Discussion Paper No.374
The First Publication of Economic Manuscripts in
MEGAand New Aspects of Marx’s Economic Theory:
Marx’s Six-Sector Model and Its Theoretical
Implications
Kenji Mori
September 2017

TOHOKU ECONOMICS RESEARCH GROUP
Discussion Paper

GRADUATE SCHOOL OF ECONOMICS AND

MANAGEMENT TOHOKU UNIVERSITY

27-1 KAWAUCHI, AOBA-KU, SENDAI,
980-8576 JAPANThe First Publication of Economic Manuscripts in MEGA
and New Aspects of Marx’s Economic Theory:
Marx’s Six-Sector Model and Its Theoretical Implications
Kenji Mori (Tohoku University)

1. Background: Multisectoral Analysis in 1858
1.1. The world economic crisis and Grundrisse
The economic crisis of 1857, the first world economic crisis in the human history, is known as thehistorical event that drove the contemporary Marx into strenuous empirical research. He collected
and systematically classified an amount of various economic data from all parts of the world. They
were classified according to the countries and topics, then ordered chronologically. They cover
countries as France, Italy, Spain, United Kingdom, Germany, Austria, United States, China, India,
Egypt and Australia, and topics as financial market data (interest rates, share and security prices,bank balance, etc.), commodity market data (prices and sales of agricultural and industrial productsetc.), and bankruptcies, unemployment, short-time working, wages, labour disputes etc. This
voluminous research on the economic crisis of 1857 reached a total of 159 manuscript pages, which
would correspond about 500 print pages, and were known as ‘Books of Crisis’. The Books of Crisis
has just been published for the first time in MEGA, Part IV, Volume 14 (in the following: MEGA
IV/14), which was edited by a team of German and Japanese editors including myself. In parallelwith these Books of Crisis and based on them, Marx published research results on phenomena andcauses of the crisis in seven articles of ‘New York Daily Tribune’ (in the following: NYDT), which
are now being edited for MEGA I/16.

As Mori (2017) examined in detail, Marx’ Grundrisse of 1857-58, the so-called rough draft ofCapital, can be seen as direct extension and generalization of research and analysis in his Books of
Crisis and articles of NYDT. In the face of the world economic crisis, Grundrisse was written under
the description plan whose goal is the explanation of ‘world market and crises’. As well known, theexplanation was planned to be made in the following five steps 1:(1) the general, abstract determinants which obtain in more or less all forms of society…
1

MEGA II/1.1, S. 43.
1(2) the categories which make up the inner structure of bourgeois society and on which thefundamental classes rest. Capital, wage labour, landed property. Their interrelation. Town andcountry. The three great social classes. Exchange between them. Circulation. Credit system(private).
(3) Concentration of bourgeois society in the form of the state. Viewed in relation to itself. The‘unproductive’ classes. Taxes. State debt. Public credit. The population. The colonies. Emigration.
(4) the international relation of production. International division of labour. International
exchange. Export and import. Rate of exchange.
(5) the world market and crises.

The description of Grundrisse was obviously limited at most to the first and a part of the second step.
However, various kinds of crisis literature were already critically reviewed there, especially in thesecond section ‘Circulation process of capital’, where J. R. MacCulloch, J. Mill and J.S. Say were
harshly criticized for ignoring specific characteristics of the capitalistic production or identifying
supply and demand so that they denied the general overproduction 2.
While criticizing the doctrines denying the possibility of general overproduction as ‘stupidity’ 3 or‘childish’ 4, Marx tried to show precisely what he meant by his ‘general overproduction’, which must
be different from the partial overproduction à la Say where ‘at most, too much has been produced of
one and too little of another” 5. In his effort to define the general overproduction unambiguously,
Marx thought up a sort of input-output table, certainly one of the oldest ones in the history of
economic thought 6. To be exact, this multisectoral analysis of Marx was motivated besides by his
effort to define the general overproduction, also by his intention to thoroughly disprove Proudhon’s
doctrine on the profit according to which the profit arises because capitalists overcharge workers inselling their products.

1.2. Marx’ first input-output table
Motivated in this way, Marx drew up the following table 7:2

In the same sense, S. Hollander summarized this section of Grundrisse in his commentary on it:‘while there is no concerted discussion of the business cycle or of the structural relations that mustbe satisfied to allow the system to reproduce itself or to expand successfully, these issues werealready on the agenda’ (Hollander, 2008, p.289).
3
MEGA I/1.2, S.336.
4
MEGA I/1.2, S. 324.
5
MEGA I/1.2, S. 336.
6
So far as we know, Maurice Potron, a French Mathematician, was the first to calculate in fact input
coefficients in 1913 prior to W. Leontief. See Mori (2008).
7
MEGA I/1.2, S.352.
2

Table 1 Marx' input-out table
For labour

Raw material

Machinery

Surplus product

20

40

20

20

=100

B)Ditto

20

40

20

20

=100

C)Machinery

20

40

20

20

=100

20

40

20

20

=100

20

40

20

20

=100

A)Raw

material

manufacturer

manufacturer
E)Workers’
necessaries
D)Surplus producer

This table has following properties:-The economy consists of 5 production sectors (A, B, C, D, E): two sectors of raw material
manufacture,machinerysector,wage-good('workers'

necessaries')

sector,capitalists'-consumption-good ('surplus product') sector.
- Means of production for each sector consists of raw material and machinery.
-Workers and capitalists consume respectively products of E (‘workers’ necessaries’) and products ofD (‘surplus products’) exclusively.

One can read the table in the following manner. Each row of A, B, C, E and D represents pricecomponents of each sector’s output; the row of A, for example, shows that the raw material of valueof 100 is produced by paying wage (i.e. buying wage good) of 20, and using raw material of 40 andmachinery of 20, and paying surplus value (i.e. buying surplus product) of 20. On the other hand,each column from ‘for worker’ to ‘surplus product’ represents the demand for each good; the columnof ‘for worker’ shows that the wage good (‘workers’ necessaries') of value of 20 is sold to each of
sectors A, B, C, E, D. In Marx’ table, contrary to the usual input-output table, rows and columns
represent respectively the buyers and sellers. The table represents an equilibrium in the sense that foreach good the output equals to the real demand. Furthermore, it stands for an equilibrium state ofsimple reproduction because all surplus value is individually consumed by capitalists.

