production – even if these are explicitly called like this only at Volume III level. However, given that we have the same set of simplifying assumptions as in [CC-1] (equations 6 and 7) we have approximations for the explanation of p, leaving some of p unexplained. (In §4.4 we shall see that Marx at some point leans to this (self-)interpretation. Fred Moseley (e.g. Moseley 2000) seems to be proposing something similar. For each I am not sure they would draw all of the consequences.) Thus p is explained in successive stages, here reduced to two. Until we reach full explanation we have an unexplained part u.

p_iq_i = k_i + s´_iv_i + u_i [partial explanation] (16)

As in CC-1 the moment of s´_iv_i incorporates the explanation represented by equations (2a and 2b). Again, we proceed by adding on the two simplifying assumptions of synchronically uniform rates of surplus-value and of synchronically uniform compositions of capital:

s´_i = s´ (for all i) [simplification/approximation] (6 =) (17)

(k/v)_i = k/v (for all i) [stage simplification] (7 =) (18)

Here, however, we have from the beginning no labour-commanded or labour-embodied proportionality because of the factor u:

(p_iq_i)/v_i = k/v + s/v + u_i/v_i [implication] (19)

(cf. 8 and 9 above). It has been argued (e.g. by Moseley) that Capital I, merely provides a partial explanation of prices (in a different way, of course, nor does CC-1 provide a full explanation).¹³

§3.5 [CP-2] Method of Completion at Volume III level: completion

Rates of profit converge to uniform rates:

r´_i = r´ (for all i) [approximation] (12) = (20)

Hence we have for (what is now explicitly thus called) prices of production:

p_iq_i = k_i + r_i [accounting identity / empirical observation] (21)

p_iq_i = k_i + r´k_i [putative explanation] (22)

This is a putative explanation pending the determination of r´. Again, we have non-uniform compositions of capital (thus the simplifying assumption of equation 18 is dropped):

(k/v)_i ≠ (k/v)_j [empirical observation] (13) = (23)

The uniform rates of surplus-value thesis of equation (17) is maintained; however, given equation (22) it is considered to play no (new) role at the current level.

s´_i = s´ (for all i) [simplification/approximation] (17)

Reordering equations (20)–(22) we have:

r´_i = r´ = r_i/k_i (for all i) [implication] (24)

Because the pq in the two price equations (16) and (22) are on the same plane, it is now methodologically legitimate to transpose quantities from the one level to the other (CP-1 and CP-2). (Cf. the Marx/Engels (in)famous aggregation tables of Capital III, chapter 9.)

Substituting (16) and (2b) into (21) we have:

r_i = s_i + u_i [implication] (25)

r_i = s´_iv_i + u_i [implication] (26)

In fact, this is a shortcut for Marx’s aggregation tables. However, it deserves a serious warning (and here the usual critiques come in), namely that it is assumed that the unexplained factor of u_i concerns, and so is to be allotted to, profits rather than (in part) any other factor in equation (16).

We now aggregate surplus-value (S) and profits (R):

S = Σs_i [aggregation] (27)

R = Σr_i [aggregation] (28)

Next it is posited that S explains, and fully determines, R:

R = S    (right to left determination) [derived explanation] (29)

(The explanation is derived from the set of equations (1)–(4) and (16)–(28).)

We also aggregate equation (25) into:

R = S + U [aggregation] (30)

Because theoretical priority is given to (29) we have:

U = 0 [theory decision] (31)

Finally, the sector u_i’s are determined via the rate of profit criterion. From (24)–(26) we have:

r´_i = r´ = (s´v_i + u_i)/k_i [implication] (32)

Hence:

u_i = r´k_i − s´v_i [implication] (33)

§3.6 Some conclusions and a preview

1. The last equation (33) is the ‘tache de beauté’ of this CP set up. The u_i’s are determined by the ‘general rate of profit’, instead of the rate of profit being fully determined by explanatory entities. Two objections may be raised against this alleged blemish. One is that we have a simultaneous determination. The other, as argued by Marx, is that, at a still lower level of abstraction, the u_i’s are determined by competition (see §4.3).

2. Remind Marx’s equation (15**) from CC-2 that I qualified as ‘methodologically illegitimate’ (a qualification that I maintain, that is within the Concretion Method). We had:

r´_i = r´ = [(s´v_i)+δ_i]/k_i (15**)

Its outward appearance is similar to (32).

