Walras, Léon (1834–1910)

Abstract

Léon Walras was the initiator of models of purely competitive general economic equilibration and equilibrium, of mathematical treatments of them, and of many aspects of microeconomic theory. In his period of maturity as a theoretician, he developed a comprehensive model that includes exchange, production of non-durable goods, production of capital goods, and monetary behaviour. That model features irrevocable disequilibrium behaviour and capital accumulation, and its equilibrium is therefore path dependent. His last theoretical effort, which was a failure but nevertheless very influential, was to try to develop a virtual and therefore path-independent model that would justify his static equation system.

Léon Walras was the founder of models of general economic equilibrium.

Biography

Walras was born on 16 December 1834 in Evreux, which is in the Department of Eure in France, and christened Marie Esprit Léon. His father was Antoine Auguste Walras, a secondary school administrator and an amateur economist; his mother was Louise Aline de Sainte Beuve, the daughter of an Evreux notary. After studying at the College of Caen from 1844 to 1850, he entered the Lycée of Douai, where he received the bachelier-ès-lettres in 1851 and the bachelier-ès-sciences in 1853. He entered the School of Mines of Paris in 1854, but, finding the course of preparation of an engineer not to his liking, he gradually abandoned his academic studies in order to cultivate literature, philosophy and social science. Although those efforts resulted in a short story and a novel, Francis Sauveur (1858), it rapidly became apparent to him that his true interests lay with social science. Accordingly, in 1858 he agreed to his father’s request to devote himself to economics and promised to continue his father’s investigations (1965, vol. 1, pp. 1–2).

During his youth in Paris, Walras became a journalist for the Journal des Economistes and La Presse from 1859 to 1862, the author of a refutation on philosophical grounds of the normative economic doctrines of P.-J. Proudhon (Walras 1860), an employee of the directors of the Northern Railway in 1862, and managing director of a cooperative association bank in 1865. He gave public lectures on cooperative associations in 1865; was co-editor and publisher with Léon Say of the journal Le Travail, a review devoted largely to the cooperative movement, from 1866 to 1868; and, during those years, gave public lectures on social topics (Walras 1868) in which he advocated Victor Cousin’s doctrine of compromise between economic classes. After the failure of the association bank in 1868, he found employment with a private bank until 1870 (1965, vol. 1, pp. 3–4). During the 1860s he tried intermittently to obtain an academic appointment in France, but he lacked the necessary educational credentials, and the 11 economics positions in higher education in France were monopolized by orthodox economists who, he complained, passed their chairs on to their relatives (1965, vol. 1, p. 3). His fortunes ultimately changed as a result of his participation in 1860 in an international congress on taxation in Lausanne, for that drew him to the attention of Louis Ruchonnet, a Swiss politician who secured his appointment in 1870 to an untenured professorship of economics at the Academy (subsequently University) of Lausanne in Switzerland. He was made a tenured professor there in 1871, and held that position throughout his teaching career.

Walras’s personal life was initially unconventional. He and Célestine Aline Ferbach (1834–79) formed a common law union in the late 1850s. She had a son, Georges, by a previous liaison, and she and Walras had twin daughters in 1863, one of whom died in infancy. In 1869 he married Célestine, thereby legitimizing their daughter, Marie Aline, and adopted Célestine’s son. A long illness of Celestine’s and the meagreness of Walras’s salary made life very difficult for him for several years. His time and energy were sorely taxed not only by the need to care for his wife but by the need to supplement his salary by teaching extra classes, contributing to the Gazette de Lausanne and the Bibliothèque Universelle, and working as a consultant for La Suisse insurance company. Five years after Célestine’s death in 1879, Walras married Léonide Désirée Mailly (1826–1900). The marriage was a happy one. Her annuity relieved his financial distress, and his situation was further improved in 1892 by an inheritance of 100,000 francs from his mother, which enabled him to pay debts incurred in publishing and disseminating his works, and to buy an annuity of 800 francs.

Influences Upon His Thought

Walras’s professional life was devoted to research and teaching. He frequently asserted that his research was a development of his father’s and that was true in some respects. It was under the influence of his father’s classification of economic studies that Léon, as early as 1862, planned the division of his life’s work into the study of pure theory, economic policies and normative goals (Walras to Jules du Mesnil Marigny, 23 December 1862, L 81; the ‘L’ stands for ‘letter’, and, like all correspondence cited in this article, the letter is in Walras 1965), the areas of study that were ultimately set forth respectively in the Eléments d’économie politique pure (1874, 1877b, 1889, 1896a, 1900, 1926, 1954), the Etudes d’économie social (1896b) and the Etudes d’économie politique appliquée (1898). Léon adopted his father’s classification of the factors of production into the services of labour, land and capital goods, regarding the source of each service as a type of capital. He adopted his father’s definitions of capital as wealth that can be used more than once and of income as wealth that can be used only once, and modified his father’s vague term ‘extensive utility’, clarifying it by defining it as the quantity-axis intercept of a market demand curve. The topic of utility had been treated in French thought by writers such as F. Galiani (a Neapolitan diplomat at Versailles) and E.B. de Condillac, and it was given further development under the name rareté by Auguste Walras, who thus bequeathed to Léon an interest in the concept of utility in relation to the value of goods and an awareness of its dependence upon scarcity, an interest that ultimately led him to define rareté as marginal utility. Auguste used the word ‘numeraire’ to mean an abstract unit of account, and Léon adapted the meaning of the word to his purposes. Auguste’s philosophy of social justice and his belief in the desirability of nationalizing land were advocated by Léon throughout his adult life. Léon’s major economic theories, however, were derived from his own original inspiration and from sources other than his father. Auguste’s greatest contributions to Léon’s development as an economist were to encourage him to study economics, to suggest that it should be a mathematical science (A.A. Walras 1831, ch. 18; Walras 1965, vol. 1, p. 493), and to give him access to a library of books on economics.

In that library was A.A. Cournot’s Recherches sur les principes mathématiques de la théorie des richesses (1838), which Léon Walras credited with having demonstrated that economics could and should be expressed in mathematical form (Walras to Cournot, 20 March 1874, L 253; Walras to H.L. Moore, 2 January 1906, L 1614; Walras 1905a). Cournot’s work introduced Walras to the mathematical formulation of exchange between two locations, the theory of monopoly and the associated conditions for profit maximization, the analysis of how prices are repeatedly changed in a search for equilibrium in a purely competitive market, and the demonstration of the effect of large numbers of traders upon the determinacy of price, all topics that Walras developed in his own work (1954, pp. 370–72, 434–40, 443). The first demand curve that Walras beheld was Cournot’s and he found it immensely suggestive. He was critical of it, however, because he perceived that Cournot’s postulate that the quantity demanded of a good is a function only of its own price is inaccurate if more than two goods are exchanged, and that Cournot did not provide a theoretical rationale for the demand function. Those perceptions, Walras observed, were the starting point for his own inquiries (1965, vol. 1, p. 5; 1905a).

Other ingredients that went into the composition of Walras’s theories were provided by Adam Smith, John Stuart Mill, François Quesnay, A.R.J. Turgot and Jean-Baptiste Say. Smith had revealed many of the consequences of unfettered competition and had formulated the concept of normal value. Mill had provided a supplement to and reinforcement of Cournot’s and Smith’s analyses of competitive pricing (Walras to Ladislaus von Bortkiewicz, 27 February 1891, L 999), and also an extension and grand synthesis of classical doctrines that served Walras as a catalyst for critical studies (Walras 1954, pp. 404–5, 411, 419, 423). Quesnay, in his Tableau économique, had expressed the concept of a circular flow of income and of the interdependence of the various parts of the economy. Turgot had clearly delineated the idea of the simultaneous and mutually determined general equilibrium of those parts. Say (1836) had suggested the distinction between the capitalist and the entrepreneur had portrayed the entrepreneur as an intermediary between the market for productive services and the market for outputs, and, in that analysis and in his law of markets, had adumbrated the interdependence between the incomes of the factors of production and the demand for goods. Walras sharpened those ideas and made them a fundamental part of his general equilibrium model.

A.N. Isnard’s Traité des richesses (1781), a book that Léon owned and that may have been in his father’s library, was probably an important source of some of Walras’s constructions (Jaffé 1969; Klotz 1994). Like Walras, Isnard was interested in determining equilibrium price ratios, set up a system of simultaneous equations of exchange showing the dependence of the value of each good upon the values of the others, stressed the necessity of having as many independent equations as unknowns, and perceived that the use of a numeraire rendered his system determinate. Anticipating Walras’s treatment of production, Isnard assumed given ratios of the inputs in a mathematical model and expressed the costs of production in equation form. Also like Walras, Isnard studied the allocation of capital among different uses, coming to the conclusion, as did Walras, that in equilibrium the net rate of income of different capital goods is the same.

Finally, Louis Poinsot’s Eléments de statique (1803) exerted a powerful influence upon Walras. He first read that book when he was 19 years old and kept it at his bedside for decades (Walras to Melle Dick May, 23 May 1901, L 1483). Poinsot painted a picture of the mutual interdependence of a vast number of variables, of how the dynamic forces in physical systems eventuate in an equilibrium in which each object is sustained in its path and relative position. Electrified by the implications of Poinsot’s work, Walras conceived a magnificent project. He would emulate Poinsot’s vision and analysis in reference to the general equilibrium of the economic universe! That he carried out that plan can be inferred from the striking similarity of the form of his work to Poinsot’s, with its careful delineation of functional dependences and parameters, its sets of simultaneous equations and its equilibrium conditions.