To translate Marx’ table into the conventional form of input-output tables by reversing the row andcolumn and separating households of workers and capitalists as final demand sectors fromintermediate (industrial) sectors, we obtain the following transformed table of Marx. Note that wecannot separate sectors A and B because we cannot know how much each sector purchases fromeach other.

3

Table 2 Translation of Marx’s input-output tableA,B) rawC)E) wageD)capitalists

workers’

capitalists’totalmaterial

machinerygood’ goodhouseholdhouseholddemand

raw

80

40

40

40

0

0

200

C) machinery

40

20

20

20

0

0

100

E)wage good

0

0

0

0

100

0

100 (dE)D)capitalists’

0

0

0

0

0

100

100 (dD)

wage

40

20

20

20

surplus value

40

20

20

20

product value

200

100

100 (xE)

100 (xD)

A,B)
material

good

Marx drew the following conclusions from his numerical example:
1) The proportion in which product value of each sector consists of wage, surplus value and thevalue of raw material and machinery on the one hand and the proportion in which the surplus value
is divided into consumption and investment in additional inputs on the other hand determine theproportion of product value among sectors. The former proportion changes with development of
productive forces 8.
2) The general overproduction takes place if both sectors E und D produce too much, i.e. the productvalue of E and D is greater than the total demand for E and D respectively so that xE > dE and xD >dD 9.
3) While capitalist in E can realize his whole profit through the direct exchange with workers,
capitalists in all the other sectors can realize their value only through the exchange among
themselves 10.
8

MEGA II/1.1, S.352-4: ‘At a given point in the development of the productive forces … a fixed
relation becomes established, in which the product is divided into one part − corresponding to raw
material, machinery, necessary labour, surplus labur − and finally surplus labour divides into one
part which goes to consumption and another which becomes capital again. This inner division,
inherent in the concept of capital … will appear in the exchange process as distribution among, say,
5 capitals’ (MEGA II/1.1, S.353-4).
9‘It is clear here that if D and E, where E represents all commodities consumed by the workers andD all those consumed by the capitalists, had produced too much− that is, to much relative to the
proportion of the part of capital going to the worker, or too much relative to the part of capitalconsumable by the capitalists…then general overproduction would take place…’( MEGA II/1.1,
S.353)
10
MEGA II/1.2, S.350, S.351-2; ‘The case we have posited, where capital E realizes the whole of its
profit in exchange with wages, is the most favourable− or expresses, rather , the only correct relationin which it is possible for capital to realize the surplus value created in production through exchangewith the workers’ consumption. But capitals A, B, C, D can realize their value in this case only4

1.3. Limits of multisectoral analysis in Grundrisse
On these conclusions, we need to make some remarks and point out also some open questions: as forthe first conclusion, Marx emphasized that the proportion of sectors depends in particular on the rateof surplus labour to necessary labour, the so-called the rate of exploitation (or the rate of surplus
value), so that the former changes as the latter changes. Marx in fact tried to calculate (in vain) how
the proportion of sectors changes as the composition of product value changes. However, according
to Marx’ input-output table, the proportion of sectors cannot be determined when the rate of surplus
value once changes. This is because, as mentioned above, it is not known how much raw material
from A the production of wage good and capitalists’ consumption good each use, and how muchfrom B. Therefore, Marx’ table was not well defined yet to allow a ‘comparative statics’ with the rateof surplus as a varying parameter.

As regard the second point, Marx regarded the general overproduction as disproportion between
supply and demand for consumption goods. Even if the demand for production goods is adequate for
their supply, this adequacy is just an illusion ('Schein') in so far as the supply of consumption goods
exceeds their demand 11, e.g. due to saving. This insight was already presented in the chapter ofmoney 12, and the overproduction (shortage of demand) was there famously linked to the existence ofmoney itself: ‘The separation of exchange into purchase and sale makes it possible for me to buy
without selling (stockpiling of commodities) or to sell without buying (accumulation of money)’ 13.
However, a closer investigation about what concrete forms the saving (money accumulation) takes in
the capitalistic production and what effects it has on the reproduction process was missing in
Grundrisse and had to wait until the eighth manuscript of Capital, volume II 20 years later, where
Marx tried, although incompletely, to explain the formation of money stock in the form ofamortization fund and accumulation fund in the framework of multisectoral analysis.

Concerning the third conclusion, Marx’ input-output table gives an effective counterexample to
Proudhon’s doctrine on profit according to which the profit arises because capitalists overcharge
workers in selling their products. Obviously, sectors A, B, C, D do not directly sell anything tolabours while their capitalists realize their profit through exchange among themselves. Then Marx
through exchange among one another, i.e. through the exchange of capitalists among themselves.’(MEGA II/1.1, S.351-2)
11
MEGA II/1.1, S.333-4.
12‘Circulation, exchange within the merchant estate, and the final stage of circulation, exchange
between the merchants and the consumers … are determined by quite different laws and motives,
and the greatest contradiction can develop between them. This separation alone can be the cause of
trade crises’ (MEGA II/1.1, S.83).
13
MEGA II/1.1, p.129.
5

went further to say that the exchange of sectors A, B, C, D with E is their indirect exchange withwage they pay to their workers 14, and the profit cannot be realized by this ‘detour’ of wage, but only
through the exchange among themselves 15.

We have reconstructed and evaluated Marx’ multisectoral analysis in Grundrisse. Marx admitted:“This example may or may not be worked out in more detail. It does not actually belong here” 16.
Marx did work out his multisectoral analysis in fact in more detail 12 years later in his second
manuscript of Capital, volume II of 1870. We can see there how Marx elaborated his input-output
table and devoted himself to the above mentioned remaining questions. First, he redefined thesectors in a suitable way and carried out a comparative statics about equilibrium proportion byvarying the rate of surplus. Second, he gave thorough consideration to the ‘detour’ of wage andformulate the ‘law of reflux’ while finding out how wage realizes on its ‘detour’ all value
components including profit. Third, he made a bridge to the analysis of accumulation fund andamortization fund in the eighth manuscript of 1878 by suggesting, although implicitly, the possibilityof money stock formation.