3. In Variant (b) of CC-2, i.e., my reconstruction of Variant (a), we have the rate of profit fully determined without any transformation ‘problem’:

r´_i = r´ = (s´_iv_i)/k_i (15*)

4. In my view CC-1 is the better interpretation of Das Kapital I, published by Marx in 1867!, and revised shortly before it. In my view CP-2 is the better interpretation of Marx’s manuscript for Part Two of Das Kapital III, written in 1864/65! (at least up to some point of the manuscript – see §4.4).

5. How is this so? This is methodologically inconsistent! Yes. Because of this problem, the following section (§4) is rather complex. My hypothesis is as follows. The 1864–65 manuscript for Part Two initially followed the path of the 1861–61 manuscript, which is in line with CP-2 – i.e., the ‘chapter 9’ transformation procedure (see §4.3). Then, reflecting on this transformation procedure in Marx’s ‘chapter 10’, Marx got increasingly worried about what he had been doing in terms of his manuscript for Das Kapital I (see §4.4).¹⁴ (Engels, however, in his editing of the 1864/65 manuscript for ‘his’ Das Kapital III polished away these worries whence we have the ‘CP-2’ result.)

6. Each of [CC-1 with CC-2] and [CP-1 with CP-2] is consistent. The usual post-Marx solutions to the transformation problem obviously took Engels’s version of Das Kapital III for granted, and then combined the ‘theoretical domain’ of CC-1 with that of CP-2, which is methodologically inconsistent. If one would want to use the ‘domain’ CP-2 (not only for a solution to the transformation problem but also for any other quantitative matters) then one is bound to CP-1.

§4.1 General outline

Regarding Part Two (originally one ‘chapter’) of the 1864–65 manuscript Engels left unchanged the main chapter structure of the text for his Capital III edition (chapters 8–12); however, there are changes of order within the chapters as well as texts that have disappeared altogether. For the reader’s convenience, I therefore refer in this section to the Capital III chapters.

At the very opening of Part Two (Chapter 8) we immediately find the crucial assumption of a uniform rate of surplus-value, for all of Part Two (M:212; E:151; F:241).¹⁵ It is repeated regularly throughout the part. According to this manuscript it is not just a simplifying assumption but rather a law. In ‘Chapter 10’ Marx writes that competition between labourers gives rise to:

a general rate of surplus-value – tendentially that is, as for all economic laws; we posit it as a theoretical simplifying presupposition; in fact it is the actual presupposition of the capitalist mode of production even if inhibited by practical frictions …; in theory we assume that the laws of the capitalist mode of production develop in their pure form {rein}. (M:250-mt; cf. E:184, F:275)

Given this assumption/law the terrain of the problematic for Part Two is defined by the prevalence of non-uniform compositions of capital (ratios of k/v) between sectors of production, or also non-uniform turnover times of capital – which offers the same problematic (M:216). Thus whereas Capital I, Parts Four, Five and Seven treat the diachronic change of the capital composition (with any divergences between sectors ‘bracketed’) the current part makes the reverse assumption: constant though diverging.

Part Two comprises three main chapters (chs. 8–10) that in my view should be considered together (the two smaller chapters (11–12) may be regarded as ‘addenda’). I begin with a brief outline of these three chapters before discussing some of the details.

In Chapter 8 Marx sets out the constellation that results from the assumptions just indicated. These together with the assumption that commodities are sold ‘at their values’ [presumably equation (5)] would result in differing rates of profit between sectors (M:223–4) [cf. equation (10)]. This constellation, however, does not exist in reality (M:229–30). Hence the presentation is insufficient.

Therefore the next chapter (Chapter 9) must widen the theoretical terrain. Marx introduces a new concept ‘production price’; “its presupposition is the existence of a general rate of profit” (GRP) (M:234). Thus Marx posits in Chapter 9 the ‘production price’ [cf. equation (11)/(22)]. He indicates that this is “a transformation of value” (“eine verwandelte Form des Werths” – M:239; E:173) but nevertheless carries out a quantitative substitution between the two levels of abstraction! This is a problem if his method is the Method of Concretion (§3.2–§3.3), which is the hypothesis from which I start.