Equipped, therefore, with ideas that he could take as building blocks and points of departure, with enough geometry and algebra to put together mathematical statements of economic relationships and conditions -his use of calculus in the Eléments came after the first edition – and with the explicit objective of developing a mathematical theory of general equilibrium, Walras began his scholarly activity in Lausanne in 1870. In a period of great creativity that lasted until 1878, he developed most of the foundations of the theory of general equilibrium that appeared in the first edition of the Eléments. Walras insisted to his publisher that the first part appear in 1874, before the second part (1877b) was completed, because he learned in May of that year that W.S. Jevons had published a mathematical theory of utility and exchange that was similar to his own (J. d’Aulnis de Bourouill to Walras, 4 May, 1874, L 267), and he was anxious to establish the independence of his discoveries and his priority in regard to most of them. For those same reasons, he published four brilliantly original memoirs containing the heart of his theory of general equilibrium during 1874, 1875 and 1876 (Walras 1877a), paid for the costs of publication of his books, and sent copies of them and of his articles to his many correspondents. From 1878 to 1889, Walras significantly extended and refined his theory of general equilibrium (section “The Mature Comprehensive Model of General Economic Equilibrium”).

Walras was an extremely conscientious teacher, but he was an uninspiring lecturer (Walras 1965, vol. 2, p. 560), and the students at Lausanne were interested in careers in law, not in economics, so he failed to develop disciples among them. Moreover, he was with increasing frequency afflicted by bouts of mental exhaustion and irritability that made it difficult for him to lecture and to read and write (see Walker 2006a, pp. 183–7). In 1892 he took a leave of absence to regenerate his strength in order to be able to continue teaching, but soon realized he would find the strain of returning to his tasks insupportable and retired in that year, being at that time 58 years of age.

Subsequently Walras’s powers waned rapidly. In 1899 and 1900, he tried unsuccessfully to develop a virtual model of general equilibration and equilibrium (section “Walras’s Last Theoretical Work”). After 1900 he completely ceased theoretical construction (Walker 2006a, p. 191), but he wrote a few articles in which he restated earlier ideas. In late 1901 and 1902, he made some inconsequential changes to the Eléments which were ultimately put into the text of the fourth edition (1900) to produce the 1926 edition, both of them unfortunately called the ‘definitive edition’ (1900, p. v; 1926, title page). The latter was chosen for translation by William Jaffé (Walras 1954) and thus became the edition that is known in the anglophone world. Walras died on 5 January 1910 in Clarens, Switzerland.

The Mature Comprehensive Model of General Economic Equilibrium

Walras’s Subject Matter

Walras recognized that there were imperfectly competitive market structures and developed a theory of monopoly to take account of an important class of such phenomena (1954, lesson 41). Realizing, however, that the incorporation of noncompetitive elements into his general equilibrium model was beyond his powers (1954, p. 256) and believing that a high degree of competition was ‘almost universal’ and deserved to be treated as the general case (Walras to Ladislaus von Bortkiewicz, 27 February 1891, L999), he devoted most of his energies to working out a comprehensive model of interrelated ‘freely competitive’ markets, the aspect of his theoretical work with which this entry is concerned. Competition is most effective, he noted, in organized markets, and he assumed that markets are of that type (1954, pp. 83–4), but he also regarded his analysis as applicable in a general way to less highly organized competitive markets (1954, p. 84).

During his period of maturity as a theoretician, Walras modified and extended the model of competitive general equilibrium that he had presented in the first edition of the Eléments, constructing what will be called his mature comprehensive model. He presented it in 1889 in the second and greatest edition (and in the third, (1896a), identical to it in regard to the main body of the text). In the following exposition of that model, all the references to Walras (1954) are to passages that appeared in the 1889 edition.

The model is comprehensive in the sense that it deals with exchange, production, capital formation and credit, and monetary behaviour. It is non-virtual: it deals with irrevocable exchange at prices that are disequilibrium ones from the point of view of the state of the entire set of markets, and with the non-virtual dynamics of production, consumption, saving and investment. Those irrevocable economic activities occur during the course of the equilibrating process and are part of it (1889, pp. 235, 280). The sub-models included in the comprehensive model, such as the models of consumer demand, of the firm, of the entrepreneur, of exchange, of production, and so on, will sometimes be called theories, because Walras had reference to the behaviour of the real economy rather than purely hypothetical schemes. Each major sub-model has four parts: structure, equilibration, equilibrium conditions and comparative statics.

Regarding the structure of each market, Walras assumed that preferences, the number of economically active individuals, the amounts of natural resources and technology are constant. He identified consumers, workers, landlords, capitalists and entrepreneurs, their economic characteristics, and their objectives and how they try to achieve them. He specified the types of goods that are traded, the institutional features and rules of the market, and the individual and market supply and demand functions for goods (material and immaterial).

Regarding the dynamic equilibrating processes by which the markets undergo adjustments when in disequilibrium, Walras called them ‘tatonnements’, which means ‘gropings’, to emphasize that the equilibrium magnitudes of prices and quantities are not known by the participants during the disequilibrium phase but are found by repeated trial and error experiments. Walras considered the exposition of tatonnement to be ‘the object and proper goal of pure economics’ because he believed that the real economy is stable (Walras to Bortkiewicz, 17 October 1889, L 927; Walras to Charles Gide, 3 November 1889, L 933). Walras gave a verbal demonstration of the stability of his model. He recognized that the dynamic functioning of markets depends on the economic agents, institutions and conditions identified in the first part of each model, and in order to portray the disequilibrium behaviour that he perceived in the real economy he accordingly discussed the activities and interactions among diverse economic agents in trade and production, the generation and elimination of profits and losses, the operation of the stock market and many other details of behaviour drawn from economic life. Most of his presentation of the model is concerned with its stability, that is, its behaviour in disequilibrium. Thus the allegation, perpetuated by generations of commentators (for example, Jaffé 1971, p. 281, 1981, pp. 252–61), that Walras devoted his attention almost exclusively to the conditions of static equilibrium in an abstract model devoid of institutional detail, economic facts and dynamic behaviour is a misrepresentation of his work.

Walras was partially responsible for that misrepresentation, because in 1900 he referred to his general equilibrium model as ‘static’ without qualification, and contrasted it with what he called ‘the dynamic point of view’, by which he sometimes meant the view taken in considering economic growth (Walras 1954, p. 318). On the other hand, he also stated on many occasions that a dynamic theory is contained in his mature comprehensive model, and his usage will be followed in this article. The ‘static theory of exchange’, he wrote, ‘may be defined as the exposition of the equilibrium formula’. The ‘dynamic theory’, in contrast, which Walras claimed to have been the first to explore, is ‘the demonstration of the attainment of that equilibrium through the play of the raising and lowering of prices’ (1895, in 1965, vol. 2, p. 630). Similarly, in responding to Irving Fisher’s criticism that he had not considered time, Walras pointed out that that was true only of his exposition of the conditions of static equilibrium, and that he gave a dynamic treatment of production in lesson 20 (1889) of the Eléments (Walras to Fisher, 28 July 1892, L 1064).

Theory of Exchange

Walras was concerned in this theory with the determination of the equilibrium prices of goods and the quantities of goods exchanged. Setting forth the structure of exchange markets, he assumed that the preferences of the traders and the aggregate amounts of the goods they hold in each market are given. He first assumed that goods (including services) are exchanged directly for each other and then that they are exchanged for money. The participants include brokers, professional traders, retailers, wholesalers, the owners of the factors of production in their capacities as demanders of consumer goods and capital goods properly speaking, and entrepreneurs, who supply and demand goods. The supply and demand functions are reciprocally related (Walras 1954, pp. 96–7). Given a trader’s demand curve for A, its price times the related number of units he wants to buy is his supply of B expressed as a function of the price of A in terms of B. Observing what happens to the areas of the rectangles under the demand curve for A as its price rises, Walras deduced that the quantity supplied of B initially rises and then falls. In the same way, a trader’s supply of A can be derived from his demand for B. Walras summed the individual demand and supply curves respectively in the market for A to obtain the market curves, and similarly for B. It will be seen that he adapted and extended this analysis of the dependence of the supply of one good upon the demand for another when he took up the question of multi-good exchange. Walras also assumed that in each market the rule is enforced that disequilibrium transactions are not allowed (1880a, p. 461; 1880b, p. 78; 1954, p. 85). He described that as being true of the nineteenth-century Paris bourse, but in fact disequilibrium transactions occurred there most of the time (Walker 2000, 2001), in recognition of which he allowed late in his career that his description is in actuality ‘a hypothesis that no scientific spirit would hesitate to concede to the theoretician’ (Walras 1895, in 1965, vol. 2, p. 630).

To explain demand and infuse his early model of exchange (1869–70) with purposive action, Walras developed a theory of preferences shortly before 1872 in which he assumed that traders want to maximize utility, that utilities are independent and additive, and that the marginal utility of a good is a decreasing function of the quantity acquired or consumed. Nevertheless, he was floundering in his attempts to relate utility to market behaviour, so he appealed for help to Antoine Paul Piccard, a professor of industrial mechanics at the Academy of Lausanne, who responded in 1872 by developing a model of utility maximization and deriving the individual demand function within it (1965, vol. 1, pp. 308–11), thus meriting a part of the credit that has previously been given exclusively to Walras for that achievement. Everything then fell into place for Walras, and he proceeded to develop the view of economizing and maximizing behaviour that he imprinted on Continental neoclassical economics. He extended the technique shown in Piccard’s model, making utility maximization the driving force in each of his models, and obtaining the equilibrium conditions of the participants in a multi-good system (1954, lesson 12).