2. Marx’s 6-sector Production Model in 1870
2.1. Overview
It is known that Marx wrote in total eight manuscripts dedicated to the Volume II of Capital, of
which only two manuscripts (Ms.II and Ms.VIII) were used by Engels for the edition of Part 3 of thisvolume 17, i.e. the chapters on the so-called 'reproduction schemata'. Although Ms. VIII was fully
utilized, the half of the relevant part of Ms. II, making up 36 manuscript pages, was totally ignored
by Engels 18 and was published after 140 years for the first time in MEGA II/11 .

In the omitted part of Ms. II, Marx presented a six-sector model of production: 1) means of workers'
consumption (sector A3 19), 2) means of production for A3 (sector A2), 3) means of production for14‘E certainly exchanges for 1/5 of the product of A, B, C and D, i.e. 4/5 of their product; but thisexchange is only a detour to get to the wages which A, B, C and D pay their own workers. …thisexchange with E is then only an indirect form of advancing the part of capital which represents
necessary labour’ (MEGA II/1.2, S.351).
15‘They cannot therefore gain thereby. The gain comes from the realization of the remaining 4/5 ofcapital a, b, c, d, and this realization consists in that each receives the labour objectified in his
product in another form through the exchange’ (MEGA II/1.2, S.351).
16
MEGA II/1.2, S.353.
17
MEGAII/12, Apparat, S.529-552, 887-934; MEW 23, S.12.
18
In Ms. II, the text related to Part 3 of Volume II of the edited Capital' begins on manuscript page130 (MEGAII/11,S.340) and ends on manuscript page 202 (MEGA II/11, S.522). The pages 167-202(MEGA II/11, S.443-522) was omitted by Engels.
19
Note that Marx used in his original somewhat awkward notations IIα, IIαα, Ia, IIβ, IIββ and Ib,
which we substitute in this paper with A1, A2, A3, B1, B2 and B3 respectively.
6

A2 (sector A1), 4) means of capitalists' consumption (sector B3), 5) means of production for B3(sector B2), and 6) means of production for B2 (sector B1). And simple reproduction was assumed.
By means of this multisectoral model, he investigated, above all, following two main issues.
First 20, Marx tried to find out equilibrium conditions for the quantity system of the 6-sector model.
Indeed, he succeeded in formulating conditions necessary for the quantity equilibrium in threeequations. Furthermore, he carried out some comparative statics in order to examine the shift ofequilibrium output, employment etc. by varying a parameter of labour share, i.e. the 'rate of surplus
value (M/V)'.
Marx's investigation of the second issue 21 leads the readers, because of its intensity and persistence,
to suppose a large amount of his effort dedicated to it. He illustrated the 'law' of monetary reflux
using numerical examples in the 6-sector model, i.e. he traced each possible route of monetarycircuit in which money advanced by capitalists returns to its starting point after realizing various
components of output of the 6-sector economy. Particularly, he concentrated his attention on the
reflux of wage paid by capitalists to workers in each sector.

As just mentioned, Marx explained the monetary circuit as the result of the reflux of wage. Wageadvanced by capitalists to workers in each sector returns to its starting points after realizing various
components of output (not only the commodity labour power but also variable-capital part,
constant-capital part and surplus value of social products). However, some rest (Überschuß) 22
remains which cannot be realized by circulating money initially advanced as wage payments. In
order to realize this rest, it is necessary for capitalists to advance additional money. In this context,
this explanation can be seen to provide us with possibilities to re-evaluate Marx's own contribution
especially to the theory of monetary circuit along the line of the so-called 'Franco-Italian circuit
school'. The crucial role that e.g. A. Graziani (1990, 1998, 2003), E. J. Nell (2004) and others attach
to wage payments in monetary circulation was de facto anticipated by Marx although the bank, oneof their three main agents of the economy besides the classes of capitalists and workers, fades into
the background in Marx's model. Furthermore, Marx’s model of monetary circuit has a suitable
framework to which the money flow computation of I-O analysis by Leontief and Brody (1993) canbe effectively applied. Since Mori (2009) already elaborated on the second issue in detail, we are
now going to concentrate on a closer examination of the first issue.

20

MEGAII/11, S.481-503.
MEGAII/11, S.443-481.
22
MEGA II/11,S.491.
21

7

2.2. Equilibrium conditions
The subject of Part 3 of Volume II of Capital, well known as ‘reproduction schemata’, was dealt
with in Ms. II as the third ‘chapter’ after the two chapters on “circuit of capital” and “turnover ofcapital”. According to Marx’s own table of contents written on the title page of Ms. II, this “thirdchapter” was structured as follows 23:

The real conditions of the processes of circulation and reproduction
1) Variable capital, constant capital and surplus value from a social point of viewA) Reproduction on a simple scalea) Described without the mediating monetary circulationb) Description with the mediating monetary circulation
B) Reproduction on an extended scale. Accumulation.
a) Described without monetary circulationb) Description with the mediating monetary circulation2) (no title)

After the description of A) a) and b) based on 2-sector reproduction schemata, Marx changed, in acontinuation of A) b), to 6-sector schemata. This part of Ms. II was neglected by Engels for theedition of Capital, Volume II. The subjects of B) a) and the following were never written in Ms. II
and to be left to the later Ms. VIII.

As mentioned above, the first issue to investigate in the omitted part of Ms. II consists in the
induction of equilibrium conditions for quantity system of the 6-sector production model. Thismodel has following properties:-The economy consists of 6 production sectors (A1, A2, A3, B1, B2, B3) 24, household (individual
consumption) of workers and that of capitalists. Each sector produces one distinct good (i.e. single
production), which is given the same notation as the sector.
-Workers and capitalists consume respectively A3 and B3 exclusively.
- Means of production of A3 is A2; that of A2 is both A2 itself and A1; and that of A1 is A1 itself.
Similarly, means of production of B3 is B2; that of B2 is both B2 itself and B1; and that of B1 are
B1 itself.
-The output value of each sector consists of value of means of production and value added, and value
added consists of wage and surplus value.
-The income of workers and capitalists come respectively from wage and surplus value exclusively.
23
24

MEGAII/11,S.3-4.
See Footnote 19.
8Both classes do not save.
-Prices are proportional to labour values.