The point is that the status of this transformation is very different from the purely conceptual one that Marx presented in Part One and in which no quantitative differences are involved. Consider e.g. the following quotation from its Chapter 2:

while surplus-value and profit are in fact the same – they are numerically identical – profit is still for all that a transformation {verwandelte Form} of surplus-value … (M:64; cf. E:58, F:139)

Or from Chapter 1:

Materially {Stoff; stofflich} considered … the profit … is not different from the surplus-value itself. Hence its absolute magnitude is not different from the magnitude of the surplus-value (…) it is however a transformation {verwandelte Form} of the latter … (M:8–9-mt)¹⁶

For the Part One transformation, no new quantitative determinations come in. However, as Marx is well aware, for his Part Two transformation we do have different quantities (π_iq_i ≠ p_iq_i and k/v ≠ k_i/v_i and s´v_i ≠ r´k_i), at least for the micro level. Thus he substitutes quantities of a theoretically insufficient constellation (πq) into that of a theoretically enriched, more concrete, constellation (pq). On the basis of the Method of Concretion this would be very awkward. Then the problem is not primarily that the two famous conditions of aggregation (of Σπq = Σpq and R=S) are mathematically incompatible (as stressed in the literature on the issue). No, the primary problem is that of conceptually incompatible quantities.¹⁷ (Note that for post-Marx solutions to ‘the’ transformation problem, there is potentially the same pitfall.)

It is especially in Chapter 10 that Marx reflects on, and questions, what he accomplished in the previous chapters, including the consequences for his self-interpretation of the concept of value set out in his manuscripts.

§4.2 Chapter 8: the problematic

I now move on to the relevant details. I begin by recapitulating the important point about Chapter 8 that was been made above. This chapter sets out the following five assumptions/theses (M:223–4, 229–30):

These five assumptions/theses are inconsistent. At least one of them must be wrong, it remains to find out which one(s). Analytically, I repeat analytically, this set up might make sense. (Chapter 8 makes sense generally – also in that there are no deviations from the later Capital I terminology.)

§4.3 Chapter 9: a cheerful accommodation for the inconsistency

Chapter 9 sets up a possible constellation accommodating for the inconsistency (this is my interpretation); however, in any case it is obvious that the text is investigative (Forschung) not presentational (Darstellung).¹⁸ As it turns out Marx’s set up is much along the lines of his earlier 1861–63 manuscript.

The chapter starts by repeating the assumption about the rate of surplus-value [B above]. Marx also introduces a number of simplifying assumptions such as the full and linear depreciation of fixed capital within the year, and equal turnover times (M:230–1).

Next he sets out the famous three schemes (M: 231–3). (The first and the second scheme apply assumptions/theses [A]-[D];¹⁹ the third scheme is the one were Marx introduces prices of production as diverging from value [equation (11)/(16)]). He drops sales at value [A], introduces (instead) production prices and so gets rid of diverging profit rates [D^b]. He does so without hardly any argument:

Their presupposition [i.e., of production prices] is the existence of a general rate of profit … In reality the very different profit rates … are by way of competition equalised into a general rate of profit … (M:234-mt; cf. E:167, F:257)

Note that he maintains the disproportional production of surplus-value [D^a] (M:234–5).

However, [A] is not really dropped. At this point in the text Marx posits one of the famous two aggregate equalities – that is, that of ‘values’ and production prices (perhaps Σpq = Σπq). The aggregate equality of profits and surplus-value (R = S) is posited throughout.²⁰

As indicated, from the point of view of the Method of Concretion (§3.3), these equalities make no sense. Not so much because of an analytical mistake (the standard critique), but because of a methodological mistake: the ‘values’ [πq] (‘values’ in Marx’s terminology at this point) have no concrete existence, hence they are quantitatively incompatible with (more) concrete existence. From the point of view of the Method of Completion (§3.5) there would not be such problem; however, at this point in the text I am still assuming Marx has adopted the Method of Concretion.

Finally, for Chapter 9 I draw attention to the passage where Marx seems quite happy about what he has achieved so far, declaring that the current presentation “reveals for the first time … the inner connection” between value and production price, and between surplus-value and profit (M:245; cf. E:178 and F:268).

§4.4 Chapter 10: gloomy reflection

In Chapter 10, however, the scene seems much more gloomy and dismal. After two pages connecting the two chapters, Marx posits two research questions, one immediately after the other (I call these Question 1 and Question 2):

{Question 1} The really difficult question here is this: how does this equalisation of profits or this emergence {Hertstellung} of the general profit rate come about, since it is evidently a result and cannot be a point of departure.

