The Analysis of Linear Economic Systems: Father Maurice Potron’s Pioneering Works [1 ed.] 0415473217, 9780415473217

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The Analysis of Linear Economic Systems

Maurice Potron (1872–1942), a French Jesuit mathematician, constructed and analysed a highly original, but virtually unknown economic model. This book presents translated versions of all his economic writings, preceded by a long introduction which sketches his life and environment based on extensive archival research and family documents. Potron had no education in economics and almost no contact with the economists of his time. His primary source of inspiration was the social doctrine of the Church, which had been updated at the end of the nineteenth century. Faced with the ‘economic evils’ of his time, he reacted by utilizing his talents as a mathematician and an engineer to invent and formalize a general disaggregated model in which production, employment, prices and wages are the main unknowns. He introduced four basic principles or normative conditions (‘sufficient production’, the ‘right to rest’, ‘justice in exchange’, and the ‘right to live’) to define satisfactory regimes of production and labour on the one hand, and of prices and wages on the other. He studied the conditions for the existence of these regimes, both on the quantity side and the value side, and he explored the way to implement them. This book makes it clear that Potron was the first author to develop a full input– output model, to use the Perron–Frobenius theorem in economics, to state a duality result, and to formulate the Hawkins–Simon condition. These are all techniques which now belong to the standard toolkit of economists. This book will be of interest to Economics postgraduate students and researchers, and will be essential reading for courses dealing with the history of mathematical economics in general, and linear production theory in particular. Paul A. Samuelson’s short foreword to the book may have been his last academic contribution. Christian Bidard is Professor of Economics at the University of Paris Ouest, France. Guido Erreygers is Professor of Economics at the University of Antwerp, Belgium.

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117 The Analysis of Linear Economic Systems Father Maurice Potron’s pioneering works Edited by Christian Bidard and Guido Erreygers

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The Analysis of Linear Economic Systems

Father Maurice Potron’s Pioneering Works

Edited by Christian Bidard and Guido Erreygers Foreword by Paul A. Samuelson

First published 2010 by Routledge 2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN Simultaneously published in the USA and Canada by Routledge 270 Madison Avenue, New York, NY 10016 Routledge is an imprint of the Taylor & Francis Group, an informa business © 2010 Christian Bidard and Guido Erreygers (this translation); Maurice Potron (original French writings). Typeset in Times New Roman by Glyph International Ltd. Printed and bound in Great Britain by CPI Antony Rowe, Chippenham, Wiltshire All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without permission in writing from the publishers. British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloguing in Publication Data Potron, M. (Maurice), 1872–1942 The analysis of linear economic systems: Father Maurice Potron’s pioneering works / translated and edited by Christian Bidard and Guido Erreygers. p. cm. Includes bibliographical references and index. 1. Economics–Mathematical models. 2. Economics, Mathematical. 3. Potron, M. (Maurice), 1872– I. Bidard, Ch. (Christian) II. Erreygers, Guido, 1959– III. Title. HB135.P678 2010 330.01 51–dc22 2010002184 ISBN: 978-0-415-47321-7 (hbk) ISBN: 978-0-203-84737-4 (ebk)

Copyrights

The following persons and institutions have granted us permission to publish translations of writings of which they hold the copyrights: – –

– – – – – –

Robert Bonfils, on behalf of the Province de France de la Compagnie de Jésus, for unpublished documents by Potron and Vittrant from the archives in Vanves (Chapters 13 and 14, and Appendix III); the Société Française de Statistique and the Société Mathématique de France for Potron’s and Barriol’s articles published in the Journal de la Société de Statistique de Paris (Chapters 6 en 17, and Appendices I and II), for Potron’s article published in the Bulletin de la Société Mathématique de France (Chapter 16), and for Potron’s article published in the Annales Scientifiques de l’École Normale Supérieure (Chapter 8); the École Normale Supérieure for Potron’s article published in the Annales Scientifiques de l’École Normale Supérieure (Chapter 8); the Université Catholique de l’Ouest for Potron’s unpublished paper conserved in the university’s library (Chapter 7); the Centre de Recherche et d’Action Sociale (CERAS) for Potron’s publications in Le Mouvement Social (Chapters 1, 5 and 10); the Union Sociale d’Ingénieurs Catholiques (USIC) for Potron’s article in the Échos de l’Union Sociale d’Ingénieurs Catholiques et des Unions-FédéralesProfessionnelles de Catholiques (Chapter 2); the International Mathematical Union for Potron’s publication in the Comptes Rendus du Congrès International des Mathématiciens, Oslo, 1936 (Chapter 12); and the publishing house Armand Colin for Potron’s publications in the Comptes Rendus de l’Académie des Sciences (Chapters 3, 4 and 15). On behalf of Maurice Potron’s family, Antoinette Salmon-Legagneur has authorized us to publish any unpublished and orphan papers of Maurice Potron. Any remaining copyright holders which have been overlooked are requested to contact the editors and the publisher.

xiv Copyrights The following persons and institutions have granted us permission to reproduce photographic material: – – – – –

the École Polytechnique (Photo 1); the École Sainte-Geneviève (Photo 2); Antoinette Salmon-Legagneur, on behalf of Maurice Potron’s family (Photos 3 and 4); the Institut Catholique de Paris (Photo 5); Georges de Charrin, on behalf of the Province de France de la Compagnie de Jésus (Photo 6).

Contents

List of tables Acknowledgements Foreword Church, society and economics: An introduction to the life and work of Maurice Potron

xvii xviii xx

1

Notes on the translations

63

1

Abstract of a study on just prices and wages

64

2

With regard to a mathematical contribution to the study of the problems of production and wages

67

3

Some properties of linear substitutions with coefficients 0 and their application to the problems of production and wages

74

4

Application to the problems of ‘sufficient production’ and the ‘living wage’ of some properties of linear substitutions with coefficients 0

78

5

Possibility and determination of the just price and the just wage

80

6

Mathematical contribution to the study of the problems of production and wages

102

7

Relations between the question of unemployment and those of the just price and the just wage

107

xvi Contents

8 Some properties of linear substitutions with coefficients 0 and their application to the problems of production and wages

110

9 Mathematical contribution to the study of the equilibrium between production and consumption

131

10 The scientific organization of labour. The ‘Taylor System’

142

11 On some conditions of economic equilibrium. Letter of M. Potron (90) to R. Gibrat (22)

166

12 On the economic equilibria

174

13 Communication made at the Oslo Congress

176

14 The mathematical aspect of some economic problems in relation to recent results of the theory of nonnegative matrices. Lectures given at the Catholic Institute of Paris

180

15 On nonnegative matrices

226

16 On nonnegative matrices and positive solutions to certain linear systems

229

17 Letter on industrial statistics

234

Appendix I

Alfred Barriol: ‘Obituary. Maurice Potron (1872–1942)’

237

Appendix II Alfred Barriol: ‘[Report on] L’aspect mathématique de certains problèmes économiques’

240

Appendix III Michel Vittrant: ‘[Report on] Le problème de la manne des Hébreux’

243

The Potron Bibliography Name index Subject index

245 256 259

Tables

1 The Potron family tree 2 Maurice Potron and his times 5.1 Coefficients of the cost price formula of a two-kilogram loaf of bread 6.1 Coefficients of the cost price formula of a two-kilogram loaf of bread 9.1 Coefficients of the cost price formula of B powder

10 58 87 104 140

Acknowledgements

The process of writing this book involved digging up loads of documents from numerous archives, tracing relatives and acquaintances of Potron, struggling with Potron’s idiom when translating his French texts into understandable English, and presenting the wealth of material we have found in a concise and clear way. We warmly thank all those who generously helped us during our long journey: Gilbert Abraham-Frois, François Audic†, Jean-Christophe Auger, Jean-Claude Baratte, Olav Bjerkholt, Claudine Billoux, Bernard de Boissière, Robert Bonfils, Michel Borjon, Marie-Madeleine Canet, Marie Cazin, Michel Chamoin, Jacques Chaumerliac, Thierry Chaumerliac, Morris Cheetham, Bati Chetanian, Hubert d’Aboville, Michel de Boisfleury, Christian de Borchgrave, Georges de Charrin, Jacques de Morel, Henri de Salins, Nadine Dechamps, Laurent Delacroix, Magali Della Sudda, Corinne Desmettre, Wesley O. Doggett, Chantal Dufour, Jocelyne Dufour, Jean Dumort, Jean-Claude Faivre, Serge Foucher, Paul-Henri Gaschignard, Pierre Gaschignard, Yves Grovalet, Thomas Hawkins, Henriette Hautemule-Durliat, Charles Huet, Marie-Hélène Idrac, Pascale Irigoyen, Bertrand Jacquet, Yves Jaigu, Pierre Jestin, Brigitte Joly-Berthier, Alain Kerrand, Kristiaan Kerstens, Heinz D. Kurz, Frank Le Blancq, Omer-Paul Le Couvreur, Nathalie Le Gonidec, Pierre Le Grignou, Louis Le Naour, François-Xavier Leclerc, Jan Leisink, Jean Lelion, Émeric Lendjel, Guy de Lirot, René Malfilâtre, François Mérand, Guy Messager, Geneviève Miard, Fabrice Millereau, Céline Mortier, Patrick O’Rorke, Wilfried Parys, Fanny Peltier, Marcel Personnic, Hubert Poupard, Anne-Catherine Putz, Elisabeth Quémerais, Laurent Quevilly, Jean Robieux, Raphaël Rogeau†, Camille Rollando, Yves Saccadas, Françoise Sagnier, Antoinette Salmon-Legagneur, Claire Salmon-Legagneur, Claude SalmonLegagneur, Denise Salmon-Legagneur (née Potron)†, Dominique SalmonLegagneur, Gérard Salmon-Legagneur, Hubert Salmon-Legagneur, Isabelle Salmon-Legagneur, Marc Salmon-Legagneur, Patrick Salmon-Legagneur, Sylvie Salmon-Legagneur, Thierry Salmon-Legagneur, Odile Sarti, Marie Sollogoub, Louis Thomas, Anne-Marie Tirand-Cuny, Pierre Tyl, Jacques Villeglé; and the staff or members of: the National Archives in Paris, the Departmental Archives of Paris, the Departmental Archives of Val d’Oise, the Municipal Archives in Vannes, the Municipal

Acknowledgements xix Archives in Epinay-sur-Seine, the Archives of the Province de France de la Compagnie de Jésus in Vanves, the Archives of the Fondation Royaumont in Royaumont, the Archives of the GRAHAL in Paris, the Archives of the DRAC in Paris, the Archives of the Mouvement des Cadres Chrétiens in Paris, the Archives of the Société Historique de Pontoise in Pontoise, the Archives of the Sisters of the Sainte-Famille de Bordeaux in Rome, the Archives of the École Polytechnique in Palaiseau, the Collège Saint François-Xavier in Vannes, the Université Catholique de Lille in Lille, the Université Catholique de l’Ouest in Angers, the Groupe de Recherches Historiques et Archéologiques in Louvres, the Municipal Council of Mours, the Pères Blancs of the Villa Saint-Régis in Mours. Without the fertile research environment offered by our respective research teams – EconomiX at the Université Paris Ouest-Nanterre-La Défense and the Department of Economics at the University of Antwerp – it would have been impossible for us to bring this project to an end. We are also very grateful to the Universitair Centrum Sint-Ignatius Antwerpen (UCSIA) for inviting Christian Bidard as UCSIA-scholar to the University of Antwerp in the period November 2008 – January 2009; this stay allowed us to make crucial progress on the book. Last but not least we very much appreciate the patience of Routledge’s editorial staff. Paris, EconomiX, Université Paris Ouest-Nanterre-La Défense Antwerp, Department of Economics, University of Antwerp