The dynamic behaviour of Walras’s exchange model is a tatonnement process in the sense that the path of the price in each market is the unplanned outcome of market forces. The process depends upon human nature (see Walker 2006a, pp. 114–39) and on the rules, institutions and conditions devised and enforced by market authorities and by government (Walras 1880a, b; 1895 in 1965, vol. 2, p. 632; and see 1954, p. 474). A price is initially cried at random (1877b, p. 127; 1954, p. 169) by any of the traders, and the suppliers and demanders subsequently follow the Walrasian pricing rule: that is, they change the price in the same direction as the sign of the market excess demand for the good. Suppliers call out progressively lower prices if it is negative, and demanders call out progressively higher prices if it is positive. Preferences are constant, and the rule against disequilibrium transactions ensures that the asset distribution remains unchanged during the equilibrating process. Therefore, the initial supply and demand functions and, consequently, the particular-equilibrium price on any given day in the temporarily isolated market, are not affected by the disequilibrium behaviour of the traders. That price equates the supply and demand quantities; it is quoted sooner or later, and the equilibrium amounts of the good are exchanged (1954, p. 106, lessons 6, 9).

Markets are not isolated, however, so Walras introduced the central feature of his contribution to economic science, namely an account, in his theory of exchange and in the other parts of his mature comprehensive model, of the interrelationships among the markets for different goods (including services). If a trader has a good that he wants to trade for several others, the amount that he offers in any market is related to the amounts that he offers in the other markets, so the amount that he wishes to purchase or sell of any good is seen to be a function not only of his preferences, his income and the price of that good but also of the prices of other goods. Consequently, the market supply and demand quantities and the price in any market are dependent in part upon the prices in other markets (1954, lesson 12).

Moreover, Walras explained that the sum of the values of a trader’s quantities demanded must equal the sum of the values of his quantities supplied. That relation is one way of stating the individual budget equation, and it is a version, on the individual level (1954, p. 165), of what has come to be known as Walras’s Law, a fundamental statement of the way that markets are interrelated. Walras was able to identify the law in part by reasoning an individual cannot demand any commodity without offering in return commodities (or money) having the same total value, so, if some of his excess demands are positive, others of them must be negative, and in part because of the device of the numeraire. The latter, a good in terms of which the values of all goods are expressed (1954, p. 161), made clear to him, as it had to Isnard, that there is exactly the right number of excess demands: in a system with n goods, there are only n – 1 independent market equations involving n – 1 price ratios, but also only n – 1 unknowns, inasmuch as the price of the numeraire, the nth good, in terms of itself is unity (1954, pp. 161–2, 241).

With reference to the market level in multi-commodity exchange, Walras affirmed that the sum of the positive or negative market excess demand quantities for each good multiplied by its price is zero (1954, p. 170), and he stated a version of Walras’s Law for the market excess demand quantities of productive services (1954, p. 248). In a Walrasian equilibrium, supply equals demand for every good, so each excess demand quantity is zero. Each excess demand quantity, multiplied by the price of the good, must therefore be zero, so the sum of the excess demands each multiplied by the price must be zero. In the case of an individual, Walras stated only, regarding the variables, that ‘there will be between them all’ the relationship that indicates that their sum is equal to zero, without addressing explicitly whether the equation is true in disequilibrium as well as in equilibrium (1889, p. 143, which differs from Walras 1954, p. 165). In the case of multi-commodity exchange, he probably implied that it is always true even though in disequilibrium the market supply and demand quantities of every commodity are not simultaneously equal (1954, pp. 169–70). In the productive services formulation, he indicated explicitly in the Eléments that it is true only in equilibrium; the functioning of the market mechanism is necessary when the economy is not in equilibrium, he stated, to drive the excess demands to zero and thus solve the equation (1954, pp. 248–9; 1889, pp. 242–6). He later declared, however, that it holds in both disequilibrium as well as in equilibrium (1898, pp. 277–8; Walker 2006a, pp. 152–4). His implicit reasoning appears to be that the sum of the excess demand quantities, each multiplied by its price, is zero in disequilibrium even though some or all excess demand quantities are not zero, thus implying that the law is an identity.

Walras asserted regarding his mature comprehensive model (and hence regarding its sub-models) that equilibrium exists, on the grounds that the number of independent equations equals the number of unknowns (prices and quantities). He was, of course, mistaken in that belief. Wilhelm Lexis had pointed out in 1881 that Walras’s equations might nevertheless not have real positive solutions or any solution at all (see Walras 1965, vol. 1, p. 747), a fact that only since the late 1920s and early 1930s became well-known (see Weintraub 1983; Van Daal et al. 1985).

The interdependence of markets, Walras explained, gives rise to the major problem of general equilibrium analysis, which is the question of the stability of the model, implicitly containing the question of the existence of equilibrium. Will a system of freely competitive markets that is initially in disequilibrium converge to a position of equilibrium? After any market reaches temporary equilibrium through the exchange of the equalized market supply and demand quantities, the traders note what has happened to the prices in other markets. Their reaction is manifested in a shift of the market demand curve, which puts the market once more into disequilibrium and initiates another series of quoted prices leading to a new market-day equilibrium. Will its readjustment aid or impede the equilibrating process taking place in other markets? Does the series of market-day prices in the set of markets move closer to an equilibrium of the entire system or further from it? Walras claimed that he had shown that the answer to those questions is that the market system converges to general equilibrium as a result of the ways that markets are interrelated and of the operation of the Walrasian pricing rule in each market (Walras 1954, pp. 172, 179–80).

Walras then specified the conditions that prevail in the static equilibrium of exchange of a multi-market system. The ratio of the raretés, or marginal utilities, of any two goods is equal to the ratio of their prices, and the price of any good in terms of another good is equal to the ratio of the prices of those two goods in terms of any third good (1954, p. 157). Those conditions are satisfied when the quantities supplied and demanded of each good are equal (1954, p. 172).

Finally, Walras briefly examined some features of the comparative statics of the exchange model (1954, pp. 147–9). He shifted the utility curves for a good and determined that its equilibrium price changes in the same direction as the shift in the curves. He then successively increased and decreased the traders’ endowments of a good and determined that its equilibrium price successively decreases and increases.

Theory of Production

In his model of production, Walras was concerned with the determination of the equilibrium prices of productive services and the equilibrium rates of output of the quantities of non-durable goods. Walras did not present this model directly and without modifications as part of his mature comprehensive model. The latter deals with durable goods as well as non-durables and therefore contains a wider model of production. Nevertheless, Walras carried over the processes of pricing and aspects of the tatonnement in the non-durables model into the comprehensive model.

Setting forth the structure of the production model, Walras first identified the markets for productive services, in which he assumed that the amounts of economic resources and therefore the maximum possible amounts of their services are given. The demanders of productive services are the entrepreneurs. Their ultimate aim is to maximize utility, which they achieve through maximizing profits. In their capacities as entrepreneurs, they combine productive services and materials in proportions that are determined by what Walras called the technical coefficients of production. The coefficients, which he assumed to be fixed in much of his general equilibrium theorization, indicate the amount of each of the inputs that is used to make a unit of output. With fixed coefficients and given prices of the productive services, the average cost is constant as the firm’s output varies. If any of those prices change, the average cost curve shifts in the same direction.

The suppliers of productive services are workers, who own personal faculties; landlords, who own natural resources; and capitalists, who own capital goods or provide capital funds (Walras 1877b, p. 218; 1954, pp. 214–15). Their aim is to maximize utility, which motivates them to offer services to the entrepreneurs in exchange for income.

Walras then identified the market for consumer goods (material and immaterial). These goods (in the production model) are consumed immediately after being produced; they are used only once and are used up in that process. The suppliers of them are the entrepreneurs. The demanders are the workers, landlords, and capitalists acting in their roles as consumers, motivated in their purchases by the desire to maximize utility. They pay for them with the incomes that they have been paid by the entrepreneurs. The only type of capital goods produced in the model is nondurables, that is, variable capital goods like raw materials. Those goods, like consumer goods, are used up in a single application as soon as they are purchased.

Of course, that is true of the services of all types of economic resources. In the model, there is no saving. The durable capital goods that are used in production do not depreciate or become obsolete, nor are they subject to accidents. There are no markets for them or for land.

The tatonnement in the production sub-model, and in the capital goods and monetary sub-models, and in the comprehensive model as a whole, is in considerable measure the outcome of the actions of entrepreneurs. Walras assumed that all resources are highly mobile. Entrepreneurs have good knowledge (but not perfect foresight) of the profitability or unprofitability of producing any particular good and accordingly enter or leave an industry. The tatonnement that occurs in the markets for inputs is a process of groping for the equilibrium amounts of resources employed in different industries. The entrepreneurs hire the factors of production, combining them in technologically determined proportions or experimenting to find optimum proportions if the coefficients are variable (1896a, pp. 490–1), and sell services and finished goods to consumers (1954, lesson 21, and pp. 426–7; Walker 1996, ch. 13). The entrepreneurs hire and use disequilibrium quantities of productive services during the tatonnement, and produce disequilibrium quantities of goods (1889, pp. 234–5, 240–1, 249–50). The payment that the entrepreneurs receive in disequilibrium for their entrepreneurial activity is profit, which Walras defined on a per unit basis as the price of output minus its average cost, with the latter including the wages of management. An entrepreneur may undertake the functions of other factors of production – he may also, for example, be a capitalist or a manager of the firm – and ordinarily he has to do so, but his multifaceted role as entrepreneur is a distinct one (1954, p. 222).