Historically, we can find a similar structure also in multisectoral models by G. Mathur and A. Lowe,
where the economy is composed of three sectors; heavy equipment, machinery and corn in Mathur(1965) while primary equipment, secondary equipment and consumer good in Lowe (1976). The
most characteristic feature of Marx’s model, however, consists in a symmetrical structure between
A-sectors and B-sectors, which we will show as basic and non-basic sectors respectively. Morerelevant in this respect is Shibata (1939) who divided the economy into four sectors according to“labourers’ consumers’ goods” and “labourers’ producers’ goods” on the one hand, “capitalists’consumers’ goods” and “capitalists’ producers’ goods” 25.

For the 6-sector model characterized above, Marx devised the following numerical example(Example 1: see Table 3).

Table 3 Marx’s 6-sector model (Example 1)
A1

A2

A3

B1

B2

B3

workers

capitalists

totaldemand

A1

200

100

0

0

0

0

0

0

300

A2

0

100

200

0

0

0

0

0

300

A3

0

0

0

0

0

0

300

0

300

B1

0

0

0

200

100

0

0

0

300

B2

0

0

0

0

100

200

0

0

300

B3

0

0

0

0

0

0

0

300

300

wage

50

50

50

50

50

50

300

surplus

50

50

50

50

50

50

300

300

300

300

300

300

300

value
output300

300

One can ‘read’ the table in the same manner as a usual input-output table. Each column from A1 to
B3 represents price components of each sector’s output; the column of A3, for example, shows thatthe good A3 of value of 300 is produced by using A2 of 200 and paying wage of 50 and surplus
value of 50. Each row from A1 to B3 represents the demand for each good; the row of A2 e.g. showsthat the good A2 of value 100 and 200 is sold respectively to the sector producing A2 and A3. The
table represents an equilibrium in the sense that for each good the output equals to the real demand.
25

On the similarity between Marx’s and Shibata’s construction, see Mori (2007).
9

Furthermore, it stands for an equilibrium state of simple reproduction because all surplus value isindividually consumed by capitalists.

On the basis of the above example, Marx traced every commodity transaction among production
sectors, workers’ and capitalists’ household to induce general conditions necessary for an
equilibrium of supply and demand. He formulated them in the following three formulae:

CA3 = (VA1+ MA1) + (VA2 + MA2)

26

(1)

CB3 = (VB1 + MB1) + (VB2 + MB2)

27

(2)

MA1 + MA2 + MA3 = VB1 + VB2 + VB3

28

(3)

where Cj, Vj, Mj are respectively constant capital, variable capital and surplus value of sector j (j =A1, A2, A3, B1, B2, B3). First, constant capital of sector A3 equals to the sum of wage and surplus
value of both sectors A1 and A2. Second, constant capital of sector B3 equals to the sum of wage
and surplus value of both sectors B1 and B2. Third, the surplus value of sectors A1, A2 and A3 (wecall them together ‘A-sectors’) equals to the variable capital of sectors B1, B2 and B3 (‘B-sectors’).
In both equations (1) and (2), one can see the well-known condition about Marxian 2-sector model
reappear, i.e. that constant capital of consumption-good sector equals to the sum of wage and surplus
value of means-of-production sector. Furthermore, the third condition anticipates the relationshipbetween subsectors of necessary consumption good and luxury which was to be considered in thelater Ms. VIII.

2.3. Rate of surplus value and comparative statics
After formulating the equilibrium conditions, he carried out some comparative statics in order toexamine the shift of equilibrium output, employment etc. by varying a parameter of labour share, i.e.
the 'rate of surplus value’ (M/V). While the rate of surplus value is unity in all sectors in Example 1,
it varied to 1/2 in Example 2 (Table 4), 5/7 in Example 3 (Table 5) and 7/5 in Example 4 (Table 6),
where input coefficients and the total number of employed workers remain constant.

26

MEGAII/11,S.483,493,495.
MEGAII/11,S.483,488,490,494.
28
MEGAII/11,S.488-91,493,495.
27

10

Table 4 Marx’s 6-sector model (Example 2) (M/V = 1/2) 29
A1

A2

A3

B1

B2

B3

workers

capitalists

totaldemand2

1

A1

266 /3133 /3

0

0

0

0

0

0

400

A2

0

133 1/3

266 2/3

0

0

0

0

0

400

A3

0

0

0

0

0

400

0

400

0

0

200

B1

0

0

0
1

0

133 /3

2

66 /3
2

0
1

B2

0

0

0

0

66 /3

133 /3

0

0

200

B3

0

0

0

0

0

0

0

200

200

wage

88 8/988 8/988 8/944 4/944 4/944 4/900400surplus

4

4

4

2

2

2

200

44 /9

44 /9

44 /9

22 /9

22 /9

22 /9

0

0

400

400

400

200

200

200

400

200

workers

capitalists

value
outputTable 5 Marx’s 6-sector model (Example 3) (M/V = 5/7) 30
A1

A2

A3

B1

B2

B3

total
demandA1

1

233 /32

116 /3
2

0
1

0

0

0

0

0

350

0

0

0

0

350

0

350

0

350

0

0

250

A2

0

116 /3233 /3

0

A3

0

0

0

0

B1

0

B2

0

B3

0

wage

0

0

0

68 /18166 /3

0

0
1

0
1

68 /180
2

1

68 /18

1

83 /3
1

0
2

0

83 /3

166 /3

0

0

250

0

0

0

0

250

250

48

48

48

0

0

350

11

11

11

0

0

250

350

250

/18

/18

/18

surplus

48

48

48

34

34

34

value

11

/18

11

/18

11

/18

13

/18

13

/18

13

output

350

350

350

250

250

250

29
30

MEGAII/11,S.481-4,494.
MEGAII/11,S.484-8,494.
11/18

Table 6 Marx’s 6-sector model (Example 4) (M/V = 7/5) 31
A1

A2

A3

B1

B2

B3

workers

capitalists

totaldemand2

1

A1

166 /3

83 /3

0

0

0

0

0

0

250

A2

0

83 1/3166 2/3

0

0

0

0

0

250

A3

0

0

0

0

0

250

0

250

0

0

350

B1

0

0

0
1

0

233 /32

116 /3
2

0
1

B2

0

0

0

0

116 /3233 /3

0

0

350

B3

0

0

0

0

0

0

0

350

350

wage

34

34

34

48

48

48

0

0

250

13

13

13

11

11

11

68 1/18

68 1/18

68 1/18

0

0

350

350

350

350

250

350

/18

/18

/18

surplus

48

48

48

value

11

/18

11

/18

11

output

250

250

250

/18

/18

/18

/18

The conclusion Marx drew from these calculations is that the output, employment and means of
production in A-sectors must increase as the rate of surplus value decreases. Theoretical implicationsof these comparative statics, however, cannot be seen far-reaching because, first, the calculations arenothing more than arithmetic exercises for adapting the equilibrium conditions and their results arealmost self-evident. Second, an assumption of his comparative statics, i.e. that all sectors have acommon labour share can sustain its meaning only in case of prices proportional to labour values (i.e.
in case of a uniform capital composition across sectors).

