1 – Introduction

1After publishing a series of strikingly original papers on the mathematical economic model which he developed out of the blue in the period 1911-1914, Maurice Potron remained silent about it for more than twenty years. Exactly why he resumed his work on the model in 1935 remains unclear, but it could be that Ragnar Frisch played a greater role in this than previously thought. What we know is that Potron’s return to economic research was signalled by a letter he wrote to his fellow Polytechnician Robert Gibrat. The letter was triggered by Gibrat’s comments on a lecture on the economic aspects of planning by Jacques Branger at the Centre Polytechnicien d’Études Économiques, also known as X-Crise, on 22 February 1935. (All of these communications were published in the monthly bulletin of the centre; Branger, 1935; Gibrat, 1935; Potron, 2010[1935].) Potron’s attention was roused by Gibrat’s reference to a “supply and demand matrix, considered by Frisch” (Potron, 2010[1935], p. 166). The matrix introduced by Frisch (1934) reminded him of “another, very interesting, matrix that I think to have been the first ever to consider, at least for its application to economic problems” (ibid.). In his letter, Potron presented not only the matrix, but also the whole economic model on which he had worked in the years just before the start of the First World War.

2By a remarkable coincidence, in 1936 Potron visited Oslo, where Frisch was Professor of Economics and Statistics at the Royal Frederick’s University. Potron attended the International Congress of Mathematicians and read two papers, one on Abelian integrals in the section ‘Analysis’ and another on economic equilibria in the section ‘Probability calculus, mathematical statistics, insurance mathematics and econometrics’ (Potron, 2010[1937a]; 1937b). Interestingly, Frisch also attended the conference, and read a paper on price indices in the same section in which Potron presented his work on economic equilibria (Frisch, 1937). So far, no evidence was available of any direct contact between Frisch and Potron before, during or after the conference. Recently, however, a document has emerged which makes it clear that Potron and Frisch met eye to eye and discussed his economic model. Before we take a closer look at the document, let us briefly reconsider the basic features of Potron’s economic model.

2 – Potron’s economic model

3Over the years Potron provided various expositions of his economic model, the core of which consists of interrelated systems of equations and inequalities. Since modern readers may find Potron’s convention of presenting these equations and inequalities by means of summations rather cumbersome, we opt here for a more compact notation involving matrices and vectors. [2] Moreover, we present the simplest version of the model, which Potron adopted in 1936. More detailed explanations can be found in our previous work. [3]

4The model is centred around four basic principles, two of which focus on quantities and two on prices. One of the originalities of Potron’s economic work is that he connected the quantity side of the model to the price side.

5Potron’s model comprises n goods and methods of production, and m social groups and labour types. Goods are produced by firms using labour and goods as inputs, and every good has the same period of production (say, a year). For each good there exists one method of production, which is of the single-product type and characterized by constant returns to scale. Labour is supplied by (the heads of) households, with each household specialised in exactly one profession, i.e. one of the m available types of labour. Under the assumption that one unit of good i requires the input of l_ih hours of labour type h and a_ij units of good j, the set of available production methods can be represented by the matrix of labour input coefficients L = [l_ih], and the matrix of commodity input coefficients A = [a_ij].

6Goods are consumed by households. Households belong to different social groups; it is the profession of the household head that determines the social group to which a household belongs. Each social group is characterized by a specific consumption profile, expressed as an annual consumption basket. Assuming that a household of social group h consumes b_hj units of good j per year, the consumption habits can be represented by the matrix of consumption baskets B = [b_hj].

7Within any social group Potron distinguished two categories of households depending on the status of the household head. A working household is one of which the head is a labourer; its income consists exclusively of the wage earned by the household head. A non-working household, by contrast, is one of which the head is a ‘rentier’ or ‘capitalist’. Non-working households are the owners of the firms’ capital and draw their incomes from the firms’ profits.

8Potron assumed that a labourer cannot work more than a well specified number of hours per year. This maximum number, N, takes into account the usual periods of rest and holidays as determined by law or tradition, and is the same for every labourer.

9The data (L, A, B, N) describe what is given in the economy. [4] These figures are used in the specification of the economic problem, which has two sides and two types of unknowns: the physical problem which involves the quantity variables, and the value problem which involves the price variables. A set of values of the quantity variables characterizes a specific physical regime, and a set of values of the price variables a specific value regime. Potron was interested in the existence of quantity and value regimes which satisfy a set of crucial conditions.