The tatonnements in the markets for productive services and for consumer goods are interrelated. If the consumers’ demand for a good increases, the price is bid up in accordance with the Walrasian pricing rule. The quantities demanded and supplied become equal at a high price because the supply function does not initially change. The price of the product then exceeds its cost of production, so the entrepreneurs in the industry make profits. Attracted by the prospect of doing the same, other entrepreneurs enter it, and existing firms increase their output. As the market supply of output function changes so that more output would be offered at each possible price, the price in the exchange market for the good is lowered by the entrepreneurs in an effort to dispose of the entire flow of output. As the demands for productive services increase, their prices are bid up, which raises the average cost of production. Thus the price falls and the average cost rises (1954, p. 253). If demand for a good decreases, its price falls below the average cost of production and the entrepreneurs make losses. This leads some of them to leave the industry and some of them to diminish the output of their firms. The prices of the productive services fall as the demand for them decreases, which lowers the average cost of production. As less of a good is offered, its price is forced up. Thus the average cost falls and the price rises (1954, p. 253). It will be noted that these are all non-virtual processes in the model. Walras concluded that the average cost of production and the price of the good become equal, whereupon equilibrium is reached and the tatonnement ends. It follows that in equilibrium the entrepreneur obtains neither profit nor loss (1877b, p. 232; 1954, p. 225).

Pursuing still further the question of the stability of the model, Walras reasoned that the interactions of traders in different markets aid the equilibrating process. The change in the output of a product, he argued, has an effect on its price that is unidirectional, whereas the unidirectional changes that it induces in the outputs of other products have only indirect repercussions on its price, and the latter more or less cancel each other because some of them change in one direction and some in another (1954, p. 246). In the non-virtual adjustment, ‘The [resulting] system of new quantities of manufactured products and of new selling prices is thus closer to equilibrium than the old one; and we have only to continue the process of groping to approach still more closely to equilibrium’ (1954, pp. 246–7). In other words, once again, ultimately the tatonnement leads to the simultaneous equalization of supply and demand in every market.

The equilibrium, Walras stressed, is the normal state of the market in the sense that it is the one to which the variables in disequilibrium perpetually and automatically tend in a regime of free competition (1954, p. 224). Since it contains the equilibrium of exchange (1954, p. 224), it is characterized by the conditions that the quantities supplied and demanded are simultaneously equal regarding each consumer good, each productive service, and each primary material. A stable circular flow is established in which the total cost equals the total revenue in each firm, the incomes received from the entrepreneurs by the owners of the factors of production equal the revenues earned by the firms, and those incomes are spent on consumer goods by the owners of the factors of production. Walras’s theory of production therefore showed ways in which input and output markets for non-durable goods are linked together.

Sales at a disequilibrium price of the items that are produced do not alter the assets held by the participants because the quantities exchanged do not have an existence after the exchange; they are used up immediately. The equilibrium of the production model is therefore not path dependent. That is of no significance, however, because the model is a hypothetical special case. It does not take account of the production of capital goods or of consumer durables which occurs in the real economy and in Walras’s mature comprehensive model.

Walras then considered variations in some of the parameters of the production model. If the marginal utility curves for a good shift up, he reasoned, its price in terms of the numeraire increases. If the marginal utility curves shift down, the opposite occurs. If the quantity of a product or service possessed by the holders changes, its price changes in the opposite direction (1954, p. 260).

Theory of Capital Formation and Credit

Walras referred to the three sources of services (labour, land, and capital) as different types of capital because they all endure through time and produce a flow of services, but in this article the unqualified word ‘capital’ or the term ‘capital goods’ will mean durable, man-made, inanimate instruments of production. Walras first examined the determination of the prices of land and personal faculties, as distinct from the prices of their services. The aggregate supply of land, a given condition, is perfectly inelastic, and its price is its gross income divided by the rate of net income (1954, pp. 270, 309). The number of workers is also a given condition. The price of a worker is equal to his gross income minus the cost of replacing and insuring him, divided by the rate of net income. Workers are not bought and sold, however, so their prices are virtual (1954, pp. 271, 311).

Walras’s theory of capital is concerned with the determination of the amounts and prices of capital goods, as distinct from their services, and the determination of their rate of net income. Capital goods are specific items of real capital; capital funds, raised by the sale of shares on the bourse, constitute fluid and mobile purchasing power which can be used to acquire economic resources to construct different kinds of physical capital (1954, pp. 270, 311). Capital is formed by capitalists saving money and, most commonly, lending it to entrepreneurs (1954, p. 270). The net saving of the capitalists as a group equals aggregate income minus aggregate consumption, minus the depreciation and insurance costs of capital goods. The entrepreneurs purchase or rent capital goods, earn revenue from their use, and repay any sums they have borrowed (1954, p. 290, §§ 190, 235). Walras’s identification of that process explains why he inserted the word ‘credit’ (1954, p. 270) into the name of his capital-goods model, but he did not develop a general theory of credit within it, because he did not introduce loans made by banks. Some capitalists prefer to own capital goods, so Walras assumed occasionally that they acquire them directly in physical form (1954, p. 289), and assumed frequently that they acquire them through buying stock certificates (for example, 1954, p. 289). In each case, the physical capital is used by entrepreneurs, so ‘the demand for new capital goods comes from entrepreneurs who manufacture products and not from capitalists who create savings’ (1954, p. 270). The entrepreneurs purchase the particular kinds of capital goods that are profitable, with the result that the kinds that are produced and used reflect the structure of demands for consumer goods (1954, pp. 225, 303).

Describing the tatonnement regarding both non-durable products and durable goods, that is, summing up the non-virtual character of the tatonnement in the mature comprehensive model, Walras explained that

After a certain rate of net income and certain prices of services have been cried and after certain quantities of products and new capital goods have been manufactured, if this rate, these prices and these quantities do not satisfy the conditions of general equilibrium, it will be necessary not only to cry a new rate and new prices, but also to manufacture revised quantities of products and new capital goods. (1889, p. 280; 1954, p. 282; and see 1889, pp. 284–94; 1954, § 258, pp. 293–4)

One aspect of the tatonnement takes place in the stock market and another in the course of the production of capital goods. In the stock market, which is the market for new capital goods, each capitalist attempts to maximize utility by saving and acquiring more stocks that have relatively high yields and less of those with lower yields (1954, p. 289), with the result that the total value of new capital goods and the excess of income over consumption both move in the same direction as all prices. It follows, Walras maintained, that the tendency of the change in prices to destroy the equality between the total value of new capital goods and the excess of income over consumption is weaker than the tendency of the change in the rate of net income to bring the total value of new capital goods and the excess of income over consumption into equality with each other. Moreover, ‘in these conditions, it is evident that the price and the average cost of the capital good (K) will have been little altered from equality by the increases in the quantities produced of the capital goods (K′),(K″),…’ (1889, pp. 292–3). ‘Thus the system involving the new rate of net income and the new prices will be closer to equilibrium than the old system; and it is only necessary to continue the process of groping for the system to move still more closely to equilibrium’ (1954, p. 288).

In the course of the production of capital goods proper, entrepreneurs acquire more capital goods that yield relatively high returns, and diminish their use of capital goods that yield lower returns. As a consequence, the rate of net income from each capital good proper tends toward the same value (1954, p. 309). If profits are being made from the production of capital goods in an industry, new entrepreneurs enter it, and those already in it increase their rate of production. That drives up the prices of productive services, which causes the average cost to rise towards equality with the price of the capital good. If losses are incurred, entrepreneurs diminish production. That drives down the prices of productive services, which causes the average cost to fall towards equality with the price of the capital good (1889, p. 293). It is ‘probable’, or, ‘it is to be presumed’, Walras maintained, that the effects of changes in the output of a new capital good that tend to cause equality between its average cost and its price will be stronger than the contrary effect of interrelated changes in the output of other capital goods, so the process converges to equilibrium (1954, p. 293). Referring again to the non-virtual character of the tatonnement, Walras explained that ‘The new system of revised quantities manufactured and of revised costs of production and selling prices of new capital goods is thus nearer equilibrium than the original system’, and it is only necessary to continue the tatonnement to approach it more and more closely (1889, p. 293; 1954, p. 293).

The tatonnement in the mature comprehensive model involving both nondurable and durable goods represents and explains what happens in the real economy:

Now, this tatonnement is precisely that which occurs by its own forces on the market for products [non-durables], under the regime of free competition, while the entrepreneurs who produce new capital goods, just like the entrepreneurs who produce products, increase their production or diminish it according to whether profits or losses are made. (1889, p. 294; 1954, p. 294)

Walras expressed the equilibrium conditions in the formation of new capital goods in the lengthy analysis that he called the theorem of the maximum utility of new capital goods, which he described as crowning and confirming his entire theoretical system (Walras to H.S. Foxwell, 16 December 1888, L 859; Walker 1984b). Although he assumed initially that new capital goods do not require amortization or insurance, Walras then made the realistic assumption that capital goods wear out and are subject to accidents. The rate of net income generated by a particular capital good is then given by the gross income it earns minus amortization and insurance costs, divided by the price of the capital good. In equilibrium each trader maximizes utility by holding the quantities of capital goods that make the ratio of the marginal utility of each capital good to its price equal for all his capital goods. Because of the adjustment of yields and capital good prices, the rate of net income derived from every type of capital good is the same (1954, p. 281). There is a single price for each type of capital good, equal to its average cost (1954, p. 280). Furthermore, the price of any given type of well-maintained old capital goods is equal to the price of the same kind of new capital goods, so the equilibrium prices of all capital goods ‘are equal to the ratios of their net incomes to the rate of net income’ (1954, p. 309). The latter is the rate of interest. Its equilibrium value equates aggregate saving and investment (1954, p. 276).

Walras believed that through this analysis he had seen behind the veil of money or numeraire and discovered the real determinants of that rate. It is manifested in the banking system, he argued, but it is determined in the stock market. It is the ratio of net profit to the price of a share of stock, which in equilibrium equals the common ratio of the net price of capital services to the price of the good that yields them (1954, p. 290).

Walras then turned to the comparative statics of the capital goods market. If the price that has to be paid for the services of a capital good increases or decreases as a result of a parametric change, the price of the capital good itself increases or decreases. Its price also varies inversely with the rate of depreciation and with the rate of the insurance premium. If the rate of net income changes, the prices of all capital goods change in the same direction (1954, pp. 309–12).

