Discussion Paper No. 249
Charasoff and Dmitriev:
An Analytical Characterisationof
OriginsofLinear Economics
Kenji Mori
January 2010TOHOKU ECONOMICS RESEARCH GROUPGRADUATE SCHOOL OF ECONOMICS AND
MANAGEMENT TOHOKU UNIVERSITY

KAWAUCHI, AOBA-KU, SENDAI,

980-8576 JAPANCharasoff and Dmitriev:
An Analytical Characterisation of Origins of Linear Economics
Kenji Mori
(Graduate School of Economics and Management,Tohoku University, Sendai, Japan)

Abstract
Georg von Charasoff was one of the first economic theorists to recognise that the price of production
is an eigenvector of the input matrix, and to determine the rate of profit using its eigenvalue. He
anticipated, at this analytical level, most of the arguments that were proposed later in the course ofthe ‘transformation problem’. This paper aims to reformulate his significant arguments in a formalmanner and to reveal their logical relationship by reproducing the mathematical reasoning, so thatthe logical characteristic of his system can be identified in comparison with Dmitriev's andBortkiewicz's linear economic system.
Key words: Charasoff, Dmitriev, Bortkiewicz, Frobenius root, Fundamental Marxian Theorem
Address for correspondence: Graduate School of Economics and Management, Tohoku University,
27-1 Kawauchi, Aoba-ku, Sendai 980-8576, Japan. email: mori@econ.tohoku.ac.jp

Introduction
Since Georg von Charasoff 1 , a Russian mathematician and economist, was rediscovered more than
70 years after his main work, Das System des Marxismus. Darstellung und Kritik (1910), he has
been acknowledged in many articles on the history of economic thought 2 as a pioneer in linear
economic theory and a forerunner of Leontief, Sraffa and von Neumann. However, reviewers tend tochoose a part of his theory and point out its pioneering role; thus, his ‘system’ as a whole has notbeen fully examined 3 .
This paper aims to reformulate all the significant arguments of Das System des Marxismus in a moreformal manner 4 and to reveal their logical relationship by reproducing the mathematical reasoningthat Charasoff must have had in mind when writing the book. We conclude the paper by comparing
Charasoff’s linear economic system with those of his predecessors, Dmitriev and Bortkiewicz.
The important elements of Charasoff’s main argument are as follows:
1) The distinction between basic and non-basic products
2) The modelling of production price and general rate of profit using the eigenvector and eigenvalue1

of the input coefficients matrix, respectively3) The duality of the price and quantity system (existence of a balanced growth path)4) The ‘fundamental theorem of Marxian profit theory’ (the so-called Fundamental Marxian
Theorem)5) The convergence theorem for the Marxian transformation procedure from values to production
prices6) The theorem of rising rate of profit (the so-called Okishio Theorem)
Although the propositions in Charasoff's main work are discussed only in a narrative form andexplained at most by numerical examples, his text allows every careful reader to unambiguously
reproduce those formal proofs that the author must have had in mind.

1. Charasoff's linear economic model and its essential condition
1.1. Definition of basic and non-basic products
Charasoff introduced a distinction between basic products/production (‘Grundproduktion’) andsubsidiary products/production (‘Nebenproduction’). He defined basic production as production that
‘reproduces its own means of production (including real wage) without others’ assistance, andproduces, in addition, the base of surplus production (i.e. subsidiary production)’ (Charasoff, 1910, p.
81, parentheses added). Basic and non-basic products were understood in Sraffa’s sense (Sraffa,1960), and extended to ‘augmented’ inputs, i.e. basic products are used directly or indirectly in all
sectors as factors of production (means of production and real wage). Non-basic products are usedneither directly nor indirectly in at least one sector. According to Marx’s distinction of social
products, basic sectors correspond to Sectors I (means of production) and IIa (necessary means ofsubsistence), and non-basic sectors correspond to Sector IIb (luxuries). The definition of basic andnon-basic products in Charasoff’s sense can be reformulated as follows.
We first introduce the following symbols:- input coefficient of good i for Sector j: aij ∈ R +- input coefficients matrix: A := (aij ) ∈ M (n × n, R + )- labour input coefficient for Sector j: l j ∈ R +- vector of labour input coefficients: l := (l1 ,", ln ) ∈ R n+
- vector of real wage (wage basket) per labour unit: d ∈ R n+
- augmented input coefficients matrix: B = (bij ) := A + dl ∈ M (n × n, R + )- activity level of Sector j: x j ∈ R +2

- activity vector: x ' := ( x1 ,..., xn ) ∈ R n+ 5
- price of good i: pi ∈ R +- price vector: p := ( p1 ,", pn ) ∈ R n+
- normalised price vector: y = ( y1 ," , yn ) :=

1

n

n

∑p
i =1

( p1 ," , pn ) ∈ Y ; Y := { y ∈ R +n | ∑ yi = 1}i =1i- general rate of profit: r ∈ RnConsider then the matrix∑ B . A good i is a basic product (‘Grundprodukt’) if and only if the i-thtt =1

row of this matrix is positive. Goods that are not basic products are non-basic products.
An equivalent definition can be given in the following manner. If B is indecomposable, all n goods
are basic products. If B is decomposable, it can be transformed into the following form by suitable
simultaneous substitutions of rows and columns.