10Let us begin by the quantity side. Potron introduced a distinction between principal and secondary unknowns. As far as quantities are concerned, the principal variables are the n activity levels of the available production methods, represented by the vector y =[y_i], and the m numbers of workers in each social group, represented by the vector x = [x_j]. All principal variables are required to be strictly positive. The group of secondary quantity unknowns consists of the m numbers of non-workers (s_h), the m hours of unemployment (z_h) per year and the n levels of excess production of goods (f_i). The hours of unemployment capture the difference between the number of hours the workers of a given type could work and the hours they actually work. Excess production refers to the difference between production and consumption. In a more compact notation these variables are represented by the vector of non-working consumers s = [s_h], the vector of unemployment z = [z_h], and the vector of excess production f = [f_i]. In contrast to the principal variables, the secondary unknowns are required to be nonnegative only. A limit case occurs when all the secondary unknowns are zero.

11What Potron called a “regime of production and labour” is defined by adequately chosen levels of the quantity variables. A regime of production and labour is considered to be satisfactory if it meets two requirements. First, the principle of sufficient production means that for each good the amount produced is at least equal to the sum of the amount used up in production and the amount consumed. Since the workers always consume according to their consumption baskets, the condition can be expressed in formal terms as:

12

13Second, the principle of the right to rest requires that for each type of labour the total amount of work to be performed does not exceed the maximum amount of labour that can be performed. In formal terms:

14

15These two inequalities can easily be transformed into equations by making use of the secondary unknowns. Since working and non-working households of a given social group have the same consumption habits, the equality which corresponds to (1) is:

16

17In a similar fashion, relation (2) leads to the equality:

18

19Potron clearly preferred to express his two principles in the form of equations (3) and (4) rather than in the form of inequalities (1) and (2). From a technical point of view, the main drawback of this is that it involves a lot of additional variables.

20Can these two requirements be satisfied simultaneously? To answer that question, Potron ingeniously used new mathematical results – the so-called Perron-Frobenius theorems on nonnegative matrices – which nobody before him had ever applied to economic problems. There are several equivalent ways in which this can be done; we concentrate here on the two approaches followed by Potron. The first consists of using (1) to obtain y ≥ xB(I – A)^–1, where I stands for the identity matrix, and then to eliminate y in (2). This leads to the condition Nx ≥ xB(I – A)^–1L. Let us then define a new matrix P:

21

22The quantity problem can now be reduced to that of finding a positive vector x such that:

23

24Alternatively we could proceed by eliminating x rather than y. From (2) we obtain y ≥ yL / N, and substituting this into (1) we derive y ≥ y(A + LB / N). We then define a new matrix Q:

25

26This means that the quantity problem can be reduced also to that of finding a positive vector y such that:

27

28Potron’s assumptions ensure that both matrix P (of which the order is equal to the number of social groups) and matrix Q (of which the order is equal to the number of goods) are indecomposable and nonnegative. [5] This means that the condition for the existence of a satisfactory regime of production and labour can be expressed in terms of the eigenvalues of these matrices. Starting from (6), the Perron-Frobenius theorem shows that a satisfactory regime of production and labour exists if and only if:

29

30where dom stands for the dominant eigenvalue. Alternatively, starting from (8) it follows that a satisfactory regime of production and labour exists if and only if:

31

32Potron showed that when any of these two inequalities is strict, there exist infinitely many satisfactory regimes. He also understood that the distance between dom(P) and N (or alternatively, that between dom(Q) and 1) could be seen as an indicator of the amount of leeway of an economy: an economy that comes too close to the danger line has virtually no margin of freedom.

33Potron realized that a straightforward economic interpretation could be given to the entries of matrix P: element p_gh represents the amount of labour of type h required to obtain a net product equal to the consumption basket of social group g. Even more interesting is the economic interpretation of condition (9): dom(P) is the minimum number of work hours compatible with the existence of a satisfactory regime of production and labour. It turns out that the dominant eigenvalue of matrix P can be taken as the measure of “the average number of normal working [hours] that a labourer must perform in order that the annual production obtained represents exactly the exclusive consumption of all workers” (Potron, 2010[1913], p. 125, n. 19). Hence, condition (9) is an essential feasibility condition: if more labour is required for the production of the goods consumed by the labourers than they can provide, there is something fundamentally wrong with the economic system.

