2.1 The Construction of a Macroeconomics

Perhaps the most important aspect of the chapters on reproduction is to be found in the opening section of Chapter 20: here we find in fact the construction of a macroeconomics, the “functioning of the social capital”, as Marx calls it, the movement of individual capitals being “an integral link in the movement of the total capital”. We have, on the one hand, the elements of production of the individual capital, “in so far as they are of the objective kind”, forming a component of the social capital; and, on the other hand,

the movement of the part of the social commodity product that is consumed by the worker in spending his wage, and by the capitalist in spending surplus-value, not only forms an integral link in the movement of the total product, but is also interwoven with the movements of the individual capitals, so that its course, too, cannot be explained by being simply presupposed. (II:469)

The problem of reproduction, then, is: “How is the capital consumed in production replaced in its value out of the annual product, and how is the movement of this replacement intertwined with the consumption of surplus-value by the capitalist and of wages by the workers?” (II:469).

Whereas Marx’s solutions to the problem are of interest – as we will see – the major achievement is the particular posing of the problem. Of course, many aspects of the problem may be obvious from the standpoint of the end of twentieth-century economics. It is therefore useful to quote three opinions from a time when Keynes’s macroeconomics had been on the scene for only a few years, and these issues were less evident:

“Marx … developed the fundamental scheme describing the interrelation between consumer and capital goods industries.” (Leontief 1938, p. 93)

“His theory is probably the origin of macro-economics.” (Klein 1947, p. 154)

“the theory adumbrated in Volume Two of Capital has close affinities with Keynes.” (Robinson 1948, p. 103)

Whilst it is perhaps arbitrary where we locate ‘the’ origin of macroeconomics (Klein) – Quesnay and Ricardo certainly also provided seminal elements – it is certain that Marx conceived the multiple dimensions of the problem: material and value, as well as production and circulation in their several aspects. In this respect we see here the culmination of both Marx’s value-form theory (Capital, I) and the theory of the metamorphoses of capital and their circuits (Capital, II, Part One). We see this in the extract from Marx given above, and it is even more obvious one page further on in the text:

As long as we were dealing with capital’s value production and the value of its product individually, the natural form of the commodity product was a matter of complete indifference for the analysis, whether it was machines or corn or mirrors …. But this purely formal manner of presentation is no longer sufficient once we consider the total social capital and the value of its product .… [The latter’s movement is] conditioned not just by the mutual relations of the value components of the social product hut equally by their use-values, their material shape. (II:470)

Thus we see the construction of not only a macroeconomics, but a particular macroeconomics emphasising the twofold conflicting guises of the capitalist economy – value and use-value – for which at least temporary modes of operation have to be established (modes which Marx shows to be ridden with contradictions, as manifest especially in economic crises). Thus we have, on the one hand, use-value, the material component of production necessary for ‘natural survival’ – however much shaped by the actual capitalist mode of production. On the other hand, we have value (ultimately money profits), driving and shaping the course of production, necessary for ‘capitalist survival’. But for capitalism the two are one; the one has no existence without the other.

This twofold macroeconomics contrasts sharply with the post-Keynes orthodox macroeconomics approaches dichotomising the problem into two separate sides, or reducing the problem to one of its sides (either monetary or physical, the latter homogenised via index numbers).

For the further construction of the macroeconomics model Marx operates in two stages. Starting in Chapter 20 with a model of ‘Simple Reproduction’, where capitalists consume all surplus-value, he considers in Chapter 21 ‘Expanded Reproduction’; that is, the realistic situation where capitalists accumulate (part of) the surplus-value. This is a very remarkable procedure, one which he had also adopted in the earlier parts of the book (see especially its Chapter 2). Marx emphasises over and over again that accumulation of capital is essential to the system. At the very end of the book he states forcefully that simple reproduction is “incompatible with capitalist production from the very start” (VIII:596). So why start with something that is alien to the object of inquiry? What kind of abstraction or kind of simplification is this? Indeed for a simplification we might expect simplification to what is essential. Or is Marx rather cutting up the problem into (non-essential) parts that can be analysed separately?

a⁸    “Simple reproduction on the same scale seems to be an abstraction, both in the sense that the absence of any accumulation … is an assumption foreign to the capitalist basis, and in the sense that the conditions in which production takes place do not remain absolutely the same in different years (which is what is assumed here) …. But since, when accumulation takes place, simple reproduction still remains a part of this, and is a real factor in accumulation, this can also be considered by itself.” (VIII:470–71)

Some pages later the point is stated again, but now in terms of the Faustian conflict between the capitalist passion for accumulation and the desire for consumption, alluded to in Part Seven of Capital I (740–1): “Simple reproduction is oriented by nature to consumption as its aim …. In so far as simple reproduction is also part of any reproduction on an expanded scale, and the major part at that, this motive remains alongside the motive of enrichment as such and in opposition to it” (VIII:487). As we will see below (towards the end of §3) simple reproduction, even if “foreign to the capitalist basis”, appears to be the sea on which accumulation moves.

The opening section of Chapter 20 contains another assumption disregarding an essential characteristic of capitalism:

b    “Moreover, we assume not only that products are exchanged at their values, but also that no revolution in values takes place in the components of the productive capital.” (II:469)

This assumption is maintained throughout the remainder of the book.⁹ Its first part (exchange at values) is not surprising: it fits into the general systematic of Capital, and is in fact dropped in Part Two of Book III. The question is whether dropping this assumption would affect the macroeconomic construct as well as the particular ‘schema’ to be developed later on. The answer is no; hence any divergence of price from value is irrelevant for the problem at stake:

In as much as prices diverge from values, this circumstance cannot exert any influence on the movement of the social capital. The same mass of products is exchanged afterwards as before, even though the value relationships in which the individual capitalists are involved are no longer proportionate to their respective advances and to the quantities of surplus-value produced by each of them. (II:469; emphasis added)¹⁰

The second part of the assumption is remarkable to the extent that in Capital I, “revolution in values” has already been shown as essential to the system. However, this part of assumption b is evidently of different status from the previous one, a. With it the very construction of the macroeconomics is at stake. Whereas the distinction related to a is relevant for the problem, it seems to be made for heuristic reasons. For b, however, Marx holds that for the problem at hand the issue of ‘revolutions in value’ is irrelevant, or neglectable:

As far as revolutions in value are concerned, they change nothing in the relations between the value components of the total annual product, as long as they are generally and evenly distributed. In so far as they are only partially and unevenly distributed, they represent disturbances which, firstly, can be understood only if they are treated as divergences from value relations that remain unchanged; secondly, however, given proof of the law that one part of the value of the annual product replaces constant capital, and another variable capital, then a revolution … would alter only the relative magnitudes of the portions of value that function in one or the other capacity. (II:469–70)

In other words, even unevenly distributed ‘revolutions in value’ – though affecting the magnitudes of the components of (social) capital – would not change the particular macroeconomic interconnections between constant and variable capital (as well as between them and surplus-value) in the way they will be seen to be set out by Marx.