Foreword

Mathematics is a flexible tool that serves diverse viewpoints. I call readers’ attention to that. Not surprising, then, that it has been used in the economic writings of the French Jesuit Maurice Potron (1872–1942). I commend to readers’ attention the reading of early Perron–Frobenius theorems that parallel the important Hawkins– Simon criterion. This demonstrates that religious altruism need not be lacking in economics. For me, the ‘Eureka Moment’ came when I realized that the same math could apply to a cynic like Voltaire or to a Sam Johnson who lived in perpetual fear of permanent hell fire forever. And at the same time can serve for a libertarian ‘Ayn Rand-Friedman’ or for a Rawlsian philosopher. When you can satisfy either Rand–Friedman anti-Keynesianism or Joan of Arc Catholicism, that is quite a stable. This math was a language that could be applied to opposites! Paul A. Samuelson [In 2007 Professor Samuelson agreed in principle to write a foreword to the edition of Potron’s economic works. When we sent the translation two years later, his health was declining rapidly. On 25 November 2009 he sent us a draft, which we have reproduced here with some minor changes. These few words may have been his last academic contribution. Paul Anthony Samuelson died on 13 December 2009. – Eds.]

Church, society and economics An introduction to the life and work of Maurice Potron

On 5 August 1868 Auguste Potron, an engineer freshly graduated from the École Centrale des Arts et Manufactures, married Cécile Frottin, the only living child of the richest notary in Paris, in the church of La Madeleine. Édouard Jean-Pierre Frottin had accumulated a considerable fortune: buildings in Paris and elsewhere in France, shares of the Banque de France (then a private institution), assets, bonds and rents. His daughter’s marriage contract, adorned by the elegantly curled signatures of about fifty family members and friends, had been signed a few weeks before the wedding. On Auguste’s side we find the autographs of his father, a barrister, of his elder sister Élise, who was married to the rich landowner Émile Leemans, and of many prominent citizens. On Cécile’s side the signatures of both parents were surrounded by those of acquaintances representing the law, the army and the Church. The contract was signed in Frottin’s apartment in the first district of Paris, 368 rue Saint-Honoré. Frottin owned all of ‘the 368’, as this four-storey building surrounding a large courtyard was affectionately known, and the newly wedded couple also settled there, in an apartment located on the third floor. Three years into their marriage Cécile and Auguste were still without child. They decided to make a pilgrimage to Lourdes, where they prayed and made vows to the Virgin Mary, to whom they were very devoted. Their first child, Maurice Marie-Jean, was born on 31 May 1872 and baptized in La Madeleine on the fourth anniversary of their wedding. In recognition to the Virgin, the parents erected an asylum just behind the castle that Édouard Frottin had bought in Courcelles, to the north of Paris. The building comprised a chapel and a primary school run by the Congrégation de la Sainte Enfance, with all expenses and salaries paid by the Potron family. A year later, Édouard Frottin died and the young couple inherited his fortune, including the 368 building in Paris and the castle in Courcelles. This book is about Maurice Potron. It may be that he was pledged to the Church when his devout parents made their pilgrimage to Lourdes; certainly he was raised in a family in which a religious life was seen as an ideal and children were actively encouraged to devote themselves to the Church. In any case, when he reached adulthood, Maurice Potron chose to become a Jesuit priest. While going through the long formation process of the Society of Jesus, he obtained a PhD in mathematics from the University of Paris, after having graduated as an engineer from the prestigious École Polytechnique. Today, however, it is not so much his religious vocation or his mathematical accomplishments which are remarkable, but

2 Church, society and economics his contributions to a field which he never properly studied: economics. Inspired by his faith, Potron used his mathematical skills to translate the principles of the social doctrine of the Church into a formal economic model. In the process he developed concepts and used tools which were well in advance of their times, and have been rediscovered later in a non-religious context. In this book we present English translations of all of Potron’s work in economics, both published and unpublished.1 (A French edition by Gilbert Abraham-Frois and Émeric Lendjel already appeared in 2004.)2 Our introduction provides elements to put Potron’s economic work into perspective. It is divided into two parts: the biographical part (part A) focuses on Potron’s life environment, whereas the methodological part (part B) clarifies and interprets his economic model. The choices made by Potron in his theoretical work cannot be separated from his life experience. The evolution of the State and the Church, as well as multiple forms of social and religious commitment of the Potron family, especially in the period between 1890 and the beginning of the Great War, influenced him deeply and gave a specific flavour to his economic writings.3

Part A Maurice Potron’s biography Our portrait of Maurice Potron mixes biographical and historical ingredients. We start with a bit of history of the second half of the nineteenth and the beginning of the twentieth century, when the relations between the Catholic Church and the French State were especially turbulent and the Church started its aggiornamento on the ‘social question’ (section 1). We then give a description of Potron’s family (section 2) and deal with Maurice’s education and religious vocation (section 3). The Potron family was at the centre of a network of Catholic organizations, in which also Maurice was personally implicated (section 4). It is by the end of that period that he elaborated and studied his economic model. The Great War interrupted the ordinary course of life, and we examine its influence on Maurice Potron and his family (section 5). We provide details about his career as teacher and researcher (section 6) and conclude with a few words about his final years (section 7).

1 The Potron Bibliography at the end of the book lists all publications by Potron, divided into several categories. It also includes details on supplementary material with regard to Potron’s life and work. Throughout the book we use a compact notation for references to works included in this bibliography, with ‘Ax’ referring to item x in category A. If a translated version of item Ax appears as Chapter y in this book, we write ‘Ax/Chapter y’. 2 Since the publication of the French edition (see E2) we had access to more material and archives, including family records, which throw new light on Potron’s thought and environment. As a result the present volume comprises papers not included in the French edition, and gives a more complete view of Potron’s life. Moreover, we disagree with the interpretation of Potron’s economic model advanced in the French edition (see below, in section 8 of the introduction). 3 For a glimpse of significant events in Potron’s life and in French society over the period 1868–1942 we refer to Table 2 at the end of this introduction.

Church, society and economics 3

1 Church, State and Society 1.1 The place of religion in society The defeat of Napoleon III in the battle of Sedan during the Franco-Prussian war led to the end of the Second Empire and the proclamation of the Third Republic on 4 September 1870.4 Nevertheless the restoration of the monarchy remained a real possibility, since the royalists dominated the French Parliament. But they were divided: the Orléanists (in reference to Louis-Philippe, the ‘bourgeois king’ of the period 1830–1848) favoured a democratic form of government, whereas the Legitimists stood for a return to the Ancien Régime. Their disagreements were highlighted by fierce discussions over the national flag, a matter with a high symbolic value. After the fall of Adolphe Thiers (1873), the new president Marshal Patrice de Mac-Mahon sent a message to the French Parliament saying that it was his aim to reestablish the ‘moral order’ with ‘the help of God’. The monarchic forces failed to reach an agreement, however, mainly due to the intransigence of the Legitimist pretender to the throne, the count of Chambord. A constitutional amendment, which passed by the narrow margin of only one vote, installed the Republic definitely in 1875. Subsequent elections confirmed and reinforced this choice, thanks to the dexterity of the leader of the republican camp, Léon Gambetta (1838–1882), who succeeded in convincing the conservative country voters that the Republic was the best guarantee against disorder and warned against the dangers represented by the Church and clericalism. With the religious forces squarely on the side of the monarchists, most republicans considered that one of their first tasks was to free French society from royalist influence. The fight started in the domain of education: the weight of the Church in teaching and other social missions (orphanages, hospitals, hospices) was great. In 1881 and 1882, Jules Ferry passed a set of laws making primary education free, compulsory and lay. They aimed especially to reduce the influence of religious congregations, in particular the Jesuits, over which the French State had much less control than over the secular clergy (after the Concordat of 1801 the bishops were appointed by the French government in agreement with Rome and the secular clergy paid by the State). Decrees issued by Ferry in 1880 had already expelled the Jesuits from France. In 1892 Pope Leo XIII (1878–1903) tried to ease the tension by means of his encyclical Au Milieu des Sollicitudes, in which he advised French Catholics to accept the legitimacy of the French Republic. Not all Catholics shared his vision, however, and the policy of le ralliement remained controversial. The relations between State and Church deteriorated again at the beginning of the twentieth century, when French political life entered into a new anticlerical period. 4 Most of the material of this section can be found in any good book on the history of France, such as Robert D. Anderson, France, 1870–1914: Politics and Society (London, Routledge and Kegan, 1977) and Jean-Marie Mayeur and Madeleine Rebérioux, The Third Republic From Its Origins To the Great War, 1871–1914 (Cambridge, Cambridge University Press, 1987).

4 Church, society and economics The 1901 law on associations, which constituted in many respects a very liberal framework, contained discriminatory measures against the congregations. They were submitted to the control of the State, which by decree decided about their authorization or non-authorization, and members of non-authorized congregations were prohibited from participating in teaching. The Jesuits, who had discreetly returned to France in the 1890s, chose to disperse the order. Many of them went into exile and entrusted their colleges to civil associations which became the official owners of their properties; the French Jesuit novices moved to centres in Belgium, England, Jersey and elsewhere. The Jesuits who remained in France sought to continue their activities under the strict legal framework imposed upon them. In 1902 the ex-seminarist Émile Combes became the head of government with a radically anticlerical program. He pushed for a rapid laicization of educational institutions and systematically rejected the authorization requests of the congregations. Their properties were sold; the proceeds (popularly known as the milliard des congrégations) were earmarked for the creation of a workers’ pension fund, but disappeared largely into the pockets of intermediaries. Combes’s policy led to a rupture of diplomatic relations between France and the Holy See. Eventually Combes had to resign shortly after it became known that the Ministry of War used the services of a masonic lodge to determine promotions in the army (affaire des fiches). The law on the Separation of the Churches and the State, voted in December 1905, intended to give a definitive solution to the religious question. This law ensured the freedom of conscience and guaranteed the free exercise of religions (article 1), but it also stipulated that the State did not recognize or subsidize any religion (article 2). Moreover it laid down provisions for the creation of cultural associations in charge of the organization of worship and of the management of Church properties. The law aimed at appeasement and reconciliation, but in the encyclical Vehementer Nos Pope Pius X (1903–1914) rejected it, and in the encyclical Gravissimo Officii Munere banned Catholics from creating such associations. The inventory of Church properties in 1906 gave rise to sporadic fights between the faithful and the army, and until the beginning of the First World War the relations between the State and the Church remained difficult. For the Jesuit order clashes with the French State were not a new experience. The Jesuits had been expelled from France (and from other European countries) already a few years before 1773, when Pope Clement XIV (1769–1774) suppressed the Jesuit order. In 1814 Pope Pius VII (1800–1823) restored it. The Jesuits returned to France after the restoration of the monarchy, but their support for the reactionary camp which dreamt of erasing the liberal aspects of the 1814 Charter (which guaranteed liberty of conscience and religion) aroused so much reaction that they were banned from teaching in 1828. This changed under president LouisNapoléon Bonaparte, the future emperor, when the Falloux law (1850) again favoured teaching by the clergy. The particular situation of the Jesuit order may be explained by its faithfulness to the orientations of its founder, Ignatius of Loyola (1491–1556). Throughout much

