Léon Walras

In 1874 and 1877 Walras published Éléments d'économie politique pure (1899, 4th ed.; 1926, éd. définitive), in English, Elements of Pure Economics (1954), trans. William Jaffé.

That work that led him to be considered the father of the general equilibrium theory. The problem that Walras set out to solve was one presented by A. A. Cournot, that even though it could be demonstrated that prices would equate supply and demand to clear individual markets, it was unclear that an equilibrium existed for all markets simultaneously. Walras's law implies that the sum of the values of excess demands across all markets must equal zero, whether or not the economy is in a general equilibrium. This implies that if positive excess demand exists in one market, negative excess demand must exist in some other market. Thus, if all markets but one are in equilibrium, then that last market must also be in equilibrium.

While teaching at the Lausanne Academy, Walras began constructing a mathematical model that assumes a “regime of perfectly free competition”, in which productive factors, products, and prices automatically adjust in equilibrium. Walras began with the theory of exchange in 1873 and then he proceeded to map out his theories of production, capitalization and money in his first edition. His theory of exchange began with an expansion of Cournot's demand curve to include more than two commodities, also realizing the value of the quantity sold must equal the quantity purchased thus the ratio of prices must be equal to the inverse ratio of quantities. Walras then drew a supply curve from the demand curve and set equilibrium prices at the intersection. His model could now determine prices of commodities but only the relative price. In order to deduce the absolute price, Walras could choose one price to serve as a unit of account, coined by Walras as the numeraire and state all other prices in units of this commodity. The term numeraire, meaning unit of account, has become part of the international vocabulary of economics and for many economists, the only French word they know. Using this numeraire he determined that marginal utility, or rarete, divided by the price must be equal for all commodities. Walras felt that because the value of what an individual consumer consumes is equal to the value of that individual's stock of goods, that the aggregate, the value of total sales must equal the value of total purchase, must hold true. This became known as Walras’ Law which held that equilibrium equations can be derived from the others until only m-1 equations in the m-1 relative prices remain. Walras then expanded the theory to include production with the assumption of an existence of fixed coefficients in said production making possible a generalization that the marginal productivity of the factors of production varied with the amount of input, making factor substitution possible.

Walras constructed his basic theory of general equilibrium by beginning with simple equations and then increasing the complexity in the next equations. He began with a two-person bartering system, then moved on to the derivation of downward-sloping consumer demands. Next he moved on to exchanges involving multiple parties, and finally ended with credit and money.

Walras wrote down four sets of equations—one representing the quantity of goods demanded, one relating the prices of goods to their costs of production, one for the quantities of inputs supplied, and one showing the quantities of inputs demanded. There are four sets of variables to solve for, namely, the price of each good, the quantity of each good sold, the price of each factor of production, and the quantity of each of those factor bought by businesses. To simplify matters, Walras added one further equation to his model, requiring that all the money received must be spent, one way or the other. But there are now more equations than unknowns. From the theory of equations, one learns that a necessary but insufficient condition for the existence of a unique solution to a system of equations is that the number of equations must equal the number of variables. Walras tackled this problem by selecting an arbitrary good, G1, whose price is designated as the standard against which the prices of the other goods shall be compared. The system of equations can now be solved for the prices of all goods in terms of G1, though not for the absolute price levels.[9]

The crucial step in the argument was Walras's law which states that any particular market must be in equilibrium, if all other markets in an economy are also in equilibrium. Walras's law hinges on the mathematical notion that excess market demands (or, inversely, excess market supplies) must sum to zero. This means that, in an economy with n markets, it is sufficient to solve n-1 simultaneous equations for market clearing. Taking one good as the numéraire in terms of which prices are specified, the economy has n-1 prices that can be determined by the equation, so an equilibrium should exist. Although Walras set out the framework for thinking about the existence of equilibrium clearly and precisely his attempt to demonstrate existence by counting the number of equations and variables was severely flawed: it is easy to see that not all pairs of equations in two variables have solutions. A more rigorous version of the argument was developed independently by Lionel McKenzie and the pair Kenneth Arrow and Gérard Debreu in the 1950s.

A significant part of the general equilibrium theory as introduced by Walras has become known as the Walrasian auction which is a type of simultaneous auction where each agent calculates its demand for the good at every possible price and submits this to an auctioneer. The price is then set so that the total demand across all agents equals the total amount of the good. Thus, a Walrasian auction perfectly matches the supply and the demand. Walras suggests that equilibrium will be achieved through a process of tâtonnement (French for "trial and error"), a form of incremental hill climbing.

Léon Walras provides a definition of economic utility based on economic value as opposed to an ethical theory of value:

I state that things are useful as soon as they may serve whatever usage, as soon as they match whatever need and allow its fulfillment. Thus, there is here no point to deal with 'nuances' by way of which one classes, in the language of everyday conversation, utility beside what is pleasant and between the necessary and the superfluous. Necessary, useful, pleasant and superfluous, all of this is, for us, more or less useful. There is here as well no need to take into account the morality or immorality of the need that the useful things matches and permits to fulfill. Whether a substance is searched for by a doctor to heal an ill person, or by a assassin to poison his family, this is an important question from other points of view, albeit totally indifferent from ours. The substance is useful, for us, in both cases, and may well be more useful in the second case than in the first one.[lower-alpha 1]

In economic theories of value, the term "value" is unrelated to any notions of value used in ethics, they are homonyms.