Theory of Money and Circulating Capital

Walras wanted to reduce the monetary mechanism ‘to its essential elements’ (1889, p. 379). He therefore carefully specified the structural and behavioural features of his mature fiat money model, drawing upon his direct experience as a young man with real financial matters and his extensive knowledge, accumulated throughout his career, of empirical monetary arrangements and problems. He explained that fiat money, like pieces of paper with an engraved image, has no utility of its own. Economic agents hold it because it enables them to purchase goods that have utility (1889, p. 378). Entrepreneurs and some consumers have net demands for cash balances because of the non-synchronization of payments and receipts (1886, pp. 40–4). The size of a consumer’s desired cash balance depends upon desires to consume and save, which depend upon his character and habits, income, the value of the commodities he wants to buy, and the rate of interest (1889, pp. 377, 268–71). The size of an entrepreneur’s desired cash balance depends upon the nature and state of his business, and on his character, habits, and the rate of interest (1889, p. 377). These are desired real balances because they represent the demand for a specific bundle of goods. The aggregate real desired cash balance is the nominal one divided by the price level, which is the inverse of the price of money. That aggregate is the demand for future consumer and capital goods (1889, pp. 378–9).

Walras incorporated a market for loans into his model. First, there are the demanders. They are consumers, entrepreneurs and savers, who go to the market and borrow. The first two groups buy the goods and services they need. A curious temporary assumption is that savers, the third group, borrow to obtain the funds they lend. Thus ‘we have the daily demand for money which is exercised on the market for money capital’ (1889, p. 381). What is happening behind the veil of money, Walras explained, is that the consumers and entrepreneurs are actually borrowing the fixed and circulating capital on which they spend the money, and the interest borrowers pay is generated by the fixed and circulating capital that the borrowed money finances. Second, there are the suppliers of funds in the loan market. In one group of them are entrepreneurs who have receipts from the previous day from sales to consumers and from sales of new fixed and circulating capital good to other entrepreneurs, and in the other group are landlords, workers and capitalists who have receipts from the previous day from sales of services (1889, p. 381)

Walras then specified how the mechanism of free competition operates in regard to monetary circulation in disequilibrium of the money market. It will be noted that the entire behaviour relating to money in the mature comprehensive model is nonvirtual. Mainly explicitly, he indicated the tatonnement that was presented later by the Cambridge cash balances theorists. He created the temporal framework for the period analysis in his model by assuming first that production and consumption ‘extend over all the moments of the entire year’ (1889, p. 316). Workers, capital goods, and money are used up and are simultaneously reproduced and replaced. He then assumed that markets function every day and that different types of related behaviour occur sequentially on a series of days (1889, pp. 381–3). Thus he developed a continuous-production and periodic-market model.

A change, say a decrease, in the quantity of money causes disequilibrium. The Walrasian period analysis then indicates that the process of equilibration takes three ‘days’. On the day that the decrease occurs, the old rate of interest still rules and the quantity of cash balances demanded exceeds the quantity supplied. The immediate result is that the rate of interest increases. On ‘the next day on the market’, a temporary equilibrium is reached, ‘at a new higher rate of interest at which the desired cash balance would be reduced’ (1889, p. 383). During that day, the prices of all goods fall proportionately to the decrease in the quantity of money and the aggregate nominal desired cash balance remains lower, but the real balance is ‘able to become what it was before’ as a result of the fall of prices. Then, ‘on the day after that’, the third day, permanent equilibrium is attained, the rate of interest falling back to its old level, which is equality with the rate of net income from capital goods (1889, p. 383). The same sequence occurs if the aggregate desired cash balance function increases. If the parameters change in the opposite direction, the opposite sequence of adjustments occurs. There are many more details, situations and variations of assumptions that Walras considered regarding the model. He was able to sum up what he had done in this way: if the quantity of money increases or the desired cash balance decreases, prices rise proportionately, and the reverse. ‘This law extends to money the principle by virtue of which value increases with utility and decreases with the quantity’ and therefore integrates the determination of the value of money into his mature comprehensive model and therefore into his general theory of value (1889, p. 383).

When there is monetary equilibrium, that model is in equilibrium in all respects. Walras summarized its aspects in the following way. There is ‘the equilibrium of exchange’ in which prices are proportional to marginal utilities and consumer satisfaction is maximized; there is ‘the equilibrium of production’ with prices equal to average costs and zero profits; there is ‘the equilibrium of capital formation’ with prices of land, human faculties and capital goods proper proportional to the prices of their services; and finally there is ‘the equilibrium of circulation, in view of the fact that the exchangers would have the cash balances that they desire at the announced rate of interest’ (1889, p. 379).

Walras therefore made significant theoretical innovations in his mature theory of money. In it, he raised basic questions about the nature of a true money, its role, the valuation of money, its place in preference functions and hence in demand and supply functions, the sequence of adjustments that occur after a change in its quantity, and the impact upon equilibrium prices and the rate of interest. His modelling of cash balance behaviour and dynamic period analysis anticipated some of the theoretical techniques used during the 1920s and 1930s by J.M. Keynes,D.H. Robertson, and J.R. Hicks, none of whom acknowledged his contributions. That was probably because the insightful analysis and potentially fruitful constructions that Walras put into the mature money model are in the forgotten 1889 edition.

Economic Growth

Walras did not develop a complete model of economic growth, but he examined some aspects of the topic in connection with his mature comprehensive model. One was endogenous variations in its parameters. He was led to speculate about that subject by the consideration that the equilibrium conditions identified in the mature comprehensive model are never reached in the real economy because tatonnement takes time, and consequently parameters such as preferences and the amount of labour change before equilibrium is reached (1954, p. 380). In order to take account of this situation, ‘we shall now suppose that the annual production and consumption, which we had hitherto represented as a constant magnitude for every moment of the year under consideration, change from instant to instant along with the basic data of the problem’ (1954, p. 380). Services and goods are used up and are produced. Net new capital goods come into existence and are put to use, and circulating capital is borrowed by entrepreneurs from the capitalists in the form of money loans (1954, p. 379). Walras did not analyse in any further detail that moving equilibrium or ‘continuous market’ economy.

A second aspect of growth that he examined was ‘the laws of the variation of prices in a progressive economy’ (1954, p. 382), that is, some of the features of alternatives paths of economic growth. For this task he first defined economic progress as the substitution of capital services in place of land services in given production functions (1954, p. 383). The substitution implies variable coefficients of production, and to introduce these Walras used the theory of marginal productivity. He did not claim to have originated that theory although he anticipated some of its features. In fact, Hermann Amstein, a mathematician at Lausanne, worked out its major principles in 1877 (Amstein to Walras, 6 January 1877, L 364; translated in Jaffé 1983, pp. 205–6). Walras did not understand or use Amstein’s work, however, and the major credit for the theory of marginal productivity that first appeared in the Eléments in 1896 (Appendix III) must be given to Enrico Barone (1895).

Walras defined technical progress as changes in production functions, including the introduction of entirely new processes, but he did not analyse it, beyond concluding that it contributes, along with economic progress, to ensuring that output increases without limit in a progressive economy (Walras 1954, p. 387). He also discussed, in a highly general way, how the prices of products and services vary with different amounts of capital and sizes of the population (1954, pp. 389–91). His principal conclusion was that the rate of net income falls as the stock of capital grows, the proximate causes of the process being rising rents and falling prices of capital services.

Walras was well aware that capital accumulation means economic growth and requires a different characterization of equilibrium, noting that ‘In order to have a supply, a demand, and prices of capital goods, it is necessary to substitute for the conception of a stationary economic state that of a progressive economic state’ (1889, p. 264). His way of dealing with this problem in 1889 was to assume that new capital goods are not used until what he represented to be the end of the tatonnement, thinking that would preserve the static equilibrium. That did not, however, remedy the problem, as will now be shown.

Walras attempted to give a mathematical proof of the stability of the tatonnement of the mature comprehensive model, spread throughout the pages of his treatise. He believed that he showed that the model is stable by working with his system of static equations of general equilibrium. He posited a disequilibrium state and then varied the prices in the equations in accordance with the Walrasian pricing rule, and claimed that equilibrium is the result of the tatonnement. That claim is logically flawed, for two reasons.

The first is that the tatonnement in the mature comprehensive model, unlike the model of the production of non-durable goods and services, is path dependent even though the new capital goods are not used during any of the phases of adjustment before equilibrium is ostensibly reached. Positive net investment has the result that individual holdings of capital goods and many of the other nominal parameters and all the variables of the model are altered. Each different disequilibrium rate of production and sales of capital goods occurring within each phase of the tatonnement changes their prices and average costs, profits, and the rate of net income. Consequently, Walras’s attempted proof was not rigorous and could not have been valid, given the static equations that he used. They have the endowments of assets in the initial disequilibrium as parameters. Their solutions therefore depend on those constants. The model, however, is not a virtual system, so the individual holdings of assets and their total amounts vary during the course of the tatonnement. The result is that the variables of the model do not converge to the solutions of Walras’s static equation system. Any equilibrium to which the prices and quantities converge cannot be the one indicated by his equations because they do not describe his model.

The second reason is that the ‘equilibrium’ that Walras asserted exists at the end of the tatonnement is factitious and cannot materialize, even if there were no problem of path dependency. The supposed equilibrium could exist only transitorily while the model is held in a state of arbitrarily suspended animation by the postulate that the additions to the capital stock are not used – a deus ex machina that interrupts the incomplete workings of its endogenous processes. The instant that Walras removed the postulate, that is, the instant that the net new capital goods are put into use, the ‘equilibrium’ is ruptured, and through dynamic processes many of the nominal parameters and all the variables of the model change, in the way just indicated in the discussion of the consequences of the use of net new capital goods. If they continue to be produced, as Walras assumed, the system follows a path of growth. The equilibrium path and any stationary equilibrium that the system may eventually reach is quite unlike the solutions to the static equations of general equilibrium that he presented in the 1889 edition of the Eléments and subsequently.