⎛ B11
⎜
0
B = ⎜⎜#⎜⎜ 0⎝

B1n0 ⎞
⎟
# ⎟
%
# ⎟
⎟
0 Bn0 n0 ⎟⎠

B12 "
B22
"

where B11 ," , Bn0 n0 (n0 <= n) denote either a square null-matrix or a non-negative indecomposable
square matrix. If B11 is a null matrix, there is no basic product. Otherwise, choose, if possible, an⎛ B1 j ⎞⎜⎟index i such that bii is an element of B11 , and 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. Then, the good i is a⎜B ⎟⎝ j −1 j ⎠basic product 6 . Note that we use inequality signs for vectors and matrices in this paper so that X > Y,X ≥ Y and X >= Y denote that X – Y is positive, semi-positive and non-negative, respectively.
The distinction of basic and non-basic products plays two roles in the theoretical system proposed by

3

Charasoff. First, it can falsify Marx’s formula of production prices and the general rate of profit byproviding an effective counterexample. Charasoff shows that the general rate of profit can be
determined only for the basic sectors, and that luxuries are irrelevant to its determination. He thenpresents a numerical example where the general rate of profit deviated from the average among all
sectors including a non-basic sector (Charasoff, 1910, pp. 93-104).
As we know, the idea of the determination of profit rate in the basic sectors can be traced back to
Ricardo. Dmitriev first provided a formal expression, which was used by Bortkiewicz to falsify
Marx’s price formula three years before Charasoff (Bortkiewicz, 1907a, pp. 15-16; 1907b, pp.
323-24). Following Ricardian tradition, Dmitriev and Bortkiewicz, however, considered basic goods
as wage goods and the inputs used to produce them directly and indirectly 7 . Charasoff generalised
the category, apart from sticking to the relationship between wage goods and basic products, which
is not logically necessary.
The second role of the distinction between basic and non-basic products in Charasoff’s system is of atechnical nature. It provides his price theory with a logical condition. In Charasoff’s text, thedistinction is assumed in such a simplified form that at least one basic product (e.g. a wage good)enters all sectors as direct input, and no non-basic products are used as input in any sector. This
simplifying assumption is expressed in Assumptions (A.3), (A.4) and (A.5) in his model as
reproduced in 1.2.1. Mathematically, these assumptions imply that the augmented input matrix has asimple Frobenius root that is the only eigenvalue of maximum modulus; this property is, as we willsee, the condition that is essential to Charasoff’s system and ensures in particular that the normalised
power sequence 8 of the augmented input matrix converges to a semi-positive matrix.

1.2. Analysis of Charasoff's linear economic model
1.2.1. Assumptions
Implicit in Charasoff's argument are several assumptions, which are as follows:(A.1) Each production process produces a unique product, which means that the output coefficientmatrix is an n-dimensional unit matrix.
(A.2) A is productive, i.e. the Frobenius root of A is less than unity.
(A.3) The real wage vector is semi-positive, i.e. d ≥ 0 .
(A.4) Labour is directly used in all sectors, i.e. l > 0.
(A.5) Non-basic products are not used as input in any sector, i.e. ∀ ∀ bij = 0 with I being the seti∉I jof indices of basic goods.

4It is useful to note the following: First, because, according to the Assumptions (A.3) and (A.4),
labour is directly used in all sectors and real wage vector is semi-positive, B has at least one positiverow, say i. According to the definition, good i is a basic product, and therefore the set of basic
products is not empty.
Second, from (A.3), (A.4) and (A.5), it follows that, if a non-basic product exists, A and B have thefollowing forms:⎛A
A = ⎜ 11⎝ 0
⎛B
B = ⎜ 11
⎝ 0

A12 ⎞
⎟
0 ⎠
B12 ⎞
⎟ , and B11 ≥ 0, B12 ≥ 0
0 ⎠

(1)

If no non-basic products exist, we have A = A11, B = B11.
Third, the Assumptions (A.3), (A.4) and (A.5) as a whole are quite strong. They can be weakened in
the following manner: the augmented input matrix B should have a simple Frobenius root that is the
only eigenvalue of maximum modulus 9 .

1.2.2. Properties of the augmented input coefficients matrix
Let I := {1,...,k}, 1 <= k <= n be the set of indices of basic products. Then, because of the form of B
in (1), all elements of the (k + 1)-th to n-th row are null. The characteristic equation of B ist n − k tE − B11 = 0 (where E is the unit matrix). All eigenvalues of B except n-k noughts are therefore
the same as those of B11, and both matrices have the same multiplicity for each identical eigenvalue.
Because B11 is indecomposable according to the definition of the basic product, the Frobenius root of
B11 is the same as that of B and a simple root of the characteristic equation of B. It is denoted as λ0.
Now, choose an index of good i0 such that the i0-th component of d is positive (which is possible
according to (A.3)). Then, the wage good i0 is used in all sectors, particularly in Sector i0. Therefore,at least one diagonal element of B11 is positive, which implies that the matrix B11 is primitive, i.e. themodulus of all eigenvalues of B11 and B except λ0 is less than λ0.
Next, we examine the eigenvectors of B associated with λ0. Because B11 is indecomposable, we canchoose a positive right-side eigenvector u of B11 and a positive left-side u associated with the 12 / λ0 ) are right- and left-side eigenvectors of B
eigenvalue λ0. Then, v := (u ', 0) ' and v := (u , uB
associated with λ0, respectively, where u > 0 ,λ0 > 0 and v > 0 . Because the eigen space of B

associated with λ0 is one-dimensional, all right- and left-side eigenvectors associated with λ0 are

5equal to v and v, respectively, up to the scalar.