Church, society and economics 5 of their history the Jesuits tended to be close both to the rulers of the countries in which they were active and to the Pope (a proximity symbolized by their fourth vow). That double position did not go without contradictions, for instance when the political power shifted or when conflicts arose between the political and the religious powers. This explains why the order often became the focal point of anti-clerical hatred and nationalist feelings. Another characteristic of the Jesuit order is its commitment to teaching, especially of the elite, and to intellectual life in general. When doing apostolic and missionary work many Jesuits actively used the solid cultural and scientific knowledge they acquired during their long training. The reputation of the Jesuit colleges was well established, and several features of ‘modern’ education were first experimented there. But, again, their dominant position in teaching made them a primary target in times of conflicts. 1.2 The social question Both the State and the Church faced the task of adapting themselves to the profound changes in the socioeconomic structure of society brought about by the process of industrialization, and in particular of finding an answer to the misery suffered by large portions of the working population. The Commune de Paris of 1871 and its bloody repression were directly related to the growing tensions between rich and poor. Around the turn of the century the demands of the workers were voiced by socialist or revolutionary parties, such as the Parti Ouvrier Français, which defended Marxist ideas, as well as by labour unions, such as the Confédération Générale du Travail (CGT), which was closer to anarchist ideas and distrusted all political parties. The reaction of the State was slow: before 1914 little progress was made on social legislation. Apart from the recognition of the trade unions (1884), the ‘workers’ laws’ concerned the limitation of child-labour, the duration of work (the Sunday was recognized as a free day in 1906) and insurance against industrial accidents (1898). The livret de travail, which linked a worker with his employer, was abolished, and the work contract provided some protection against arbitrary lay-off (1890). A few precautionary institutions were founded at the local level and in particular industries, like the coalmines, sometimes at the initiative of the employers. The social question came to the forefront of political life as the religious conflicts eased. Strikes, often violent and sometimes revolutionary, reached a peak in the period 1906–1910. In spite of a rise of nominal wages, real wages decreased as a result of the high cost of living (la vie chère). Seasonal employment and precarious work were widespread. The creation of a Ministry of Labour in 1906 and the vote of a law which organized a pension scheme for industry workers and peasants in 1910 can be seen as the first steps in the development of a ‘welfare state’ and the integration of the working population into society. Many Catholics realized that the Church was losing its influence on the working class, especially in the urban centres. Under Popes Gregory XVI (1831–1846)

6 Church, society and economics and Pius IX (1846–1878) the position of the Church (shared by the French Jesuits) consisted of defending a compassionate approach, urging the poor to accept their fate and the rich to practise charity in order to alleviate the worst forms of destitution. This approach was called into question by a series of Catholic intellectuals who advocated various forms of ‘social Catholicism’. Some of these, like Félicité Robert de Lamennais (1782–1854) and Frédéric Ozanam (1811–1851), embraced the ideals of the French Revolution. The ‘social question’ was also central for a conservative Catholic stream, exemplified by Frédéric Le Play (1806–1882), which called for a return to traditional family values as a basis for social order. A counter-revolutionary stance was taken by the counts Albert de Mun (1841–1914) and René de la Tour du Pin (1834–1924): in their analysis of the causes of the Commune de Paris they attributed the responsibility of the physical and moral distress of the working class to the so-called Voltairian and hedonist bourgeoisie, that had broken the historical link between France and the Church. Both counts started animating the Cercles Catholiques d’Ouvriers, where they eventually teamed up with Léon Harmel (1829–1915). De Mun failed in his attempt to create a great Catholic party but, for years, argued in favour of social laws in the Parliament. By contrast, La Tour du Pin rejected le ralliement, became the theoretician of corporatism and inspired the royalist movement Action Française, which attracted many French Catholics. Harmel was an industrialist famous for the social experiments at his textile factory in Le Val des Bois, where he tried to develop a non-paternalist approach of the social question. These three have played a significant role in the elaboration of the encyclical Rerum Novarum, issued on 15 May 1891 by Pope Leo XIII, which defined the new social doctrine of the Church. Against the socialists, the encyclical justifies private property and rejects the idea of a class struggle, but it also condemns unrestricted liberalism. It recognizes ‘the misery and wretchedness pressing so unjustly on the majority of the working class’ (article 3) and it admits that a workman ‘is made the victim of force and injustice’ if the conditions induce him to accept a wage which is ‘insufficient to support a frugal and well-behaved wage-earner’ (article 45).5 This means that the principle of justice has to prevail over that of charity in the organization of social relationships, and that every worker must receive a decent wage. In more practical terms the encyclical recommends a corporatist organization of society, but leaves room for flexibility. For instance, it does not specify whether workers and employers of a given industry have to join a single or separate associations. Sharing the Christian values should eliminate major social conflicts; if employers or workers feel injured in any way, the dispute must be settled according to the decision of ‘prudent and upright men’ of the same body (article 78). The publication of the encyclical triggered new types of action from the Church and its members, even if ancient forms of charity subsisted, as exemplified by

5 We have used the English version of the encyclical which can be found on the official website of the Holy See (http://www.vatican.va).

Church, society and economics 7 the tragic fire of the Bazar de la Charité (1897) which caused the deaths of more than one hundred women of the nobility and high society. Two initiatives should be mentioned here. In 1903 the Jesuit Henri-Joseph Leroy (1847–1917) founded Action Populaire, which rapidly gained fame for its series of bestselling yellow booklets on social questions. The movement started to flourish when it moved to Reims in 1905 and was put under the direction of another Jesuit, Gustave Desbuquois (1869–1959).6 In 1909 Action Populaire took over the journal L’Association Catholique, founded by de Mun and de la Tour du Pin, and transformed it into Le Mouvement Social. The second inititiative is that of the Semaines Sociales de France, launched by Marius Gonin (1873–1937) and Adéodat Boissard (1870–1938).7 These annual one-week meetings in the summer, each year in a different town, were large-scale events in which laymen and clergy listened to conférences and discussed about the social doctrine of the Church. The first meeting was held in Lyon in 1904.

2 The Potron family environment 2.1 Paris and Courcelles The Potron family lived both in their apartment in ‘the 368’ in Paris and in the castle of Courcelles. The Parisian building saw a lot of coming and going, especially after 1900, since it served as meeting place and seat for various organizations, invariably of Catholic inspiration, which Auguste Potron supported financially as well as morally (see section 4). The castle was a more private residence. Courcelles is a small town to the north of Paris, administratively attached to Presles, then in the Seine-et-Oise department. At the end of the nineteenth century the Courcelles–Presles town was apart from the Parisian hustle and bustle without being completely isolated from it, as the train from Paris had a stop there. The castle, an eighteenth century stone building of the neoclassical style described by Honoré de Balzac in the novel Un Début dans la Vie (‘A Start in Life’, 1842), has a grand entrance adorned with a gate in wrought iron, in which the curled initials E and F remind us of its former owner Édouard Frottin. The main curiosity of the property is its seventy-acre park that Auguste commissioned to lay out in the romantic style. The landscape gardener Louis-Sulpice Varé (1803–1883), famous for his role in the creation of the Bagatelle park (Bois de Boulogne, Paris), developed the theme of water using a small river that runs through the parc before flowing into the Oise in Mours. He created ponds, wooden bridges and a pier, paths winding the woods, a false ruin, and a grotto with 6 On Desbuquois, see the two-volume biography by Paul Droulers, Politique Sociale et Christianisme. Le Père Desbuquois et l’Action Populaire (‘Débuts, Syndicalisme et Intégristes (1903–1918)’, Paris, Editions Ouvrières, 1969; ‘Dans la Gestation d’un Monde Nouveau (1919–1946)’, Paris, Editions Ouvrières, 1981). 7 On the Semaines Sociales, see Jean-Dominique Durand (ed.), Les Semaines Sociales de France. Cent Ans d’Engagement Social des Catholiques Français. 1904–2004 (Paris, Parole et Silence, 2006).

8 Church, society and economics the Virgin. Several dependencies of the castle were built in the park. In one of them, ‘Les Tilleuls’, Auguste and Cécile installed a school for young boys. That school was run by the Salesians, an order specialized in teaching. Later it served as a place for spiritual retreats, organized by the sisters of Notre-Dame du Cénacle, a female congregation of Ignatian inspiration. At the beginning of the twentieth century Auguste came into possession of another property in the region. His sister Élise and her husband Émile Leemans lived in Mours, a village on the railway line just after the Courcelles–Presles station. Émile Leemans owned three quarters of the lands of Mours, and also part of the lands of the nearby Abbaye Notre-Dame du Val, a Cistercian abbey founded in the twelfth century, which had been sold as a bien national and transformed into a farm. (The ‘national goods’ were those which had belonged to the nobility and the clergy under the Ancien Régime and were sold in the aftermath of the French Revolution.) He joined it to a 200 acre domain he had inherited from his father. When Émile Leemans died childless in 1873, his wife inherited everything. In his will he asked her to build an orphanage. Élise sold the abbey and its lands and used the proceeds to erect a home for the education of forty to fifty orphan girls between the age of 8 and 21 years, of legitimate birth and Catholic religion. The construction of the Marie-Émilie orphanage in Mours started in 1882, and although the tense relations between the Church and the State threatened the project, the chapel was consecrated on 6 November 1886. Élise Leemans entrusted the orphanage to the congregation La Sainte-Famille de Bordeaux, which had acquired the abbey of Royaumont, not far from Mours, to develop its activities. One of the seven branches of the congregation was specialized in the intellectual, moral and religious education of orphans. In her first will Élise intended to perpetuate the orphanage by leaving it to the congregation. But afraid that the orphanage would be taken over by the State if it belonged to a congregation, she changed her mind and decided to bequeath it to a man of absolute confidence, her own brother Auguste. He inherited it when Élise died in December 1903. 2.2 The Potron family Genealogical research has enabled us to find traces of the Potrons up to the middle of the eighteenth century.8 During the nineteenth century, there were several notaries and lawyers in the family. The origin of their fortune is unknown but a likely hypothesis is that it came from the sales of the biens nationaux, which would explain the importance of the landed property. The wealth of the family also

8 The information which follows comes from official records and from the family archives. We are especially grateful to Chantal Dufour and Jocelyne Dufour for digging up lots of biographical material from various archives, and to Isabelle Salmon-Legagneur for her extensive research in the family archives. The vivid memories of Denise Salmon-Legagneur (1908–2006), née Potron, have breathed life into these documents (see I1 in the Potron Bibliography).

