Applying mathematics to economics according to Cournot and Walras

ABSTRACT

This article examines the reasons that led Cournot to refuse Walras's request that he writes an article defending mathematical economics. From a reading of Cournot's works on philosophy and economics we show three reasons which explain the Cournot's refusal. First, Cournot does not attach the same importance to the theorems proposed by Walras. Second, these theorems enable Walras to defend an economic system that he considers to be truer than any other while Cournot believed that the economy could be subject to various forms of mathematical representations. Third, Cournot does not refer to the same conception of mathematics of Walras.

KEYWORDS:

• History of economic analysis
• epistemology
• mathematics
• Cournot
• Walras

JEL CODES:

• B13
• B16
• B40
• B41

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ Fisher (1898), Moore (1905), Roy (1933, 1936), Nichol (1938), Schumpeter (1954), Theochatris (1961), Clifford and Ekelund (1971), Jorlan (1978), Ménard (1980), Rhodes (1978), Guitton (1978), Mathiot (1984), Zilberberg (1990), Vatin (2000), Friedman (2000), Negishi (2001), Ekelund and Hébert (2002), Lallement (2000, 2004), Dos Santos Ferreira (2004, 2006), Sigot (2005).

² Walras and Cournot exchanged 11 letters between 12 August 1873 and 10 February 1875.

³ The article is archived IS 1927 by the Fond Walras de la Bibliothèque Cantonale de l'Université de Lausanne with the publication date on the brochure of 1875. Walras gave up submitting the text to the Journal des Économistes, and was rejected also by the Revue des Deux-Mondes and the Journal des actuaires; it was not published until 1876 in the Giornale degli Economisti. The difficulties encountered by Walras while attempting to get his article published in a French journal are testimony to his foresight in requesting Cournot's assistance to carry out this project.

⁴ In 1874, supporters of mathematical economics faced opposition from French liberal economists even more than during the first half of the nineteenth century which explains why Walras asked Cournot did this.

⁵ The goal of the 2011 article was to examine the methodological differences between Walras and Cournot in order to show why the author of Researches did not abandon the use of mathematics after 1836. We also discuss the differences between the two authors regarding the theory of rent, free trade and protectionism, the theory of value or competition. Our objective was to show why Walras’ criticism of Cournot with regard to the previous concepts did not persuade this author to support Walrasian mathematical economics. In this work, we do not discuss this viewpoint. We delve further into certain arguments and highlight others by focusing more on the exegesis and chronology of Walras’ works to explain why Cournot refused to write an article in support of using mathematics. We show why Walras’ mathematical exchange theory is essential to the construction of his general equilibrium model. We assess the implications and methodological foundations regarding articulation in a single triptych, Eléments d’économie politique pure, Etudes d’économie sociale and Etudes d’économie appliquée. We thus demonstrate that Cournot gives barely any place to the exchange theory. However, we stress the arguments that Walras develops in the three “Tentatives” of “Applying mathematical methods to political economics” that he devotes to the theory of exchange and that of supply and demand in which he refers to Cournot. The three “Tentatives” attest to the importance of mathematising exchange as the keystone of Walras’ pure economics model and, beyond that, his entire economics system. This argument is put in perspective with other writings in which Walras criticises Cournot. Finally, we reflect on the epistemological foundations that Walras borrows from Vacherot's philosophy as he refers differently to Cournot (that which was not considered in the previous article).

⁶ Cournot, Letters to Walras, from 3 September 1873 and 23 August 1874, Jaffé 1965, Vol. 1, 331 and 423.

⁷ The term “geometrician” was the terminology used to describe mathematicians in France at that time.

⁸ Alcouffe and Freyssé (1992).

⁹ Application des mathématiques à l’économie politique (2nd tentative), in (Walras 1893, 341–359).

¹⁰ La théorie mathématique de la richesse sociale (1883) contains seven dissertations, the first four were published in 1877. Walras refers to Cournot in the first dissertation. It was read in 1873 at the Académie des sciences morales et politique and reproduced in 1874 in the Comptes rendus des séances from this Académie (Walras 1993).