Much more interesting about Marx’s comparative statics, however, is the fact that he raised somecomplementary research questions which are anything but trivial and he never posed anywhere else.
We take up particularly two of such important research questions here. The first is how to carry out acomparative statics by taking the capital composition (C/V) as a varying parameter instead of therate of surplus value as before. He asked not only how the quantity equilibrium would change but
also what would be a new equilibrium price after varying the capital composition. He was namely
well aware that the prices could not remain proportional to labour values when the sectors havedifferent capital compositions. As we will show in the next section, this question confronted Marx
with a serious theoretical problem.

Another important research question Marx posed complementarily to the comparative statics withregard to varying rates of surplus value was how a dynamic process of transition from one
31

MEGAII/11,S.492-4.
12equilibrium to another is possible. Marx was namely confronted with the problem of “traverse”
triggered by changing wage rates.

In each of the following two sections, we will focus respectively on the first and second researchquestion, i.e. the comparative statics with regard to the capital composition and the dynamic processof transition of equilibria.

3. The Problem of Non-Basic System in Marx’s 6-Sector Model
3.1. Definition of basic and non-basic products
According to Sraffa (1960), if “a commodity enters (no matter whether directly or indirectly) into the
production of all commodities”, it is basic, and “those that do not, non-basic products” (Sraffa 1960,
8, parenthesis and italics in original). This definition can be in principle adapted to Marx’s model
although the concept must be extended to ‘augmented’ inputs in that basic products are used directly
or indirectly in all sectors as means of production or wage good. Non-basic products are used neither
directly nor indirectly in at least one sector.

Consider any single production, where each sector produces one distinct good (i.e. there is no joint
production). The definition of basic and non-basic products in the augmented sense can be formallyreformulated as follows.

We first introduce the following symbols:- input coefficient of good i for sector j: aij ∈ R +- input coefficient matrix: A :=(aij ) ∈ M (n × n, R + )- labour input coefficient for sector j: l j ∈ R +=l : (l1 , , ln ) ∈ R n+
- vector of labour input coefficients:- vector of wage goods (wage basket) per labour unit: d ∈ R +n- augmented input coefficient matrix: B = (bij ) := A + dl ∈ M (n × n, R + )- activity level (or output) of sector j: x j ∈ R +- activity (or output) vector:=
x ' : ( x1 ,..., xn ) ∈ R n+ . Note that the prime applied to a matrix or avector denotes, as usual, their transposition.
13

- price of good i: pi ∈ R +=p : ( p1 , , pn ) ∈ R n+
- price vector:- general rate of profit: r ∈ R

Definition 1.
nConsider then the matrix∑ B . A good i is basic if and only if the i-th row of this matrix istt =1

positive. Goods that are not basic are non-basic.

An alternative equivalent definition can be given in the following manner. If B is indecomposable,
all n goods are basic. If B is decomposable, it can be transformed into the following form by suitable
simultaneous substitutions of rows and columns.

 B11

0
B = 


 0


B12 
B22


B1n0 

 

 

0 Bn0 n0 (4)where B11 , , Bn0 n0 (n0 ≤ n) denote either a square null-matrix or a non-negative indecomposable
square matrix. Let Ii be the set of columns whose component is included in Bii, i.e.

I=j 1, , n | b jj ∈ Bii } . Obviously, all goods in I2, …, In0 are non-basic. If B11 is a null matrix,
i : {=
 B1 j there is no basic good. Otherwise, assume that for each j = 1, ..., n0,     0 if B jj isB  j −1 j  B1 j indecomposable, and ι    > 0 (ι := (1,...,1)) if B jj is a null-matrix. If and only if thisB  j −1 j 
assumption is true, any good in I1 is basic 32. Note that we use inequality signs for vectors and32

To prove the equivalence of both definitions, use the property of a non-negative indecomposablen × n matrix M:n∑M > 0.
t

t =1

14

matrices in this paper in the manner that X > Y, X  Y and X ≥ Y denote that X – Y is positive,
semi-positive and non-negative, respectively.

3.2. The problem of non-basic system
We assume in the following that there are both basic and non-basic goods. Non-basic goods can bealways eliminated from the balanced growth output vector (or standard system) as Sraffa suggested 33.
On the other hand, they can be sometimes included in the balanced growth path depending on their
input coefficients. It can be shown that a balanced growth output includes a non-basic good in Ii (i>1) if the Frobenius root of Bii is larger than that of B11, …, Bi-1,i-1, i.e. λi > λh for all h < i, where λj
denotes the Frobenius root of Bjj 34. In this case, the Frobenius root of Bii determines the growth rate,
which is now equal to gi:= (1/λi) – 1, lower than the previous rate of g1:=(1/λ1) – 1. Then, we cantake as a balanced-growth path the vector x’:=(x’1, ..., x’i, …, x’n0), where x is partitioned into n0
sub-vectors whose length corresponds to the dimension of B11, …, Bn0n0, with xi being a positive
eigencolumn of Bii, and =x j (λi I − B jj ) −1i

∑ B x for j < i and x = 0 for j > i.

h= j +1

jh h

jIt is known that the above mentioned possibility of a balanced growth path including a non-basic
good (we call it “non-basic system” in this paper) caused a theoretical problem. If there is a group of
non-basic goods Ii (i >1) such that the Frobenius root of Bii is not less than that of B11, i.e. λi ≥ λ1,
which case is implied by the “non-basic” system, then one of the following two alternatives mustoccur. Either basics’ prices are zero while non-basics’ prices in Ii are positive with a (possibly) lower
rate of profit ri = (1/λi) – 1. Or basics’ prices are positive while some prices in Ii are negative with ahigher rate of profit r1 = (1/λ1) – 1. This is the problem we mean by the problem of non-basic system,
which we will be dealing with, i.e. choice between two evils: either zero prices for all basics ornegative prices for some non-basics.