Éléments d’Économie Politique Pure

The Éléments of 1874/1877 are the work by which Walras is best known. The full title is

• Éléments d'Économie Politique Pure, ou Théorie de la richesse sociale.

The half title page uses only the title (‘Éléments d'Économie Politique Pure’) whereas inside the body (e.g. p. 1 and the contents page) the subtitle (‘Théorie de la richesse sociale’) is used as if it were the title.

The work was issued in two instalments (fascicules ) in separate years. It was intended as the first of three parts of a systematic treatise as follows:

• 1^re partie:– Éléments d'Économie Politique Pure, ou Théorie de la richesse sociale.
  • Section I. Objet et divisions de l’économie politique et sociale.
  • Section II. Théorie mathématique de l’échange.
  • Section III. Du numéraire et de la monnaie.
  • Section IV. Théorie naturelle de la production et de la conommation de la richesse.
  • Section V. Conditions et conséquences du progrès économique.
  • Section VI. Effets naturels et nécessaires des divers modes d’organisation économique de la société.
• 2^e partie:– Éléments d’Économie Politique Appliquée, ou Théorie de la production agricole, industrielle et commerciale de la richesse.
• 3^e partie:– Éléments d’Économie Sociale, ou Théorie de la répartition de la richesse par la propriété et l’impôt.[12]

Works with titles echoing those proposed for Parts II and III were published in 1898 and 1896. They are included in the list of other works below.

• First (1874/1877). Most readily available. Described by Walker and van Daal as a ‘brilliant expression of pure originality, containing many theoretical innovations’ which ‘needed alteration and development in a variety of important respects’.[13]
• Second (1889). Revised, corrected and enlarged.
• Third (1896). A minor revision with new appendices. This is considered the best edition by Walker and van Daal.
• Fourth (1900). Revised and extended. According to Walker and van Daal, ‘these changes resulted in an incomplete, internally contradictory, and occasionally incoherent text’.
• Fifth (1926). Posthumous; published by his daughter Aline. ‘Édition définitive, revue et augmentée’.[14] Follows the fourth.

The ‘Théorie Mathématique de la Richesse Sociale’ included in the list of other works (below) is described by the National Library of Australia as ‘a series of lectures and articles that together summarize the mathematical elements of the author's Élements ’.[15]

• William Jaffé (1954) of the fifth edition as Elements of Pure Economics.
• Donald A. Walker and Jan van Daal (2014) of the third edition as Elements of Theoretical Economics.

Walker and van Daal describe Jaffé's translation of the word crieur as ‘a momentous error that has misled generations of readers’.

• Online: Walras, Léon (1874). Éléments D'économie Politique Pure, Ou, Théorie De La Richesse (in French). Retrieved 17 August 2018.
• Facsimile: cheap photographic reprints are produced by facsimilepublisher.com.

Both of these are made from the first edition and are defective in respect of illustrations. The original figures were included as folding plates (presumably at the end of each fascicule ). The online edition contains only Figs. 3, 4, 10, and 12 whereas the facsimile contains only Figs. 5 and 6.

• Francis Saveur, 1858.
• "De la propriété intellectuelle", 1859, Journal des économistes.
• Walras, Léon (1860). L'économie Politique Et La Justice: Examen Critique Et Réfutation Des Doctrines Économiques De P.j. Proudhon, Précédés D'un Introduction À L'étude De La Question Sociale (in French). Paris: Guillaumin. Retrieved 17 August 2018.
• "Paradoxes économiques I", 1860, Journal des économistes.
• "Théorie critique de l'impôt", 1861.
• De l'impôt dans le Canton de Vaud, 1861.
• Walras, Léon (1865). Les associations populaires de consommation, de production et de crédit (in French). Paris: Dentu. Retrieved 17 August 2018.
• "La bourse et le crédit", 1867, Paris Guide.
• Walras (1868). Recherche de l'idéal social: leçons publiques faites à Paris (in French). Paris: Guillaumin. Retrieved 17 August 2018.
• "Correspondance entre M. Jevons, professeur a Manchester, et M. Walras, professeur a Lausanne", 1874, Journal des économistes.
• Walras, Léon (1882). De la fixité de valeur de l'étalon monétaire (in French). Paris. Retrieved 17 August 2018.
• "Un nuovo ramo della matematica. Dell' applicazione delle matematiche all' economia politica", 1876, Giornale degli economisti.
• Théorie mathématique de la richesse sociale, 1883.
• "Notice autobiographique de Léon Walras", 1893.
• Études d'économie sociale; Théorie de la répartition de la richesse sociale, 1896.
• Études d'économie politique appliquée; Théorie de la production de la richesse sociale, 1898.
• "Théorie du crédit", 1898, Revue d'économie politique.
• "Sur les équations de la circulation", 1899, Giornale degli economisti
• "Cournot et l'Économique Mathématique", 1905, Gazette de Lausanne.
• "La Paix par la Justice Sociale et le Libre Échange", 1907, Questions Pratiques de Legislation Ouvrière.
• L'état et le chemin de fer (1875).
• "Leone Walras, Autobiografia", 1908, Giornale degli Economisti.
• "Un initiateur en économie politique, A.A. Walras", 1908, La Revue du Mois.
• "Économique et méchanique", 1909, Bulletin de la Société Vaudoise de Sciences Naturelles
• Correspondence of Léon Walras and related papers (ed. by William Jaffé, 3 vols.), 1965.