34When dealing with the value side of the economic system, Potron again made a distinction between principal and secondary unknowns. The principal value variables are the n prices of the produced goods (p_i) and the m annual wages of the various types of labour (w_h). The price vector is then p = [p_i], and the wage vector w = [w_h]. These variables must all be positive. The secondary value unknowns comprise the n unit profits (π_i), which measure the difference between the selling price and the cost price per unit of good produced, and the m household savings (e_h), which capture the difference between a household’s earnings and its cost of living. In vector terms we have the vector of unit profis π = [π_i] and the vector of savings e = [e_h]. The secondary unknowns are required to be nonnegative.

35Two cases must now be distinguished. Let t_h designate the number of hours a worker of type h actually works. (i) If t_h = N (i.e. if he works the maximum number of hours), his earnings amount to Nw_h; (ii) if t_h < N, his earnings are equal to t_hw_h, and the out-of-work hours N – t_h correspond to what Potron called unemployment, which he assumed to be shared equally among all workers of a given profession and is therefore partial unemployment. Since the total yearly demand for labour of type h is equal to ∑_iy_il_ih, the number of hours effectively performed by a worker of type h is t_h = ∑_iy_il_ih / x_h. In matrix representation, these m scalars can be considered as the diagonal entries of a diagonal matrix T, the matrix of effective working hours. It is defined implicitly by the equality:

36

37The principle of the right to rest, which was introduced above, can therefore be written alternatively as:

38

39A “regime of prices and wages” is defined by adequately chosen levels of prices and wages variables. It is considered to be satisfactory if it meets two requirements. First, the principle of justice in exchange means that the selling price of a good is sufficient to cover its costs of production. Formally, the condition holds if the prices and wages are such that:

40

41The second principle, or right to life, stipulates that the wages earned by labourers must be sufficient to cover their costs of living. If all workers worked N hours per year, the condition would be written as:

42

43As they effectively work t_h hours per year, the exact expression of the right to life is:

44

45Condition (14) is necessary in any case. For Potron, the difference between these two versions of the right to life led to the distinction of a simply and an effectively satisfactory regime of prices and wages, with the first satisfying (13) and (14) and the second (13) and (15). Probably the main reason why Potron made the distinction is that condition (14) can be checked independently of the quantity variables. By contrast, condition (15) can be verified only if one knows the effective per capita labour matrix T, which depends upon the vector of workers x and the activity vector y.

46As in the case of the quantity side, the inequality conditions relating to the value side can be expressed as equations by making use of the secondary unknowns π and e. Conditions (13) and (15) then become:

47

48By another stroke of genius, Potron realized that the structure of the value problem closely mirrors the structure of the quantity problem, and that this similarity offers the possibility of linking the solution of the value problem to that of the quantity problem. He began by looking at the existence condition for a simply satisfactory regime of prices and wages. The P-approach reduces the value problem to the existence of a positive vector of wages w such that:

49

50Alternatively, following the Q-approach the problem can be reduced to the existence of a positive vector of prices p such that:

51

52Remarkably enough, the inequalities (18) and (19) are nothing but the transposes of the conditions (6) and (8). This allowed Potron to apply once again the Perron-Frobenius theorems to deduce that any of the two conditions (9) and (10) is also necessary and sufficient for the existence of a simply satisfactory regime of prices and wages. Potron repeatedly stressed this property, and thereby established what in modern parlance is called a duality of quantities and values. This is Potron’s basic duality result, which constitutes the first duality result ever stated and proved for disaggregated economic models.

53As a remarkable extension of this basic duality result Potron demonstrated that the same conditions also guarantee the existence of an effectively satisfactory regime of prices and wages. In other terms, there exists a positive vector of prices p and a positive vector of wages w such that conditions (13) and (15) are satisfied. In Potron’s own words: “If a satisfactory regime of production and labour is possible, it is always possible to associate with it an effectively satisfactory regime of prices and wages.” (Potron, 2010[1911], p. 79) This property is what may be called Potron’s strong duality result. We will return to it below.