2.2 A two-sector macroeconomic model

The next phase for constructing the model is central to Marx’s approach. He constructs a two-sector macroeconomics model – as far as is known, the first in the history of economics, even if the inspiration for thinking in similar abstract categories may have come from Quesnay (1759). The model is composed of two ‘departments’. Department I is the sector producing means of production, department II the one producing consumption goods. At the same time this composition fits Marx’s particular value-theoretical distinction between constant capital and variable capital.

c    “The society’s total product, and thus its total production process, breaks down into two great departments:

1. Means of production: commodities that possess a form in which they either have to enter productive consumption, or at least can enter this.

2. Means of consumption: commodities that possess a form in which they enter the individual consumption of the capitalist and working classes.

In each of these departments, all the various branches of production belonging to it form a single great branch of production … The total capital applied in each of these two branches of production forms a separate major department of the social capital.” (II:471)

In the text there follow definitions of variable and constant capital (471–2) which emphasise again the twofold character of capital: its material constituent and its value constituent.

So we have three sets of abstractions (retained throughout this volume – Book II – as well as Book III): First the abstraction of the macroeconomic categories of total product, total production process and social capital; second, the division of these categories into two material functional forms (means of production and means of consumption) – which is a generic abstraction, applicable in principle to all modes of production; third, we have the determinate abstraction, particularly applicable to the capitalist mode of production, of the division of the same categories into their value constituents (constant capital, variable capital, surplus-value) and which, at the same time, reflects the class division in this society.¹¹ Together these constitute a major analytical and synthetical achievement.

2.3 Further Assumptions

d    Apparently so as to reduce the problem to its bare elements, Marx next assumes temporarily (that is, throughout the earlier sections of Chapter 20, as well as in most of Chapter 21) that there is no fixed capital or, equally, that all fixed capital is used up during the production period (VIII:473). Note that we still have a flow both in value (constant capital) and in the ‘natural form’ of means of production.¹²

e    It is further assumed that for both departments the rate of surplus-value (s/v) is equal, constant and given (100 per cent). This assumption is maintained throughout this Part. Although it is not commented upon (it is treated at length in both Book I and Book III of Capital), it seems a simplifying device without particular relevance to the problem at hand.

f    The next assumption concerns the value composition of capital (c/(c + v)), which is, for each department, taken as equal, constant and given. This assumption is maintained throughout Chapter 20, but relaxed several times in Chapter 21. Marx comments:

What is arbitrarily chosen here, for both departments I and II, is the ratio of variable capital to constant capital; arbitrary also is the identity of this ratio between the departments … This identity is assumed here for the sake of simplification, and the assumption of different ratios would not change anything at all in the conditions of the problem or its solution. (VIII:483)

In fact both simplifications e and f can be made because their possible departmental divergences do not fundamentally affect the problem. This is related to the more severe assumption b: the possible divergences at hand would not affect the interconnection between the departments – yet to be developed. (From the point of view of method, all this is most important: the transformations in Capital are systematic, not historical. Thus, for example, the value–price transformation in Book III is conceptual and cannot be said actually to affect the size of the departments.)

A final assumption, which is maintained throughout the Part, is made explicit much further on in the text:

g    “Capitalist production never exists without foreign trade …. Bringing foreign trade into an analysis of the value of the product annually reproduced can … only confuse things … We therefore completely abstract from it here.” (VIII:546)

This is again an assumption of simplification of the type ‘neglectable’ for the current problematic.

2.4 The schema of simple reproduction and the condition for simple reproduction

Schema ch. 20 presents the departmental schema and the numerical example, that is used throughout this chapter – in the dimension of money, that is £ or $ and so on – (VIII:473).

where:

Although Marx does not comment on the numbers in the schema, they do not seem arbitrary. In an earlier chapter (Ch. 17, II:397–8) Marx quotes an estimate of the ratio of the total capital stock and the total consumption for Britain and Ireland (as reported by Thompson 1850). This ratio amounts to 3.¹⁴ A similar ratio in the schema above is 2. However fixed constant capital has been excluded for the time being.

Generalising the schema, Marx uses the notation:¹⁵

In what follows, I adopt the notation that has become conventional in modern Marxian economics:

For simple reproduction, then,

x₁ = c    or equally, (D)

x₂ = v + s (E)

Analysing at length the mutual exchange between the departments, which is “brought about by a money circulation, which both mediates it and makes it harder to comprehend” (VIII:474), Marx derives the following proportionality condition for simple reproduction (VIII:478):

v₁ + s₁ = c₂ (F)

He does not use the term equilibrium, but talks of “proportionate part”, and holds that the proportionate part on the left side of (F) “must be equal” to the proportionate part on the right side (VIII:474, 478). The result is:

The new value product of the year’s labour that is created in the natural form of means of production (which can be broken down into v + s) is equal to the constant capital value c in the product of the other section of the year’s labour, reproduced in the form of means of consumption. If it were smaller than IIc [that is, c₂], then department II could not completely replace its constant capital; if it were larger, then an unused surplus would be left over. In both cases, the assumption of simple reproduction would be destroyed. (VII1:483–4)

Note that condition (F) and the conditions (D) and (E) each imply each other. Representation (F) specially emphasises the interconnection between the two departments as revealed in their mutual exchange.

2.5 The value of the total product and the value product of labour

In an alternative formulation the concept of value-added is brought to the fore:

On the premise of simple reproduction … the total value of the means of consumption annually produced is equal to the annual value product, i.e., equal to the total value produced by the labour of the society in the course of the year, and the reason why this must be the case is that with simple reproduction this entire value is consumed … for the capitalists in department II, the value of their product breaks down into c + v + s [that is, c₂ + v₁ + s₂], yet, considered from the social point of view, the value of this product can be broken down into v + s. (II:501–2)

Marx formalises this as:¹⁶

x₂ = (v₁ + s₁) + (v₂ + s₂) (G)

which has condition (F) at its base.

