Charasoff and Dmitriev: An analytical characterisation of origins of linear economics

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 of the ‘transformation problem’. This paper aims to reformulate his significant arguments in a formal manner and to reveal their logical relationship by reproducing the mathematical reasoning, so that the logical characteristic of his system can be identified in comparison with Dmitriev's and Bortkiewicz's linear economic system.

Keywords:

• Charasoff
• Dmitriev
• Bortkiewicz
• Frobenius root
• Fundamental Marxian Theorem

Notes

¹On Charasoff's biography and contemporary reception, see Mori (2007, 118–41)

²See Egidi and Gilibert (1989, 59–74), Kurz (1989, 11–61), Howard and King (1992), Kurz and Salvadori (1995; 1998, 25–56; 2000, 153–79); Egidi (1998, 96–100); Stamatis (1999).

³As critical comments to the literature on Charasoff, we refer to Mori (2007, 118–41)

⁴All propositions in section 3 to 7 were first formulated and commented in Mori (2007, 118–41)

⁵Prime applied to matrices and vectors denotes, as usual, their transposition.

⁶To prove the equivalence of both the definitions, use the property of a non-negative indecomposable n × n matrix M: Σⁿ _t=1 M ^t >0.

⁷In 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 a state 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, 66–9).

⁸To be precise, this means the power sequence of the augmented input coefficients matrix divided by its Frobenius root. By the normalised augmented input matrix, we mean the augmented input coefficients matrix divided by its Frobenius root. We consider this terminology valid for the rest of this paper.

⁹To be sufficient for all propositions, pν > 0 and pd > 0 would have to be then additionally postulated.

¹⁰Proposition 2, Corollary 1, Corollary 2 and Corollary 3 follow obviously from Proposition 1 and Lemma 1 because the vector of the dimensions and the original type v* are left-side and right-side eigenvectors of B associated with the Frobenius root, respectively (see Equations (10) and (11) in Appendix A.1 and A.2).

¹¹The proof of Proposition 3 can be also provided in a usual manner: i.e. considering that the general rate 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 Proposition 6 given below). Because w > 0, v ≥ 0, l > 0, d ≥ 0 and therefore wdlv > 0, we obtain: r > 0 ⇔ 1−wd > 0 ⇔ (1- wd)/wd > 0.

¹²Maurice Potron, a French mathematician, proved de facto Fundamental Marxian Theorem 48 years earlier than Morishima, Seton and Okishio by adapting the Perron-Frobenius theorems to economic problems, and he proved it by considering heterogeneous labours 65 years earlier and even more generally than Bowles and Gintis (1977, 173–92; 1978, 311–4). See Potron (1913, 53–76) and Mori (2008, 511–28).

¹³On the other hand, in his example, where the basic assumption of ‘Austrian’ process is suspended, Dmitriev (1974, 63–6) stated that the profit could exist without any labour input in the economy. This amounts to an invalidation of the theorem.

¹⁴According to Dmitriev's definition, the organic composition of capital is equal in all sectors if and only if lA ⁿ (n = 0, 1, … ) are linearly dependent in pairs.