3 – Potron’s Oslo Paper

54The second phase of Potron’s economic research was initiated by three separate, but nevertheless clearly related, expositions of his model. The first can be found in the letter to Gibrat (Potron, 2010[1935]), the second in the proceedings of the International Congress of Mathematicians (Potron, 2010[1937a]), and the third in a manuscript conserved in the archives of the French Jesuit order (Potron, 2010[1936]). While the first is rather extensive and provides information on how the equations and inequalities should be interpreted, the second is very dense and almost incomprehensible. Probably space constraints forced Potron to make tough choices and to focus on the mathematical aspects. The third document, an unpublished typescript to which Potron added the handwritten title “Communication faite au congrès d’Oslo”, contains a more accessible version of the model published in the proceedings of the Oslo conference.

55Let us zoom in on this last paper. Potron began by writing down systems (3) and (4), but in slightly more complicated way. Instead of (4), where vector z refers to the amount of unemployment for every type of labour in the economy as a whole, he opted for a version which specifies the amount of unemployment for every type of labour in every sector. [6] He then moved to state the problem by means of matrix Q, but in terms of equations rather than inequalities. Instead of (8), he therefore obtained:

56

57As far as the existence of satisfactory solutions to the problem is concerned, he pointed out three results: (1) if dom(Q) > 1, no satisfactory solution exists; (2) if dom(Q) < 1, the vectors f, z and s can be given arbitrary nonnegative values (but not all equal to zero) and the system will have a positive solution y; and (3) the dominant root is an increasing function of the elements of Q, i.e. ∂dom(Q) / ∂q_ij > 0. According to Potron, this last property imposes an upper limit on the values of the consumption requirements represented by matrix B.

58Next, he considered the price side. He formulated systems (16) and (17), but as before he complicated things by allowing that wage and saving rates differ from sector to sector. Combining these two expressions, he stated the problem of an effectively satisfactory regime of prices and wages, again in terms of equations rather than inequalities. This comes down to finding economically relevant solutions to the system:

59

60where Q′ = A + LT^–1B. Comparing (21) to (20), Potron observed that by analogy the same three results hold, provided dom(Q) is replaced by dom(Q′). From (12) it follows that Q′ > Q, and therefore that dom(Q′) > dom(Q). Hence, dom(Q) > 1 implies dom(Q′) > 1, which means that if no solution exists to the quantity system (20), there certainly exists none to the price system (21). But what if dom(Q) < 1 ? Here Potron ingeniously observed that the existence of a positive solution to (20) implies the existence of a positive solution to the system:

61

62This follows from the property that zB / N = yQ′ – yQ. Therefore, dom(Q) < 1 implies dom(Q′) ≤ 1 (and dom(Q′) < 1 if f and s are not both zero). Potron concluded that exactly the same conditions determine whether a solution exists to the quantity problem and to the price problem – what we have called Potron’s strong duality result.

63He ended his paper by showing that the sum of the profits earned by firms ((y–f)π) and the savings made by households (xe) is equal to the sum of the cost prices of the overproduced commodities (f(p–π)) and the cost of living of those who do not work (sBp). The identity can be deduced after multiplying both sides of (21) on the right by p and both sides of on the left by y, and observing that yQ′p = yQp + zNp / N and yLT^–1e = xe. In the case in which there is no overproduction (f = 0), the cost of living of those who do not work is exactly equal to the sum of profits and savings (Potron, 2010[1937c], Lecture 2, §5).

4 – Potron’s Letter to Frisch

64The typescript version of Potron’s Oslo communication was never published. However, the fact that it was typed might indicate that it was meant for wider circulation. As it turns out, a carbon copy of the typescript was found among Frisch’s papers at the University of Oslo by Olav Bjerkholt. [7] The copy lacks the title and the footnote which Potron had added in pencil to the typescript, but for the rest the two documents are identical. Very interestingly, however, the copy was found together with a short letter by Potron to Frisch (see the facsimile next page) and a six-page handwritten document, both of which were hitherto unknown.