Church, society and economics 9 increased by a strategy of matrimonial alliances illustrated by Élise and Auguste, but Auguste could impossibly have married Frottin’s daughter if he did not come from a rich family himself. Auguste and Cécile had five sons (Maurice, Henri, Émile, Robert and Édouard) and one daughter (Marie-Élisabeth, or Marie), all educated at home (see Table 1.1). Education by private tutors had been the rule in rich families but that tradition was declining. Cécile herself had been educated by a pious priest, and in all probability her children were also tutored by a cleric. Before we deal with Maurice, let us say a few words about his brothers and sister (see Table 1 for a family tree of the Potrons). Henri studied law in Paris. In 1906 he married Marthe Baldé. The Baldés lived in Louvres, a small town to the north-east of Paris. Originally a family of farmers, they became rich thanks to their holding of a post house and bought a castle by the middle of the nineteenth century. Henri Potron and Marthe Baldé had two daughters, Denise and Élisabeth. The family lived in Courcelles most of the year but moved to ‘the 368’ from February to May. In 1904, at the age of 28, Henri was elected mayor of Presles. He remained mayor for 40 years, with a break during the Great War when he was mobilized in the cavalry. Émile first followed his father’s path and graduated as an engineer of the École Centrale des Arts et Manufactures. Auguste advanced Émile’s career by buying for him the ironworks factory of Persan, known as Les Forges. Persan, a town on the river Oise not far from Courcelles and Mours, had been transformed into an industrial centre after the arrival of the railways in the 1840s, as the creation of the railway station and a workshop of the Compagnie des Chemins de Fer du Nord triggered a rapid industrialization. Existing factories such as Les Forges, La Soierie (silk) and India Rubber developed rapidly, and new industries such as Les Tapis (carpets), La Carroserie (coachwork), La Blanchisserie (cleaning), La Teinturerie (dyeing), L’Électricité, La Papeterie, were founded. The ‘Potron ironworks’ worked mainly for the railways, but their catalogue also showed the diversity of their production in coachwork. During the Great War, the reserve officer Émile Potron was appointed to his own factory, which had a strategic role for its production of car pieces. In 1930, the society had become a limited company, with Émile as director of the board. After a period of under-investment, the Persan ironworks definitely closed in 1954. Édouard was Maurice’s youngest brother. He entered the novitiate of the Society of Jesus at the age of 18 in Italy, then went to Jersey and taught English and German in Belgium. He was first mobilized as a nurse during the Great War, but finished the war as a translator attached to the American army. He pronounced his solemn vows in 1921, with his brother Maurice leading the ceremony. Édouard mainly served as an administrator in several Jesuit colleges. At the end of his life he negotiated, on behalf of the Jesuit order, with the Rothschild family about the purchase of the domain Les Fontaines near Chantilly. One brother and one sister of Maurice did not reach maturity. Robert died from diphtheria at the age of 6, while Marie, who intended to enter a religious order, succumbed to tuberculosis at the age of 15.

Robert Potron (1880–1886)

Auguste Potron (1844–1926) × Cécile Frottin (1849–1918)

Émile Potron (1878–1955)

Élisabeth Potron (1912–1938)

Henri Potron (1876–1954) × Marthe Baldé (1877–1945)

Denise Potron (1908–2006)

Maurice Potron, S.J. (1872–1942)

Table 1 The Potron family tree

Édouard Potron, S.J. (1883–1950)

Marie-Élisabeth Potron (1884–1899)

Élise Potron (1838–1903) × Émile Leemans (1829–1873)

Church, society and economics 11

3 Maurice’s education and vocation 3.1 Engineering studies The Catalogus Universalis Nostrorum, the book in which the Society of Jesus summarizes basic information on the lives and careers of its members, mentions that Maurice Potron was educated until the age of 16 ‘cum praeceptore domi’.9 In June 1889 he obtained the degree of baccalauréat in philosophy, and during the school year 1889–1890 he attended the Sainte-Geneviève school, a Jesuit-run institution where students prepared for the entrance exams at the Grandes Écoles. Maurice passed the entrance exam for the École Polytechnique after only one year of preparation, when two years are considered as the minimum, and ranked 91st in a class of 265 students. The École Polytechnique is the most prestigious French school for engineers; its initial purpose was military and it has conserved a military status. After two years of study, Maurice obtained the 14th place in the exit ranking. To complete their training students then had to choose an ‘application school’, with the highest ranking students getting first choice. Maurice did not follow the established hierarchy of these schools and opted for a barely prestigious institution, the École d’Application des Poudres et Salpêtres (Powder school): he was the only one in his class to make that choice.10 Very little is known about the way Maurice experienced his two years at the Polytechnique. Apart from the official photograph (Photo 1) with the uniform and the cocked hat, another traditional photograph shows the students in groups of ten, lying in eccentric positions. Maurice stands there, with crossed arms and no smile. He may have suffered from not having led a normal social life during his childhood and from being hampered by a stutter. In the political climate of that time, his religious convictions can also have been a source of mockery. It must be noticed, however, that Potron’s later social ties were mainly limited to Polytechnic engineers and Jesuits. His choice for the low profile Powder school may have been motivated by his decision to opt for a religious life: in 1891 and 1892 he attended spiritual retreats at the Villa Manrèse, located in Clamart, to the south of Paris.11 After three years of military service, two at the Polytechnique and one at the Powder school, Maurice was a second lieutenant and free of further military obligations, except periodic military exercises. The time had come to follow his vocation. 3.2 Becoming a Jesuit The training of a Jesuit is extremely long and alternates periods of religious education, apostolic work and general studies. The scheme is derived from Ignatius 9 Source: Catalogus Universalis Nostrorum, Archives Jésuites de la Province de France. 10 Source: Archives of the École Polytechnique. 11 Source: Folder ‘Retraites Manrèse – A: 1870–1893’ in the Archives Jésuites de la Province de France.

12 Church, society and economics

Photo 1 Potron’s official photograph as a student of the École Polytechnique, 1890. Source: École Polytechnique.

of Loyola’s Constitutions (1547). Two years of novitiate are followed by the first vows of poverty, chastity and obedience. After the novitiate, the standard path comprises two years of juvenate (language and literature), three years of scholasticate (philosophy), two years of regency (teaching or supervision in a Jesuit college) (see Photo 2) which are a training for apostolic activity, and again four years of scholasticate (theology). Then comes the Tertianship, or ‘third year of the novitiate’. Eventually, the solemn vows (including the fourth vow concerning special obedience to the pope with regard to missions) mark the full admission into the Society.

Church, society and economics 13

Photo 2 Potron as a young teacher among some ‘taupins’ at the École Sainte-Geneviève, 1898. Source: École Sainte-Geneviève.

Maurice’s personal trajectory began in October 1893, when he entered the Canterbury novitiate.12 On 13 November 1895 he took his first vows. He then followed a course in rhetorics in Canterbury (October 1895–October 1896), after which he returned to France to teach mathematics in the Sainte-Geneviève school (1896–1899). He went back to Jersey for two years of philosophy (teaching

12 The information comes mainly from the Catalogus Universalis Nostrorum.

14 Church, society and economics

Photo 3 Potron as a photographer, probably 1904. Source: Family archives. Note on the date: The photograph is not dated, but Potron seems to be about thirty and wears a mourning band. His aunt Elise died in December 1903.

mathematics at the same time), followed by one year of theology in Canterbury (1902) and three years at the Catholic Institute of Paris (1903–1906) (see Photo 3). In February 1905 he was ordained deacon; the ceremony, celebrated by Mgr Étienne-Marie Potron, took place in the chapel of Courcelles and was rather intimate.13 It was on 29 June 1905, day of the Saint Apostles Peter and Paul, that he was ordained priest in a more abundant ceremony in the church of Presles directed by a Jesuit bishop in the presence of fifteen priests. At that time Maurice was 33 years old. It was apparently to show respect for some tradition that he

13 There is no family relationship between Maurice Potron and Etienne-Marie Potron (1836–1905), titular bishop of Jericho; most probably he was invited to Courcelles because of the homonymy.

Church, society and economics 15 waited for this highly symbolic age to become a priest (there is no rule of this type in the Constitutions). According to Barriol (W1/Appendix I), ‘this was the goal he wanted to reach above all’. Maurice Potron spent his third year in Canterbury in 1906–1907. After an unusually long delay, he professed his final vows on 2 February 1912 (Candlemas, or day of the feast of the presentation of Jesus at the Temple, is a traditional day for this ceremony), and hence became a ‘professed of the four vows’ in the Society of Jesus after almost 19 years of training. On this occasion he made a significant gift to the Church, for which he was acknowledged by cardinal Léon-Adolphe Amette, archbishop of Paris, who regularly attended meetings in the 368. 3.3 Mathematics studies In parallel with his religious commitment, Maurice undertook a career as a teacher and researcher in mathematics. He had obtained a license degree in mathematics at the end of his years at the Polytechnique, and taught maths in France and Jersey. It is during his stay in Jersey that he wrote his first published scientific paper (see A1), which is directly related to the activities of the Society of Jesus. In 1893 the Swiss Jesuit Marc Dechevrens (1845–1923) had taken the initiative of building a meteorological observatory at the Maison Saint-Louis, the Jesuit college in St Hélier, Jersey.14 Dechevrens had lived for many years in China, where he had been director of the observatory of Zi-Ka-Wei, in the French Jesuit mission of Xujiahui, Shanghai; the quality of his forecasts had earned him the nickname of ‘father of typhoons’. He developed several new instruments and mechanical devices, one of which was a contrivance to draw mathematical curves. Similar devices had already been used by engineers and technical drawers, but Dechevrens’s campylograph (from the Greek καμπυλoς , ‘bent’ or ‘curved’) was remarkably simple and elegant: it was basically made of two wheels.15 In 1900 Dechevrens reported the invention in a short note presented to the Academy of Sciences;16 in 1901–1902 Potron published a thorough mathematical study of the tool, and wrote down the equations of the curves it could draw: these are Lissajous curves, which include in particular the family of the cycloids. Potron continued his research in mathematics by preparing a PhD thesis on the theory of finite groups at the University of Paris, under the supervision of Émile Picard (1856–1941). The theory of groups started around 1800 with the works of Carl Friedrich Gauss (1777–1855) and Évariste Galois (1811–1832). A group is a set of abstract elements on which one has defined an operation which admits 14 For more details, see Frank Le Blancq, ‘A brief history of weather observing and forecasting in Jersey’ (Weather, October 2005, 60: 284–90). 15 See Robert J. Whitaker, ‘Harmonographs. II. Circular design’ (American Journal of Physics, February 2001, 69: 174–83); a representation of the campylograph is on p. 175. 16 Marc Dechevrens, ‘Le campylographe, machine à tracer des courbes’ (Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences, 11 June 1900, 130: 1616–20).

