The elements of pure economics

Five editions: 1874-77,1889,1896,1900 and 1926

Leon Walras published four editions of the Elements d'economie politique pure and in 1902 he prepared some corrections for the final edition that his daughter Aline published in 1926.^[107] Let us first briefly introduce the contents of the first edi­tion of the Elements (1874-77).

(i) Section I, “Object and divisions of political and social economy”, deals with

the different definitions of political economy and the distinctions between sci­ence, art and ethics. Then it presents the definition of social wealth as a set of rare things (i.e. useful and available in limited quantities) and focuses on the general fact of exchange value.

Walras presents three models of general equilibrium successively: one pure exchange model, a second model with production, and a third with capitalisa­tion and credit. Why begin with exchange rather than production? In his “Notes d’humeur”, the author observes in this regard: “economists generally think that since things are produced before they are traded, production must be studied before exchange. Mistake. Things are exchanged during production or before as produc­tive services” (2000, 564). General equilibrium models with production therefore constitute extensions of pure exchange theory.

(ii) Section II, “Mathematical theory of exchange”, considers the exchange of two commodities for each other and the exchange of several commodities for each other (general equilibrium). It should be noted that Walras, unlike William Stanley Jevons (Theory of Political Economy, 1871), does not focus on marginal utility theory and rejects the calculations of pleasure and pain of those whom he calls the “utilitarians” [utilitaires]. He does not focus on the individual, on his satisfaction and consumption, but on the social phe­nomenon of the market.

Walras acknowledged the precedence of Jevons in the theory of exchange of two commodities, and more specifically the “equation of maximum utility in exchange” (1900, Preface to the 4th edition, 5; 1954, 37). However, he considered that Jevons did not draw the “effective demand equation” from the maximisation of utility and therefore did not solve the problem of the determination of equilibrium prices.

(iii) Section III, “Of numeraire and money”. In 1874, the theory of money contains an equation that anticipates Irving Fisher’s equation of exchange (1874-77, 30th lesson, “Problem of the value of money”, 468-72).^[110]

The second part of the Elements was not published until September 1877. It com­prises the following three sections: section IV, “Natural theory of the production and the consumption of wealth”; section V, “Conditions and consequences of eco­nomic progress”; and section VI, “Natural and necessary effects of the various modes of the economic organization of society”.

Walras placed his theory of production, which sets out the second general equi­librium model, in section IV. Unsatisfied with his hypothesis of “fixed coefficients of production”,^[111] he consulted Hermann Amstein, professor of analysis and mechan­ics at the Academy of Lausanne on the problem of their variability.

In section V of the Elements, whose content is very heterogeneous, Walras started by presenting his theory of capitalisation and credit, which contains the third general equilibrium model. In this model, a market for new capital proper is introduced. However, by claiming to no longer ignore the “accessory phenom­ena”, Walras was taking a step towards the integration of time: “we pass from the hypothesis of a market that is held continuously, to that of a market that is held periodically, we could say once per day, we will say rather once per year to take better account of the renewal of the seasons” (1877, 35th lesson, 576; 2014, 315). But we must go even further: “in order to approach closer and closer to the reality of things, we must still pass from the hypothesis of a periodic annual market to the hypothesis of a permanent market” (1877, 579; 2014, 317). He would later clarify that this was the transition from “the static to the dynamic state” (1900, 579; 1954, 380). This “permanent market” always tends to equilibrium, but never reaches it, just like a “lake agitated by the wind, where the water is incessantly seeking its level without ever reaching it” (1877, 580; 1954, 380). But it is necessary to place ourselves within the framework of a “progressive society”, that is to say a society in which capital and population increase. In this perspective, Walras introduces the important distinction between “economic progress” (when the value of the coef­ficients of production varies through the substitution of capital proper for land) and “technical progress” (when the coefficients of production change because of the withdrawal and arrival of certain productive services) (1877, 36th lesson, 585; 2014, 321).

Finally, in section VI, Walras deals with monopoly theory, but also with differ­ent modes of taxation.

One of the main reasons that led Walras to prepare a second edition of the Ele­ments (1889) was the evolution of his monetary thought, as set out in his Theorie de la monnaie (1886), the substance of which he incorporated in the second edi­tion. His monetary theory was now based on the concept of demand for “desired cash balances” [encaisses desirees], with the use of an equation similar to the Cambridge equation. Consumers and producers want cash. Money is considered as a circulating capital that provides consumers and producers with a “service of availability” [service d’approvisionnement] whose price is the interest rate. The order and number of sections were also reorganised in the second edition: the subject and divisions of political and social economy, exchange, production, capi­talisation and credit, and then money. The author introduced improvements in the theories of exchange, production, capitalisation and credit.