On the same theme (remember that the numerical schema for department II runs: 2000_c + 500_v + 500_s = 3000_x) Marx writes:

As far as the constant value component of this product of department II is concerned … it simply reappears in a new use-value, in a new natural form, the form of means of consumption, whereas it earlier existed in the form of means of production. Its value has been transferred by the labour process from its old natural form to its new one. But the value of this two-thirds of the value of the product, 2000, has not been produced by department II in the current year’s valorisation process. (II:503)

Hence, again, the importance of formula G.¹⁷

Conversely, for department I (4000_c + 1000_v + 1000_s = 6000_x) the 4000 constant capital

is equal in value to the means of production consumed in the production of this mass of commodities, a value which reappears in the commodity product of department I. This reappearing value, which was not produced in the production process of department I, but entered it the year before as constant value, as the given value of its means of production, now exists in that entire part of the commodity mass of department I that is not absorbed by department II. (II:498)

Thus we have c₁ + v₁ + s₁ = x₁ = c₁ + c₂. Or, in terms of the circuits of Capital II, Part One:

These distinctions gain even more force when explicitly linked to the twofold character of capitalist economic entities, central to Marx’s theory (cf. Capital I, Ch. 1):

Thus the difficulty does not lie in analysing the value of the social product itself [c + v + s = 9000]. It arises when the value components of the social product are compared with its material components.

The constant portion of value, that simply reappearing, is equal to the value of the part of the social product that consists of means of production, and is embodied in this part. The new year’s value product = v + s is equal to the value of the part of the annual product that consists of means of consumption, and is embodied in this. (II:506; cf. 504)

This is even more forcefully expressed in a later notebook:

The overall annual reproduction [c + v + s = x], the entire product of the current year is the product of the useful labour of this year [L^u → x]. But the value of this total product is greater than the portion of its value which embodies the annual labour, i.e., the labour-power spent during this year [L^v → v + s = y]. The value product of the current year, the value newly created during the year in the commodity form [y], is smaller than the value of the product, the total value of the mass of commodities produced during the year [x]. (VIII:513)

Here we see the distinction related to the twofold character of the labour process as technical and valorisation process.

2.6 Money circulation and ‘the widow’s cruse’

Throughout the text much emphasis is on the money circulation within and between the two departments (see also Campbell 1998); a recapitulation is on pp. 491–2; cf. Ch. 17 on the same issue. Especially here we may notice similarities with Quesnay’s ‘zigzag’ in his Tableau economique.¹⁸

In the course of outlining money circulation, Marx formulates the so-called ‘widow’s cruse’ argument – derived in Keynes’s Treatise on Money (1930) and in Kalecki’s ‘A macroeconomic theory of business cycles’ (1935) and which in Kaldor’s (1955/56, p. 85) well-known paraphrase of Kalecki runs: “capitalists earn what they spend, and workers spend what they earn”. Marx: “it is the money that department I itself casts into circulation that realises its own surplus-value” (VIII:495; his emphasis). And in more general terms (cf. Chapter 17, II:409):¹⁹ “In relation to the capitalist class as a whole, however, the proposition that it must itself cast into circulation the money needed to realise its surplus-value … is not only far from paradoxical, it is in fact a necessary condition of the overall mechanism” (VIII:497).

2.7 Maintenance of fixed capital and disproportionate production

In Section 11 of Chapter 20, Marx drops assumption d and considers the effect of the incorporation of fixed capital for his model. Thus in terms of annual reproduction he incorporates constant capital components whose life is longer than a year (cf. VIII:525). For the individual capital, “the part of the money received from the sale of commodities, which is equal to the wear and tear of the fixed capital, is not transformed back again into … productive capital … it persists in its money form”, that is, hoard formation, to be expended when the fixed capital components have to be replaced (VIII:526). Thus the commodity value “contains an element for depreciation of … fixed capital” (VIII:528).

For simple reproduction, then, as a “precondition”, the annual total of fixed capital to be renewed “has to be equal to the annual wear and tear”. “Such a balance accordingly appears as a law of reproduction on the same scale” (VIII:540). Next Marx discusses the two cases in which this equality does not hold. In the first case, fixed capital has to be renewed, for which there has been insufficient production; thus “there would be an insufficient amount of reproduction, quite independent of the monetary relations” (VIII:543). “The reverse happens in the second case, where department I … has to contract its production, which means a crisis” (VIII:544). Marx emphasises that such “disproportionate production of fixed and circulating capital” (“a factor much favoured by the economists in their explanation of crises”) can “arise from the mere maintenance of the fixed capital”, that is with simple reproduction. “Within capitalist society … it is an anarchic element” (VIII:545).

2.8 Conclusions to the model for simple reproduction

The first major achievement of the chapter on simple reproduction is the construction of a macroeconomics generally, with its particular emphasis on the twofold character of the capitalist mode of production. This leads Marx to the – now familiar – distinction between “value of the product” (production value) and “value product” (value-added), The second major achievement is to grasp the macroeconomic relations in terms of a two-sector system fitting Marx’s approach of general and determinate abstractions. And the third is the general thread in Marx’s analysis: to search for the necessary interconnections between the two departments of production. Therefore, rather than the two equations x₁ = c, or x₂ = v + s, it is the equation v₁ + s₁ = c₂ that is central to the analysis [for these equations see p. 230 above]. We will see in §3 that a similar equation also provides the guiding thread for Marx’s analysis of the macroeconomics of expanded reproduction.