16 Church, society and economics the same basic properties as the addition for integers. It is finite if the number of its elements, called the order of the group, is finite. If, inside a group, a certain subset has itself a group structure, that subset is called a subgroup and, under certain conditions, one can define a quotient group obtained by ‘dividing’ the group by its subgroup. In that way, a group can be decomposed into elementary basic pieces: the operation has some analogy with the decomposition of an integer into a product of prime numbers, with the primes as basic pieces. As a matter of fact, the groups of a prime order p constitute the elementary bricks of the decomposition. In the nineteenth century an ambitious mathematical programme was launched to classify all finite groups by using similar decompositions and more complex operations. A first step in this direction was accomplished by the Norwegian mathematician Ludwig Sylow (1832–1918). The groups of a prime order p ( p = 2, 3, 5, . . .) being elementary, a natural task consisted in studying the groups of order p2 ( p2 = 4, 9, 25, . . .), then of order p3 , and so forth. At the end of the century, the Italian mathematician Giuseppe Bagnera (1865–1927) had characterized the groups of order p5 . Following a proposition by Picard, Potron studied those of order p6 : the Potron groups have 64, 729, 15 725, etc. elements. It is hardly necessary to say that the question falls within the realm of ‘pure’ mathematical theory. Potron submitted his PhD thesis in June 1904 (see B1). The jury consisted of three very distinguished mathematicians: besides the thesis supervisor Émile Picard, its members were Paul Appell (1855–1930), president, and Henri Poincaré (1854–1912). In his thesis Potron showed that he mastered intricate mathematical techniques; some equalities required several lines to be written down. The jury evaluated his work in the following terms: In spite of a speech defect, Mr. Potron has exposed the results he has obtained in his thesis in a clear and interesting way. He has shown that he has perfect knowledge of the theory of groups, and in particular of the recent works on this difficult subject, of which the applications require really off-putting calculations which he has completed with a consciousness and a patience worthy of praise.17 It gave him the grade of honorable, which is one below the highest of très honorable.18 In the first pages of his thesis Potron duly referred to a few authors on the theory of groups. Two of them deserve our particular attention. The first is Jean Armand de Séguier (1862–1937), a Jesuit who was working on the same topic 17 ‘Rapport sur la Soutenance’, 28 June 1904 (file ‘AJ16 5538’, Archives Nationales de France, Paris). 18 On the meaning of these grades, see: Murad S. Taqqu, ‘Bachelier et son époque. Une conversation avec Bernard Bru’ (in: Jean-Michel Courtault and Youri Kabanov (eds), Louis Bachelier. Aux Origines de la Finance Mathématique, Besançon, Presses Universitaires de Franche-Comté, 2002: 87–110).

Church, society and economics 17 and had published the book Théorie des Groupes Finis. Éléments de la Théorie des Groupes Abstraits (Paris, Gauthier-Villars, 1904) just shortly before Potron completed his thesis. Obviously, they were in close contact: de Séguier thanked his ‘devoted friend’ Potron for reading the whole manuscript with the greatest care (see preface, p. ii), whereas Potron made use of a procedure that de Séguier had perfected. Though these hints leave little doubt as to their close collaboration (years later, they also wrote together a booklet on group theory, see B5), we have found no material details on it. The second author we have to mention is Georg Frobenius (1849–1917), the famous German mathematician who made a significant contribution to the theory of groups at the end of the nineteenth century. Frobenius’s idea was to find an analogy between an abstract group and the multiplicative group of square matrices with nonzero determinants. If it is possible to associate a matrix with every element of a group (that matrix then ‘represents’ the element) in such a way that the matrix associated with the product of two elements is the product of their associated matrices, the abstract group is basically reduced to that of the matrices, which is much better known. That ‘theory of representation’ proved to be an efficient tool; but as we shall see (see part B), it is a later contribution of Frobenius which had a profound influence on Potron’s economic work.

4 Faith and action In the encyclical Rerum Novarum Pope Leo XIII called upon the Catholics to take steps to implement the Church’s programme. The activities of the Potron family around 1900 are to a large extent part of the wave of initiatives generated by the encyclical in France. In this section we present some of the associations and projects in which the Potron family was heavily involved, and which are at the core of the faith-based network that Maurice Potron’s father maintained with the help of a very active associate, the Jesuit Henri-Régis Pupey-Girard. 4.1 The Union Sociale d’Ingénieurs Catholiques In 1891–1892 Henri Pupey (1860–1948), then still studying to become a Jesuit, spent his regency at the Sainte-Geneviève school in Paris, where lots of future engineering students were preparing for the entrance exams.19 Inspired by similar

19 The information on Pupey and his activities comes from the folders ‘Pupey-Girard’ (H. Pu 50 to H. Pu 62) of the Archives Jésuites de la Province de France; Henri-Régis Pupey-Girard, ‘Les origines du Centre U.S.I.C. Souvenirs de son fondateur’ (Echo de l’Union Sociale d’Ingénieurs Catholiques, 1943, 34: 83–92); Henri du Passage, Soixante Ans d’Effort Allègre. Le Père HenriRégis Pupey-Girard. 1860–1948 (Paris, Alsatia, 1949); Hugues Beylard, ‘Pupey-Girard, Henri’ (in: Jean-Marie Mayeur and Yves-Marie Hilaire (eds), Dictionnaire du Monde Religieux dans la France Contemporaine. vol. 1: Les Jésuites, Paris, Beauchesne, 1985: 220–1); André Grelon and Françoise Subileau, ‘Le Mouvement des Cadres Chrétiens et La Vie Nouvelle: des cadres catholiques militants’ (Revue Française de Science Politique, 1989, 39: 314–40).

18 Church, society and economics initiatives of the Jesuit Henri Watrigant (1845–1926) in the north of France, Pupey first organized a closed spiritual retreat for workers,20 but soon transposed the idea to his own environment. It was at the end of a retreat for students of the École Centrale des Arts et Manufactures at the Villa Manrèse in Clamart that he launched the project of founding the Union des Ingénieurs Catholiques (UIC), an association of engineers with both a spiritual and a professional aim. While retreats and conferences would ensure that the engineers remained good Catholics, an employment office and other professional services would advance their more material interests. The success of the UIC manifested itself in the increase of its members (140 in 1895, about 400 in 1900). A Lettre de l’Union was created and, in 1902, the UIC became L’Abeille, a kind of trade union for engineers of the École Centrale, of which Pupey was counsellor-chaplain and de facto strong man (Pupey had an extensive conception of the counsellor’s role). Pupey also founded and inspired similar professional associations of employers, like the Unions Fédérales Professionnelles de Catholiques (UFPC). At a time when almost no professional structures existed in France, Pupey’s projects could count on the approval and encouragement of employers such as Léon Harmel. This was their way of making engineers, and employers in general, part of the solution of the social question as advocated by the encyclical Rerum Novarum.21 The Potron family played an important role in these ventures. In 1905 L’Abeille transformed itself into the Union Sociale d’Ingénieurs Catholiques (USIC). Its first president was an engineer who never practised, Auguste Potron, and up from 1909 the assistant-counsellor-chaplain was another engineer, Maurice Potron. The new association opened itself to a larger category of engineers: after years of perseverance and in spite of the opposition of the direction of the school, Pupey finally managed to get access to the students of the École Polytechnique, who became the privileged subjects of his attention. Auguste Potron also offered accommodation to Pupey and his organizations. Already in 1901 Pupey himself had moved into ‘the 368’, and in 1907 the secretary of USIC was transferred from its previous headquarters at 76, rue des Saints Pères, a property of the Jesuits, to the same place.

20 Source: Folder ‘Œuvre des retraites fermées (action des PP. Watrigant et Pupey-Girard)’, (I. Pa 78) of the Archives Jésuites de la Province de France. Pupey wrote three papers entitled ‘L’apostolat de l’ouvrier par l’ouvrier’ (Lettres de Jersey, December 1888, 7: 439–51; December 1889, 8: 331–41; and May 1892, 11: 90–7). 21 An early articulation of this position can be found in A.B., ‘L’ingénieur et son rôle social’ (Etudes Religieuses, Philosophiques, Historiques et Littéraires, 15 February 1895, 64: 193– 219). The initials are those of the Jesuit Auguste Belanger (1860–1905); see: ‘Le P. Auguste Belanger’ (Etudes, 1905, 104: 102–4). In a document entitled ‘Origine de l’USIC – Souvenirs épars à assembler’ (folder ‘Pupey-Girard’ (H. Pu 50) of the Archives Jésuites de la Province de France), which is a collection of typewritten shorthand notes by Pupey-Girard, one reads: ‘During philosophy at Jersey, collaboration with Father Bellanger urged to write and sign a paper for Etudes: social role of engineer’. This explains why Henri du Passage (o.c.: 32–3) wrote that Belanger did little else than transmit the ideas of his friend Pupey.

Church, society and economics 19 4.2 The Ligue Patriotique des Françaises The adoption of the 1901 law on associations enraged many Catholics in France. Inspired by the Jesuit Antonin Eymieu (1861–1933), a group of women issued from the conservative bourgeoisie and traditional aristocracy, and headed by Jeanne Lestra (1864–1951) and countess Octavie Thomas de Saint Laurent (1845–1940), founded the Ligue des Femmes Françaises (LFF) in order to mobilize against the anticlerical reforms.22 The League started its crusade in Lyon; enthusiastically supported and directed by the nobility, it soon spread everywhere in France. But its activities also led to frictions and rivalries within the Catholic camp. The fear that the royalist hardliners from Lyon might endanger the more republican-minded approach of the Catholic hierarchy made cardinal Richard, archbishop of Paris, decide to send someone to put the house in order. What he needed was a man with exceptional organizational skills, and he found him in ‘the 368’: Henri Pupey. In 1902 Pupey took matters in hand at the Parisian branch of the LFF. By transforming the name of this branch into Ligue Patriotique des Françaises (LPDF) he broke away from the LFF and effectively set up a rival organization with headquarters in Paris.23 At the direction of the LPDF he put women he trusted. The first president, baroness de Brigode, who remained in office until 1906, and the secretary, Marie Frossard (1869–1954), who served until 1933, were members of the Société des Filles du Cœur de Marie, a secular institute close to the Jesuit order and the Holy See; Pupey himself became chaplain, which he remained until the war. Also in this case Pupey involved the Potron family: the seat of the new league was located at ‘the 368’, and Cécile Potron was its treasurer (an important position since the league secretly subsidized the Action Libérale Populaire, a political party of Catholic inspiration). Although the ‘dictatorship’ of Frossard and Pupey plunged the LPDF into a crisis in 1914, the league continued to grow fast and claimed to have up to 3.5 million members in the inter-war period, making it the most important women’s movement in France at that time.24

22 See Hugues Beylard, ‘Eymieu Antonin’ (in: Jean-Marie Mayeur and Yves-Marie Hilaire (eds), Dictionnaire du Monde Religieux dans la France Contemporaine. Vol. 1: Les Jésuites, Paris, Beauchesne, 1985: 110–11); Claudie Brunel, ‘Lestra Jeanne’ (ibid. Vol. 6: Lyon, Paris, Beauchesne, 1994: 276); Christian Ponson and Anne Cova, ‘Thomas de Saint-Laurent comtesse Henry, née Marie Adeline Octavie Bolot d’Ancier’ (ibid.: 398); Bruno Dumons, ‘Mobilisation politique et ligues féminines dans la France catholique du début du siècle. La Ligue des Femmes Françaises et La Ligue Patriotique des Françaises (1901–1914)’ (Vingtième Siècle. Revue d’Histoire, January–March 2002, 73: 39–50); Bruno Dumons, Les Dames de la Ligue des Femmes Françaises (1901–1914) (Paris, Editions du Cerf, 2006). 23 Only in 1933 the two organizations would merge again, forming the Ligue Féminine d’Action Catholique Française. 24 For details, see Odile Sarti, The Ligue Patriotique des Françaises (1902–1933): A Feminine Response to the Secularization of French Society (New York, Garland, 1992) and Magali Della Sudda, Une Activité Politique Conservatrice avant le Droit de Suffrage en France et en Italie. Socio-histoire de la Politisation des Femmes Catholiques au Sein de la Ligue Patriotique des Françaises (1902–1933) et de l’Unione fra le Donne Cattoliche d’Italia (1909–1919) (PhD thesis, Paris, EHESS and Rome, Università La Sapienza, 2007).