The third edition of the Elements (1896) does not contain any major changes. However, it is marked by the withdrawal of four lessons on money (lessons 37 to 40 in the 2nd edition), which would be partly reused in the Etudes d’economie politique appliquee, and by the inclusion of three appendices. Appendix III, “Note on Mr Wicksteed’s refutation of the English theory of rent” (written in 1894-95), contains Walras’s first consideration of the theory of marginal factor productivity, where the variability of the coefficients of production reappears. This text was written after reading Philip Wicksteed’s book, An Essay on the Co-ordination of the Laws of Distribution (1894) and a note written by Enrico Barone, “Sopra un recento libro del Wicksteed”, and after exchanging letters with Enrico Barone and Vilfredo Pareto.

The fourth edition of the Elements (1900) reveals substantial changes. Walras now integrated desired cash balances and circulating capital equations into a fourth general equilibrium model. The introduction of cash demand in the individual util­ity functions marks, in his view, the completion of “economic statics”. Appendix III disappears and the theory of marginal productivity is taken into account beyond the four general equilibrium models in section VII (“Conditions and consequences of economic progress”) in the 36th lesson (1900, § 326, 586-91; 1954, 384-6),^[113] where the author discusses the possibility in a progressive society of using less and less land and more and more personal capital and capital proper. Indeed, Walras admitted that he preferred not to introduce marginal productivity into his “general theory of economic equilibrium, for fear that the general theory, which was already complicated enough, might then be too difficult to grasp in its entirety” (1900, 36th lesson, 588; 1954, 385). In the fourth edition of the Elements, he further modified the sections. He duplicated his presentation of the theory of exchange: first two commodities, then several commodities. But he also revised the tatonnement pro­cedure in models with production in order to achieve, as in the exchange model, a virtual equilibrium.

General equilibrium in a pure exchange economy: the role of the price system and tatonnement

For the setting of market prices in the theory of exchange, Walras describes an oral tatonnement (trial and error process), performed by crying out prices and associated quantities. This procedure of establishing market equilibrium prices by trial and error then enables economic agents to carry out their buying and selling. Indeed, in the context of this tatonnement, no transaction can take place before the equilibrium prices are obtained. Let us summarise his argument.

Walras defines the market in a very simple way: “The market is a place where commodities are exchanged” (1874, 5th lesson, 70; 2014, 42).

The markets that are best organized in regard to competition are those in which purchases and sales are made by the crying out of prices, through the intermediation of agents such as floor traders [agents de change], commercial agents, criers, who centralize transactions in such a way that no transaction takes place without the conditions being announced and known and without the sellers being able to lower their prices and buyers to raise them.

(1874, 5th lesson, 70; 2014, 42)

In these organised markets, such as the Stock Exchange, the physical presence of real buyers and sellers is not necessary, because the stockbrokers or other market agents centralise the purchase and sale orders. Walras gives some examples: Stock Exchange, commercial Bourse, grain Bourse, fish markets.

Walras points out that there are other markets in which competition is still work­ing fairly well, even if it is “less well regulated”: these include fruit and vegetable markets and poultry markets (1874, 5th lesson, 70).

Third, there are other markets where the organisation turns to be “a little more flawed”, where the competition is less direct: the streets of the cities where you will find bakers, butchers, grocers, etc. (1874, 5th lesson, 70-1). Walras also points out that “[i]t is similarly incontestable that competition presides in the determina­tion of the value of doctors’ and lawyers’ consultations, of musicians’ and singers’ recitals, etc.” (1874, 5th lesson, 71; 2014, 43). Before, he delivered a first key definition:

Value in exchange left to itself occurs naturally in the market under the regime of competition. As buyers, the traders make offers to buy at higher prices; as sellers, they make offers to sell at lower prices, and their competi­tion thus leads to a certain value in exchange that is sometimes rising, some­times falling, sometimes stationary.

(1874, 5th lesson, 70; 2014, 42).

However, in order to achieve the “ideal type” of the market of “absolutely free competition”, the “real type” must first be established. To find it, Walras turns to the best-organised market, the Stock Exchange, which is in fact the “typical market [marche type]” in his eyes. He takes the example of French government bonds in the Paris Stock Exchange [Bourse de Paris], but his representation of a session a la criee of an important quotation in the nineteenth century does not aim at a rigorous description of its functioning. Walras invites us to attend a session of the Paris Stock Exchange, for the cash market, where 3% French government bonds (rente a 3 %) are negotiated. The 3% French government bond has an opening price of 60 francs. On the one hand, the stockbrokers have orders to sell “at best possible price” (whatever that may be) and “at limited price” (60 F, 59 F, 58 F, and so on), and therefore offer a certain quantity of bonds, and on the other hand, they have orders to buy “at best possible price” and “at limited price” (60 F, 61 F, 62 F, and so on), and therefore demand a certain quantity of bonds.

