Why didn't Charasoff and Remak use Perron–Frobenius mathematics?

Abstract

At the beginning of the twentieth century the German mathematicians Perron and Frobenius published their powerful theorems on non-negative matrices. For many decades these tools were overlooked by all pioneers of linear economics (except Potron in France). I concentrate on Charasoff and Remak, the two pioneers in the German-language literature. Both were mathematicians, but both failed to use Perron–Frobenius mathematics in their economics. I discuss possible reasons for this neglect, and I also draw attention to the communication between different protagonists, the connection between Perron's forgotten Limit Lemma and Charasoff's economics, Remak's bizarre prices, and some interesting archival material.

Keywords:

• Perron–Frobenius
• Charasoff
• Remak
• Leontief
• Sraffa

JEL Classification:

• B3

Acknowledgements

I would like to thank my discussant Arrigo Opocher and other participants of the 2013 ESHET Conference in London for useful comments on a longer version of this paper. I also benefited from remarks by participants of the January 2013 colloquium on ‘The Pioneers of Linear Models of Production’, University of Paris Ouest, Nanterre. Special thanks are due to Thomas Hawkins for his stimulating communications on Perron–Frobenius since 2005, and for his comments on my January 2013 text. I am also grateful to Christian Bidard, Olav Bjerkholt, Nerio Naldi, Hans Schneider, the late Paul Samuelson, and two very competent anonymous referees. Last but not least, I want to express my gratitude to the very helpful staff of various archives (see more details in footnotes of my paper).

Notes

1 Good introductions to Perron–Frobenius in linear economics are supplied by Kurz and Salvadori (1995) and Bidard (2004).

2 See the Perron bibliography by Frank (1982), and the authoritative book on Frobenius by Hawkins (2013).

3 See Bidard et al. (2009) for more details on Potron.

4 Vectors and matrices are called non-negative if all their elements are non-negative; positive if all their elements are positive. A non-negative matrix or vector with at least one non-zero element is often called semipositive.

5 He signs his first economic book in 1909 as Dr. Georg von Charasoff, but in his other publications, and in all his letters that I have seen, he dropped his ‘von’.

6 Several extracts from his two books were later reprinted in Die Aktion or in Der Gegner: one extract even in both journals around the same time (Charasoff 1920).

7 For many years, biographical details on Charasoff were scarce. Recently, more information became available, thanks to Klyukin (2008) and Gehrke (2013). The latter presents a wealth of new material on Charasoff's activities in Heidelberg, Zurich, and Lausanne, and also on his later years in Tbilisi, Baku, and Moscow.

8 On Gumbel and Lederer in Heidelberg, see Brintzinger (1996). The Gumbel biography by Brenner (2001, pp. 32–4) described how the pacifist Georg Friedrich Nicolai tried to resume his university lectures in Berlin in 1920, but was shouted down by hundreds of demonstrators. When the Faculty Senate accused Nicolai of desertion and treason, only three persons signed a letter to protest against Nicolai's banning from the university: the publisher Curt Thesing, Emil Gumbel, and Otto Buek. Note that Buek was Charasoff's most important helper, the only person that received thanks in the preface of Charasoff's (1909) book. Gumbel is not an important innovator in the present story, but in the 1930s Gumbel seemed to be the first scholar who came into contact with the works of four pioneers of linear economics: Charasoff (see Klimpt's dissertation), Remak (see Section 3.3), Leontief (see again Section 3.3), and Potron (who presented his economic model at the International Congress of Mathematicians in Oslo in 1936, in the same session as Gumbel).

9 Consider, for example, the equation x² − 5 = 0. It is called reducible if its left-hand side (a polynomial of the second degree) can be ‘reduced’ to polynomials of lower degree. If we consider all real numbers, this is possible, because x² − 5 = 0 can be written as , but if we consider only rational numbers, the equation is irreducible. More sophisticated cases of irreducibility exist, of course, for example, in the differential equations in Charasoff's dissertation.

10 After Hilbert's rejection, it remained unpublished and unfound. The name Charasoff is not mentioned in the comprehensive bibliographies on continued fractions compiled by Wölffing (1908) and Brezinski (1991).

11 See Nachlass David Hilbert, Cod. Ms. D. Hilbert 59, Abteilung Handschriften und seltene Drucke, Niedersächsische Staats- und Universitätsbibliothek, Göttingen. I thank Dr Hunger and his staff for providing me copies of the Hilbert correspondence with Charasoff and Remak.

12 In 1905 Charasoff enrolled as an ‘auditor’ for Heinrich Burkhardt's lectures on elliptic functions at the University of Zurich (Gehrke 2013, p. 17), one of the specialties of Georg Frobenius.