20 Church, society and economics 4.3 ‘The 368’ as Pupey’s headquarters Pupey transformed the building in the centre of Paris into the nerve centre of his activities. A photograph of ‘the 368’ taken around 1910 shows a four-storey building organized around a rectangular courtyard. At the third floor were the rooms occupied by the USIC, including two rooms for the library and a meeting room for about one hundred persons. During his period as archbishop of Paris (1908–1920), cardinal Amette went there to conclude some of the meetings organized by the USIC. At the fourth floor was the so-called jésuitière (Jesuits’ dwelling, or more humorously Jesuits’ breeding) where Pupey and a collaborator, the Jesuit Joseph de Baudicour (1856–1936), were accommodated, and also Maurice Potron himself when he was counsellor-chaplain of the USIC. In the courtyard, a big iron and glass structure resembling a greenhouse was the meeting room for the women of the LPDF. On the first floor, one can discern a number of flags and, above them, the name of the League in big stone letters running all along the façade. Some of these rooms were used as offices by the LPDF, others by the USIC, the UFPC, the Union des Retraites Régionales, and so forth; all of these organizations operated under the strange collective name of Union par le Secrétariat Central,25 the initials of which coincide with those of the Union du Sacré-Coeur, which Pupey considered as the spiritual core of his network. The working of these organizations was the cause of incessant comings and goings by students, engineers, employers, women, true Jesuits and false secular priests. Yet all were mindful to have a legal cover for their activities, as the police monitored the actions of the man they once designated as public danger No. 1.26 4.4 The Villa Saint-Régis In 1908, to the astonishment of many, Auguste Potron decided to close the Mours orphanage which he had inherited from his sister Élise, and to transform it into a centre of spiritual retreats under the supervision of the Jesuits.27 Many people were shocked by this apparent betrayal of his promise to Élise and tried to make him change his mind. Auguste maintained his decision, arguing that his sister had given him orally some freedom in the interpretation of her will. Pupey’s influence on this transformation is unmistakable. Spiritual retreats were Pupey’s favourite means of doing apostolic work, allowing him to reach carefully selected audiences, such as engineering students. Between 1896 and 1909 he used

25 One can also find references to Union par le Syndicat Central and to Union par le Service Central. 26 Source: ‘Dossier des archives secrètes du Ministère de l’Intérieur’, undated (around 1935), folder ‘Pupey-Girard’ (H. Pu 50 B2) of the Archives Jésuites de la Province de France. 27 The information in this section is mainly based on the folder ‘Mours’ of the Archives Jésuites de la Province de France.

Church, society and economics 21 to organize his retreats at the Villa Saint-Joseph in Épinay, a town 15 km north of Paris, where Maurice acted as a paymaster for several years. In 1901 he had founded the Union des Retraites Régionales, an organization meant to provide a legal basis for centres of retreat in several French towns. A setback occurred when a court ruled that, contrary to what some documents suggested, the Villa Saint-Joseph had remained de facto property of the Jesuits and must therefore be closed. Transferring the retreats to Potron’s property in Mours made it possible to continue these essential activities. In a climate where the State snatched social and educational tasks from Catholic organizations and proceeded to direct attacks against religion, Pupey must have convinced Auguste Potron that an orphanage was much less effective than a centre of spiritual retreats directed by the Jesuits. A strong indirect hint of Pupey’s hand in all this is the change of name signalling the transformation. When taking his first vows he had added Régis to his first name, apparently because he aspired to do apostolic work among the working population of Paris (the Jesuit Jean-François Régis (1597–1640) is known as the apostle of the Velay and Vivarais regions, and as the patron saint of lacemakers).28 When the orphanage became a retreat, its name changed from Marie-Émilie to Villa Saint-Régis. Spiritual retreats for men, according to Saint Ignatius’s Spiritual Exercises, were organized in Mours from Easter 1909 onward. To make sure that the participants reached their destination, many documents issued at ‘the 368’ had a stamp showing the way from the railway station to the Villa. The success of the retreats led to an extension of the Villa: a new aisle was added, more bedrooms were fitted out, and a promenade gallery with glass windows was built for use in case of bad weather. There were 1,000 participants in 1910 and this number increased significantly in the years thereafter, requiring an important logistical support for accommodation. The privileged organization formula consisted of retreats for homogenous groups based on profession (peasants, railway workers, teachers, engineers), region or nation (e.g., Poles working on French farms). For members of the clergy, who in general came for longer periods of time, specialized lectures were organized on theological subjects like the doctrine of the Mystic Body, or on more practical matters like the organization of apostolic work in a diocese. Pupey was especially keen to organize retreats for Sainte-Geneviève students, for engineers, and most of all for engineers from the Polytechnique (in the announcements he drew attention to the presence of an ancien, Maurice Potron). The Polytechnicians were numerous enough for the organization of two groups, first-year and second-year, and it cannot be doubted that these retreats contributed significantly to the strong Catholic influence in the Polytechnique in the 1930s.

28 See Bernard Deponcin, Madame Pupey-Girard, 1834–1893. Son Rayonnement Surnaturel dans la Vie et les Œuvres de ses Enfants (Paris, Bloud et Gay, 1943: 159).

22 Church, society and economics

5 The war years 5.1 Maurice’s involvement Maurice Potron had completed his three years of military service by studying at the Polytechnique and the Powder school.29 At regular intervals he had been called up for training sessions lasting from one week to one month, which he followed as requested. He was promoted to lieutenant and appointed at the powder factory of Le Bouchet, to the south of Paris, where he was appreciated for his scientific knowledge and the work he did. A report noticed that, during a one-week training period in July 1911, he ‘followed with zeal the operations of fabrication of the B powder’. Two years later, he ‘swiftly learned the improvements in fabrication since his last period’. The reasons of his interest in the production process of that explosive were not merely military: he was then working on his economic model which assumes that the technical coefficients of fabrication are known. At the outbreak of the war in August 1914, Maurice was 42 years old. He was mobilized and sent to the powder factory, but he immediately asked to be transferred to the artillery; in September 1914 he joined the 2nd Régiment d’Artillerie Lourde (heavy artillery regiment). He was thrilled that the officers showed an inclination for religion and celebrated a ‘quasi-official’ mass for the departure to the front. In 1915 and 1916 he was on the battle front in the Champagne region as part of the officer staff of the 82nd Régiment d’Artillerie Lourde Tractée. In 1916 and 1917 he served as captain of the 8th battery of the 106th Régiment d’Artillerie Lourde Hippomobile in Verdun, and in 1917 he also fought in the fierce Chemin des Dames battle. In July 1917 he joined the staff of the 6th army, and in June 1918 he was nominated in Saumur where he became an artillery instructor for the 1st US army. A few days before the armistice he was again on the front, commanding a battery of the 136th Régiment d’Artillerie Lourde Hippomobile. Maurice Potron was repeatedly praised for his military activities. On Christmas 1916 he became knight of the Légion d’honneur with the commendation: ‘Energetic and competent battery commandant. Gets excellent results from his men’. The US army also spoke highly of him: ‘Excellent officer, working hard and methodically’. What the military reports fail to mention is that he had also been gassed: his health remained deteriorated for the rest of his life.30 5.2 In Courcelles The war was also felt in Courcelles.31 In August 1914 the sisters who used to organize secret retreats in a dependency of the castle had to cease their activities with immediate effect. Henri, though mayor of Presles, was sent to a cavalry

29 The information on Maurice Potron military career comes from the folder ‘Potron’ (5YE.139069) at the Service Historique de la Défense. 30 Source: Interview of Denise Salmon-Legagneur (see I1). 31 This section is based on information from the family archives and on recollections of Denise Salmon-Legagneur (see I1).

Church, society and economics 23 corps, and the dependencies of the castle were requisitioned for the installation of two military hospitals (the one at Les Tilleuls was named Marie-Élisabeth, in remembrance of Maurice’s sister). Maurice returned to Courcelles during military leaves. After one such leave in 1916, he wrote to Pupey to complain about his father’s inefficient management of the hospitals and his mother’s hostility to the chief nurse. He urged Pupey to send a serious warning to his parents, threatening to spend his future leaves elsewhere if nothing happened. In the park of the castle temporary buildings were erected for refugees from invaded regions; during the raids of the German aviation on Paris these had to be evacuated. After the Chemin des Dames battle, four barracks were built for the wounded. The front was not so far away, and once a shell landed in the park, with no serious damage. The family tradition holds that Maurice located the place of the Paris Gun (the German cannon known as Big Bertha) after having examined the impact of the bomb on the ground. Pupey also served in the army, as military chaplain attached to the 37th division. The presence of chaplains near the soldiers had been developed after Albert de Mun’s interventions on behalf of national reconciliation, or ‘Sacred Union’, proclaimed at the beginning of the war. Pupey, aged 54 when the war broke out, was present on many exposed battlefields, received the Légion d’honneur and the War cross for his ‘absolute disdain of danger’ but was never wounded.32 He had made arrangements with Auguste in case he was killed: this was necessary since the president of the Unions Fédérales did not know they were financed by the Potron family. In a report to a superior in the Society he expressed his concern about the way some of the organizations he had founded were managed, as decisions were taken without the advice of the counsellor-chaplain. He was especially worried about his initiatives in the direction of the students and alumni of the École Polytechnique, insisting that these should be continued. During his military leaves he spent much of his time developing his apostolic work towards them. 5.3 The Potron family after the war Cécile Potron, Maurice’s mother, died after an illness in February 1918. When the war was over, Auguste entrusted the management of his affairs to his son Henri. A photograph taken in the 1920s shows Auguste aged and serene, in his office, with a snapshot of his prematurely dead son Robert on the wall. He then lived at ‘the 368’ with his son Émile and Henri’s father-in-law. In 1926 Auguste was 82 years old and tired. Questions were raised about the future of his fortune and the organizations he had so generously supported. The conversion of the Mours orphanage into a centre for retreats remained a delicate subject. At some point the bishop of Versailles seemed convinced that the best solution would be to restore the original situation, but after discussions with Henri he changed his mind and considered it preferable to leave things

32 Folder ‘Pupey-Girard (1860–1948), Éléments biographiques’ (H. Pu. 50) of the Archives Jésuites de la Province de France.