(i) First case: if demand is higher than supply, the initiative will come from the long side of the market, from the demand. The agents who have orders to pur­chase at best price will overbid. The purchasers limited to a price at 60 F, 61 F, etc., will gradually withdraw; sellers attracted by the rise will offer bonds dur­ing the session. The result will be a gradual reduction in the spread between the quantities offered and demanded and the establishment of an equilibrium at a higher price than at the opening of the stock exchange.

(ii) Second case: if supply is higher than demand, the initiative will come from the supply side. The agents who have orders to sell at best price will underbid. The sellers at 60 F, 59 F, etc. gradually withdraw; purchasers attracted by the fall will demand bonds during the session. The result will be a gradual reduction in the spread between the quantities offered and demanded and the establishment of an equilibrium at a lower price than at the opening of the stock exchange.

When the agents do not find their counterpart in the market, “theoretically, trading must be suspended” (1889, 5th lesson, 72; 2014, 44) - this clarification, added by Walras to the 2nd edition of the Elements, is a response to the objection formu­lated by Joseph Bertrand in his review of Cournot’s Recherches sur les principes mathematiques de la theorie des richesses (1838) and Walras’s Theorie mathe- matique de la richesse sociale (1883), published in the Journal des savants, in 1883.^[114]

For Walras, the market tends towards the “stationary state or equilibrium of the market” (1874, 5th lesson, 71; 2014, 44) which is maintained as long as market conditions do not change. The price is known to all agents, who then carry out their transactions. The closer one gets to a perfectly organised market, the closer one gets to the true exchange value, and reasoning on a typical real market is a step towards this determination that will be only perfect in the ideal.

The Walrasian model is therefore a price adjustment model. Buyers overbid [demandent a l'enchere] when demand is higher than supply, while sellers under­bid [offrent au rabais] when supply is higher than demand. The price changes in the same direction as the sign of excess demand. Therefore, Walras does not define “free competition” in terms of market structure, nor by a list of conditions of “pure and perfect competition”, as in traditional microeconomics, but as a specific kind of behaviour by buyers and sellers confronted with a maladjustment between sup­ply and demand. Contrary to what the commentators say, there is no question of an “auctioneer” to achieve the adjustment between supply and demand. So how did the “auctioneer” figure emerge with regard to the bidding mechanism in Walras? As Paul A. Samuelson explained to Donald A. Walker:

Around 1935 Schumpeter’s lectures at Harvard used to speak of “an angel of the marketplace” or “a Walrasian angel” or “a Walrasian auctioneer”, who lowered price systematically in some proportion to the excess of supply over demand. In my 1941 MIT lectures, I continued the rhetoric (26 September, 1991).

(Walker 2019, 152)

In the literature devoted to the Walrasian general equilibrium, the figure of the “auctioneer” does not appear before World War II. Why did it then emerge in the 1950s? One can mention the impact of the English translation of the Elements by William Jaffe (Walras 1954, 84, 106), in which Walras’s “crieurs” are sometimes translated as “criers” and sometimes as “auctioneers”, and the expression “a la criee” is translated as “by auction” (See Walker 1996, 82 and 84-5; Translators’ introduction, in Walras 2014, XXXV-XXXVI).¹³

From real typical market to ideal typical market: the role of formalisation

However, one cannot conduct analytical reasoning on the real typical market (pub­lic bonds stock exchange). It may be a synthesis, but it remains empirical, and the phenomena that are played out therefore have both the extreme complexity of concrete reality and its necessary imperfection. So the real must be analysed, that is to say, broken down into simple ideal elements; we must define them and reason on them in a deductive way by mathematical formalisation, and rebuild by succes­sive complexifications in order to find again in the world of ideas the equivalent of the complex real type, but ideal, perfect. Thus, Walras asserts: “Let us, therefore, retrace our steps..., let us take any two commodities, say oats and wheat, or, designate them even more abstractly as (A) and (B)” (1874, 5th lesson, 74; 2014, 46). In a market, in the absence of money (and of numeraire), agents are willing to sell commodity (B) to obtain commodity (A). At the closing price of the previous market, an agent agrees to supply n units of (B) for m units of (A). Here Walras poses the individual “equation of exchange”:

where v_a is the exchange value of one unit of (A) and v_b the exchange value of one unit of (B). In other words, the value of the quantity demanded of (A) is equal to the value of the quantity supplied of (B). Today, one would refer to budgetary constraint.