65The letter and the note are in French; an English translation can be found in Appendix 1. The letter was written immediately after the end of the International Congress of Mathematicians. Although Potron used the conference’s stationary, he wrote the letter from the Jesuit institution ‘Maison St. Louis’ on Jersey, one of the Channel islands. The letter is very brief, but nevertheless reveals two important pieces of information. First, we learn that Potron personally met Frisch and his wife, since he asked Frisch to “be so kind as to convey [my regards] to Mrs Frisch”, and thanked them “for the warm welcome you have given us”. Second, Potron apparently discussed his model with Frisch. It was, in fact, as a result of that discussion that he decided to write a note of clarification: “The interest which you have been so kind to show in my paper incites me to send you herewith a few additional explanations, and some remarks on practical applications which may perhaps be possible.”

Potron’s letter to Frisch

Potron’s letter to Frisch

66The note opens with a list of matrices and vectors considered by Potron. For this he adopted a “spatial representation” which he found “convenient”, but which is unusual in his work. He then moved to the formulation of the two conditions characterizing satisfactory economic regimes, i.e. the existence of a positive vector of activity levels y satisfying (8) and the existence of a positive vector of prices p satisfying. While presenting these conditions Potron singled out the crucial role played by the consumption baskets, represented by matrix B. It is absolutely necessary that this matrix is such that dom(Q) < 1, automatically ensuring that dom(Q′) ≤ 1.

67A substantial part of the note is devoted to a discussion of the tension between what mathematics has to say with regard to the existence question and how things proceed in real life. Let us assume that B is such that dom(Q) < 1. Potron noted that from a mathematical point of view, we can say that in the quantity space to any set of values of the secondary unknowns (f > 0, z > 0, s > 0) we can always associate a set of values of the principal unknowns (y > 0, x > 0) characterizing a satisfactory regime of production and labour. Likewise, in the price space to any set of values of the secondary unknowns (π > 0, e > 0) we can always associate a set of values of the principal unknowns (p > 0, w > 0) characterizing a satisfactory regime of prices and wages. This follows from the fact that (I – Q)^–1 > 0 and (I – Q′)^–1 > 0. However, the reverse does not hold: starting from a set of values (y > 0, x > 0) satisfying condition (2), there is no guarantee that there exists a set of values (f > 0, z > 0, s > 0) such that (20) holds; and likewise, starting from a set of values (p > 0, w > 0) such that (15) is satisfied, there is no guarantee that there exists a set of values (π > 0, e > 0) such that (21) holds.

68Unfortunately, Potron observed, in the real world one tended to follow this second, much more risky path. In his letter to Gibrat he had already warned about the possible dire consequences of starting from the ‘wrong’ variables:

69

To sum up, all the evil comes from the failure to recognize that distinction between principal unknowns and secondary unknowns. The mathematics say: if a certain condition is met by your standard of living requirements (…) and the state of your industry (…), [viz.] the condition ‘characteristic root < 1’, you can assign to all secondary unknowns totally arbitrary positive values; for the principal unknowns, there will follow positive values such that everything goes at best. But you cannot assign to the principal unknowns totally arbitrary values, such that everything goes fine. Despite this warning, one stubbornly gives arbitrary values to the principal unknowns, because it is easier. It is very likely that one does not draw one of the favourable combinations.

70In the note which Potron sent to Frisch he went a bit further in explaining how difficult it could be to find a satisfactory solution. He saw a chain of events beginning with producers consulting amongst themselves and fixing their production levels (y) based on previous levels of demand. With some luck, these production levels are compatible with the distribution of labour (x), which means that the principle of the right to rest is satisfied, and that the unemployment levels (z) are nonnegative. Given y and x, the effective working time (T) can be determined. Negotiations between bosses and labourers then fix the wage levels (w), based upon prices (p) which are taken as given, e.g. those of the previous period. At these prices and wages, savings (e) are positive for all types of labour, meaning that the right to life is guaranteed. However, it may very well be that the agreed upon combination of prices and wages entails losses in some sectors, i.e. a non-positive profit vector (π). In an attempt to restore their profitability, firms in these sectors increase their prices, but that may lead to negative savings for households, which would mean that the right to life is no longer guaranteed. That would call for other changes, and possibly set in motion a long and difficult process of price and wage adaptations. It is possible, but far from certain, claimed Potron, that these “successive trials and errors” would lead to prices characterizing a satisfactory regime of prices and wages.