24 Church, society and economics as they were. The decision satisfied the Jesuits, as the retreats had become an important instrument of their apostolic activity (the yearly number of participants to retreats attained 3,700 in 1937). It was only in the aftermath of the Second World War that a final settlement was reached: after the departure of the German troops that had occupied the Villa, it was used for a short span of time by a congregation for the education of young women and then sold in 1950 to the Pères Blancs, the Society of the Missionaries of Africa. Auguste Potron died on 23 November 1926. He was the father of two Jesuits, Commander of the Pontifical Equestrian Order of Saint Gregory the Great (one of the Papal orders of knighthood bestowed to lay persons in recognition of their eminent contributions to the Catholic cause), generous donor of many organizations and charities, and according to a former bishop of Versailles ‘the greatest Catholic in the diocese’.33 His sons had been raised in the same spirit and were expected to continue to support his initiatives. To a large extent they did so, but it cannot be denied that they acted more prudently, maybe out of fear of getting drawn into affairs like that of the orphanage. In 1927, for instance, Émile ceased to finance the health centre in Persan, and Henri interrupted the payment of the salaries of the Courcelles school. Actions like these surprised the Jesuits and the Church authorities, and one wondered whether Maurice and Édouard, the two Jesuits who had their part in the inheritance, agreed with all the decisions of their brothers. A judgement of a French tribunal about the financing of the Villa Saint-Régis (the question was whether it should have a commercial status) indicates that in the 1930s the expenses of the Villa continued to be at the charge of its owner, then Maurice Potron. ‘The 368’ in Paris gradually lost its status as the headquarters of the Potron– Pupey network. Pupey–Girard left the building in 1920. In 1928 the USIC needed larger premises and moved to new headquarters in 18, rue de Varennes, but it did not leave without paying tribute to its first president, whose financial generosity and hospitality had been crucial for the organization in the first years of its existence. Émile moved to another district of Paris, and only Henri kept an apartment, where his daughter Denise lived until her marriage in 1930. The building was then sold to an insurance company.

6 Maurice Potron’s career as teacher and researcher 6.1 Teaching mathematics In 1912 Potron was nominated as professor in the Facultés Catholiques de l’Ouest in Angers and put in charge of the course on differential and integral calculus.34 He remained nominally in charge of this course until 1919, but he effectively 33 Folder ‘Mours’ of the Archives Jésuites de la Province de France. 34 The information on Maurice Potron’s career as a teacher comes from the Archives de l’Institut Catholique de Paris, the Archives de l’Université Catholique de Lille, the Archives de l’Université Catholique de l’Ouest, and the Archives Jésuites de la Province de France.

Church, society and economics 25

Photo 4 Potron as a Jesuit, around 1932. Source: Family archives. Note on the date: The photograph is not dated, but Potron is about the same age as on another photograph from Sainte-Geneviève, dated 1932.

taught only until the beginning of the war. In 1919 he joined the Sainte-Geneviève school in Versailles,35 where he lectured mathematics and helped students in their preparation for the entrance exams of the Grandes Écoles. In the period 1921–1924 he also taught at the Facultés Catholiques de Lille, where he was responsible for two courses, one on differential and integral calculus and another one on astronomy. Correspondence from that period indicates that it was his ambition to get appointed to a chair in the Institut Catholique of Paris, but doubts over his qualities as a teacher clouded his prospects. Mgr Alfred Baudrillart, rector of the

35 The school was originally in Paris. A long dispute, about whether it was Jesuit property or not, had led to its closing in 1913, but it immediately reopened in Versailles.

26 Church, society and economics Institute, strongly objected to Potron’s nomination.36 Eventually he did join the Institute in 1928 he was offered a position as chargé de la conférence d’analyse et éléments de mécanique, which he accepted even though it was a position of lecturer, not professor. In the following years he made several attempts to obtain a chair, but every time he failed. In 1934 a kind of compromise solution was reached when he became professeur hors cadre. During his time at the Institut Catholique he lectured on mechanics, differential and integral calculus, group theory and its applications, advanced geometry,37 etc. During the Second World War he taught at the Collège Saint François-Xavier in Vannes, Brittany. A substantial part of Potron’s scientific output is directly linked to his teaching activities. At regular intervals he published solutions of exercises and problems which had been used in exams at institutions of higher education. He did this first in 1913 for a competitive exam organized by the Facultés Catholiques de l’Ouest (see S1), then for the entrance exams of the period 1901–1921 at the Polytechnique (in collaboration with François Michel, see B2),38 and finally for the competitive exams organized at the national level (the concours d’agrégation) and for the final exams at all French faculties of mathematics (the examens de licence) of the period 1933–1937 (see S2 to S10, and also X1 to X6). Moreover, he issued two volumes of exercises with solutions on differential and integral calculus (see B3). In his very last published paper he dealt with new orientations in the teaching of mathematics (see A45). In the course of his scientific activities, Potron also wrote book reviews for the French Jesuit journal Études (see R1 to R8). 6.2 Mathematical research Potron kept a life-long passion for the topic of his PhD thesis, the theory of groups. In 1904 he published summaries of his results in the Bulletin de la Société Mathématique de France (see A2 and A3) and in the Comptes Rendus de l’Académie des Sciences (see A4 and A5). In the 1920s he returned to the subject with publications about a group of order 25 920 (see A16 and A17) and about the theory of continuous finite groups of transformations (see A19, A23 and A24). In the 1930s he explored further aspects of the theory of groups (see A32 and A42), and he published two books in the Mémorial des Sciences Mathématiques series: one on the theory of Lie groups (see B4) in 1936, and the other jointly

36 In 1930, complaints of his students, concerning in particular his mathematical notations, led to an affaire Potron at the École Sainte-Geneviève; cf. folder ‘Sainte-Geneviève’ of the Archives Jésuites de la Province de France. 37 One of Potron’s colleagues at the Institute, the renowned philologist Louis Mariès (1876–1958), dedicated his book Hippolyte de Rome. Sur les Bénédictions d’Isaac, de Jacob et de Moïse (Paris, Les Belles Lettres, 1935) ‘to my master Maurice Potron’. 38 François Michel (1866–1931) was a Polytechnic engineer who worked for a railway company; in 1900 he published Recueil de Problèmes de Géométrie Analytique. Solutions des Problèmes Donnés au Concours d’Admission à l’École Polytechnique de 1860 à 1900 (Paris, Gauthier-Villars, 1900).

Church, society and economics 27 with de Séguier on the theory of abstract groups in 1938, just after de Séguier’s death (see B5). In the same period Potron also lectured on the relation between the theory of groups and quantum mechanics, and he wrote an unpublished paper in which he applied the theory of groups to the decoration of a plane (see U4). His publications on algebra (see A26 and A37), analysis (see A6, A33 and A34), differential geometry (see A28, A29, A30 and A36), and differential and integral calculus (see A22, A27, A31 and A38) reveal an interest for research on the connections between algebra, analysis and geometry. Potron may have been induced to link the theory of groups to these other fields by his teaching duties, since analytical geometry constituted a basic course for many students in Grandes Écoles, universities and preparatory schools. Potron also wrote a few papers about number theory (see A20, A21 and A25). The network of mathematicians in which he was active was not very extensive. We have already mentioned his collaboration with de Séguier and Michel. Two other mathematicians with whom he had contacts were the Jesuits Raymond Alezais (1860–1940) and Élie Pinte (1883–1968); he thanked these ‘two devoted friends’ in the book he published in 1926–1927 (see the preface of Vol. 2 of B3). In the 1920s he maintained an extensive correspondence with Alezais, who was in charge of the course of differential and integral calculus at the Facultés Catholiques de Lyon.39 In 1925 he proposed Pinte for membership of the Société Mathématique de France (the only time he ever did),40 but by the end of the 1920s he complained about his teaching of mathematics at the Sainte-Geneviève school. Potron himself had been elected as member of the Societé Mathématique de France at the end of 1904, after the completion of his PhD thesis, on the proposition of de Séguier and d’Ocagne.41 In the period 1936–1938 he served on the Council of this association, and in 1936 he attended the International Congress of Mathematicians in Oslo as a member of the delegation of the association. Until the very end of his life he kept on doing mathematical research. Although his discoveries lacked spectacular results or radically new ideas, it is fair to say that he has contributed to the development of the mathematics of his times. 6.3 Science and philosophy Apart from his incursion into economics, Potron also made sporadic contributions to other disciplines. Between April 1905 and April 1906 (he was then a student of theology at the Catholic Institute of Paris) Potron spent many hours at the Paris Observatory. Astronomers working there had noticed that one of its ‘meridian 39 This correspondence is conserved in the folder ‘Potron’ of the Archives Jésuites de la Province de France. 40 See ‘Comptes rendus des séances’ (Bulletin de la Société Mathématique de France, 1925, 25: 19–20). 41 See ‘Comptes rendus des séances’ (Bulletin de la Société Mathématique de France, 1904, 32: 317); curiously, in all subsequent lists of members published by this journal Potron’s year of admission is given as 1894.

28 Church, society and economics circle telescopes’ performed poorly. Potron was one of four persons with a scientific training invited by the observatory to identify the origin of the problem and to test a new instrument (it is unclear whether these highly qualified guests were paid for their collaboration). Potron completed five observation notebooks of about 40 pages.42 Each double page corresponds to the observations of one night and is filled with cryptic notations, a few words (like ‘Hydra’) and calculations concerning the required corrections.43 For completeness we also mention three contributions which belong to the realm of philosophy of science. The first is a lemma on ‘Systems of the universe’ for the Dictionnaire Apologétique de la Foi Catholique Contenant les Preuves de la Vérité de la Religion et les Réponses aux Objections Tirées des Sciences Humaines (see A18). This paper leans heavily on the work of the French physicist and philosopher Pierre Duhem (1861–1916), author of the monumental work Le Système du Monde. Histoire des Doctrines Cosmologiques de Platon à Copernic (10 volumes of which only 4 were published before his death). The second is a short article on the foundations of arithmetics (see A43), one of Potron’s last papers, and the third an unpublished paper on the philosophy of mathematics (see U5). 6.4 Contributions to economics Potron started working on economic issues during the last stages of his formation as a Jesuit (see Photo 4), when he was active in Pupey-Girard’s network of organizations. Maybe the discussions of economic and social problems in the Action Populaire movement headed by Gustave Desbuquois and Joseph Zamanski triggered Potron’s interest in economics, but at a deeper level his main source of inspiration seems to have been Pope Leo XIII’s encyclical Rerum Novarum. The first summary of Potron’s economic research can be found in a footnote of an article on the law of just prices published by Desbuquois in the October 1911 issue of Le Mouvement Social (see A7/Chapter 1). Immediately thereafter Potron presented his economic model in his very first economic paper published in the 15 October 1911 issue of the Échos de l’Union Sociale d’Ingénieurs Catholiques et des Unions-Fédérales-Professionnelles de Catholiques (see A8/Chapter 2), and sketched its mathematical aspects in two notes published in December 1911 in the Comptes Rendus de l’Académie des Sciences (see A9/Chapter 3 and A10/Chapter 4). A more developed version, with all the technical details relegated to a very long footnote, was published in the 15 April 1912 issue of Le Mouvement Social (see A11/Chapter 5); at about the same time he also drafted a short note on his model, which remained unpublished (see U1/Chapter 7). The details of the extension of the Perron–Frobenius theorem and its application to

42 These notebooks are conserved in the Complément à l’inventaire Bigourdan, folder ‘A-F 14’, of the Archives de l’Observatoire de Paris. 43 Potron followed astronomy courses at the observatory during that period. He taught astronomy during one year in Angers and then signed a contract for the publication of a book, but the project never materialized (see folder ‘Potron’ of the Archives Jésuites de la Province de France).