As mentioned above, the problem of non-basic system must always occur if non-basics’ input
coefficients for their own use, Bii, are sufficiently large. This will be typically the case when the
non-basics concerned are not just luxuries but means of production which reproduce themselves
quite costly. The problem was recognized by several authors besides or even before Sraffa (1960,
90-91, Appendix B). Gautam Mathur (1965) treated as a non-basic good a heavy equipment (e.g.
dam) in its accumulation process which is being expanded before it can be employed as a means of
production for other (machinery) industries in future.
Sraffa (1960), 25.
This is a sufficient condition for the possibility of a eigenculumn (balanced growth output vector)
including non-basics in Ii. A necessary condition would only need the weak inequality, i.e. λi ≥ λh for
all h < i.
33
34

15

“The heavy equipment in this state of affairs is thus a non-basic good, i.e., it does not enter directly
or indirectly into the production of all goods … All that the heavy equipment does is to reproduce
itself as fast as it can by feeding on the surplus produced by the rest of the economy” (Mathur 1965,
115-116, italics in original).

Indeed, the input coefficients Mathur exemplified for an accumulation process of the heavyequipment exhibit the above mentioned problem of non-basic system 35. In the consequence,“no pricing system can be discovered which would give an equal profit rate for each process in these(basic and non-basic) subeconomies”. Therefore, “the H-making process (heavy equipment industry)would be a loss-process and has to be subsidised. Because the dependent processes can exist only
after subsidies are given, the prices can be calculated only after subsidies. After the subsidy is given,positive prices can be found which equalise the rate of profit to the rate of growth. … Thus in perfectcompetition this subeconomy would not exist” 36.

More than 30 years before Sraffa (1960), a mathematician Robert Remak (1929) was well aware of
the problem of non-basic system. Adapted to the assumptions and notations in this section, his rather
abstract algebraic analysis can be summarized in the following way. First, he assumed an input
coefficient matrix whose quantity (column) eigenvector is the aggregation vector ι and the associatedFrobenius eigenvalue is unity, i.e., Bι = ι. Then, for this input matrix B, he proved that there exists a
non-negative price eigenvector p such that p = pB, and examined how zeros are located in p.
Defining In0 as “the highest group (höchste Gruppe)”, he concluded that all prices except the highest
group must be zero 37. Obviously, this implies that the prices of all basic goods in I1 must be zero.
This result is true because in Remak’s matrix B, we have 1 = λn0 ≥ λj for all j < n0.
The problem of non-basic system can be further traced back to Marx. In 1870, Marx attempted in his
Manuscript II of Capital Volume II to calculate production prices in a six-sector production model
whose setting exhibits the problem of non-basic goods.

Since Marx regards not only physical means of production but also wage goods as inputs based on
which the production price is calculated, we have to use an augmented input coefficient matrix
35

Strictly speaking, Mathur’s model deviates from the above explanation of the problem because heuses a joint-production model instead of the single production based on which the problem of
non-basic system was analyzed. However, the problem of non-basic system occurs there as well.
36 Mathur 1965, 123-124, parentheses added.
37 Remak 1929, 730.
16

including wage goods to determine the production price. The (augmented) input coefficient matrix(simply “input matrix” in the following if the distinction from a usual input coefficient matrix is notnecessary) implied in his six-sector model can be calculated from Table 3 in the following way:

 23

0
 16
=
B (b=
)

ij
0
0

0

1
1
1

3
3
6

0

0

0

2

0

0

1

1

1

3
6

6

6

0
0

0
0

2

0

1

0

0

0

0

3

1

0

0
1 
6

0
2 
3

0 

3
3

(5)

bij denotes the input of good i to produce one unit of good j (i, j =1, …, 6), where good 1, 2, 3, 4, 5
and 6 stand for A1, A2, A3, B1, B2 and B3, respectively.

As we can easily see from (4) and (5), 23
I1 = { A1, A2, A3} , B11 =  0
1
 6

1
1
1

3
3
6

0

3  and its Frobenius root is λ1 = 5 / 6
1 
6

2

I 2 = {B1} B22 = ( 2 3 ) and its Frobenius root is λ2 = 2 / 3
I 3 = {B 2} B33 = ( 1 3 ) and its Frobenius root is λ3 = 1/ 3
I 4 = {B3} B44 = ( 0 ) and its Frobenius root is λ4 = 0

The basic good is defined as a good that is used directly or indirectly in every sector’s production
process. In Marx’s model, A1, A2 and A3 are basic goods because A1 goes into the production of A2,
and A2 goes into A3, while A3, i.e. workers’ wage good, goes into all six production processes.
Therefore, A1, A2 and A3 enter all six production processes, A3 directly while A1 and A2 indirectly.
All goods that are not basic are called non-basic goods; in Marx model, B1, B2 and B3 are allnon-basic. Obviously, B3 is the “highest group” according to Remak’s definition.
Furthermore, since we have λ1 > λj for all j > 1, all goods can have positive production prices, say pj= 100 for all j and r = 1/5=20% as Marx’s price setting based on labour values yields. We know,
17however, such is not always the case. A characteristic feature of the non-basic system is, as wealready saw, that basic goods cannot have positive prices if λ1 ≤ λj for some j > 1. And this will be
exactly the case in the following example. Marx namely asked himself the following researchquestion. He suggested that the value composition (C/V) can differ between A-sectors andB-sectors, e.g. 4 in A-sectors and 10 in B-sectors, and “Let’s see temporarily how the things
turn out to be under the condition of the general profit rate” 38. He used to set the prices so far equal
to labour values because the value composition was the same in all sectors. Now he asks what
happens with production prices representing a general profit rate if different value compositions
between A-sectors and B-sectors are assumed. This was to be a tricky question for himself.