3.1 The general frame for the analysis: general assumptions and abstractions

The chapter on expanded reproduction starts with an analysis of fixed constant capital and the addition to it, which from the side of individual capitals runs in gradual lumps of hoarding (depreciation allowances) and discrete dishoarding (investment); within a department and its branches, one section of capitalists will be engaged in stages of the former (“one-sided sale”), while another section actually buys additional elements of constant capital (“one-sided purchase”) (VIII:565–70).²⁰

The fact that the production of commodities is the general form of capitalist production already implies that money plays a role, not just as means of circulation, but also as money capital within the circulation sphere, and gives rise to certain conditions for normal exchange that are peculiar to this mode of production, i.e., conditions for the normal course of reproduction, whether simple or on an expanded scale, which turn into an equal number of conditions for an abnormal course, possibilities of crisis, since, on the basis of the spontaneous pattern of this production, this balance itself is an accident. (VIII:570–1)

However, Marx’s aim for this chapter is not the analysis of crises, but rather the accidental balance. (In this respect the point of application is similar to that of the ‘equilibrium’ growth models of Harrod and Domar.) To this end he assumes, even for the case of expanded reproduction, that

h    “balance exists … that the values of the one-sided purchases and the one-sided sales cover each other.” (VIII:570)

i    In the same vein, Marx assumes a sufficient monetary accommodation for expanded reproduction (VIII:576).

j    A further delimitation of the problematic is revealed in the assumption of a sufficient labour force; that is, that “labour-power is always on hand” (VIII:577).

The last assumption, however, is not an analytical one, as Marx for its explanation refers back to Capital I.

Nevertheless a problem of potential imbalance – or, rather, of potential overproduction – is central to reproduction on an expanded scale insofar as we consider either a transition from simple to expanded reproduction or a transition to further expansion, that is, to a higher growth path. Marx states: “in order to make the transition from simple reproduction to expanded reproduction, production in department I must be in a position to produce fewer elements of constant capital for department II, but all the more for department I” (VIII:572). In effect, then, department I would substitute spending part of surplus-value (s₁) to means of consumption (some equivalent part of c₂) for spending it on additional means of production (which are now to that equivalent available in commodity form from department I). Department II would thus be stuck with a commodity stock to that equivalent: “There would thus be an overproduction in department II, corresponding in value precisely to the expansion of production that took place in department I” (VII1:580).

The ‘normal’ reaction would be for department II to cut back production, which would be fine if it were to the extent of the means of production they could not get from department I anyway. However, given their overproduction, they might want to cut back production more than that, and thus buy even less means of production: “The over-production in department II might in fact react so strongly on department I … [that the] latter would thus be inhibited even in their reproduction on the same scale, and inhibited, moreover, by the very attempt to expand it” (VII1:580). We thus have a real paradox. Marx brings up the problem and refers back to it several times, but does not analyse it any further: from the text it is clear that he purposefully wants to abstract from any crisis elements so as to set out the situation of accidental balance (assumption h).

3.2 The schemes for expanded reproduction

In setting out expanded reproduction, Marx proceeds on the basis of – apart from the assumptions h to j just mentioned – the earlier assumptions b to g [pp. 225 and 227–29 above] (assumption a was the one of simple reproduction). However, assumption f, about the composition of capital, is sometimes relaxed so as to allow for divergent compositions as between the departments; nevertheless, within a department it remains constant. Apparently, Marx does not aim to set out the transition from simple to expanded reproduction. Indeed, he assumes that:

k    there has “already been reproduction on an expanded scale” (VIII:566).

For the analysis of expanded reproduction, Marx uses three numerical schemes, which I refer to as Schemata A, B and C.²¹ Marx treats Schema A very briefly, and its analysis is apparently a preliminary one (omitted here). Below I present an outline of Schema B, which is also the best worked out case in Marx’s text. Towards the end of this section I make some remarks on Schema C.

Once again these schemes are in numerical form; each with different starting values. For all schemata it is at first sight unclear why these specific starting values in particular have been chosen – only towards the end of the chapter does it become clear that they are meant to be representative cases for three particular circumstances. (Quite apart from this it is also obvious from the text that Marx tried to employ ‘easy numbers’ for his calculations.)

Each schema (A, B, C) is presented for a sequence of periods, each representing the production in that period. At the end of each period capitalists in each department make plans (‘arrangements’) to accumulate capital for an expanded production in the next period (= intended exchange arrangement). Thus they aim to use more means of production (c) and labour-power (v) than they did in the running period. However, these plans may not match, for example, the means of production that have actually been produced in the running period, thus there might be over- or underproduction in comparison with these plans. Thus especially for the case of underproduction there may be bottlenecks preventing steady growth. At the end of each period then the confrontation of the realised production and the intended exchange arrangement gives rise to some actual exchange arrangement which is the basis for the next round of production.

Once we are in a situation that the intended exchange arrangements match the actual arrangements (and therefore also production), and no new changes in parameters occur, we are on a steady growth path. I will call a situation of a fixed set of parameters a ‘regime’. Marx then analyses the transition from one regime to another by varying just one parameter, which is the rate of accumulation out of surplus-value for department I (α₁). Particularly he assumes that in department I half of surplus-value is being accumulated; the rate for the other departments stays, as intended, initially at the old rate (in the proportions of the existing compositions of capital in each department).²²

In the way Marx makes his model work (at least for Scheme B, as we will see) there is only one period of transition from the old regime to the new one. Hence starting from a steady state regime in period 1, and changing the regime at the end of that period (intended), a new steady state will already be reached in period 3.

Thus we have the sequence as set out in the table on p. 239.

Although I interpret the starting situation (period 1) of each schema as one of proportionality for a specific steady state growth path, Marx does not say this explicitly. Nor does he calculate the steady state parameters for the starting situation (as I will do below). (And as we see later on, his omission to do this may have put him on the wrong track for his conclusions from the model.)

The schemes of production (Ba, Be, Bh) that I present below are identical to the ones that Marx gives. The other schemes (Bb, Bc, Bd, Bf, Bg) are presented by Marx in different and varying formats.

The following notation is used:

Thus we have for surplus-value (s)

The actual rate of accumulation out of surplus-value (α) is defined as:

The parameters for Marx’s scheme (old regime) are only explicit by his numbers. These are for the composition of capital:

For the rate of surplus-value:

For the rate of accumulation out of surplus-value:

Where ∆c and ∆v have the same proportions as in (7) and (8):

Thus there is no technical change – at least no change in the value composition of capital (assumption b).

The remainder of (potential) surplus-value is the ‘unproductive consumption’ (u) by or via capitalists:

Thus ‘hoarding’ is set aside, that is all incomes are expended – at least in the aggregate. (In his text, however, Marx devotes considerable attention to hoarding, for example in the opening section of Chapter 21. Indeed, he conceives of hoarding as crucial to the circulation and reproduction process – see Campbell 1998.)