Now, define a matrix B as B := B / λ0 , and consider the limit of Bt

for t→∞. As discussed

above, the Frobenius root of B is unity and simple, and v and v are its right- and left-side
eigenvectors, respectively. It is also clear that the modulus of all eigenvalues of B except for unity
is smaller than unity. Therefore, for t→∞, B
Considering B*tconverges to a limit that is a semi-positive matrix.

as the limit, we have:B * = lim B t = lim B t +1 = lim BB t = BB* = B* B
t →∞

t →∞

t →∞

We can see that each column and row of B*

is either a right- and left-side eigenvector of B

associated with the Frobenius root, respectively, or the null vector. Therefore, there is a row vectorq ∈ R n+ such thatB * = vq(2)

 = v , we also have:
Because of vB
 = vB 2 = vB
 3 = " = vB *v = vB(3)

From (2) and (3), we obtain: * = vvqv = vB(4)

 > 0 . Then, from (2) and (4), we obtain:
Since v ≥ 0 and v > 0 , vv
B * = vq =

1
vv

vv

(5)

We can see that each column of B*

is the multiplication of the vector v by a scalar, and that the

proportion of the scalars to each other is equal to that of the components of v .

2. Production price and general rate of profit
After Charasoff effectively falsified Marx’s price formula (Charasoff, 1910, pp.93-104), he then tried
to formulate his own ‘completely correct’ price theory. To this end, he dedicated three chapters (VIII
to X), which undoubtedly formed the climax of his book. The method of deduction of production
prices is marked by his highly original ideas, represented by the notions ‘production series’, ‘original
capital’ and ‘dimensions’, which characterise his theory as a whole. Formally speaking, his approachconsists of finding the semi-positive eigenvector of the normalised augmented input matrix by6

deducing the convergence of its power sequence in the same manner as the Markov chain.
As we know, Dmitriev had, prior to Charasoff, proposed correct equations and provided a rational
solution to determine the production prices. However, as shown in Section 7, the logical condition of
his linear economic system does not coincide with that of Charasoff's system; therefore, the validityof each price equation is complementarily limited.

2.1. Production series
Let X be an arbitrary bundle of goods and X’ be the input for X, i.e. both the means of productionused for the production of X and the means of subsistence of labourers employed for the productionof X. Similarly, let X’’ be the input for X’ and X’’’ be the input for X’’, etc. Charasoff called X’‘capital of the first order’, X’’ ‘capital of the second order’, etc. The sequence X, X’, X’’, X’’’… wascalled‘production series (Produktionsreihe)’ of X. In order to confine his attention to the

proportional composition, he normalised the capital of each order in that sequence and called thenormalised composition the ‘type (Typus)’ of capital of each order. If we now introduce new
symbols, Z, Z’, Z’’… for normalised X, X’, X’’…, then we have a new sequence Z, Z’, Z’’… as anormalised ‘production series’ or sequence of ‘types’.

2.2. Original type and original capital
Based on the above definitions, Charasoff deduced some important ideas that are representative of
his theoretical system as a whole. They can be reformulated as follows.
Proposition 1. For any bundle of goods, the normalised production series converges to a common
limit (See Appendix A.1).
The limit was called ‘original type’, and the capital that has the original type as its composition was
called ‘original capital’.