Indeed, he changed the value composition in B-sectors from 4 to 10 leaving A-sectors as before.
Then, the new equilibrium in value terms will be the following.

Table 7Marx’s 6-sector model (Example 5) (C/V=4 in A, 10 in B)
A1

A2

A3

B1

B2

B3

workers

capitalists

totaldemand

A1

200

100

0

0

0

0

0

0

300

A2

0

100

200

0

0

0

0

0

300

A3

0

0

0

0

0

0

300

0

300

B1

0

0

0

625

125

0

0

0

750

B2

0

0

0

0

500

250

0

0

750

B3

0

0

0

0

0

0

0

300

300

wage

50

50

50

62.5

62.5

25

0

0

300

surplus

50

50

50

62.5

62.5

25

0

0

300

300

300

300

750

750

300

300

300

value
output

And the augmented input coefficient matrix corresponding to this new state will be

38

MEGA II/11, 494-495.
18

 23

0
 16
B=
0
0

0

1
1
1

3
3
6

0
0
0

0
2
1

3

0
0

0
0

6

1
12

1
12

5

5

0
0
0

6

0
0

5

12
12

0

0

0
1 
12

0
5 
6

0 

Note that we have now λ=λ=
5 / 6 and, therefore, this new state exhibits the problem of
1
2
non-basic system. In fact, we can easily verify that any non-negative solution must have zero prices
for all three A-sectors, i.e. the non-negative solution must be p1=p2=p3=0 and p1=p2=p3=c, where c isany non-negative number. And the profit rate is uniquely r = 1/11.

In this way, Marx set a numerical example that exhibits the problem of non-basic system, i.e. a
choice between zero price for all basics and negative price for some non-basics. Once we have thisproblem of non-basic system, there will be two possibilities to treat the problem and keep all prices
positive. First, non-basic sectors, i.e. B-sectors, should be closed. If every producer moves from
B-sectors to A-sectors, A-sectors can regain positive prices, and the profit rate will return from 1/11(≈9%) to 20%. This is the only possibility the free market economy would allow. The second
possibility is just what Mathur (1965) suggested: non-basic sectors, i.e. B-sectors, should bemaintained by receiving subsidies from the government. The profit rate for A-sectors only is 20%
while it is 9% in the whole economy including B-sectors. To balance this inequality of profit,
B-sectors must be given a subsidy, e.g. 75 to B1 and B2 each, and 30 to B3, leaving each price at
100 and the profit rate at 20% as before.

4. The Problem of Traverse
As already mentioned, Marx posed another research question concerning a dynamic process oftraverse from one equilibrium to another. He assumed that after wage increase by 1/3, the rate ofsurplus value decreases from the original level of 1 to 1/2. As shown in Table 4, he could calculate
without any problem a new equilibrium state, where each of A-sectors must expand its output from
300 to 400 while each B-sector must shrink from 300 to 200. Here, not only the proportion between
A-sectors and B-sectors is changed, but this change must be also accompanied by a growth inproduction and employment in A-sectors. Then, Marx began to consider how to start such a growth
process from the stationary state, or how to move from the old equilibrium (300) to the new one(400).

19

Marx was clear what occurs after such wage increase.

“A large part of products [A3] … consist of foodstuffs which must be produced (at least in their raw
material) beforehand for an annual consumption. The demand for them would increase very much.
The profit made in [A3] etc. would rise so far very much. In short, capital and workforce would be
drawn from [B3] etc. to [A3] etc” 39.

He was, however, well aware that “the mechanism of the bourgeois society brings with it that suchchanges … are accompanied by circumstances which paralyze their effect and break the changes
themselves” 40. The circumstance Marx meant by the paralysis can be interpreted as follows: Despite
the high profit of A3, “the transfer to [A3] is not so fast as the dismissal under [B3]” so that an
ensuing unemployment lowers the wage and annuls in the end the initial effect of wage increase.
Due to such paralysis, he concluded with “difficulty and relative impossibility of the change on thisway” 41.

Marx, however, did not elaborate in more detail on how such paralysis must occur and why the
expected expansion of A-sectors does not take place. A century later, a comparable study on the
problem of traverse was resumed by Lowe (1974) 42. There, he identified four phases of the efficient(speediest) course of the expanding traverse 43.

1. Partial liberation of existing capacity
2. Augmentation of output of primary equipment (which corresponds A1 in our definition)
3. Augmentation of output of secondary equipment (A2)
4. Augmentation of consumer-good output (A3).

The first phase, in particular, means that we must release means of production from sector A2 andemploy them in sector A1 to expand A1 as fast as possible. After analyzing these four phases of theefficient traverse, Lowe came to the same conclusion as Marx 44. It is, therefore, not surprising that
39

MEGA II/11, 501-502. Sector indices in square parentheses were changed from the original inorder to be consistent with our definition.
40
MEGA II/11, 501, italics added.
41
MEGA II/11, 502.
42
Another pioneering work on the problem of traverse was carried out by Hicks (1973) almost
simultaneously. See Mori (2017).
43
Lowe 1974, 109-114.
44
It must be noted that Lowe treated unlike Marx a traverse between steady growth equilibria(instead of stationary state equilibria) and a traverse due to a sudden increase of labour supply(instead of a wage increase). Therefore, the situation of the traverse is symmetrically different
between both authors: Lowe assumed wage fall and decreasing demand for consumption goods
20both use the same word “paralyze/paralysis”: The motorial and behavioral conditions in the freemarket system are “goal-inadequate” and “diverting action in the wrong direction if not blocking it
altogether”. “[I]ts consequences are not just under- or overbuilding but a total paralysis of theadjustment process” 45.

Let us now reconstruct Marx’s traverse problem in order to make sense of his “paralysis”. First, we
define the efficient traverse as the path of each sector’s output which moves the most speedily fromthe old to the new equilibrium. Formally, the efficient traverse in A-sectors 46 is defined by thesolution of the following minimizing problem.