Schemas B: Expanded Reproduction

I reiterate that for the model below the ratios c/(c+v) and s/v are given and constant. Thus once we have a starting value for e.g. c the numerical values for the other variables follow. Scheme Ba sets out the starting point.

Since (x₁-c)/c = (6000–5500)/5500 = 9.1%, Schema Ba might be a schema of proportionality for a steady growth path of g = 9.1%, if α₁´= 45.5%; α₂´= 27.3%; with ∆c₁/s₁ = 36.4%; ∆c₂/s₂ = 18.2%; and for both departments ∆v/s = 9.1%. (Marx does not calculate these ratio’s.) Equivalently: for such a steady state growth the ratio c₁/c₂ is fixed so that we can find ∆c = ∆c₁ +∆c₂. Next, given c/(c + v) we also find ∆v₁ +∆v₂ = ∆v From these values then we derive the necessary rates of accumulation α₁´= (∆c₁ + ∆v₁)/s₁ = 45.5% and α₂´= (∆c₂ + ∆v₂)/s₂ = 27.3%.

Accordingly, had the old regime continued, we would have had the Scheme Bb’s intended exchange arrangement at the end of period 1 (Marx does not mention this).

In Schema Bb, the u, ∆v and ∆c are the (intended) destination of the total of profits s. Schema Bb matches Schema Ba, so the intended exchange arrangement can also be the actual exchange arrangement (x₁ = 6000 = c + ∆c and x₁ = 3000 = v + u + ∆v).

The part of the surplus product that is accumulated (∆c and ∆v) seems to have a different status from the other components (c, v, u). Although ∆v in particular is materially produced within the period under consideration, this part of (potential) surplus-value is only realised within the next, when the extra labour-power is hired (VIII:580–l). The realisation of ∆c can be conceived of in the same way (VII1:575). Thus the realisation of these components of scale increase, in a way lags behind. Of course, it applies to all components, and not just the last-mentioned, that their production and circulation – even within a period under consideration – involves complex intertemporal processes:

The continuous supply of labour-power on the part of the working class in department I, the transformation of one part of departments I’s commodity capital back into the money form of variable capital, the replacement of a part of departments II’s commodity capital by natural elements of constant capital IIc [that is, c₂] – these necessary preconditions all mutually require one another, but they are mediated by a very complicated process which involves three processes of circulation that proceed independently, even if they are intertwined with one another. The very complexity of the processes provides many occasions for it to take an abnormal course. (VIII:571)

Nevertheless the lagging behind of realisation, Marx concludes, is not the vital point of difference between simple and expanded reproduction:

Just as the current year concludes … with a commodity stock for the next, so it began with a commodity stock on the same side left over from the previous year. In analysing the annual reproduction – reduced to its most abstract expression – we must thus cancel out the stock on both sides … and thus we have the total product of an average year as the object of our analysis. (VII1:581)

Now instead of carrying on at the old regime (Schema Bb) at the end of period 1, department I decides to increase the rate of accumulation (department II intends to maintain the old rate). Thus Marx fixes α₁ = 50% per cent and then analyses the transition numerically. For this he takes as starting point the condition for simple reproduction (v₁ + s₁ = c₂), gradually developing this in the course of the examples into a condition for expanded reproduction.

It is self-evident that, on the assumption of accumulation, I_{(v + s)} [that is, v₁ + s₁] is greater than II_c [that is, c₂], … since (1) department I incorporates a part of its surplus product into its own capital and transforms … [∆c₁] of this into constant capital, so that it cannot simultaneously exchange this … for means of consumption; and (2) department I has to supply the material for the constant capital needed for accumulation within department II [∆c₂] out of its surplus product. (VIII:590).

Thus we have:

(v₁ + s₁) − ∆c₁ = c₂ + ∆c₂ (I)

or

(v₁ + u₁) + ∆v₁ = c₂ + ∆c₂ (J)²⁴

In further presenting the numerical schemes, I will indicate for each schema whether it satisfies this condition. Marx does not do this. Again he derives generalisations from his numerical schemes. Thus they are not illustrations, but rather heuristic tools. So, for schema Bb, above, we have the condition satisfied, as 1000 + 545 + 91 = 1500 + 136. Following on from the change in the rate of accumulation (α₁ = 50%) we get, instead of this schema, the following intended arrangement at the end of period 1 (Scheme Bc).

With these plans (Scheme Bc) there is imbalance, the intended arrangement does not match production (Ba):

(v₁ + u₁) + ∆v₁ < c₂ + ∆c₂ [1600 < 1637]

This situation cannot be. There are fewer means of production on offer (6000) than there is intended demand for (5500 + 537). Conversely there are more means of consumption on offer (3000) than the intended demand (1750 + 1045 + 168). So what happens? In fact Marx lets the course of development be dictated by department I as they hold the means of production. (Note that it is assumed there are no price changes.) Thus department I fulfils its plans and department II is stuck with a shortage of means of production (37), plus an equivalent unsold stock of commodities for consumption. However, it will then hire proportionally less extra labour-power (from 68 to 50) giving rise to an extra stock of 18. (Thus we have the paradox for department II: eager to expand at overcapacity. If department II were to react to its overcapacity by decreasing demand for means of production from department I, we would have the same paradox for department I. In sum, a downward spiral would be plausible. Cf. §3.2.) Marx shortcuts the transition, apparently because he wants to make the strongest possible case for ‘balance’, by assuming that department II capitalists absorb the stock of means of consumption (37 + 18) by consuming it unproductively, thus realising their surplus-value to that extent. (We see the ‘widow’s cruse’ in effect – §2.6.) Thus we get the arrangement of Scheme Bd (the differences from the previous Scheme Bc are in italics).

(where condition (J) is met: 1000 + 500 + 100 = 1500 + 100).

Scheme Bd is the ‘rational’ reaction for department II to have; α₂ = 20% being the result. In effect the plan for department I to increase the rate of accumulation out of surplus-value, results in a decreased rate for department II (and this, according to Marx, is the only way in which an (extra) expansion can come about: VIII:572). For the next period the production is as in Scheme Be.

Consequently in Scheme Be the rate of growth for department I has increased, to 10% and that for II has decreased to 6.7% (both initially at 9.1%).

For the end of period 2 (Scheme Bf), Marx then (implicitly) assumes that department II intends to reach the old rate of accumulation out of surplus-value (α₂´ = 27.3%; ∆c₂/s₂ = 18.2%, that is, 146) and moreover to catch up with the former level of accumulation (in means of production 36). Thus the intended ∆c₂ becomes 146+36=182. Department I maintains α₁ = 50%.

