All but one: How pioneers of linear economics overlooked Perron-Frobenius mathematics
In the period 1907-1912 the German ‘pure mathematicians’ Oskar Perron and Georg Frobenius developed the fundamental results of the theory of nonnegative matrices. Today Perron-Frobenius mathematics enjoys wide applications in many fields, for example in economics, probability theory, demography and even in Google’s ranking algorithm. In linear economic models of the Leontief-Sraffa type it is often the crucial tool to solve many mathematical economic problems. My paper concentrates on the history of Perron-Frobenius in linear economics, and some related stories. In the 1910s and 1920s, several pioneering publications in linear economics could have benefited from applying Perron-Frobenius results, but failed to do so, even the economic publications authored by the mathematicians Georg Charasoff, Hubert Bray and Robert Remak. Either they didn’t know Perron-Frobenius, or they didn’t realize its usefulness. The only exception was the French Jesuit mathematician Maurice Potron, who used Perron-Frobenius mathematics in the core of his economic model, in many of his writings, as early as 1911. He constructed a sort of disaggregated open input-output system, formulated duality theorems between his quantity system and his price system, and anticipated the Hawkins-Simon conditions. Potron’s economic or mathematical contemporaries didn’t recognize his originality. A general treatment of Charasoff’s economic system needs Perron-Frobenius mathematics, especially Perron’s Limit Lemma. Although some of Charasoff’s mathematical interests (irreducibility, continued fractions) were close to those of Perron or Frobenius, the theory of nonnegative matrices is never explicitly used in Charasoff’s work. It is doubtful whether Charasoff knew the relevant matrix theorems. Probably he just assumed that the properties of his numerical examples with three commodities also hold in the general case with n commodities. Frobenius had been Remak’s doctoral supervisor in 1911. After a forgotten non-mathematical paper in 1918, on the repayment of the national debt, Remak presented his mathematical system of ‘superposed prices’ in 1929, twelve years after Frobenius’ death. With suitable units of measurement, Remak’s system can be handled by Perron-Frobenius tools. However, Remak failed to normalize his units, and provided lengthy proofs of his own. Moreover, he spent most of his mathematical efforts on freak systems in which the most important commodities have zero prices. A few years earlier, in 1922, Bray also had overlooked Perron-Frobenius in a mathematically similar model that studied Cournot’s equations of currency exchange. Contrary to Dorfman’s well-known article on Leontief’s Nobel Prize in 1973, I provide archival evidence that Leontief knew Remak’s results already in the early 1930s, before he submitted a paper containing ideas of input-output theory to Keynes for the Economic Journal in 1933. Keynes quickly rejected Leontief’s paper; a few months later Leontief submitted it to Frisch for Econometrica. Frisch formulated a lot of critical remarks on Leontief’s first and revised version in 1933-34. In the light of this criticism, it is highly probable that Leontief simplified and linearized his mathematics, and a few years later he finally started publishing his Nobel Prize winning empirical and theoretical results in American journals. Just like Leontief, Sraffa started related research in the late 1920s. He didn’t discuss his mathematical problems with competent economic colleagues in Cambridge, nor with the specialists of the Econometric Society, but preferred mathematical help from three non-economists: Ramsey, Watson and especially Besicovitch. I suggest that Besicovitch in his early mathematical research in Russia ‘came close’ to Perron-Frobenius results, but it is well-known that he didn’t know Perron-Frobenius, and tried to invent his own proofs for Sraffa in the 1940s. In the first half of the twentieth century, abstract algebra started to flourish and became a more prestigious and widely researched subject than the ‘old-fashioned’ Perron-Frobenius matrices. In this context, it is less surprising that for many decades even the mathematicians (except Potron) overlooked the usefulness of Perron-Frobenius in linear economics. Results, connections or applications that seem evident after the fact, were not obvious to the original pioneers.
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- Christian Gehrke & Heinz Kurz, 2002. "Keynes and Sraffa's 'Difficulties with J. H. Hollander'," The European Journal of the History of Economic Thought, Taylor & Francis Journals, vol. 9(4), pages 644-671.
- Olav Bjerkholt & Mark Knell, 2006. "Ragnar Frisch and the Origin of Input-Output Analysis," Economic Systems Research, Taylor & Francis Journals, vol. 18(4), pages 391-410.
- Georg Stamatis, 1999. "Georg Charasoff: A Pioneer in the Theory of Linear Production Systems," Economic Systems Research, Taylor & Francis Journals, vol. 11(1), pages 15-30.
- Kurz,Heinz D. & Salvadori,Neri, 1997.
"Theory of Production,"
Cambridge University Press, number 9780521588676.
- Kurz,Heinz D. & Salvadori,Neri, 1995. "Theory of Production," Cambridge Books, Cambridge University Press, number 9780521443258, February.
- William Baumol & Thijs ten Raa, 2009. "Wassily Leontief: In appreciation," The European Journal of the History of Economic Thought, Taylor & Francis Journals, vol. 16(3), pages 511-522.
- Amartya Sen, 2003. "Sraffa, Wittgenstein, and Gramsci," Journal of Economic Literature, American Economic Association, vol. 41(4), pages 1240-1255, December.
