Economists, mathematicians and engineers

Nowadays, there is little doubt that Leon Walras’s analysis represents the main contribution of French-speaking economists to economic analysis during the period 1870-1914. However, his contemporaries in France were not aware of its importance.

During Walras’s lifetime, law faculty professors generally showed little interest in his work. From 1880 to 1910, Gide was nevertheless his main ally in the facul­ties, publishing in the Journal des economistes a review of his Theorie mathema- tique de la richesse sociale (1883). He involved Walras in the creation of the Revue d’economiepolitique in 1887 as a mere collaborator, and from then on Walras pub­lished most of his articles in this journal. But their relationship remained ambigu­ous, since Gide never understood the relevance of a mathematical demonstration in economics and concluded his 1883 review by wishing that Walras could put his equations into ordinary language. The reception of mathematicians was not much better, as they did not see the value of using mathematics in economic analysis: even engineers like Cheysson and Colson remained reluctant.

Albert Aupetit

Among the economists who relied on mathematics in their analyses, Aupetit occu­pied a singular place for two reasons. He was the foremost disciple of Walras, and he was the only one who had not received full mathematical training. In spite of this, he appeared to be a better mathematician than Walras himself. He was the first economist in France to systematically use Lagrange multipliers.

His thesis Essai sur la theorie generale de la monnaie (Aupetit 1901) consisted of a systematic analysis of monetary theory that combined theoretical research and empirical verification. Walras had shown that, if money is considered as a com­modity, the quantity theory is only roughly verified. Aupetit clarified this conclu­sion by showing that the effect on prices of a variation in the quantity of metallic money depends on the characteristics of the utility function.

To integrate international trade into his analysis, Aupetit, drawing on Pareto, did not rely on the classical theory of the distribution of precious metals but on the law of one price: this led him to assume as an equilibrium condition that the prices of each commodity, expressed in the same currency, should be the same in all coun­tries. This international levelling of the prices of goods and their services implied that the rates of return must be identical in the various countries. It thus broke with the Ricardian tradition, which considered, based on the hypothesis of immobility of factors, that these rates could even differ at equilibrium between countries.

Aupetit admitted that if money is a pure instrument of circulation with no utility of its own, then the prices of goods should be proportional to the quantity of money. But this result is only achieved in static equilibrium: he assumed that the interest rate is equal to the net rate of return. Aupetit’s analysis suggested the existence of a dynamic process: an increase in the quantity of money leads to a decrease in the price of its service, and since commodity prices do not adjust instantaneously, the interest rate falls during the transition phase. We thus end up with the traditional view where a change in the money supply first affects the interest rate and only then prices.

The empirical validation of these results raised some problems, which were dif­ficult to overcome because Aupetit did not have estimates of the money stock and income.

The engineer-economists

Among the economists who used mathematics, the largest group was that of the engineers. Gradually, several schools - the Ponts et Chaussees, the Mines, Poly­technique - established chairs of economics, and a small but active group of professors ensued, who played an increasingly influential role in the development of economic analysis in France.

Walras tried to persuade Emile Cheysson to adopt his views, but their approaches to political economy were too different for any meaningful dialogue to take place. Cheysson was a liberal, who presented himself as a disciple of Le Play and a sup­porter of social Christianity. Most of his works were empirical studies devoted to the description of family budgets. Among these texts, one stands out, in which Cheysson (1886) advocated the use of “geometric statistics” as a tool for the entre­preneur to set the best prices. Cheysson’s idea was that the managers of a firm are able to gather sufficient information to draw a curve showing the quantities sold at different prices and a production cost curve. From this, they can determine gross and net product curves, from which they will deduce the optimal price for their product. This method could be employed to determine the wage rate, he argued, based on empirical evidence.

Clement Colson

Colson’s work marked a crucial turning point. He was appointed professor in 1892 at the Ecole des Ponts et Chaussees, where he renewed the teaching of econom­ics.

To do this, it was first necessary to dismiss Cheysson’s ideas on mathematics. Colson’s argument, however, was a curious one: the reason why calculus can be employed to explain human actions is that the law of large numbers applies to what men will do in most cases. Each individual is free to do what he or she wants, but the actions of all obey general laws that mathematics helps to explain. The framework in which Colson developed his analyses is that of partial equilibrium.^[132] He did not attempt to anchor the demand function in choice theory. This function is decreasing because the need to pay more for a good never increases its demand: if high prices are a cause of attraction for some atypical purchasers, the number of those they exclude is so much greater that this law is absolute. On the other side, a change in price may lead to an increase or decrease in supply. The issue is thus the uniqueness of the equilibrium and its stability. To demonstrate this result, Colson did not invoke the existence of scarce resources or economies of scale. For example, a lower price encourages the seller to increase his offer when he has to sell at all costs to avoid bankruptcy. But if the seller has sufficient liquid assets, he can instead reduce his offer and accumulate stocks. Similarly, a wage cut forces

Tradition and innovation at the turn of the twentieth century 185 workers to extend their working day to the point of exhaustion in order not to starve. On this basis, he investigated the uniqueness and stability of equilibrium.

