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.