t

Let E j be the capital of the t-th order of product unit j (i.e. the (t + 1)-th term of the productiontttseries of product unit j), and w j be the sum of all components of E j . Let w be the vector( w1t , w2t , ..., wnt ), and wt be the normalised wt .
Lemma 1: For t → ∞ , w converges.
t7Charasoff then called the i-th component of the limit the ‘dimension’ of product unit i. He thendeduced the following from Proposition 1 and Lemma 1 10 :Proposition 2: The dimension of each product provides the normalised production price of thisproduct.
Corollary 1: The original capital is produced using an original capital (i.e. the input of the original
capital is an original capital).
Corollary 2: The rate of growth of the original capital provides the general rate of profit.
The above propositions are slightly replenished formalisations of the following original text ofCharasoff.
‘If X is a certain commodity, and X* and X*' are capitals of two succeeding sufficiently highorders ..., then both capitals are of almost the same type (Proposition 1). Therefore, it follows
that the original type to which all capitals of lower orders tend as their common limit has theproperty of growing without any qualitative change in the production process (Corollary 1). It
follows that the rate of its growth must provide the general rate of profit (Corollary 2)’. The
original capitals ‘are all of the same composition, and therefore are no longer different fromeach other qualitatively; however, they differ quantitatively, i.e. only according to theirdimensions (Lemma 1). They are different quantities of the same capital. ... and the production
prices of their final products comply with these dimensions ...; therefore, their prices must be
proportional to their dimensions (Proposition 2)’ (Charasoff, 1910, pp. 124, 111, 123,
parentheses added).
The analytical meanings of Proposition 2 and its corollaries are as follows: Considering the formulaof production prices in the matrix form, i.e. p = (1 + r ) pB , Proposition 2 implies that the n-tupleof ‘dimensions’ of product units is a left-side eigenvector of the augmented input coefficients matrix
B. On the other hand, Corollary 1 means that the original capital (original type) is a right-side
eigenvector thereof. From Corollary 2, it follows that the general profit rate and the balanced growthrate can be calculated as dual phenomena of the same eigenvalue.
Indeed, Charasoff then tried to find a state in which all acquired profit is accumulated and all sectors
grow in a common tempo, i.e. the state now known under the name of von Neumann’s balanced

8

growth. Charasoff obtained the same result as von Neumann did later, i.e. the equality of the rate ofprofit and growth.
‘One should think of a capitalistic society where all surplus labour is directed to theaccumulation or to the production of new capitals. At the same time, the accumulation process
should be globally synchronised so that all enterprises grow annually in the same tempo, and all
private entrepreneurs acquire a profit that is proportional to their capital and use it entirely for
expanding their operation. Under these conditions, the social capital bears the original type, andthe profit rate ... will provide the growth rate of the original capital in the process of the annual
production' (Charasoff, 1910, pp. 126-7).
We paraphrase this content as follows:
Corollary 3: In case of balanced growth, the quantities of products in an economy bear the original
type, and the general rate of profit equals the growth rate of the economy.

3. ‘Fundamental theorem of Marxian profit theory’
Discovering the ‘original capital’ and the ‘dimensions’, Charasoff considered the price problem as‘finally solved’. He solved the price problem without considering labour value, contrary to the
Marxian approach. However, he stated that it did not mean the bankruptcy of the labour theory ofvalue (Charasoff, 1910, p.104). He was convinced that labour values ultimately regulate the prices(Charasoff, 1910, p.112). He then tried to base his price theory on Marx's value theory. This is said
to occur in two ways: by the ‘fundamental theorem of Marxian profit theory’ (elsewhere, the‘fundamental theorem of Marxian surplus value theory’) (Charasoff, 1910, pp. 13, 98) on the one
hand, and on the other, by what is currently known as the convergence theorem for Marxiantransformation procedure from values to production prices.
Using this ‘fundamental theorem’, Charasoff tried to ‘prove the relationship between the level ofprofit rate and the provided surplus labour’ explicitly (Charasoff, 1910, p. 104). His idea is asfollows:‘If there is a profit for capitalists, then it can be explained as that the labour is sold under thevalue, or that the wage, as price of labour, falls short of the cost prices of labour -- i.e. the actual
quantity of labour provided by the labourer. ... The main condition is and remains thedisproportion between the wage and the actual provided labour’, and ‘this profit rate depends
on the surplus labour because the ability of the original capital to grow in the production

9

process and to provide a surplus is determined by surplus labour, i.e. by prolonging the working
day over the necessary extent’ (Charasoff, 1910, pp. 11-13, 112-13).
We reformulate his ideas as follows:
Proposition 3: The rate of profit is positive if and only if surplus labour is positive 11 .
We know that more than a half century had passed until this theorem was ‘discovered’ by Morishima
and Seton (1961) and Okishio (1963) without any mention of Charasoff's prior contribution and wasformalised under the name ‘Fundamental Marxian Theorem’ (Morishima, 1973, p. 53; Morishima
and Catephores, 1978, p. 30) 12 . However, it is not claimed here that the first proof was provided byCharasoff, since it is also possible to assert that a de facto proof had already been provided by
Dmitriev (Dmitiriev, 1904/1974, p.63, 77) 13 . However, Charasoff showed the equivalence of thepositive exploitation and the positive profit rate by recognising the price vector as an eigenvector of
the input coefficients matrix and determining the profit rate using the eigenvalue.

4. Convergence theorem for Marxian transformation procedure
As discussed in Section 3, Charasoff tried to base his price theory on the value theory in both ways.
After discussing the ‘fundamental theorem of Marxian profit theory’, he then considered theconvergence theorem for Marxian transformation procedure from values to production prices.
Marx's transformation procedure begins with considering labour values as initial prices andcalculating the average of profit rates of all sectors to obtain the new prices. This is, however, only
the first step of the transformation, and the obtained prices are the prices of the first order. Charasoff
proceeded with the second step: re-calculating the cost prices according to the new prices andfinding the intersectoral average of profit rates to obtain the prices of the second order. The sameprocedure should be repeated endlessly (Charasoff, 1910, pp. 134-39). If we normalise the prices ofeach order and call the normalised prices of the k-th order pk, we can reformulate Charasoff’s idea asfollows:
Proposition 4 (convergence theorem for Marxian transformation procedure): pk converges and thelimit provides production prices. At the same time, it is indifferent to the limit what kind of prices aretaken as initial prices, especially whether the labour values are the initial prices (See Appendix
A.3).
This proposition, as well as the convergence to the original type (Proposition 1), is based on the10