71In the closing paragraphs of his note, Potron expressed his scepticism concerning the capacity of existing economic systems to find satisfactory solutions for quantities and prices. The sheer number of equations constitutes a formidable obstacle, and it would be very difficult to reduce this number substantially. What should be done “in order to avoid dreadful deceptions” is to make “the highest possible number of verifications”. When fixing their production levels, producers should check as much as possible whether their production plans are compatible and feasible. More importantly, however, Potron suggested that “each agreement which determines wages, should mandatorily determine prices, at least of the goods which are consumed by households”. In this way, it seems that he hoped to split the process by which the economic system tried to find satisfactory prices into two separate problems, which could be solved sequentially. The first would consist of the determination of wages and the prices of consumption goods, the second of the determination of the prices of industrial goods. However, the indications provided by Potron are too scanty to figure out exactly what he had in mind.

72All in all, these observations confirm Potron’s pessimistic view on the economic system and echo the “discouraging” considerations he had voiced in his letter to Gibrat (Potron, 2010[1935], p. 173). The difficulty of finding prices which guarantee both the principle of justice in exchange and that of the right to life preoccupied Potron from the very beginning of his work (Potron, 2010[1912], §1). In his later work it resurfaced in caveats about “the blind race to the rise between prices and wages” (Potron, 2010[1935], p. 171; see also Potron, 2010[1937c], Lecture 6, §8).

5 – A Missed Opportunity

73We do not know whether Frisch ever replied to Potron. There are no traces of any further contacts between the two. Frisch stored Potron’s letter and notes along with other documents he collected in connection with the International Congress of Mathematicians, but apparently never retrieved them. [8] For all it seems, the discussion between Potron and Frisch ceased immediately after it began. We can only speculate about the reasons for this dialogue manqué, but we suspect that Potron’s esoteric style of writing and lack of knowledge of what was going on in the international community of (mathematical) economists played a major role.

74Reading the work of Potron is not easy going. It is probably fair to say that his prose lacks elegance, and that most of his texts are as dull as they come. Even in the few occasions when he touched upon the Christian roots of his work and his deepest convictions might have come to the surface, his words are hardly inspiring. On top of that, Potron spent very little effort to make his mathematics digestible for a non-mathematical audience. His poor pedagogical skills, amplified by an idiosyncratic notation, must have made many of his papers impenetrable, except for the most tenacious of readers. Only very few would have been able to grasp the full meaning of Potron’s arcane economic model.

75The letter to Frisch highlights Potron’s deficiencies with regard to getting his message across. Without suggesting that he deliberately created obstacles, we have to acknowledge that he did make it hard to understand what he was trying to show. In the handwritten “additional explanations” he began by introducing a notation which differs from the one he used in the typescript note. Instead of explaining the meaning of the new symbols at the beginning, he did so only when dealing with the interpretation of the main results. His description of how the economic system works in the real world remains very abstract, and moreover his analysis is marred by two typos. And finally, his practical suggestions at the end are opaque and incomprehensible.

76We must also take into account Potron’s isolated position vis-à-vis the economic profession. He had no training in economics, showed almost no awareness of the economic literature, and knew very few economists. Some of the economic terms he used are awkward: ‘benefits’ instead of ‘profits’, ‘economies’ instead of ‘savings’, and so on. He never discussed ‘markets’ or ‘competition’, and ‘supply and demand’ occurs only once in his work. Even economists with a solid background in mathematics and a keen interest in abstract models – such as Frisch – would have had a hard time when first confronted with Potron’s writings.

6 – Concluding Remarks

77In this paper we have explored the connections between Frisch and Potron. It was already known that Frisch, through Gibrat, exerted an indirect influence on Potron’s later work on economics. Newly discovered documents show that there was also a direct influence. How far and how deep the Frisch effect went, is difficult to ascertain. While the contact with Frisch did not lead to a major shift in Potron’s economic model, it is not impossible that it convinced Potron of the usefulness of resuming his economic research in the second half of the 1930s and of getting economists interested in it. The Frisch episode may have given him the confidence to organize a series of six lectures at the Institut Catholique in Paris in 1937, despite the resistance from his superiors. Not insignificantly, he noted the presence of two prominent mathematical economists during these lectures, René Roy and François Divisia, and he corresponded with Divisia about an illuminating (albeit controversial) numerical example inspired by the Bible. Moreover, in 1938 he became a member of the Econometric Society, of which Frisch was the driving force. Unfortunately, the advent of the Second World War abruptly ended all of this.