In Scheme Bf, again (v₁ + u₁) + ∆v₁ < c₂ + ∆c₂ [1760 < 1782], and again department I can dictate the course and again department II absorbs the potential overproduction (22 plus 11, since labour-power hired decreases proportionally). Accordingly we have the exchange arrangement of Scheme Bg (differences from schema Bf in italics).

(Where condition J, i.e., (v₁ + s₁) − ∆c₁ = c₂ + ∆c₂ is met: 1100+550+110=1600+160.)

In Scheme Bg Department II has recovered part of the former level of accumulation, but not all. As a result the schema for the next period is that of Scheme Bh.

With Schema Bh we are at the new steady state growth path. From now on all entries can increase at a growth rate of 10% (g = 10% for both departments). Department II cannot catch up with accumulation any further, so α₂ stays at 30%. (Though for this example it will have caught up in effect after two more periods, since the growth rate has risen.) Marx calculates the schema for three more periods (VIII:589). So much for Schemas B.²⁵

As has been said above, Marx’s schemes are not illustrations; they are tools for arriving at a generalisation. He (implicitly) applies the formula (v₁ + s₁) − ∆c₁ = c₂ + ∆c₂ (condition J) in all his examples, and explicitly derives it from them (pp. 590 and 593). Nevertheless, at the very end of the text (595–7), when Marx is preparing to draw general conclusions from his schemes, he once again falls back on the modified simple reproduction condition v₁ + s₁ = c₂ (condition F). Why? The easy answer is to refer to the unfinished shape of the text: it was perhaps meant to be followed by a piece indicating the relevant difference between the conditions for simple and expanded reproduction.

However, there is another explanation, which directly relates to Marx’s examples. Note that his generalisations (595–7) follow just after setting out the schema that is set out below, Schema C (590–95). The problem is not so much that he takes the formula v₁ + s₁ = c₂ for a starting point of the analysis. Indeed, with Schema C, Marx takes an example for which this formula does not apply in the initial situation – as it did for Schemata B and A.²⁶ The point is that Schema C is an unlucky example (though, since Marx neglects to calculate the relevant initial properties of his schemes – especially the rates of accumulation and growth – he seems unaware of this). In fact, with his Schema C, he describes the transition to a decreasing rate of accumulation and growth, whilst it is apparently meant to describe (further) expansion, taking off with a rate of accumulation of 50% for department I as in all his examples.

Schema C might be a schema of proportionality for a steady growth path of g = 8.9% if for both departments ∆c/s = 44.3%; ∆v/s = 8.9%; hence α´= 53.2% (Marx does not calculate these ratios). The new rate of accumulation decreases to α₁ = 50%.

For our purposes we do not need to go through this example any further (in the end, the new growth rate will slow down to 8.3%). Indeed, for the new situation, (v₁ + u₁) < c₂ (that is, 1500 < 1430). What is relevant, however, and whence we have potential overproduction in department I, is that (v₁ + u₁) + ∆v₁ > c₂ + ∆c₂ (that is, 1000 + 500 + 83 > 1430 + 127, thus 1583 > 1557).

3.3 A formal recapitulation of the model for expanded reproduction: conclusions

Marx’s main tool, as has been indicated, is numerical schemes with some elementary formalisation. Thus, although we do not find the formalisation given below in the text, this type of formalisation may be said to be in their spirit.

Apart from the properties of the model for expanded reproduction described below, we have the following assumptions, as discussed earlier on:

• prices do not change (or prices are equal to values) (assumption b);

• there is no fixed capital (or it is used up within the production period) (assumption d);

• there is no foreign trade (assumption g);

• monetary accommodation is sufficient (assumption i);

• sufficient labour-power is available (assumption j).

Below the assumptions and equations marked * are identical to the ones for simple reproduction. (Symbols are defined at the beginning of §2.4 and in §3.2 below the table ‘Periodisation of the transition between steady states’ [see p. 239].)

We have the system:

c₁ + v₁ + s₁ = x₁ (1)*

c₂ + v₂ + s₂ = x₂ (2)*

c + v + s = x (3)*

There are three definitions for aggregation:

c₁ + c₂ = c (4)*

v₁ + v₂ = v (5)*

s₁ + s₂ = s (6)*

We have four equations fixating the dynamics of the structure of production: in each department, one for the value composition of capital (c/(c + v)) and one for the rate of surplus-value (s/v):

c₁ : (c₁ + v₁) = γ₁ (7)*

c₂ : (c₂ + v₂) = γ₂ (8)*

s₁ : v₁ = ε (9)*

s₂ : v₂ = ε (10)*

(These 10 equations, together with the condition v₁ + s₁ = c₂ (condition F) comprise the model for simple reproduction analysed in §2.)

The ratios γ and ε may in principle be estimated; here, however, they are fixed, for analytical purposes.

For the model of expanded reproduction the crucial element is α, the rate of accumulation out of surplus-value (commented upon below), which is defined as follows:²⁷

(∆c₁ + ∆v₁) : s₁ = α₁ (11)

(∆c₂ + ∆v₂) : s₂ = α₂ (12)

Where ∆c and ∆v have the same proportions as in (7) and (8):

∆c₁ : (∆c₁ + ∆v₁) = γ₁ (13)

∆c₂ : (∆c₂ + ∆v₂) = γ₂ (14)

Thus there is no technical change – at least no change in the value composition of capital (assumption b).

The remainder of (potential) surplus-value is the ‘unproductive consumption’ (u) by or via capitalists:

u₁ = (1 − α₁) s₁ (15)

u₂ = (1 − α₂) s₂ (16)

Thus ‘hoarding’ is abstracted from.

The rates of accumulation, α₁ and α₂, may in principle be estimated (elsewhere Marx further theorises α as a necessary force in capitalism). Here, however, α₁ is fixed, for analytical purposes; α₂, on the other hand, is taken for a semi-variable. Its starting intended value is that of the previous period (see below), but within the period it acts as a result. Unproductive consumption u₂ varies accordingly. In this way, Marx’s account short-cuts adaptation after any changes in the system (α, γ, ε); it also precludes downward spiral effects: effective overproduction is ruled out. Any potential overproduction (given a rate of accumulation α₁) is absorbed via the adaptation in α₂: either by unproductive consumption (for means of consumption) or by accumulation (for means of production).²⁸

Finally expanded reproduction and proportionality is defined by the condition:

(v₁ + s₁) − ∆c₁ = c₂ + ∆c₂ (17) = (J)²⁹

It centres the analysis on the interconnecting exchanges between the two departments.