converging power sequence of the normalised augmented input matrix; therefore, the duality of bothproblems can be observed. Charasoff considered the iterated procedure stated above as Marx's own
contribution. However, he saw it as a problem that Marx stopped continuing the procedure after the
first step, and that Marx thought that he should begin with labour values as initial prices (Charasoff,
1910, p. 138).
Charasoff first showed the convergence of Marxian transformation procedure in the same manner as
the Markov chain. Twenty-three years later, Shibata (1933) illustrated the convergence with anumerical example without referring to Charasoff, and Okishio (1972, 1973, 1974) then provided aformal proof of the convergence theorem.

5. Theorem of rising rate of profit
It is well known that starting from Tugan-Baranowsky (1901) to Okisiho (1961), a series of major
objections to Marx's law of the falling rate of profit were raised. While Tugan-Baranowsky falsified
the law by providing counterexamples, Bortkiewicz, based on his price formula, proved that the
profit rate must rise if a technical change lowers the cost price of at least one wage good
(Bortkiewicz, 1907a). For Charasoff, the law means the negative part of Marxian theory whichcannot be saved more even after his reconstruction. He brought forward two new points into thedebate: first, a new cost criterion for the technical progress, i.e. an advanced production method
lowers the cost price of the original capital (composed of basic products according to hisassumptions) measured by the current prices. Second, he proved the theorem of the rising rate ofprofit by deducing implicitly the change (decrease) of the Frobenius root of the augmented input
matrix.
‘The general rate of profit is always intermediate among all partial profit rates that arecalculated on the basis of an arbitrary price system’ (Charasoff, 1910, p. 190).
By introducing a new production method, a capitalist can lower his cost price and raise his
individual profit rate. Then, we have:
The new general rate of profit ‘will lie between the profit rate R’ of our capitalist and the profit
rate R of all the other capitalists that are calculated on the basis of the old normal system. It cannever fall under the earlier profit rate R, and will always rise over this rate when the basic
production is considered’. ‘In order to obtain a falling rate of profit, we must assume a newmode of production that promises in advance a lower profit rate to the capitalists applying it’(Charasoff, 1910, pp. 190-192).

11

We paraphrase this idea as follows:
Lemma 2: The general rate of profit lies always between the minimum and maximum among all
individual profit rates that are calculated in terms of any arbitrary positive price system (SeeAppendix A.4).
Proposition 5: The general rate of profit rises if a new production method lowers the cost price of atleast one component of the original capital in terms of the current prices, with the cost price of theother products remaining constant (See Appendix A.5).

6. Calculation of labour value
For proposing the 'fundamental theorem of Marxian profit theory', Charasoff had first assumed theexistence of labour values and the well-defined rate of exploitation as given. In order to complete theproof, he dedicated chapter XII to the analysis of labour value.
Let X be an arbitrary bundle of goods, and X, X', X'', X''' ... be the production series of X. It is
implicitly assumed here that the production technique is able to produce a surplus, i.e. the augmented
input matrix is productive. Charasoff then called the series X' + X'' + X''' +... ‚ ‘reproduction capital’of X. The ‘reproduction base’ of X is defined as the reproduction capital of X obtained by neglecting
labourers’ subsistence.
Let X be the reproduction base of X. Then, Charasoff proposes the following (Charasoff, 1910, p.
147)
Proposition 6: The sum of labour used directly to produce X + X is the labour value of X (SeeAppendix A.6).
−1

It is easy to see that X + X is a series equivalent to ( E − A) X , and that Charasoff’s value
formula amounts to the well-known value equation w = l ( E − A)−1

(w and E denote thelabour-value vector and a unit matrix, respectively). However, it is Dmitriev, not Charasoff, whomust be acknowledged as the originator of the value equation.

7. Analytical characteristics of Charasoff’s and Dmitriev’s system
It is claimed in this paper that the issues listed in the introduction comprise Charasoff's originalcontributions to linear economic analysis. On the other hand, as a matter of fact, we acknowledge