13 To our knowledge, the oldest use of the term “auctioneer” in the English-language literature about Walrasian general equilibrium appears in the handbook Microeconomic Theory by Henderson and Quandt (1958, 95 and 113). They speak about an “auctioneer” on each market. According to Clower and Howitt (1995, 31 note 1), the concept of “auctioneer” could have been invented by Richard Quandt in the United States; it was used in Northwestern University in 1959 and 1960.

When the price increases in the market, each agent reduces his demand, which can be written as a function of the price. The demand of agent 1, d_a1, is a decreasing function f_a1 of the price of the good: d_a1 = f_a1 (p_a ).

By aggregating, the total demand for commodity (A) in exchange for (B) is written (1874, 6th lesson, 85; 2014, 56):

Similarly, total demand for commodity (B) in exchange for (A) is:

The corresponding total supply equations can be then written:

To ensure the coherence of the price system, the Walrasian solution is to intro­duce a numeraire, which in the absence of money allows relative prices to be stated in the same units. The price of the numeraire is 1 by convention. The numeraire is any good (e.g. wheat) and it serves as a unit of account. This solution reduces the number of relative prices to m - 1. We have:

Let us consider the exchange of m commodities, A, B, C, D,... whose prices in numeraire are p_b,p_c,p_d,∙∙∙ if A is the numeraire good (p_a = 1 ). Because of the individual budgetary constraints, the sum of each individual’s excess demand is equal to zero, regardless of the price system. Let X, Y, Z, W,... denote the total excess demands of A, B, C, D,... (the algebraic sum of the demands of certain individuals and the supplies of others). One can then state what Walras names the “equation of the equivalence of quantities exchanged” (1874 and 1889, 12th les­son, 176 and 179; 2014, 133):

For all his models, Walras indicates:

to demonstrate that commodity prices, which are quantities... result effec­tively from such and such data or conditions, it is absolutely indispensable: 1° to formulate, on the bases of those data and conditions, a system of equa­tions, rigorously equal in number to the number of unknowns, of which the quantities in question are the roots; and 2° to prove that the interlinked phe­nomena in the real economy constitute indeed the empirical solution of this system of equations.

(1877, 40th lesson, 651; 2014, 372-3)

Without trying to analyse Walras’s thought in light of the developments in gen­eral equilibrium theory from the 1930s onwards, it must be noted that the equality between the number of unknowns and the number of independent equations is no longer a sufficient condition for the existence of an equilibrium.

Let us note that in Walras’s models, the different markets are successively bal­anced, but each market once balanced will be again unbalanced by the variations of other prices. However, Walras asserts that these variations only generate “indirect effects” of weak extent which partially cancel each other out. In the end, the “direct effects” outweigh the “indirect effects”, so each new price system is closer to the equilibrium than the previous system (1889, 12th lesson, § 130, 195; 2014, 143).^[115]

Far from being a descriptive and realistic process, the tatonnement, realised without regard to time, is a technique of iteration allowing to reach the establish­ment of ideal equilibrium prices. This would be confirmed with the introduction in 1900 in the Elements d’economie politique pure of the tatonnement on written pledges [tatonnement sur bons] procedure.

General equilibrium with production

In the different general equilibrium models with production, Walras introduces dif­ferent capital and economic agents. There are three types of capital: landed capital, [capitaux fonciers], capital proper (or artificial capital) and personal capital (or personal faculties). Landowners dispose of the land, provide the land services (or rente) and are remunerated by the ground rent [fermage]. Capitalists dispose of capital proper, provide the capital services (or profit) and are remunerated by inter­est. Workers, who dispose of personal capital, provide personal services (or travail) and are remunerated by wages. Among the services, we must distinguish between “consumable services” that are bought by landowners, capitalists and workers for public or private consumption (use of housing, lawyers, doctors, etc.) and “pro­ductive services” that are bought by entrepreneurs and will be transformed into products (land fertility, labour of workers, use of machines, etc.). Only productive services are included in the general equilibrium models.