It is clear first of all that an assessment of commodity values, e.g. in money, can only be the result of their exchange, and that, when we presuppose such assessment, we have to consider them as the outcome of actual exchanges of commodity value against commodity value. {Question 2:} But how could this exchange of commodities against their actual values have come about? (M:250-mt; cf. E:183–4, F:274–5)

It is especially at this point in the text that the reader (I) may start doubting if Marx indeed aims to adopt (remember this is a research manuscript) the Method of Concretion. Note the “actual exchanges” (“wirklicher Austausche”), which especially for a Hegel inspired scholar cannot leave room for compromises. So do we bid ‘πq’ farewell?

Marx devotes about 20 pages to Question 2, before he gets to the first. The answer to Question 1 (M:269–70; E 205–7; F:297–8) is rather limited. We learn mainly that capital moves from low to high profit rate spheres and that the thus affected supply in relation to demand establishes the transformation of values into production prices. This answer is far more problematical than it might perhaps seem at first sight. (The process briefly described here is not problematical – such movement of capital and labour is also part and parcel of classical political economy, and of many economics paradigms after it.) The problem is that Marx must rely, so it seems, on a historical process to set out a systematic problem! The movement of capital is a continuous systemic process. However, the implication of Marx’s set up is that the GRP ‘transformation’ is of the past. Marx is aware of this, as we can infer from the fact that he realises that Question 1 cannot properly be answered before Question 2.

By this time – that is, when he finally gets to Question 1 (page M:269) – I guess that Marx had run out of steam as a result of being disillusioned by the consequences of his Chapter 9 outline (1864–65) – that is, the answer to Question 2.

On to Question 2: how could the actual exchange of commodities against their actual values {πq} have come about? Marx repeats (M:250) that such exchange (given the equalised rate of surplus-value assumption, as he once again stresses) would result in unequal rates of profit – which is counterfactual.²¹ Obviously, the production prices of the research manuscript for Chapter 9 put the (current) Capital I Chapter 1 presentation – of commodity exchange according to value – into question in a rather disastrous way.²² It is clear from the text that Marx was much bothered by this. My reading is that Marx sets out, in a unsystematic way, a number of analytical consequences of where he has got to, together with some possible ways out. We find, for example, a model-like case in which workers own the means of production and exchange products according to their value; then a move to a ‘historical transformation’; next a long détour on market value and supply and demand generally, without coming to the point (in his work of 1896, all this was rightfully ridiculed by Böhm-Bawerk in this context).²³

The ‘try out’ of the historical transformation especially is inconsistent with the Chapter 9 procedure.²⁴

On page M:267 (E:203; F:294) Marx finally arrives at a systematically and thus methodologically relevant statement. Note that he uses the kind of 1859 (or 1867) terminology of exchange (Engels puts the following in the past tense suggesting an even more direct reference to Capital I, Chapter 1):

In considering money, it is assumed that commodities are sold at their values, because there is no foundation {Grund} to consider prices deviating from value since the concern is just the changes in form that commodities have to undergo when they are turned into money and then transformed back into commodities again. … it is completely irrelevant for them as such [the commodities] whether the realised commodity price is below or above their value. The value of the commodity as groundwork {Grundlage} remains important, since money can only be developed conceptually from this foundation {Fundament}, and price, in its general concept {seinem allgemeinen Begriff nach}, is only valeur monetisie [monetised value; the two words appear in French]. (M:267-mt; cf. E:203, F:294–5)

Methodologically this is fine. I believe this sheds light on what Marx (with the 1859, or later, text in his mind) intended to do in the 1867 text. It is a conceptual presentation in stages of complexity. So, finally, it seems that Marx leans back to the Method of Concretion?

In fact, Marx shows here himself (implicitly that is) that the Chapter 9 procedure makes no methodological sense. An abstract magnitude of value cannot be put into quantitative equality with the magnitude of some concretum.

I round off this section with a comment on Engels’s editorial work. Quite apart from all my methodological critique on Chapter 9 as addressed above, Engels seems to have misjudged Marx’s own critique on that chapter in his Chapter 10 manuscript (and Engels’s polishing work made all this worse). Thus Engels provided intelligent people like Böhm-Bawerk with plentiful opportunities to point out inconsistencies, inconsistencies that Marx himself in fact laid bar.