12that some of those topics had been addressed by Dmitriev and Bortkiewicz. However, because of thedifference in their logical conditions, Charasoff's contributions cannot be absorbed into theachievement of his predecessors.
In general, the following achievements are considered as those of Dmitriev (see Nuti, 1974; Kurz &Salvadori, 1995; Gehrke, 1998; Kurz & Salvadori, 2000): determining the labour-value equation(Dmitriev, 1904/1974, pp. 44-45), formulating the price equation as the sum of the wage of dated
labour and profit (Dmitriev, 1904/1974, p. 49), determining the necessary and sufficient condition
for the equality between labour values and prices (i.e. the equality of ‘organic composition of capital’in all sectors 14 or r = 0) (Dmitriev, 1904/1974, pp. 55-56, 71-73), formulating the reciprocal relation
between wage and profit (Dmitriev, 1904/1974, p. 57), and determining the profit rate using priceequations for wage goods (Dmitriev, 1904/1974, p. 59-61, 73). In addition, as mentioned in Section
3, he could prove the ‘Fundamental Marxian Theorem’ implicitly (see Appendix A.6). Based on
Dmitriev’s price equations, Bortkiewicz was able to provide a counterexample to Marx’s formula ofproduction prices by distinguishing basic products from non-basic products and prove the rising rateof profit following technical progress (see Appendix A.7).
We can, however, identify the essential condition of Charasoff's linear economic system, i.e. thecondition of the system that distinguishes it from the system of his predecessors. It consists of the
postulate that the augmented input coefficients matrix has a simple Frobenius root that is the only
eigenvalue of maximum modulus. His assumption of basic and non-basic products in the simplified
form, i.e. (A.3), (A.4) and (A.5), fulfils the essential condition as shown in 1.2.2.
On the other hand, the essential condition of Dmitriev’s and Bortkiewicz’s System is included in the
postulate that the profit rate is lower than the so-called ‘maximal profit rate’ and the price of thewage basket is positive. That can be seen as follows. We know Dmitriev’s price equation (Dmitriev,
1904/1974, p.49) as∞

p = (1 + r ) pd ∑ lAt (1 + r )t
t=0

(6)

By post-multiplying both sides by d and cancelling them by pd, we obtain Dmitriev’s equation of
profit rate (Dmitriev, 1904/1974, p.60) as∞

1 = (1 + r )∑ lAt (1 + r )t d
t=0

(7)

For the derivation of both the basic equations (6) and (7) of Dmitriev, which were taken over by

13

Bortkiewicz (1907a, Formulas (20) and (30)), it must be postulated that the Frobenius root of(1 + r ) A is less than unity (in other words, r is less than the ‘maximal rate of profit’), and that pd is
positive. These postulates can be therefore seen as the essential condition of their system.
Now, we can show that Dmitriev’s and Bortkiewicz's assumption of the so-called ‘Austrian’ process
fulfils these postulates. For their linear economic analysis, they assume explicitly that a series of
dated labour is finite:‘ascending ever higher and higher to “production goods of higher orders” (the Productivgüter
höherer Ordnung of the theoreticians of marginal utility), let us finally arrive at a capital good(or capital goods) produced solely by current labour’ (Dmitriev, 1904/1974, pp. 53-54).
Just as suggested by the German words 'Productivgüter höherer Ordnung' and the reference to themarginal utility school, this property is named 'Austrian' by his reviewers (Kurz & Salvadori, 1995;
Gehrke, 1998; Kurz & Salvadori, 2000).
First, the ‘Austrian’ process, i.e. a finite series of dated labour, can be formulated as follows:n −1

A = 0 and l ∑ At > 0
n

t =0

(8)

Then, A = 0 implies that the Frobenius root of A, and therefore that of (1+r)A is nought. On the
n

n −1

n

n

t =0

t =1

t =1

t
t
t
other hand, l ∑ A > 0 implies that ∑ B = ∑ ( A + dl ) has positive rows for all wage goods,

which means that the wage goods are all basic products. If B is decomposable, B can be transformedby suitable simultaneous substitutions into the form shown in A.1.1, where B11 is indecomposable
and its columns involve all wage good sectors, and where

B22 ," , Bn0 n0

are null matrices. As we

can confirm, all semi-positive price vectors (left-side eigenvectors) with a well-defined profit rate
have positive components for all basic products including wage goods. Therefore, we obtain pd > 0.
Charasoff's essential condition and Dmitriev's (and Bortkiewicz's) do not coincide with each other;
therefore, the price equation of one system has an unsolvable problem that the other is able to solve,
as we can show by the numerical examples below. Correspondingly, the validity of the ‘Fundamental
Marxian Theorem’ that was proved explicitly by Charasoff and implicitly by Dmitriev and14

Bortkiewicz and the theorem of rising rate of profit is limited to the specific area for which eachsystem is able to provide a solution.
(i) Counterexample of Dmitriev’s and Bortkiewicz’s system
Set⎛1
A := ⎜ 4
⎝0

1
8
1
2

⎞
⎟ , l := (1 1) , and⎠⎛1⎞d := ⎜ 8 ⎟ . Then, we have
⎝0⎠

⎛3
B=⎜8
⎝0

1
4
1
2

⎞
⎟ and
⎠

⎛3 1⎞
⎛0 2⎞
B = 2 B = ⎜ 4 2 ⎟ . According to Charasoff’s procedure, we can show that lim B t = ⎜⎟ , and
t →∞
⎝0 1⎠
⎝0 1⎠
that (0 1) is the unique normalised semi-positive production price. The corresponding profit rate isunity (100%). On the other hand, Dmitriev’s equation of profit rate provides the unique solution r =5/3. His price equation, however, cannot provide any meaningful solution, i.e. we could either have p= (0 0) or p2 = ∞.
(ii) Counterexample of Charasoff’s system⎛0⎞⎛ 0 12 ⎞⎟ , l := (1 0 ) , and d := ⎜ 1 ⎟ . According to Dmitriev’s equation of profit rate, we
⎝0 0⎠
⎝2⎠