Despite Colson’s commitment to partial equilibrium, he was well aware of the interdependence between markets, which he illustrated by discussing the relation­ship between the wage rate and the interest rate.

The other key contribution of Colson belongs to transportation economics (and public economics in a broader sense). He showed that in a monopoly market, a concession was preferable to a public operator. His analysis of tolls (Colson 1890) relied on his reading of Dupuit: he used the surplus theory to show that the intro­duction of custom duty results in a deadweight loss even when it increases domes­tic production (1901, vol. 4). He lastly confirmed Dupuit’s views on the need for price discrimination in order to maximise utility.

Marcel Lenoir

Polytechnician engineer Marcel Lenoir (1881-1927) published in 1913 his the­sis, Etudes sur la formation et le mouvement des prix, in which he used the latest developments in statistics. He devoted this work to the formation and movement of prices; it is a crucial step in the long evolution that led to the emergence of econo­metrics. Based on Aupetit, he considered that statics (price formation) and dyna­mics (price movement) relate to two different methods: the formation of prices is made along a logical synthesis of the neoclassical price theories, while the laws of their movement are to be found in statistical observation. Thus, he did not see statistical research as a means of verifying the conclusions of the theory, but as a method of obtaining new findings.

His study of price formation is a remarkable synthesis of the analyses on which economists could draw in the early twentieth century. The key point is the com­bination of the French and English traditions. He used the Edgeworth diagram to explain contract formation between a buyer and a seller and the Marshallian assumption of a constant marginal utility of money to study individual demand functions and to determine consumer and producer profit.

As this curve is defined assuming constant tastes and economic conditions, all that can be revealed by observation is the shape of the curve or what Marshall called the elasticity of demand. On the producer side, the marginal cost of goods

determines their supply of goods. The equilibrium price of a commodity depends on three factors: its supply, its demand and the general economic conditions. These three elements are never fixed and prices follow their variations. This observation, which may seem trivial, was the basis for Lenoir’s central question in any econo­metric study: the question of identification. Supply and demand curves cannot be observed directly: if a time series of quantities exchanged and prices is available, the path taken can be plotted on a graph, but generally, the graph cannot be inter­preted as a supply or demand curve.

The second part of his thesis, which deals with price movements, starts with a presentation of the concepts of covariation and regression that Lenoir used to study the price evolution of four products: coal, wheat, cotton and coffee. To distinguish between long movements and cyclical variations, Lenoir calculated 9-year moving averages and the deviations of the observations from this average. It was then pos­sible to interpret price changes as the effect of shifts in supply, demand or general conditions. Lenoir did not attempt to determine a law of supply and demand: he merely referred to the shifting of these curves as possible causes of price evolution. He did not claim to test a theory. On the contrary, he regarded the theory as a means of explaining the evidence he observed.

Maurice Potron

The example of Polytechnicien Jesuit Maurice Potron^[133] (1872-1942) shows how difficult it is for a researcher who has obtained new results to be heard and understood. Generalising the Perron-Frobenius theorem, he showed how it could be used to study the ability of an economy to reproduce itself (Abraham-Frois and Lendjel 2001; Bidard and Erreygers 2016). Oskar Perron (1907) and Georg Frobenius (1908, 1909) demonstrated that if every element, a_j, from an irreduc­ible square matrix A, is positive, the matrix has a dominant eigenvalue, l, which is unique, real and positive, and associated with a positive eigenvector x. One can then write lx = Ax. The positive value l is a simple root of the characteristic equa­tionThe moduli of the other eigenvalues of A are not greater than λ.

Frobenius showed that the equations

J

held for all positive values of x_l,...,x_n. Potron (1911) generalised this result to the cases where a_jj are not negative and not all zero. This generalisation was necessary to deal with economic problems, particularly within the framework of input-output

Tradition and innovation at the turn of the twentieth century 187 matrices in which a large number of coefficients are zero because the production of a commodity does not directly require all other goods.