¹¹ Walras dedicated the 16th lesson of the Elements to the “Exposition et réfutation des doctrines d'A. Smith et de J-B. Say sur l'origine de la valeur d’échange” (Elements, 2–4 eds.). This lesson corresponds to the 27th in the first edition; it was followed, in 1874, by a lesson dedicated specifically to the Examen critique de la doctrine de M. Cournot sur les changements de valeur, absolus et relatifs (28th lesson) (Walras 1874, 255–260).

¹² This reasoning is repeated in Researches, Principles, and Review. It has its origins in Cournot's translation of A Treatise on Astronomy by John Hershel, a portion of which is repeated in L'exposition de la théorie des chances et des probabilités (Cournot 1843). This is additional proof that Cournot did not abandon his references to Newton's mechanics after 1838 to develop a measurement standard for prices.

¹³ It is lesson 28 in the first edition of Elements. Walras had proposed a first version of Elements to the Département de l'instruction publique et des Cultes of the Canton of Vaud for publication in January 1873. Department representative Charles Estoppey wrote to Walras on 18 January 1873, that he “agreed to have 100 copies of his Traité d’économie politique pure published” (Jaffé 1965, Vol. 1, 317–318). Walras apparently sent the manuscript on 15 January 1873 (Jaffé 1965, 315) so it was most likely written in 1872 (Walras [1872] 1992, Vol. XI, 19–24).

¹⁴ Walras confirmed the above criticism in 1888, in his Théorie mathématique de la richesse sociale: “The mathematical theory of exchange is finished. It allows the rebuttal of popular doctrines on the origin of value (27th lesson), and criticism of one of Mr. Cournot's important doctrines on changes in absolute and relative value” (Walras 1883, 493).

¹⁵ Walras, “Application des mathématiques à l’économie politique” (1st tentative, 1860), (2nd tentative, 1869–70), (3rd tentative ‘said’ “La bonne”, 1871) (Walras 1993, 342–409).

¹⁶ W. Jaffé proposed a comment on Walras's Tentative(s) (Letter from Walras to Ruchonnet, 6 September 1870, Jaffé 1965, note 33, Vol.1, 147–148).

¹⁷ Les Principes mathématiques de l’échange was read to the Académie des sciences morales et politiques. It was summarised and edited in the April 1874 edition of the Journal des économistes in 1873, and then in the Bulletin de la société vaudoise des sciences naturelles in 1877.

¹⁸ Lesson 11 of the definitive edition of Elements (Walras ([1874] 1988)).

¹⁹ Lesson 12 of the definitive edition of Elements (Walras ([1874] 1988)).

²⁰ Lesson 6 of the definitive edition of Elements (Walras ([1874] 1988)).

²¹ Walras refers to Cournot on these themes. Cf. Elements (Walras, 1874, lessons 41 and 28); “Economie et mécanique” (Walras, 1908, 333); “Une branche nouvelle des mathématiques” (Walras, 1876, 292, 321–323).

²² Lessons 15, 19 and 22 in the first edition of Elements, to which Cournot had access.

²³ Esquisse d'une doctrine économique et sociale was published for the first time in 1898 in the Gazette de Lausanne (Walras, Etudes d’économie politique appliquée, Foreword from the publishers; Dockès alii are: P., Goutte P-H., Hébert C., Mouchot C. Potier, J-P and Servet, J-M. 1992, XIX and XX).

²⁴ It is recognized that Walras borrowed his methodology from the philosophy of Etienne Vacherot (1858) [Arena and Ragni (1994), Dockès (1996), Lallement (1997), Potier (1998), Rebeyrol (1999), Baranzini and Tatti (2002), Baranzini and Bridel (2005)]. On Walras's use of mathematics, see Jolink and Van Dall (1989, 1993).

²⁵ Cf. Walker (1996), Baranzini (2011). Mäki (1998) uses the term realism to designate his philosophical meaning, and the term realisticness to designate common meaning.

²⁶ For more on the normative aspects of Walras's philosophy, see Jaffé (1977) and Koppl (1995).

²⁷ Cournot is referring to an “idealist” concept of mathematics (Barrow, 1992), according to which mathematical objects exist outside the mind and are the result of an imperfect discovery process.

²⁸ Letter from Walras on 27 July 1898 to Penjo (Jaffé, 1965, Vol. 3, 25–26).

²⁹ Walras had certainly not read La métaphysique et la science since he does not quote this work when he mentions that La métaphysique et la science is on his bedside table (Walras ([1898] 1992), 413).