Set A := ⎜can show that the unique non-negative solution is r = 1. Then, his price equation provides the unique⎛0

normalised semi-positive production price p = (0.5 0.5). On the other hand, because B = ⎜ 1

⎝2

⎞
⎟
0⎠
1
2

⎛0 1⎞
t
⎟ , B is cyclic, and B does not converge; therefore, Charasoff’s
10⎝⎠

and B = 2 B = ⎜procedure cannot be applied to this example.
Even after relativising considerations mentioned above, we can indeed say as follows: Georg vonCharasoff modelled the production prices and the general rate of profit de facto using the eigenvector
and eigenvalue of the input matrix, and brought forward, on this analytical level, all significant
arguments that were destined to be discussed in the course of the ‘transformation problem’. In thissense, he can be seen as the initiator of mathematical Marxian economics. Considering that thePerron-Frobenius theorems were published no earlier than 1907/8, Charasoff excelled his
contemporaries, Dmitriev and Bortkiewicz, in the sense that he could present all the relevant results
on an algebraically advanced level on which just the following generation of linear production
theory was going to discuss.

15

Appendix
A.1. Proof of Proposition 1
Let x ≥ 0 be an arbitrary vector of goods. B t x is the ‘capital of the t-th order ’ of x according to
Charasoff’s notation. The normalisation of B t x is the ‘type’ of capital of the t-th order of x. Next,tnormalise B x, and let ( B t x ) i be the i-th component of the vector B t x . Then, we have:1n∑ ( B x)Bt x =

t

i =1

i

1
n

∑ (λ B x )
t
0

i =1

λ0t B t x = nt1∑ ( B x)

i

i =1

Bt x

(9)

t

i

We can see that the normalisation of B t x and B t x are identical. Because of (5), for t→∞, they
converge to the following limit:

1n∑ ( B x)i =1

B* x = n

*

1

∑v

i

i =1

v =: v*(10)iNote that the limit v* is the normalised right-side eigenvector of B associated with the Frobenius root,
and that it is independent of x. Therefore, the limit is common to all bundles of goods and is called
the ‘original type’ by Charasoff. (qed)

A.2.Proof of Lemma 1
The proof of Lemma 1 is provided in an analogous way to Proposition 1 because they can be seen as
dual problems. The procedure of normalisation and convergence in (9) and (10) is now applied to theaggregation of rows instead of columns of Bt in order to obtain:

lim wt = n
t →∞1∑ v jv =: v*

(11)

j =1

A3. Proof of Proposition 4
Choose a normalised price vector y0 > 0 arbitrarily (it can also be the labour-value vector). Let x > 0
be an arbitrary activity vector. With y0 being the current price system, y0Bj is the cost price of
product unit j, and y0(Ej − Bj)/ y0Bj is the individual profit rate of Sector j, where Bj and Ej are thej-th column of B and the unit matrix E, respectively. The individual profit rates need not be equalamong the sectors. Calculating their weighted average r0 according to the Marxian procedure, we

16have:
r0 = y0(E − B)x/y0Bx
In the first step of the transformation procedure, we substitute all individual profit rates with theaverage rate r0. We then obtain a new price vector y1. After normalisation, it can be written as
follows:
y1 = (1 + r0)y0B /(1 + r0)y0B ι = y0B/y0B ι
s.t. ι := (1, ...,1)’
With y1 being the current price system, the individual profit rates need not be equal among thesectors. Calculating the average r1, we have:
r1 = y1(E − B)x/y1Bx
In the second step, we substitute all individual profit rates with the average r1 to obtain a new(normalised) price system y2:

y2 = y1 B / y1 Bι = y0 B 2 / y0 B 2ι
By following the same procedure, we obtain the price vector yt in the t-th step as follows:

yt = yt − 1 B/ yt − 1 Bι = y0 B t / y0 B tι
According to (5), yt converges for t→∞ as follows:

lim yt = lim y0 λ0−t Bt / y0 λ0−t Btι = lim y0 B t / y0 B tι = v/vι = v*
t →∞

t →∞

t →∞

According to Lemma 1 and Proposition 2, this limit is a normalised production price. (qed)A.4. Proof of Lemma 2
Choose a price vector p > 0 arbitrarily. With p being the current price system, pBj is the cost price of
product unit j, and p(Ej − Bj)/ pBj is the individual profit rate of Sector j. Let rm and rM be the lowest
and highest individual profit rates, respectively. Then, we have:11
p <= pB <=p
1 + rM
1 + rm

(12)

From pv > 0, we obtain:1
1
pv <= pBv = λ0 pv <=pv
1 + rM
1 + rm1
1<= λ0 <=
1 + rM
1 + rm17