13 For more historical details on the mathematics of Perron–Frobenius, see Hawkins (2008, 2013).

14 Egidi and Gilibert (1984) presented the first sound mathematical treatment of Charasoff's general case, and attributed the limit propositions to Nikaido (1968). Neither Nikaido nor Egidi and Gilibert mentioned Perron's Limit Lemma of 1907.

15 On the one hand, the word ‘Gleichung’ (equation) is sometimes used by Charasoff in unexpected places. On the other hand, Charasoff failed to show us explicitly the simple system of linear equations that was used a few years earlier by the non-mathematicians Mühlpfordt or Dmitriev to determine the labour values.

16 Divide A by its dominant eigenvalue λ = 0.8, and obtain the matrix . For the sequence converges to a limit matrix that is proportional to the matrix A^∞.

17 I add the following to avoid criticism from mathematical readers: Perron's (1907b) results can be directly applied to the core of Charasoff's system, i.e. the subeconomy of his Grundprodukte, here the 2 × 2 submatrix of corn and bread; however, I handle the complete 3 × 3 matrix A here, because this approach is mathematically allowed by a theorem of Egidi (1992, pp. 249–50). For more details, see Parys (2013).

18 This journal was founded in Berlin in 1916, and its weekly issues contained many contributions on political and economic problems, written by among others Edgar Jaffé (the editor), Carl Ballod, Eduard Bernstein, Lujo Brentano, Karl Diehl, Robert René Kuczynski, Adolf Löwe, Otto Neurath, Franz Oppenheimer, Max Weber, etc.

19 Numerous stories on Remak's unusual behaviour exist. See, for example, the memories by Pinl (1969), Behnke (1978), and Fenchel (1980). More references can be found in Merzbach (1992), Hagemann and Punzo (2007), and Siegmund-Schultze (2009).

20 David Hilbert Nachlass, Cod. Ms. D. Hilbert 326, Göttingen (I translated from German).

21 At the start of his argument, Remak assumes a system with n different individuals, each producing a different commodity, but we can also interpret his equations as a system with n different sectors.

22 Such a normalisation is a routine operation in modern economics; see, for example, the Remak system in Gale (1960, p. 261) or Kurz and Salvadori (1995, p. 398).

23 Another eigenvalue system is Fx = x, with F = D⁻¹T, but in general F is not stochastic.

24 Some of these submatrices on the main diagonal can be 1 × 1 matrices (zero or non-zero). I follow the same tradition as Hawkins (2008, p. 686) and I call every 1 × 1 matrix irreducible.

25 Observe the different terminology of Charasoff and Remak. In a system with basics and non-basics, Remak's ‘highest’ groups never contain basics, whereas Charasoff's Urkapital (his capital of an infinitely ‘high’ order) contains all the basics.

26 My simple corn–banana–bean example assumes that some ‘own’ input coefficients are equal to one, but it is possible to construct examples of zero prices in Remak systems where all input coefficients are smaller than one. See Parys (2013) for a 7 × 7 matrix example, related to Gale (1960, pp. 267–71), with two basics and five non-basics, forming one group of basics and two groups of non-basics. Note that Remak and Gale did not use the notion of a basic commodity (or something similar), and they raised no alarm about the bizarre economic content of the zero prices in their systems. Some interesting mathematical results about basics and non-basics in a no-surplus economy are given by Opocher (2006).

27 Bray (1922) studied Cournot's linear equations of currency exchange. Bray (from Rice Institute, Houston) is another example of a mathematician who neglected Perron–Frobenius in his economics. See Parys (2013).

28 Wassily Leontief Papers, Harvard University Archives, Accession 12255, Box 10, Folder: Ancient Correspondence 1930–1932 (I translated from German).

29 Leontief often preferred to work out his economic theories on his own. After visiting Harvard in 1933, Richard Kahn wrote to Joan Robinson that Leontief was really brilliant and doomed to isolation (Rosselli 2005, p. 265).

30 Note that the important publications by Bortkiewicz on Marx were already finished before the Perron–Frobenius story started.

31 Remak claimed that high profits depress the purchasing power of the workers, and lead to a sort of underconsumption. I doubt whether Remak saw the following problem: suppose equal value composition of capital and uniform rates of profit exist in all sectors, then relative prices would not be changed by high or low rates of profit (of course, the income distribution would be different).

32 See Kurz and Salvadori (2001, p. 264) and Salvadori (2011).

33 I owe thanks to Jonathan Smith, archivist at Trinity College, Cambridge, and to John Eatwell, the literary executor of the Sraffa Papers.

34 I refer to Parys (2013) for more details.