Walras’s Last Theoretical Work

The Written Pledges Sketch

In 1899 Walras changed his work in two major ways, and put the changes into the Eléments in 1900. One was to devise a new model of money and circulating capital (see below). The other was to try to construct a virtual model. The motive for the latter was that Walras had come to realize by 1899 that his equation system is compatible only with such a model. Instead of trying to develop a different equation system, however, one that would represent the non-virtual mature comprehensive model, he chose to abandon the latter, to retain his static equations, and to try to construct a virtual model that would serve as their foundation and justification. In the 1900 revision, he eliminated much of each of two forceful and lengthy statement of the non-virtual tatonnement (Walras 1889, pp. 234–5, 280), which consequently appear in Jaffé’s translation only in very abbreviated form (1954, pp. 242, 282). He retained, however, crucial parts of those statements and retained elsewhere throughout the revision most of the other language describing the non-virtual behaviour of the economy and of the mature comprehensive model. That explains why the reference ‘Walras 1954’ can be cited to document many the features of the 1889 mature comprehensive model. It is also one of the principal causes of the 1900 and 1926 editions (and therefore Jaffé’s translation of the latter) being a chaotic mixture of incompatible language and sub-models.

To construct a virtual model, Walras conceived the device of written pledges (engagements écrits, as they were and are called in the Paris bourse). He asserted that the model has three phases, made identifiable, he believed, ‘by means of the hypothesis of written pledges’. First, there is ‘the phase of preliminary gropings towards the establishment of equilibrium in principle’, the purely virtual phase (1954, p. 319). When a price is cried in any market, suppliers of goods and services write out the amounts that they pledge to produce and sell at that price, but only if it turns out to be the equilibrium value, that is, the one that is part of the set of prices that equates the market supply and demand quantities simultaneously in every market (1899, p. 103; 1900, pp. 215, 260; 1954, pp. 242, 282). Thus Walras adopted the device in order to eliminate changes in the holdings of assets before the entire system of markets has reached equilibrium, changes which would otherwise occur as a result of trade occurring in a market when it reaches market-day equilibrium while other markets are still in disequilibrium, or as a result of disequilibrium production, which changes the aggregate amounts of goods held before general equilibrium has been reached.

In the first phase, entrepreneurs are supposed to plan to move from unprofitable to profitable industries and to plan to create firms or to expand or contract their existing firms, and to predict accurately the financial results of their plans, without actually moving or creating or hiring or spending or producing at all. Owners of productive services are imagined to offer their services repeatedly at disequilibrium prices without actually earning any income or consuming any goods or services. The entire system of interrelated markets is imagined to go through complex costless processes of information acquisition, price changes and changes in the supply quantities that are pledged, all without anyone being allowed to agree to a single actual transaction or undertake any production or consumption, until the equilibrium set of prices has been found.

It is symptomatic of Walras’s significantly diminished ability to concentrate and pursue lengthy chains of reasoning after 1898 that his words ‘the hypothesis of written pledges’, although followed by dozens of pages of modelling and theorization, including his immediately following account of the three phases which he introduced into the fourth edition of the Eléments at the same time as writing those words, is the last sentence in which Walras mentioned them in any of his writings. His references to written pledges had become fewer and fewer in the successive pages of the 1900 revision. Finally he either forgot about them or decided they were not an idea worth pursuing any further, abandoned his attempts to change the older constructions in his treatise to accord with their use, and introduced new sub-models in which they are not used.

Second, there is ‘the static phase in which equilibrium is effectively established ab ovo as regards the quantity of productive services and products made available during the period considered, under the stipulated conditions, and without any changes in the data of the problem’ (1954, p. 319). This means that the economy ‘remains [for the time being] static because of the fact that the new capital goods play no part in the economy until later in a period subsequent to the one under consideration’ (1954, p. 283). In this postulated static equilibrium, services and nondurable consumer commodities are produced, sold and used. Walras was asserting that the result of the tatonnement in the sketch is that the market supply and demand quantities are equated in every market simultaneously, whereupon the non-virtual activities of exchange, production, consumption, saving and investment take place. He therefore believed that none of the parameters (‘the data of the problem’) of his system of equations of general equilibrium undergoes endogenously induced changes during the tatonnement. He believed that a static equilibrium is consequently the one given by the solutions to his static equation system, and that his new version of tatonnement converges to that equilibrium for the same general reasons as he had adduced in 1889.

Third, continuing to write as though the sketch were a complete functioning system, Walras indicated that it undergoes ‘a dynamic phase in which equilibrium is constantly being disturbed by changes in the data and is constantly being reestablished’ (1954, p. 319; 1900, p. 302). The new capital goods, both fixed and circulating, Walras wrote, ‘are made available during the second phase’ but ‘are not put to use until the third phase’. When they are used, however, ‘the first change in the data of our problem’ occurs (1954, p. 319). The result of changes in the data is that the ‘fixed equilibrium will then be transformed into a variable or moving equilibrium, which re-establishes itself automatically as soon as it is disturbed’(1954, p. 318). The use of the new capital goods generates economic growth.

Of course, Walras asserted that the three phases and all the behaviours and outcomes that he wanted the sketch to have, such as the equalization of supply and demand, do in fact occur in it, but in actuality the sketch does not support his claims. Those accounts are not descriptions of the sketch. They are simply postulates; they cannot be deduced from its structure and procedures, which is why Walras’s work on virtuality is properly described as a sketch rather than a model. He made the mistake of assuming that potential demanders, whether consumers or entrepreneurs, do not write pledges to buy, so they have no way of making their desired demands known. For that reason alone (although there are many others; Walker 1996, section I), the sketch cannot function. There are no individual or market demand functions, Walrasian pricing, transactions or production. Equilibrium does not exist because the number of unknowns (prices, the quantities of goods and services) exceeds the number of independent equations (the market supply functions). Moreover, the device of written pledges is so inherently flawed that there are no conceivable revisions that can transform it into a functioning system (Walker 1996, section I). Finally, it is evident that the sketch has no explanatory value in reference to the real economy.

Despite the sketch’s shortcomings, its aim of excluding irrevocable disequilibrium behaviour from a general equilibrium model, achieved simply as a postulate, nevertheless became a central and indispensable part of Walras’s intellectual legacy for certain of his successors, including Gustav Cassel, Abraham Wald, John von Neumann, Kenneth Arrow, Frank Hahn and Gérard Debreu (Walker 2006a, pp. 288–312). It is a pity that they adopted his virtual notion and were unaware of or disregarded his robust and more realistic mature comprehensive model, for through the development of many of its characteristics lies the way to a more useful general equilibrium theory, as recent research has shown (Walker 2006a, pp. 334–6; 2006b).

Commodity E in the Last Model of the Production and Pricing of Capital Goods

In 1900 Walras invented a fictional good (E) constituted of perpetual annuity shares, another example of the deterioration of the quality of his modelling after 1899. It was by means of that concept that he chose to deal with a positive, zero or negative excess of aggregate income over aggregate consumption in his capital-goods model. It appears that he wished to express aggregate savings as a single homogenous quantity – the demand for E. Each perpetual share pays one unit of numeraire per year, and its price, determined by supply and demand, is the reciprocal of the rate of interest (Walras 1954, pp. 274–6, 308–9). If people want an additional amount of interest income, they provide savings through purchasing new perpetual shares. The numeraire-capital that capitalists pay for units of E is used by entrepreneurs to buy productive services and materials that are transformed into new capital goods).Walras viewed the capital-goods markets as reaching equilibrium through adjustments of the goods’ rates of return (1954, pp. 275–6, 308).

Walras’s device of the E commodity, which he frankly described as imaginary (1954, p. 274), has not been adopted by economists who succeeded him, because of its remoteness from the realities of capital accumulation and the distortions that it creates in a model of that process. He did not realize that it is incompatible with the written pledges procedure. If the latter is assumed to occur, the capitalists have no way of expressing their demand for E. Moreover, as it happens, Walras retained his references to the purchasing of stock certificates and private and government bonds which appear in the mature capital goods model (for example, 1954, §§ 255, 270), and he introduced short-term loans into his new money model, although the incompatibility of E with those financial instruments increases the incoherence of the last two editions of the Eléments. His treatment of saving, investment and the capital goods market in the mature comprehensive model is superior to his final thoughts on the subject.

Walras’s Last Model of Money and Circulating Capital

In his revision of the Eléments in 1900, Walras stated that he wanted to design the structure of this model on ‘exactly the same terms and in precisely the same way’ as in the 1900 models of exchange, production and capital formation (1900, p. 42). He did not in fact do that because he mentioned written pledges only twice in the first half of the exposition of the model and not at all in the second half of it. In fact, he constructed the new model of money and real circulating capital before he thought of written pledges. He first presented it in an article in 1899. After the last page of the article, he added an afterthought, a note of 37 lines introducing the device of written pledges (1899, p. 103). He subsequently inserted most of the note into the text of the article, and inserted the result almost verbatim into the 1900 edition of the Eléments (1954, lessons 29, 30), completely eliminating the treatment of money that he had presented in the mature comprehensive model.

In contrast with his mature comprehensive model, Walras described the functions of money and the formation of money prices in his last model of general equilibrium on the assumption that there is no uncertainty in equilibrium, and consequently that the dates and monetary value of future purchases and sales are known (1954, pp. 317–18). Money is one type of circulating capital, he explained; the other is circulating physical capital. Replacing his formulation of an equation of exchange that had anticipated Irving Fisher’s (1877b, pp. 180–81), Walras asserted that circulating physical capital yields utility from its ‘service of availability’ -that is, by being readily available -and money provides, by proxy, the same service of availability as the good that it is destined to purchase and yields the same utility as that service. All economic agents try to hold utility-maximizing amounts of money and circulating physical capital (1954, pp. 320–1). The latter, held by consumers and entrepreneurs, is acquired with money, so the essential concern of Walras’s model of circulating capital reduces to the question of the demand for and supply of money and its price.