- Samuelson, Paul, 2012. "Understanding the Marxian Notion of Exploitation: A Summary of the So-CalledTransformation Problem Between Marxian Values and Competitive Prices," Economic Policy, Russian Presidential Academy of National Economy and Public Administration, pages 182-202, August.
- Samuelson, Paul A, 1971. "Understanding the Marxian Notion of Exploitation: A Summary of the So-Called Transformation Problem Between Marxian Values and Competitive Prices," Journal of Economic Literature, American Economic Association, vol. 9(2), pages 399-431, June.
- Kenji Mori, 2008. "Maurice Potron'S Linear Economic Model: A De Facto Proof Of 'Fundamental Marxian Theorem'," Metroeconomica, Wiley Blackwell, vol. 59(3), pages 511-529, 07.
- Redlich, Fritz, 1957. "Academic Education for Business: Its Development and the Contribution of Ignaz Jastrow (1856–1937): In Commemoration of the Hundredth Anniversary of Jastrow's Birth," Business History Review, Cambridge University Press, vol. 31(01), pages 35-91, March.
- J. E. Stiglitz, 1970. "Non-Substitution Theorems with Durable Capital Goods," Review of Economic Studies, Oxford University Press, vol. 37(4), pages 543-553.
- J. Jastrow, 1923. "The New Tax System of Germany," The Quarterly Journal of Economics, Oxford University Press, vol. 37(2), pages 302-341.
- J. Jastrow, 1920. "The German Capital Levy Tax," The Quarterly Journal of Economics, Oxford University Press, vol. 34(3), pages 462-472.
- Pollitt, Brian H, 1988. "The Collaboration of Maurice Dobb in Sraffa's Edition of Ricardo," Cambridge Journal of Economics, Oxford University Press, vol. 12(1), pages 55-65, March.
- Takayama,Akira, 1985. "Mathematical Economics," Cambridge Books, Cambridge University Press, number 9780521314985, February.
- Heinz Kurz & Neri Salvadori, 2000. "'Classical' Roots of Input-Output Analysis: A Short Account of its Long Prehistory," Economic Systems Research, Taylor & Francis Journals, vol. 12(2), pages 153-179.
- Bjerkholt, Olav, 2014. "Promoting Econometrics through econometrica 1933-39," Memorandum 28/2013, Oslo University, Department of Economics.
- Maria Cristina Marcuzzo & Nerio Naldi & Eleonora Sanfilippo & Annalisa Rosselli, 2008. "Cambridge as a Place in Economics," History of Political Economy, Duke University Press, vol. 40(4), pages 569-593, Winter.
- Christian Bidard & Guido Erreygers & Wilfried Parys, 2009. "'Our daily bread': Maurice Potron, from Catholicism to mathematical economics," The European Journal of the History of Economic Thought, Taylor & Francis Journals, vol. 16(1), pages 123-154.
- BIDARD, Christian & ERREYGERS, Guido & PARYS, Wilfried, 2007. "“Our daily bread”: Maurice Potron, from Catholicism to mathematical economics," Working Papers 2007007, University of Antwerp, Faculty of Applied Economics.
- Arrigo Opocher, 2006. "Exchange and Production Equations: Understanding a Debate between Harrod and Sraffa," History of Political Economy, Duke University Press, vol. 38(1), pages 175-184, Spring.
- J. A. Mirrlees, 1969. "The Dynamic Nonsubstitution Theorem," Review of Economic Studies, Oxford University Press, vol. 36(1), pages 67-76.
- Bidard,Christian, 2011. "Prices, Reproduction, Scarcity," Cambridge Books, Cambridge University Press, number 9780521180191, February.
- Bidard,Christian, 2004. "Prices, Reproduction, Scarcity," Cambridge Books, Cambridge University Press, number 9780521472838.
- W. W. Leontiep, 1946. "Exports, Imports, Domestic Output, and Employment," The Quarterly Journal of Economics, Oxford University Press, vol. 60(2), pages 171-193.
- E. Roy Weintraub, 1987. "Stability Theory via Liapunov's Method: A Note on the Contribution of Takuma Yasui," History of Political Economy, Duke University Press, vol. 19(4), pages 615-620, Winter.
- P. Klyukin., 2008. "Towards the G. von Charasoff’s Legacy as a Crucial Point in Evolution of Russian Tradition Based on the Circular Flow Concept," VOPROSY ECONOMIKI, N.P. Redaktsiya zhurnala "Voprosy Economiki", vol. 2.
- Christian Bidard & Guido Erreygers, 2007. "Potron and the Perron-Frobenius Theorem," Economic Systems Research, Taylor & Francis Journals, vol. 19(4), pages 439-452.
- Gilbert Abraham-Frois & Emeric Lendjel, 2006. "Father Potron's Early Contributions to Input-Output Analysis," Economic Systems Research, Taylor & Francis Journals, vol. 18(4), pages 357-372.
- Oskar Morgenstern, 1959. "International Financial Transactions and Business Cycles," NBER Books, National Bureau of Economic Research, Inc, number morg59-1, June. Full references (including those not matched with items on IDEAS)
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