So we have 17 equations and 19 unknowns, leaving two degrees of freedom. Similarly as for simple reproduction it is within the logic of Marx’s reasoning to start from a given accumulation of capital in each department, from which follow numerical values for the other variables (given some initial value for α₂, that is, intended accumulation in department II). However, as α₂ is a semi-variable (its intended value may not be equal to its realised value, or its ‘ex ante’ value may not be equal to its ‘ex-post’ value), condition (17) may be violated.

Thus, in the face of the pattern for α, γ and ε, the starting values c₁ and c₂ – or (c₁ + v₁) and (c₂ + v₂) – determine the course of things, notably smooth adaptation or potential overproduction in department I or department II, with their potential downward spiral effects. Each time condition (17) may turn out to be an inequality ‘at the end’ of the period, the resulting accumulation of capital (‘ex post’) thus determining the course for the next period. The following three cases can be distinguished:³⁰

(1) Potential overproduction in department II (cf. Schemata A and B), if:³¹

(v₁ + u₁) + ∆v₁ < c₂ + ∆c₂ (Marx has v₁ + u₁ = c₂)

(2) Smooth adaptation, if:

(v₁ + u₁) + ∆v₁ = c₂ + ∆c₂

(3) potential overproduction in department I (cf. Schema C), if:

(v₁ + u₁) + ∆v₁ > c₂ + ∆c₂

In effect the process of adaptation runs as follows. Ensuing upon a (positive) change in the rate of accumulation from a previous α´ to a new intended increased α (requiring a relative increase of department I), (new) proportionality is established via a readaptation of the rates of accumulation α₁ and α₂. In Marx’s model the period of transition is short-cut by a pre-emptive readaptation for especially α₂, thus absorbing any overproduction and evading downward spirals.

In other words, upon the change of α₁´ to α₁ the ∆c₁ (that is, α_1*γ_1* s₁) is a constant fraction of c₁, whence we have a constant rate of growth for department I. However (v₁ + u₁) + ∆v₁ [that is, v₁ +(1–α₁)s₁] is also a constant fraction of c₁; at the same time it determines c₂ + ∆c₂ [that is, c₂ + α₂ + α₂(γ_2*s₂)]: the extra production of means of production in department I that it does not use up itself – department II cannot have more, only less; however, given the α₂ planned, it absorbs what is available. Therefore department II becomes chained to the growth rate of department I. (In this process of adaptation, department I thus dictates the course. The ownership of means of production for producing means of production is thought of as crucial: department II cannot expand unless I does.)

More so than the chapter on simple reproduction, the chapter on expanded reproduction reveals the defects of an unfinished draft and an unfinished analysis. Guiding Marx’s generalisations is an adjustment of the condition for simple reproduction. However, the adjustment is not carried through to its full extent; it is nevertheless effected in the numerical schemes. Even if unfinished, the power of the model is revealed very well. Heuristically it also leaves plenty of room for further analysis of dynamic processes. At the core of the model are the same fundamental macroeconomic abstractions, developed into a two-sector approach, as those of simple reproduction (equations (1) to (3)). Generally Marx succeeds in showing convincingly that, even abstracting from all sorts of further complications, proportionality between the two sectors – or generally, steady-state growth – is most unlikely. In the process of transition from one growth path to another, we saw in effect, as an interesting digression (?), the ‘widow’s cruse’ mechanism: ‘capitalists earn what they spend, and workers spend what they earn’.

4.1 Precursor to the modern conventional economic modelling

Marx’s text abounds with elements demonstrating similarities to modern economic modelling approaches. We find a set of explicit assumptions delineating the problematic in its – purposefully – case elements. We are then left with a set of variables and parameters ready for analysing the properties of their interconnection. Generalisations concerning the problematic can be drawn from this analysis. Although the main tool for the analysis is a numerical schema, we also find an elementary formalisation.³² The approach also contains a heuristic: the findings of an earlier model – simple reproduction – can be carried over to be adjusted for a model dealing with different or more complex phenomena: expanded reproduction.

If we add to this that a dialectics, at least a dialectical jargon, is almost absent from this text, at least apparently so (see below), it is no wonder that of all of Marx’s economics this Part especially has much influenced orthodox economics. Of course, that is not just a matter of method. It is also the case that the content of the approach, the construction of a particular macroeconomics, was seen to be fruitful, especially for the theory of the business cycle and of economic growth.

So is this a decisive case for defending the thesis that Marx adopts a method of inquiry in line with formal logic, that is, not different in principle from modern orthodox economics approaches? The textual evidence certainly favours this view (this may, of course, be different for other cases).³³

A next question is whether there are any important differences distinguishing Marx’s modelling approach from the conventional. In this respect I may emphasise that Marx, as we have seen, adopts as a methodological requirement a particular abstraction procedure: the particular designation of his representations at an early phase of the exposition is intended to anticipate later expositions, earlier abstractions remain in force at later stages, albeit in modified form. We have seen this prominently in the carrying over of the condition for simple reproduction to expanded reproduction. This is in fact the case for many of the representations in Capital II: they are still applicable, in modified form, when their underlying simplifying assumptions are dropped (for example, V:162).

From the perspective of a systematic-dialectical methodology (see below) this requirement is no surprise. Most of the reviewers of Marx that question his systematic dialectics have at the same time no doubt that he adopts a systematic in his work. Even if dialectics and its particular way of logical proceeding are suppressed, the methodological requirement for abstractions (in anticipation of later exposition) enforces a systematic for presentation, as well as an order for the process of model building. In this case, as with a systematic dialectics, the process of discovery cannot be the same as the process of presentation (an issue much stressed by Marx; see his 1873 Postface to the 2nd edition of Capital I).

4.2 First and second thoughts on systematic-dialectics

Let us now consider arguments stemming from this case for the view that Marx adopts a systematic-dialectical method. Two relevant issues will be discussed: first the general point of the (in)compatibility of ‘model building’ within a systematic-dialectical approach; second the specific point of the notebook status of the text.