Savers make some of their balances available as loans through buying or perpetual annuity shares (1954, pp. 318–20). The aggregate gross supply of money is the total stock issued by the monetary authority in the case of a fiat money economy, and is the amount of circulating coin in the case of a commodity-money economy (1954, pp. 321–4). The price of money, the numeraire, is unity and the price of its service is the rate of interest (1954, pp. 320, 327). Given the flows of receipts and purchases, the individual gross and net demand for cash balances and the individual net supply of them are functions of the rate of interest. The sum of the individual net demands for money is the aggregate demand function, and the sum of the individual net supplies of money is the aggregate supply function (1954, pp. 320–1).

The tatonnement in the money market, Walras contended, explains how the rate of interest and the equilibrium aggregate net quantities of cash balances supplied and demanded are determined (1954, pp. 325, 327). The rate of interest changes according to the Walrasian pricing rule. When the excess quantity demanded of cash balances is positive, the rise in the rate decreases the quantity demanded of cash balances by consumers and entrepreneurs by increasing the cost of the service of availability that money provides, and also decreases the quantity demanded by entrepreneurs by causing a fall in profits and hence in the desired rate of production. The rise in the rate of interest also causes the net quantity of cash balances that savers want to supply to increase. If the desired supply of cash exceeds the desired demand at the current rate of interest, the opposite effects occur (1954, p. 333). The tatonnement continues until the equilibrium price of the service of availability of money is found -namely, the price that equates the net and therefore the gross quantities supplied and demanded of cash balances – whereupon the money market reaches equilibrium (1899, p. 96; 1900, pp. 297–319; 1954, pp. 315–33).

The equilibrium prices of all goods in terms of money are given by its role as the numeraire and by the workings of the entire model that determine the ratio of exchange between each good and the numeraire. In general equilibrium, the price of the service of all money held by different individuals for different purposes is the same (1954, p. 326). Moreover, because an underlying influence upon the rate of interest on money is the value productivity of physical capital, an influence exerted through variations in the volume of funds invested, the equilibrium rate on money is the same as the equilibrium rate of net income determined in the market for capital. There is therefore equality in the rate of net income from all capital goods and real and monetary circulating capital (1954, p. 323).

Walras then considered the comparative statics of the model. He changed the utility functions for the service of money and deduced that the marginal utility and value of the service of money changes in the same direction. He changed the quantity of money and deduced that the marginal utility and value of the service of money changes in the opposite direction, and that all prices change in the same direction without any alterations in relative prices (1954, p. 333). He noted that, if the utility curves for net income shift up or down, the rate of net income changes in the opposite direction. If the quantity of net income varies, the rate of net income varies in the same direction. If utility functions and the quantity of net income both vary in such a way that the marginal utilities remain unchanged, the rate of net income also remains unchanged (1954, p. 307).

An aura of unreality is imparted to Walras’s 1900 edition of the Eléments by his abstracting from money through much of his exposition of exchange, production and capital formation, and then by introducing it in such a way that it does not change their characteristics (1954, pp. 319–24). In particular, by postulating that there is no uncertainty in his last model of money and circulating capital, without the slightest explanation of how that would be possible, he eliminated consideration of much of the behaviour associated with money in the real economy. Money does, however, influence a great deal of real economic behaviour in special ways associated with uncertainty, a fact of which Walras’s extensive writings on real monetary arrangements, problems and policies reveal him to have been perfectly cognizant. Moreover, his concept of fictional perpetual annuity shares is a superfluity that further detracts from the verisimilitude of his models of capital formation and money. He should instead have retained his mature model of the money market in which he dealt with the behaviour related to some of the major financial assets in which people actually invest.

Economic Policies

Walras developed all his policy proposals during the years prior to his last theoretical efforts. He never mentioned the latter formulations in connection with real empirical matters. In particular, the written pledges sketch did not influence him to modify or innovate policy proposals, necessarily so because the functioning and hence the problems of the economy are not virtual.

Walras was greatly interested in the economic problems of his day and in socioeconomic reform, guided in his major policy proposals by his normative convictions, which were derived from his father’s philosophy of society and justice. Those convictions were a mixture of conventional nineteenth-century liberalism and notions of the rightness and efficacy of state interventionism (1896b). Like many scholars, each with different views, Walras bestowed the title of ‘natural law’ upon the principles of justice that he considered desirable, and so he might be called a natural-law philosopher or casuist. Nevertheless, he was not a natural-law economist. He did not believe that there is, behind observable facts, a structure of economic laws that are divinely ordained, or that are peculiarly in tune with the structure of the universe and human aspirations, or are ceaselessly at work so that violations of them can only result in chaos or frictions. Nor did he construct his economic model with the conscious intention of expressing his normative views. Sharply distinguishing normative and positive economics, he stated that he designed his theories for the purpose of understanding economic reality (Walker 1984a, 2006a) and presented his normative work explicitly as such and carefully segregated it (Walras 1896b) from the publications presenting his economic theories.

Walras’s policy recommendations ranged over natural monopolies, which he believed should be nationalized; prices, which he believed should be stabilized by a monetary authority; bimetallism, which he believed had both advantages and disadvantages; the stock market, which he believed should be regulated by the state in order to improve its organization and ensure its integrity; taxes, which he believed were unjust and confiscatory and should be abolished; and land, which he believed should be purchased by the state and rented to private users, thereby providing it with revenue (1905b, pp. 272–3). Arguing that his advocacy of nationalization of land and natural monopolies was based upon a scientific understanding of the functioning of the economy, Walras called himself a ‘scientific socialist’.

Contributions

Criticisms of Walras’s work cannot obscure the greatness of his contributions. When he began his investigations in 1868, economics on the Continent was hardly a scientific pursuit but rather a mixture of normative prescriptions, classical theories expressed alongside protectionist doctrines, and commercial law. In England it was in the state exemplified by the work of J.S. Mill –with much that could be used as a basis for future investigations, but also without a clear view of the relationships of distribution and production, limited by a cost-of-production theory of value, and lacking a theory of supply and demand in multiple markets. The attitude of most of Walras’s contemporaries was that, since economic behaviour involves preferences and the human will, it cannot be expressed in a rigid and deterministic set of algebraic relations. Walras changed all that, transforming economics and propelling it forward in a gigantic intellectual leap.

His contribution can be divided into two interrelated parts. One is that, in his mature comprehensive model, he constructed or refined or adapted to his purposes many of what became the fundamental building blocks of modern economic theory. In this effort he accomplished an enormous amount of highly creative economic analysis, brilliantly analysing the structure of economic reality to bring many of its essential features into clear relief, in eight major original contributions. First, he went far beyond the work of the other developers of the marginal utility theory by using it to analyse the disequilibrium and equilibrium behaviour of a variety of participants undertaking different economic functions in multiple markets, rather than confining the theory to the investigation of consumer demand and of exchange in an isolated market. Second, he had clear priority in constructing the theory of exchange in multiple competitive markets. In that regard, his work was greatly in advance of his predecessors’ and was replete with fruitful constructions, theorems and postulates, like the reciprocal relation of supply and demand, the device of a numeraire, the individual budget equation, Walras’s Law, the theorem of equivalent distributions, and the laws of change of prices. Third, he constructed a theory of the firm and of market supply in which he appears to have developed independently the modern idea of a firm’s production function, derived the equation for a firm’s average cost, expressed the firm’s offer of output mathematically, and aggregated the firm’s supply functions to obtain the market supply in a particular industry. Fourth, he was the first to examine the question of the existence of equilibrium in a competitive multi-market system of exchange and production. Fifth, in his work on tatonnement he initiated the study of the stability of competitive general equilibrium and contributed significantly to its understanding, with his most successful theorizing on the topic relating to his mature comprehensive model. There is nothing in the literature before Walras’s time or until the time that his work was used by others that is even remotely like or on the level of his reasoning regarding the process of convergence to equilibrium of a non-virtual competitive multi-market system. Sixth, he developed a theory of the entrepreneur, of profits, and of the allocation of resources that became the basis of Continental work on those topics (Pareto 1896–7, passim; Pareto 1906, passim; Barone 1896, p. 145; Schumpeter 1912/1926, p. 76; Schumpeter 1954, p. 893; Walker 1986). Seventh, Walras created a fruitful theory of capital, achieving an early formulation of the conditions for a Pareto optimum in capital markets. As in a number of his other investigations, his characteristic contribution in that regard was not to be the first to think of the problem but to be the first to offer an account of those markets’ disequilibrium interrelationships and equilibrium conditions in a model of the general equilibrium of an economy. Eighth, he developed a cash-balances theory of money in his period of maturity as a theoretician which had great originality and has stimulated much valuable research (Marget 1931, 1935; and see Walker 1970, p. 696; 1996, pp. 235–55). Those eight areas of analysis were the core of neoclassical microeconomic theory and thus constituted much of the structure of knowledge that was the starting place for twentieth-century economics.

The second part of Walras’s contribution was not the idea of the general equilibrium of a freely competitive economic system, which other economists had suggested; it was his implementation of that idea. Other economists had helped in fashioning the building blocks that Walras used. His achievement, however, was not only to develop them but also, through incisive analysis and constructive thought, to weave them into an account of the equilibrating processes of a complex, non-virtual, multi-market economy. Those building blocks dealt with real economic behaviour, and it is his use of building blocks with that essential quality that gives his work its richness, robustness and relevance. Walras was also the first economist to try to set up a system of equations to describe the conditions of static equilibrium of a general equilibrium model.