In order to apply mathematical results to the economy, Potron first had to define the inputs needed to produce goods or provide services. His solution was different from those of his successors, for example, Wassily Leontief or Piero Sraffa. Rather than talking about goods or products, he preferred to use the expression “out­comes of work”, which consists of either work or work products (raw materials, machines). The relationship between results and means forms the technical condi­tions. The various types of work correspond to different social categories (foremen, employees, workers, labourers), which he called administrative conditions. Each social category has its own way of life, which implies the consumption or use of this or that outcome of work (food, clothing, housing, etc.). Once these conditions have been set, we can imagine that the distribution of workers across the various jobs changes, that their social category changes, and that prices and wages also change. Three types of problems can then be identified: (i) Is it possible to assign workers to jobs and social categories in such a way that consumption equals pro­duction without any worker having to work on non-working days? Potron called any system that satisfies this double condition “satisfactory” (satisfaisant); (ii) Is it possible to determine prices and wages in such a way that prices equate to costs and that, for any worker, the wage corresponding to the maximum work done is at least equal to the individual’s “cost of living”, that is, the value of the basket of goods usually consumed by people belonging to his social category? If so, Potron called the regime “simply satisfactory”; (iii) If the first condition is fulfilled, is it possible that the price of each good covers its cost and that, for any worker, the wage cor­responding to the work actually required from him is at least equal to the cost of living? If so, then the regime will indeed be “effectively satisfactory”.

Thus, in this analysis, the conditions for a regime to be satisfactory - there is no question of optimality - are technical, economic and social. Coefficients a_ij are not just technical coefficients, but socio-technical coefficients which depend on the way of life of the respective social categories. Actually, one can write three different matrices, reflecting the three previously defined states, according to the definition of coefficients a_jj. And, to determine whether the regime is satisfactory in the various senses of the term, the dominant eigenvalue must be calculated. The system is satisfactory if the dominant eigenvalue is less than or equal to 1. It should be noted that Potron, unlike the neo-Ricardians later, did not address the question of profits and, in particular, did not question the relationship between the dominant eigenvalue and the rate of profit.

Mathematicians

Besides the group of engineers, French-speaking mathematicians of the nineteenth century showed little interest in political economy and regarded the attempts of Cournot, Dupuit and Walras with scepticism. The exceptions are, however, quite notable because their contributions, which were completely ignored during their lifetime, appear today to be fundamental.

Joseph Bertrand

The case of Joseph Bertrand, probably the best known, is somewhat confusing. His contributions are remarkably rare and brief: a review of the works of Cournot and Walras (Bertrand 1883), and a short excerpt from his Calcul des probabili­ties (1889). However, they were, at least for the first one, widely quoted and discussed.

On the one hand, he seemed to share the opinion of most of his colleagues. After presenting Cournot’s analysis of monopoly, he concluded that “[his] calculations... are not clear... the results seem to be of little importance; sometimes, I must confess, they even seem unacceptable” (Bertrand 1883, 503). Cournot’s work was therefore, in his view, understandably neglected. On the other hand, he did make the effort to read Cournot and Walras, and the questions this reading suggested were important.

Cournot’s reasoning on duopoly was as follows. Let there be two owners of two sources of the same quality supplying concurrently, at zero cost, the same market (1838). It would be in their interest to determine their production to maximise their overall profit. But this equilibrium is unstable, with each producer having an incentive to modify his production in order to make a temporary profit. To this reasoning, Bertrand objected:

With this hypothesis, no solution is possible, the reduction would have no limit; whatever the common price adopted, if one of the competitors alone lowers his price, and if we neglect exceptions without importance, he attracts to himself all sales, and he will double his revenue if his competitor lets him do it.

(Bertrand 1883, 503)

It has often been pointed out that, in the process described by Cournot, the pro­ducers fix the quantity produced, whereas Bertrand assumed that they fix their price. Bertrand’s oligopoly is thus contrasted with Cournot’s. Note that both pro­cesses lead to the conclusion that the situation where the price is equal to the monopoly price is an unstable equilibrium. From this result, however, Bertrand drew an opposite conclusion to Cournot, who argued that this situation could only persist if a formal link existed between both producers: the equilibrium will be achieved when each entrepreneur fixes his production to maximise his profit for a given production of his competitor. Bertrand dismissed this hypothesis as implausible: the competitor’s output is variable. The adjustment process thus leads to an equilibrium where profit is zero. Bertrand’s conclusion was that we must abandon Cournot’s hypothesis that owners, each on their own, seek to max­imise their profits. If they do so, their profits will be zero. They must therefore cooperate.

Bertrand also criticised the Walrasian analysis of price formation (1883, 505). Suppose that the supply and demand curves for a good intersect at a price p. If this price is spontaneously proposed, transactions will be easily completed, each supplier finding a buyer and each buyer finding a supplier. If a price p' > p had been offered, supply would quickly have been found to exceed demand, and prices would have been lowered. Bertrand disagreed. If exchanges are made at price p', the supply and demand curves are modified: the resulting curves are constantly changing and it is easy to show that the equilibrium price, the one at which they intersect, necessarily varies. Accepting this argument, Walras specified in the sec­ond edition of the Elements d'economie politique pure that if the suppliers or buy­ers do not find a counterparty at the price announced, “theoretically, the exchange must be suspended” (1889, 72).