Problem 1.

arg min T !
x ( t )≥0s.t.
2 3 1 3 0 
x (t ) ≥  0 1 3 2 3  ( x (t ) + x (t ))2 9 2 9 2 9
 300  400 

x (0) ≤  300  , x (T ) ≥  400 
 300  400 
where x(t):=(x1(t), x2(t), x3(t))’ and xi(t) denotes the output of sector i at time t.
Assuming that time is measured continuously, and there is no storage to be carried over (i.e. theoutput must be used immediately or disposed), the efficient traverse can be determined uniquely (seeAppendix) and depicted in Figure 1.

while Marx assumed wage hike and increasing demand for consumption goods. Despite thisdifference, they reached the same result that the efficient traverse is impossible.
45
Lowe 1976, 151, 161. Italics added.
46
The efficient traverse in B-sectors can be treated in the analogous way, which elaboration we omitin this paper.
21

Figure 1.
Outputnew equilibrium

400

A1

A2

A3

350

old equilibrium

300

250

200

1

2

3

4

time

5

As we can see from Figure 1, the duration of the efficient traverse is about 5.65, and its path can becharacterized by following remarkable features.

1. Sector A1 starts from a higher point while sectors A2 and A3 must instead start from a lower point
than the old level (300).
2. A1 is the first to reach the new level (400), A2 is the second, and A3 the last.
3. The growth of A1 and A2 temporarily overshoots the new level to keep sustaining the growth ofrespectively A2 and A3 to the end.
4. A3 follows a balanced growth path (with the rate of growth g=1/8) while A1 and A2 aredecelerating the growth.

Comparing with Lowe’s efficient traverse, we can see that the first feature here corresponds to thefirst phase of Lowe’s traverse, and the second feature represents the order of augmentations of thethree sectors characterizing the second, third and fourth phases 47. Note that Lowe’s first phase isinvisible in the continuous-time model in Figure 1 because the “liberation” takes place there
instantly at time 0. If we magnify the neighborhood of time 0 by measuring time discretely (seeFigure 2), we can see the first phase (from time 0 to time 1) more clearly, where sector A1 isgrowing rapidly at cost of sector A2 (while sector A3 shrinks by leaving their capital idle).

47

Lowe admitted that the three phases may not necessarily proceed successively (or “horizontally)
but parallel (or “vertically) as shown in Figure 1.
22

Figure 2.
A1

400
350
300

A2

250

A3

200
150
100

0

1

2

Lowe regarded this process of “liberation” in the first phase as “paradoxical” in the sense that “inorder ultimately to increase the output of consumer goods [A3], such output must, to begin with, be
reduced” 48. It might be possible to make sense of Marx’s “paralysis” by referring to Lowe’s“paradox”: Sector A3, which must be reduced at first in order to finally expand effectively to thegoal, enjoys instead increasing demand and, therefore, high profit so that it draws capital andworkers from other sectors. Obviously, if sector A3 expands, sector A2 as provider of their means of
production must expand as well. Then, the liberation of resources from A2 to A1, i.e. the firstrequirement of the efficient traverse, fails.

5. Conclusion
In his comparative statics with regard to the capital composition, Marx must have been confronted
with an aporia because such a situation as zero or negative prices is impossible according to Marx’s
theory of production price. His formula of profit rate, r = M/(C+V), yields namely positive prices inall these cases. It is symptomatic of this aporia that Marx could not manage to solve the prices andinterrupted this tricky example by changing assumptions and finally saying: “This must be
investigated later (Dieβ später zu untersuchen)” 49. We know that the announced investigation never
happened.

On the other hand, the research problem on the dynamic process of traverse is a good proof for
Marx’s awareness that in general the dynamic analysis is a theoretically different question from thecomparative statics and a much trickier one because such changes are “paralyzed” by “hindrancesthat are immanent in the mechanism of capitalistic production” 50. And we hope to have been able to

Lowe 1976, 110.
MEGA II/11, 495.
50
MEGA II/11, 501-503.
48
49

23

show that Marx de facto anticipated there the problem Lowe tackled with more than one hundredyears later.

These interesting theoretical episodes in his Capital manuscript which might have lead readers toquestion the validity of Marx’s price theory on the one hand, and to open new perspectives to further
develop his multisectoral analysis on the other hand, was made invisible in the Capital edition afterEngels eliminated this part of manuscript in his edition of Capital Volume II. As already stated, this
6-sectoral analysis by Marx as a whole was completely ignored by Engels when he edited Capital
Volume II. Can Engels’s omission of this part of Marx’s manuscript be justified? The considerationof this paper would tend to suggest denying the question.

It is true that most of Marx’s essential results reappear either in Engels’s edition of “Reproduction
Schemes”, or they might be no more than trivial arithmetic exercises which would not deserve being
published. However, some research questions Marx posed for his own further investigation suggest
profound theoretical problems although Marx obviously could never give any answer to them. Ifthey had appeared in the Capital edition, it would have been at least clear to the readers that Marx
was confronted with critical and significant problems in economic theories. It could be possible thatthey would have exerted an influence on the course of posthumous controversies on his economic
theories.

24

Appendix: The solution to Problem 1
The solution of Problem 1 can be found by solving the following differential equations.

 x3 (t ) = x3 (0)et 8 x (t ) 2  x (t ) − 9 x (t ) 
=
3
 2

 2
8



1

( x1 (t ) − x2 (t ) − x2 (t )) x1 (t )=
2

1

300
 x1 (0) + 2 x2 (0) =

x=
x=
400
1 (T )
2 (T )
3 (T )
 x=
The solution can be determined uniquely as follows:

x1 (t ) ≈ −23.71628et 2 + 0.00099e 2t + 355.28684et 8x2 (t ) ≈ −0.00099e 2t + 236.85789et 8x3 (t ) ≈ 197.38158et 8T ≈ 5.65061

25References
Collected Works of Marx and Engels
Marx-Engels-Werke, Band 23, Dietz Verlag, 1979 (abbr.: MEW 23).
Marx-Engels-Gesamtausgabe, II. Abteilung, Band 1, Teil 1, Dietz Verlag, 1976
(abbr.: MEGA II/1.1).
Marx-Engels-Gesamtausgabe, II. Abteilung, Band 1, Teil 2, Dietz Verlag, 1981
(abbr.: MEGA II/1.2).
Marx-Engels-Gesamtausgabe, II. Abteilung, Band 11, Akademie Verlag, 2008(abbr.: MEGA II/11).
Marx-Engels-Gesamtausgabe, II. Abteilung, Band 12, Akademie Verlag, 2005
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