For the first point I start with a contentious thesis: even if Marx’s method were systematic-dialectical, it would not prevent the conceiving of Capital as a model of the capitalist economy.³⁴ In this view, the term ‘model’ is itself neutral as to a particular logic and method of constructing models. However, since the capitalist system entails contradictory entities, relations and processes, a dialectical logic is most appropriate, as it is able to grasp contradictions. Hegel’s logic, in this view, is the proper logic of and for capitalism.³⁵ Several layers (Parts) of Capital can next usefully be seen as ‘sub-models’, the one presented in this article being a case of such a sub-model. In dialectical jargon it would be called a moment; that is, “an element considered in itself, which can be conceptually isolated, and analysed as such, but which can have no isolated existence” (Reuten and Williams 1989, p. 22). Indeed the great advantage of a systematic-dialectical method is that it is called upon to connect its ‘sub-models’ within the systematic whole.³⁶ If my initial thesis of conceiving the whole of Capital as a model is for some unacceptable, we may restrict the matter to conceiving particular moments as dialectical models, our case being a possible example.

This view, however, if useful at all, seems not particularly illuminating for the case at hand: a systematic-dialectical logic seems largely absent from it. Undoubtedly that is the first impression one gets from the text, but rather than leaving the point at that, let us list what one might expect for a systematic-dialectical text.

(1) An abstract-general starting point. Of course, for the case at hand this cannot be an all-embracing starting point, as we are already under way (Part Three of Capital II). However, the case as a ‘moment’ may have its own relatively abstract-general starting point. This can be well defended by the macroeconomic abstractions that Marx starts with.

(2) The positing of contradictions. Absent (but see below).

(3) The transcendence of contradiction. Consequently absent.

(4) Along with 2–3, a gradual conceptual progress, in layers of abstraction, towards concretisation, distinguishing necessary from contingent moments. Although apparently not along with 2–3, one can show that aspects of this are happening in the text: notably the very move from simple to expanded reproduction (even if we were not to agree with Marx that the former is in some way essential – he does argue for it); and along with it there is obviously conceptual progress on the notions of reproduction and circulation, including money (even if this has not been emphasised in the present article); indeed, after Part Three, we have a better grasp of Part One. A possible distinction between necessary and contingent moments, however, is awkward in the text, especially if we consider ‘balance’ and the ‘normal imbalance’ or even crisis. The text is unclear on this point. On the one hand, Marx convincingly shows the ‘knife edge’ of balance, whereas on the other at least a degree of balance must prevail for the system to exist at all (necessity). Of course, this would have been an obvious point for grasping dialectically. So perhaps we can grant this point, though, to say the least, with a dialectics suppressed.

(5) Along with 2–3, showing the systematic interconnection of what is theorised, within the whole of the object of inquiry. Again, although apparently not along with 2–3, the interconnection is shown: first that with the earlier parts of Capital II, as well as with Capital I (see Arthur and Reuten 1998), secondly within the theory at hand (Part Three) the interconnection of the elements theorised ranks high.

(6) Points 1–5 together determine the systematic for the proceeding. Generally transcendence of contradiction and the new problems created by it show the insufficiency of the previous theorisation, and hence the way to proceed. Given the absence of contradiction and transcendence, at least explicitly, this kind of systematic seems absent from the text (even if there is the systematic of ‘anticipative abstraction’ referred to above).

Thus, on second thoughts, considering the six points together, perhaps the case is not that clear-cut methodologically? It is even less so if we bear in mind the emphasis in the text on the twofold character of the entities (material, value – cf. §2.1 and §2.5). This, in retrospect, seems very much to guide Marx’s approach in this part, at least as far as the positing of the problems is concerned (in my view, the citations given in §2.5 are the most thought-provoking of the whole text). The twofold character seems after all central to Marx’s schemes (which is no surprise in the face of the rest of Capital, especially Book I, Chapter 1).³⁷ Unfortunately, and this is perhaps misleading, the theme is not carried through systematically – at least not in a clear way. Manifestly so, not only do the major entities discussed (c₂, v₁, and so on) have a twofold character (value, material), but there is also a ‘redoubling’ in that they stand for two material guises, and their two value forms (for example, c₂ is means of production as well as means of consumption – emphasised in the guises it goes through in the capital circuit). This might have been expressed in a different notation, perhaps akin to the circuit models of Part One.

It must be emphasised that none of this affects the fact that within a dialectical presentation one can build in analytical ‘moments’. Within its restrictedness there is nothing wrong with formal logic or a formal model. They are indispensable tools in research practice; formal logic and formal modelling can have a proper place within systematic dialectics (cf. Reuten and Williams 1989, p. 27). Rather it is the other way around that is difficult.

So where does this leave us? From point, 1–6 above we saw that, dialectically, a main defect of the text is that contradictions and their transcendences are not made explicit, and do not explicitly lead the systematic conceptually. However, at the same time, there is the emphasis in the text on the twofold character of entities, which is the major contradiction of the system. In the text it is perhaps too often expressed abstractly, rather than at the level of concreteness that we have already attained. Nevertheless this is an obvious anchor for a systematic-dialectical presentation. All this, however, does not lead to the conclusion that this is a systematic-dialectical text. It is not. However, there are arguments for conceiving it as compatible with a systematic-dialectical method.

This takes us to the second point, which can be dealt with briefly: the notebook status of Book II of Capital. It is rather speculative to argue about something that might have been if … Nevertheless, to answer the leading question of this article, this notebook status must be taken into account. All the more so since it is not only that, as we have just concluded, the Part Three text we have considered is compatible with a systematic-dialectical approach, but we also have the textual evidence of Capital I, and of Part One of Capital II (see Arthur 1998) which are written in a dialectical vein, even if perhaps not perfectly from several points of view. I have no doubt whatsoever (partly because of personal experience) that a dialectical presentation is often preceded by an analytical stage of inquiry: even more so for the study of new problems. The dialectical hard work lies in the way of systematising the material one has at hand. Indeed empirical inquiry and analytical inquiry are the building stages and material for a systematic-dialectic. From this we cannot answer the question whether Marx intended a systematic-dialectical presentation, let alone that of how the kind of analysis we find in the Book II manuscripts might have been incorporated in a dialectical presentation. It is rather that this notebook status strengthens the conclusion that the text we have considered is compatible with a systematic-dialectical methodology.³⁸