Bertrand’s third contribution is perhaps less well known. It relates to his analy­sis of a baccarat game¹¹ in his book Calcul des probabilites (1889, 38-42). This game is of particular interest: the outcome does not depend solely on chance but also on the skill of the players, on their ability to mislead their opponent. Bertrand thus took a first step towards what is now called game theory.^{[134] [135]}

Louis Bachelier

The mathematician Louis Bachelier (1870-1946) had all the characteristics of the unrecognised scholar. After studying under difficult conditions, with a poorly rated thesis, he only became a professor at the University of Besanςon at the age of 57. However, the thesis he defended in 1900, Theorie de la speculation, is the origin of mathematical finance. It contains the starting point of the theory of efficient markets and the premises of the theory of Brownian motion which he developed in a paper published in 1913. Bachelier refrained from “undertaking an analysis of the causes that may affect stock market prices; this research would be futile and could only lead to errors” (1914, 176-7) and he believed that the evolution of stock prices is unpredictable: it obeys the law of chance. He did not seek to dismiss determinism, which he took for granted: according to him, price variations can be attributed to causes, but these are unknown, or worse, unanalysable. It is therefore necessary to reason as if randomness were acting on its own. Giving up the idea of forecasting trends, the object of his research was to determine the law of prob­ability which they obey.

Bachelier applied three principles: mathematical expectation, indifference and uniformity. Before him, Jules Regnault (1834-1894), in his Calcul des chances (1863), had already stated that, in the stock market:

the whole mechanism of the game consists... in two opposite possibilities: the rise and the fall.... At any time, there is never more advantage to one opportunity than to another; and, in our ignorance of the future effect, it is absolutely immaterial whether we bet for or against... whether we buy rather than sell, or sell rather than buy.

(1863, 34)

This idea was taken up and developed by Bachelier: at a given stage in the market, some speculators expect prices to rise, and others expect them to fall. But, as the equality of supply and demand determines the price, “it seems that the market, that is, all the speculators, must believe at a given moment neither in the rise nor in the fall, since, for each price listed, there are as many buyers as sellers” (Bachelier 1900, 31-2). He expresses this idea mathematically by claiming that the math­ematical expectation of any operation is zero.

Regnault argued that on the stock market:

the player is always tempted to conjecture what must happen from what has happened, so that after three or four days of decline, he will more readily believe in a rise the next day, or at other times will see in it, on the contrary, a reason for continuing the trend, although after all there is complete independ­ence between these various effects.

(1863, 38)

This idea is probably the origin of what Bachelier (1912, 279) called the princi­ple of independence: the stock price variations that occur at a certain moment are independent of previous variations and of the price that is listed at that moment. Prices follow a random walk, a random process without memory in which the future depends on the past only through the present.

The principle of uniformity states that price volatility does not vary over time. Bachelier (1912, 286) argued that the market has no reason to believe that the probability of a price change of a given magnitude will be different tomorrow from today.^[136] On these grounds, Bachelier set up the probability distribution of stock prices. Let p dx be the probability that at time t the price of a security is between x and x + dx. Bachelier looked for the probability that the market price is listed at z at time t₁ + t₂ given that this stock price was listed at x at time t₁. According to the principle of compound probability, this probability is equal to the probability p_xtγd^χ that the stock price is x at t₁, multiplied by the probability that it is z at t, given that it was quoted at x in t₁, that is to say, p_z~_x⅛dz. It is thus possible to conclude that p_zh,dx = p_xtp_z__xtdxdz. By an integration calculation, Bachelier showed that the probability that at time t_l + t₂ the stock price is z is given by:

Then, Bachelier established that if the principle of uniformity is verified, one solu­tion (where k is the coefficient of instability) is:

Thus, the stock price follows a Laplace-Gauss centred normal distribution, at time t. The probability is a function of time: it increases until a certain time and then decreases. To study the distribution of probability followed by the stock prices, Bachelier admits that they are influenced by a large number of unanalys­able factors.

Many scholars before Bachelier had applied mathematics to economics, but the tools they used were well known. Antoine-Augustin Cournot was, in this respect, a typical example. With Bachelier, the situation changed. He introduced new math­ematical notions to analyse questions that had certainly been addressed by some of his predecessors, such as Regnault, but which had yet to be elucidated. He even wrote that:

the theory of speculation has been useful above all from the point of view of pure science It has given rise to the theory of the radiation of prob­

abilities and the theory of continuous probabilities If speculation did not exist, we would have to imagine it in order to better conceive the laws of chance.

(Bachelier 1914, 177-8)

The profusion of discoveries he made, both in economics and mathematics, is astonishing. The originality of his thought was so strong that it is not surprising that it remained misunderstood for a long time.