Entrepreneurs own neither capital nor services, nor do they have any spe­cific remuneration. Their role consists in buying the three productive services and bringing them together in various economic activities. They can cumulate their function with those of landowners, capitalists, workers, thus ensuring up to four roles, “in real life”, Walras said. They then credit themselves with a ground rent, interest and wage, which are their only normal remuneration. Entrepreneurs then “make their living not as entrepreneurs, but as landowners, workers, or capitalists in their own firms or in others” (1874, 18th lesson, 284; 2014, 210). In a letter to the American economist Francis Amasa Walker, Walras makes an interesting point:

The definition of the entrepreneur is, in my opinion, the heart of the whole economics. I consider him exclusively as the person who buys productive services in the service market and sells products in the product market, thus making a profit [benefice] or a loss.

(in Walras 1965, II, l. 800, 12 June 1887, 212)

He adds that “if he takes part as a director” in the trans­formation of services into products, he is a “worker” and no longer an “entrepre­neur”. Leon Walras distinguishes here the activity of the director in charge of the management routine and the activity of the entrepreneur. Elsewhere, he gives the example of individuals forming a limited company who “become entrepreneurs” and who nominate “one or several directors to carry out the operations of the firm” (1897b, 246-7).

In general equilibrium models with production, the entrepreneurs are at the interconnection between two markets: the market for productive services (rente, profit and travail) and the market for consumer goods (1877, 18th lesson, 281-2). The conditions of production equilibrium include the exchange equilibrium, that is to say: (1) equality between supply and effective demand of productive services and the fixing of the price for each of them; (2) equality between supply and effec­tive demand of products and the fixing of the selling price for each of them; (3) equality between the selling price of the products and the cost price of productive services.

The third condition is a characteristic of the production equilibrium, “an ideal and not a real state”, as Walras specifies. But it is the “normal state”, because in “a regime of free competition” things spontaneously tend towards this equilibrium (1877, 18th lesson, 283; 2014, 209).

If the equilibrium is not achieved, we find ourselves in one of two cases:

(i) if, for certain entrepreneurs, the selling price of the goods is higher than the cost price of productive services, a profit [benefice] appears. The branch’s entrepreneurs will then increase their production and there will be an influx of entrepreneurs to this activity, which will lead to the reduction of the initial gap.

(ii) if, for certain entrepreneurs, the selling price of the goods is less than the cost price of productive services, a loss appears. The branch’s entrepreneurs will reduce their production or even leave it, leading to a reduction in the initial gap.

Walras concludes that, in the state of production equilibrium, entrepreneurs make “neither profit nor loss [ni benefice, niperte]”; the entrepreneur’s function ceases and then we can:

make abstraction from the intervention of entrepreneurs, and consider not only the productive services as being exchanged for products and products as being exchanged for productive services, but, when all is said and done, even consider the productive services as being exchanged for each other.

(1877, 18th lesson, 284, 2014, 210)

So we return to J.-B. Say’s idea that production is a “great exchange”.

A new conception of the tatonnement

In the first three editions of the Elements, the models with production operate by means of an oral tatonnement, by crying out on the two groups of market, as in the pure exchange model. In these models, the author again asserts that the “direct effects” outweigh the “indirect effects” and thus that the system of new manufac­tured quantities and new selling prices is closer to equilibrium than the previous one (1889, 21st lesson, 318; 2014, 236). But if the product prices are not in equi­librium, not only must other prices be cried, but other quantities of products must also be manufactured. So, in the 2nd edition of the Elements, Walras explains that, given the productive-service prices that are cried, the entrepreneurs borrow the necessary quantities of these services to produce quantities of products, and then sell them on the market. As long as the general equilibrium is not achieved, the tatonnement must continue (1889, 20th lesson, 308; 2014, 229). Disequilibrium transactions are therefore possible during the tatonnement with production.

In the fourth edition of the Elements, Walras introduces a tatonnement on writ­ten pledges [tatonnement sur bons] in his models with production. Through this system of pledges for the supply of products and productive services, the agents achieve a virtual market equilibrium before starting the production process:

After certain prices for services have been cried and certain quantities of products have been manufactured, if these prices and quantities are not the equilibrium prices and quantities, it will be necessary not only to cry new prices but also to manufacture revised quantities of products. In order to work out a rigorous tatonnement... we have only to imagine, on the one hand, that entrepreneurs use pledges [bons] to represent the successive quan­tities of products which are first determined at random and then increased or decreased according as there is an excess of selling price over the cost of pro­duction or vice versa, until selling price and cost are equal; and, on the other hand, that landowners, workers and capitalists also use pledges to represent the successive quantities of services [which they offer] at prices first cried at random and then raised or lowered according as there is an excess of demand over offer or vice versa, until the two become equal.

(1900, 20th lesson, § 207, 309; 1954, 242, revised translation)

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