Because the general rate profit is r =1

λ0− 1 , we obtain:

rm <= r <= rM
In particular, if the inequality in (12) is strict for a positive component of v, we obtain:rm < r < rM(13) (qed)A.5. Proof of Proposition 5
If a new production method is able to lower the cost price of at least one component of the original
capital in terms of hitherto current prices, the individual profit rate of the sector producing it rises,
while the individual profit rates of the other sectors remain constant. Let p and r be the hitherto
current production prices and the hitherto current general rate of profit, respectively. Denote the newaugmented input matrix as Bˆ , the new right-side eigenvector of B̂ associated with the Frobenius
root as v̂ and the new general rate of profit as r̂ . Assume that v and v̂ have positive values forthe same sectors. By substituting B and rm in (12) with B̂ and r, respectively, the second inequalityin (12) is valid and strict for the sector where the technical change occurs. Then, the first inequalityin (13) is valid, which means that r < r̂ in the present terms. (qed)A.6. Fundamental Marxian Theorem by Dmitriev
By considering Dmitriev’s equation of profit rate (7) and the assumption of the ‘Austrian’ process(8), we can show the following inequality in the case of r > 0:
∞

∞

t=0

t =0

1 = (1 + r )∑ lAt (1 + r )t d > ∑ lAt d = l ( E − A) −1 d = wd > 0
On the other hand, r <= 0 implies that
∞

∞

t=0

t=0

1 = (1 + r )∑ lAt (1 + r )t d <= ∑ lAt d = l ( E − A) −1 d = wd
Therefore, we can establishr>0⇔

1 − wd>0
wd

A.7. The rising rate of profit by Bortkiewicz
Assume that measuring by hitherto current prices, a technical change lowers the cost price of at least
one wage good, while the other cost prices do not rise. Denote the new input coefficients matrix andlabour coefficients vector as Â and lˆ, respectively. In addition, assume that the assumption ofthe ‘Austrian’ process is valid for the new technology. According to Dmitriev’s price equation (6),18

we obtain:∞

At (1 + r )t d
1 > (1 + r )∑ lˆˆ
t=0Therefore, the new profit rate r̂ that satisfies the following equation of profit rate must be higher
than r.
∞ˆˆ t (1 + rˆ)t d
1 = (1 + rˆ)∑ lA
t=019

Notes1

On Charasoff's biography and contemporary reception, see Mori (2007).
2
See Egidi and Gilibert (1989); Kurz, 1989; Howard and King (1992); Kurz and Salvadori (1995,
1998, 2000); Egidi (1998); Stammatis (1999).
3
As critical comments to the literature on Charasoff, we refer to Mori (2007).
4
All propositions in Section 2 to 6 were first formulated in Mori (2007).
5
Prime applied to matrices and vectors denotes, as usual, their transposition.
6
To prove the equivalence of both the definitions, use the property of a non-negative
indecomposable n × n matrix M:n∑M > 0.
tt =1

7

In Dmitriev’s system, the profit rate is determined in the subsystem to which all wage goods belong.
However, this subsystem does not need to be that of basic products. In a supplementary example
where his basic assumption of the ‘Austrian’ process is suspended, Dmitriev himself illustrated astate in which the economy can be divided into two separate subsystems, in one of which the wage
goods are used neither directly nor indirectly. See Dmitriev (1974, pp. 66-9).
8
To be precise, this means the power sequence of the augmented input coefficient matrix divided by
its Frobenius-root. By the normalised augmented input matrix, we mean the augmented input
coefficient matrix divided by its Frobenius-root. We consider this terminology valid for the rest of
this paper.
9
To be sufficient for all propositions, pν > 0, pd > 0 and wv > 0 must be additionally postulated.
10
Proposition 2, Corollary 1, Corollary 2 and Corollary 3 follow obviously from Proposition 1 andLemma 1 because the vector of the dimensions v * and the original type v* are left-side andright-side eigenvectors of B associated with the Frobenius root, respectively (see (10) and (11) inAppendix A.1 and A.2).
11 The proof of Proposition 3 can be also provided in a usual manner: i.e. considering that the gereralrate of profit is determined as r =1

λ0− 1 , and using the equation λ0 v = Bv and the value

equation w = wA + l, where w > 0 is the vector of labour values (whose existence is proved in the
proof of Proposition 6 given below). Because w > 0, v ≥ 0, l > 0, d ≥ 0 and therefore wdlv > 0 , weobtain:r > 0 ⇔ 1 − wd > 0 ⇔12

1 − wd>0 .
wd

Maurice Potron, a French mathematician, proved de facto Fundamental Marxian Theorem 48

years earlier than Morishima, Seton and Okishio by adapting the Perron-Frobenius theorems toeconomic problems, and he proved it by considering heterogeneous labours 65 years earlier and evenmore generally than Bowles and Gintis (1977, 1978). See Potron (1913) and Mori (2008).
On the other hand, in his example, where the basic assumption of ‘Austrian’ process is suspended,
Dmitriev stated that the profit could exist without any labour input in the economy (Dmitriev, 1974,
pp. 63-66). This amounts to an invalidation of the theorem.
14
According to Dmitriev’s definition, the organic composition of capital is equal in all sectors if andonly if lAn (n = 0,1,...,) are linearly dependent in pairs.
13

20

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