Walras thus accomplished by the mid-1870s far more than any other economist had done in regard to constructing a model of an economic system as a whole, and more single-handedly in that regard than any other economist in the history of the discipline.

Influence

Walras’s work was not given the recognition that it merited in France during the 25 years after 1874, and his centennial, in 1934, elicited no conference on his work there. By the 1950s, however, the French attitude toward Walras had changed, as was ultimately symbolized by the creation in 1984 of the Centre Auguste et Léon Walras (but no longer symbolized in that way, for the research group has a new name; it is now a part of the organization known as ‘TRIANGLE, Unité mixte de Recherche 5206 du Centre national de la Recherche scientifique’) at the université Lumière Lyon 2. With the English, Walras’s experience was also disappointing. His initial cordiality towards W.S. Jevons, as a fellow pioneer in mathematical economics, was dissipated by Jevons’s failure to recognize Walras’s contributions to the theory of exchange and to the construction of a relatively complete model of a competitive economy, and eventually Walras, quite unreasonably, came to regard Jevons as a plagiarist of his work (Walras to M. Pantaleoni, 17 August 1889, L 909). Similarly, Walras’s relations with P.H. Wicksteed began well (Wicksteed to Walras, 1 December 1884, L 619) but deteriorated sharply when Wicksteed failed to give credit to those whom Walras considered to be the true originators of the theory of marginal productivity (Walras 1965, L 1220, n. 3; 1896a, pp. 490–2). Walras justly felt neglected by Alfred Marshall, who mentioned him only thrice in the briefest of comments in the Principles (Marshall 1890, 1920) and wrote not a word about Walras’s development of general equilibrium theory. Walras also came to dislike Edgeworth for criticizing his theories of tatonnement, capital goods and the entrepreneur (Walras to Gide, 3 November 1889, L 933, and 11 April 1891, L 1000; Walras to Pantaleoni, 5 January 1890, L 953). In general, Walras believed, the English had closed their minds to his theories and had become spiteful in their treatment of them (see Walker 1970, pp. 699–70).

The extremity of the language with which Walras characterized the English was unjustified, because, although he had reason for disappointment with their neglect of his general equilibrium theory, Jevons (1879, preface) and Edgeworth (1889) had recognized valuable elements in his work, and he was the only living economist included in the first edition of Palgrave’s Dictionary of Political Economy (Sanger 1899). The fact is that Walras grew hypersensitive about the motives of his critics, the failure of the majority of economists to recognize the value and priority of his contributions, and the possibility of plagiarism of his ideas during the 1880s and 1890s. There had been two periods in his life, he complained, ‘one during which I was a madman, and one during which everyone made my discoveries before me’ (Walras, undated, in Jaffé 1983, p. 203, n.54).

This account of Walras’s disappointments should be balanced by a realization that his scientific labours had afforded him ‘up to a certain point, pleasures and joys like those that religion provides to the faithful’ (Walras to Marie de Sainte Beuve, 15 December 1899, L 1432), and a recognition of the professional satisfactions that he increasingly experienced in the last two decades of his life. Maffeo Pantaleoni (1889), Enrico Barone (1895, in Jaffé 1983, p. 186; 1896), and Vilfredo Pareto (Pareto to Walras, 15 October 1892, L 1077) contributed greatly toward giving Walras’s work a secure place in Continental economics and thus ultimately in economics everywhere. In 1895 Pareto’s appointment as Walras’s successor to the chair of economics at Lausanne assured Walras that his doctrines, expressed in his treatment of a non-virtual competitive economy, would be perpetuated and developed, and the accessible literary presentations of Walras’s ideas in Pareto’s books (1896–7; 1906) began their widespread dissemination. Pareto borrowed most of the ideas of Walras that have been mentioned in this article, using them as the basis for his contributions to the theories of non-virtual general equilibrium, the monopolistic entrepreneur, and welfare economics. Wilhelm Lexis, Ladislaus von Bortkiewicz and Eugen von Bohm-Bawerk gave Walras’s models serious attention. Knut Wicksell based his theory of price determination squarely upon Walras’s work (Wicksell to Walras, 6 November 1893, L 1168), and Karl Gustav Cassel was inspired in the construction of his models (1903, 1918) by Walras’s equation system and idea of virtuality expressed in the written pledges sketch. Walras was given recognition in the United States: in 1892 he was made an honorary member of the American Economic Association, Irving Fisher praised his work (Fisher 1892, p. 45; 1896), H.L. Moore became his avowed disciple and explicator (Moore to Walras, 19 May 1909, enclosure to L 1747; Moore 1929), and Henry Schultz taught Walras’s economics and, like Moore, undertook theoretical and econometric studies of general equilibrium relationships (for example, Schultz 1929, 1932, 1933). His mature comprehensive model was the starting point for the work of Joseph Schumpeter, who, throughout his entire career, devoted himself to the study of disequilibrium and general equilibration of non-virtual economic phenomena.

The manifestations of acceptance led Walras to believe he would ultimately triumph, and that enabled him to achieve a mental calmness (Walras to Marie de Sainte Beuve, 15 December 1899, L 1432; Walras to A. Aupetit, 28 May 1901, L 1485). ‘Be assured of my serenity’, he wrote to old friends in 1904, ‘I have not the least doubt about the future of my method and even of my doctrine; but I know that success of this sort does not become clearly apparent until after the death of the author’ (Walras to G. and L. Renard, 4 June 1904, L 1574). Walras’s prediction of great success was accurate. An indication of what the future would hold for his theories was given by the celebration of his jubilee in 1909 by the University of Lausanne, in the course of which he was honoured as the first economist to establish the conditions of general equilibrium, thus founding the School of Lausanne (Walras 1965, L 1696, n. 5). His achievements were praised in a statement signed by 15 leading French scholars, including Charles Gide, Charles Rist, Georges Renard, Alfred Bonnet, Albert Aupetit and François Simiand (Walras 1965, enclosure to L 1747), and in communications from many others (Pareto to the Dean of the Faculty of Law of the University of Lausanne, 6 June 1909, L 1755; Schumpeter to Walras, 7 June 1909, L 1756).

Walras’s contributions inspired and provided a substantial beginning for the branches of general equilibrium theory and applications as they have developed since the nineteenth century. Indeed, the filiations of his ideas have become so numerous and dense as to be an integral and central part of the mainstream of modern economics. His achievement of developing particular theories and binding them together in a model of an entire economic system has given his work an influence on the verbal, mathematical and econometric study of the interrelationships of all parts of economic systems that has been durable and immense. For sheer genius and intuitive power in penetrating the veil of the chaos of immediately perceived experience and divining the underlying structure of fundamental economic relationships and their extensive interdependencies and consequences, Walras has been surpassed by no one.

See Also

• General Equilibrium

• Tâtonnement and Recontracting

Selected Works

• 1858. Francis Sauveur. Paris: E. Dentu.

• 1860. L’Economie politique et la justice. Paris: Guillaumin.

• 1868. Recherche de l’idéal social: Leçons publiques faites à Paris. Paris: Guillaumin; and in Walras (1896a).

• 1869–70. 2^e tentative, 1869–1870. In Walras (1965), vol. 1, 218–219.

• 1871. Discours d’installation. Séance académique du 20 Octobre. Lausanne: Académie de Lausanne.

• 1877a. Théorie mathématique de la richesse sociale: quatre mémoires. Paris: Guillaumin.

• 1874, 1877b. Eléments d’économie politique pure; ou théorie de la richesse sociale. 2 parts. Lausanne/Paris/Bâsle: L. Corbaz/Guillaumin/H. Georg.

• 1880a, 1880b. La bourse, la spéculation et l’agiotage. Bibliothéque Universelle et Revue Suisse 5: March 1880a, 452–476; 6: April 1880b, 66–94.

• 1886. Théorie de la monnaie. Lausanne: Corbaz.

• 1889. Eléments d’économie politique pure, 2nd edn. Lausanne/Paris/Leipzig: F. Rouge/Guillaumin/Duncker & Humblot.

• 1893, 1904, 1909. Notice autobiographique. In Walras (1965), vol. 1, 1–15.

• 1895. Enclosure by Léon Walras to Vilfredo Pareto. In Walras (1965), vol. 2, 628–632.

• 1896a. Eléments d’économie politique pure, 3rd edn. Lausanne/Paris/Leipzig: F. Rouge/F. Pichon/Duncker & Humblot.

• 1896b. Etudes d’économie sociale (Théorie de la réparation de la richesse sociale). Lausanne/Paris: F. Rouge/F. Pichon.

• 1898. Etudes d’économie politique appliquée (Théorie de la production de la richesse sociale). Lausanne/Paris: F. Rouge/F. Pichon.

• 1899. Equations de la circulation. Bulletin de la Société Vaudoise des Sciences Naturelles, June, 85–102 and note, 103.

• 1900. Eléments d’économie politique pure, 4th edn. Lausanne/Paris: F. Rouge/F. Pichon.

• 1905a. Cournot et l’économique mathématique. Gazette de Lausanne, 13 July; and, except for a few deletions, in Revue d’Economie Politique, January–February, 1962.

• 1905b. Draft of a recommendation for a Nobel Peace Prize. In Walras (1965), vol. 3, 270–274.

• 1926. Eléments d’économie politique pure. Definitive edn. Paris/Lausanne: R. Picon & R. Durand-Auzias/F. Rouge.

• 1954. Elements of pure economics. Trans. and annotated by William Jaffé. Homewood/London: Richard D. Irwin/Allen & Unwin.

• 1965. Correspondence of Léon Walras and related papers, 3 vols, ed. W. Jaffé. Amsterdam: North-Holland.