Church, society and economics 29 the economic model were spelled out in a long paper in the 1913 volume of the Annales Scientifiques de l’École Normale Supérieure (see A13/Chapter 8). In two related papers Potron concentrated on the empirical aspects of his model. The first stressed the necessity of sufficiently detailed industrial statistics in order to build the generalized input–output table which is at the core of the production side of the model; it was published in the 1912 volume of the Journal de la Société de Statistique de Paris (see A12/Chapter 6). The second linked the model to the question of unemployment; it appeared as an appendix to the proceedings of a conference held in 1913 in Ghent (see A14/Chapter 9). A bit of an outlier is Potron’s last paper in his first period of economic research (1911–1914): a study of the ‘Taylor system’ published in two parts in the issues of 15 June and 15 July 1914 of Le Mouvement Social (see A15/Chapter 10). By that time Frederick W. Taylor’s ideas on scientific management had already been disseminated in France by influential engineers such as Henry Le Chatelier, Charles Fremont and Charles de Fréminville. Potron borrowed heavily from their writings for the description of the system. A more personal contribution concerns the link he established between the search for higher productivity and his economic model. On a more philosophical level, and in contrast to some of the reservations voiced in Catholic journals, he defended the Taylor system by arguing that the discipline it imposes on the workers helps them to develop Christian virtues. At that time Potron’s writings received little attention from economists; we have not found any references to them in economics journals of that period. The awkward mix of loose economic wording and hard mathematical formalization certainly did not ease their understanding. An obvious reason for the lack of recognition is that in general economists do not read mathematics journals or Catholic periodicals: the very choice of these publication outlets reveals Potron’s isolation. There is a sharp contrast between the ambitious aims of Potron’s model and the ill-directed attempts of getting his message across. The outburst of the war brought his economic research to a brutal stop. Perhaps as a reaction to the failure of finding an audience for his views, he completely abandoned economics in the 1920s and published exclusively on mathematical topics. The second period of Potron’s economic research (1935–1942) began more than 20 years after the first had ended. By then the École Polytechnique had started to organize courses in economics, and many engineers believed they could contribute to the development of techniques to better understand and manage the complex economic machinery. The deep and prolonged economic crisis provided a strong incentive to study economics. It is no surprise that the Centre Polytechnicien d’Études Économiques, commonly known as X-Crise, was founded in that period. Mainly constituted of engineers of the Polytechnique (as indicated by the X in its name),44 this group pioneered the mathematical and econometric study

44 The name X is often used to designate the Polytechnique. It apparently refers both to the two crossed cannons in the emblem of the school and to the mathematical symbol, as an indication of the importance of mathematics in its training programme.

30 Church, society and economics

Photo 5 Announcement of the 1937 lectures at the Institut Catholique de Paris. Source: Institut Catholique de Paris.

of economic issues in France.45 Potron, who had remained close to the school (he published twice in its Journal, see A21 and A34) drew attention to his previous work in economics when he heard about a discussion mentioning the use of matrices by Ragnar Frisch. He wrote a letter to Robert Gibrat, which was published in the July–August 1935 bulletin of X-Crise (see A35/Chapter 11). He revisited his economic model in a very concise paper presented at the 1936 International Congress of Mathematicians in Oslo (see A39/Chapter 12), of which he also drafted a more extended version (see U2/Chapter 13). In 1937 he organized an ambitious series of six lectures on ‘The mathematical aspect of certain economic problems in relation to recent results of the theory of nonnegative matrices’ at the Institut Catholique in Paris (see Photo 5), an initiative for which he received only lukewarm support from the institute itself. The texts of these lectures, together with an appendix entitled ‘The Hebrew Manna Problem’, were gathered in a typescript (see U3/Chapter 14); he announced the publication of the booklet by SPES in Paris, but that project seems to have failed.46 One of the novelties which Potron introduced in the lecture series concerned a new way of finding positive solutions of certain systems of linear equations, based upon an idea of Cornelius Lanczos.

45 On the X-Crise group, see: X-Crise. Centre Polytechnicien d’Études Économiques. De la Récurrence des Crises Économiques. Son Cinquantenaire, 1931–1981 (Paris, Economica, 1982) and Émeric Lendjel’s 2002 habilitation thesis (see W6). 46 The publisher SPES was closely associated to Gustave Desbuquois’s Action Populaire movement.

Church, society and economics 31

Photo 6 Caricature of Potron and other teachers of the Collège Saint François-Xavier, around 1941 (excerpt). Source: Collège Saint François-Xavier in Vannes. Note: The author of the caricature is André Bouler, then pupil at the college, who later became a Jesuit himself and a painter. The caricature represents a train with several coaches. The coach regrouping the mathematicians ‘Potiron’ (pumpkin), ‘Blum’ (for his physical resemblance with the French politician Léon Blum) and ‘Vincent’ is headed for Lesvellec, the closest psychiatric asylum.

He presented this procedure in two separate papers, one in the Comptes Rendus de l’Académie des Sciences (see A40/Chapter 15) and the other in the Bulletin de la Société Mathématique de France (see A41/Chapter 16). Potron’s final economic contribution appeared in the 1942 volume of the Journal de la Société de Statistique de Paris: it was a letter to Alfred Sauvy in which he returned to the issue of the collection of statistical data (see A44/Chapter 17). In spite of Potron’s considerable – and original – work in economics, it seems that he interacted very little with economists. He never published in economics journals and almost never referred to the professional literature. It was only late in life (in 1938) that he became a member of the Econometric Society.47

47 Alfred Barriol, a close friend of Maurice Potron, mentioned in his obituary that Potron was also a member of the Société d’Économie Politique (see W1/Appendix I). We have not been able to find a confirmation of this.

32 Church, society and economics

7 The last years In September 1940, Potron, then aged 68, was sent to the Saint François-Xavier college in Vannes, a town in the south of Brittany. In this Jesuit secondary school he taught mathematics to pupils preparing the baccalauréat exam, in replacement of a teacher who had been summoned for military service. More than 60 years later, some of his former students remembered Potron – nicknamed potiron or pumpkin – as a devoted teacher unable to adapt his lessons to the needs of his audience48 (see Photo 6). The winter of 1941–1942 was extremely cold. Since all the fuel was requisitioned by the German army which occupied a part of the college, the students had to break the ice if they wanted to have a wash in the morning. Maurice Potron caught pneumonia and died there on 21 January 1942. Due to the circumstances of the war, only his brother Édouard managed to travel to Vannes and attend the funeral service in the chapel of the college. Maurice Potron was buried in the common tomb of the Jesuits at the Vannes cemetery. The text of the memorial address of the Rector of the college was reproduced in the school’s newsletter, Xavier. Only one obituary was published: a glowing tribute written by his friend Alfred Barriol for the Journal de la Société de Statistique de Paris (see W1/Appendix I). In the same issue of the journal Barriol also published a favourable report on Potron’s model (see W2/Appendix II). About ten years later, however, Michel Vittrant, a fellow Jesuit, penned down a ruthless critique of ‘The Hebrew Manna Problem’ and harshly concluded: ‘This work is good to keep as a remembrance of the defects of its author, a talented mathematician, unable to understand reality and to adapt himself to it’(see W3/Appendix III). The second part of our introduction will show that this assessment of Potron’s work is wide off the mark.

Part B Potron’s economic model We now turn our attention to Potron’s mathematical-economic model. Our purpose here is to present its essential features in a simple and transparent way, in an attempt to clarify Potron’s idiosyncratic formulation and make the model more comprehensible to present-day readers. We begin with a brief explanation of the gist of Potron’s approach (section 8). We then present his economic model, which consists of two parts. First we deal with the physical side of the model: we specify the principles imposed upon the production and consumption of goods and upon the provision of labour, and examine which condition guarantees the existence of a ‘satisfactory regime of production and labour’ (section 9). Next we move to the value side, where we look at the principles imposed upon prices and wages. The identification of the condition which guarantees the existence of a ‘satisfactory regime of prices and wages’ brings us to Potron’s remarkable duality results (section 10). After that we turn to Potron’s analysis of systems involving decomposable matrices (section 11) and explore the mathematical criteria he

48 Recollection of ancient pupils of Maurice Potron in Vannes (see I2).

Church, society and economics 33 developed to check whether satisfactory regimes can exist, thereby anticipating what is known today as the Hawkins–Simon condition (section 12). We also touch upon Potron’s proposals for a more efficient organization of society in line with the views of the Church (section 13), and assess the hints he gave about the dynamic aspects of his model (section 14). We conclude with an appreciation of his contribution as an economist (section 15).

8 Potron’s approach to economic problems It may be useful to recall that Potron was a mathematician with no training in economics and social sciences. Initially, his main source of inspiration seems to have been the encyclical Rerum Novarum, considered as an update of the old scholastic doctrine. It is therefore no surprise that concepts such as ‘just price’ and ‘just wage’ figure prominently in his first publications. Potron gave a formal expression to these notions and examined the consequences of their joint implementation. In this process he became aware of the intimate connections and potential conflicts between the rights and principles of justice an economic system ought to realize. He took it for granted that mathematics constitutes the appropriate tool to study these interdependencies; translating the issue into equations allowed him to understand the origin and nature of the economic problem and to make suggestions for its solution. Even if Potron was convinced that his model gave valuable insights into the causes of real world problems such as the ‘miseries and ruins’ and the ‘blind race to the rise between prices and wages’,49 he did not develop it in order to explain the functioning of the existing economic system. His approach was mainly normative: what interested him in the first place was the question of the mere existence of quantities and values compatible with a specified set of rights and principles of justice. He never studied the implications of competitive behaviour for the economy and, in fact, never referred to either market or competition (which may explain why he introduced the assumption of the uniformity of wages across firms for a given profession only in 1937). Potron showed that, if the condition for the existence of a ‘satisfactory’ regime is met (which he considered to be likely), the corresponding quantities and values must lie within certain limits. Problems are bound to occur if the actual quantities and values do not lie in the adequate region, a distinct possibility given that they are chosen ‘in a haphazard manner’ by independent entrepreneurs.50 Potron believed that the corporatism proposed in Rerum Novarum and advocated by leading Christian intellectuals would give society the means to solve these problems, especially if enough efforts were made to systematically collect socioeconomic data. With the help of a Bureau de Calculs the complex interdependencies between economic magnitudes could be taken into account, and a satisfactory regime implemented.

49 A35/Chapter 11. 50 ibid.

34 Church, society and economics Potron’s economic writings come from two distinct periods, the first covering the years just before the beginning of the Great War (1911–1914) and the second running from the mid-thirties to the end of his life (1935–1942). In spite of this two-decade hiatus Potron’s model remained basically the same, although he did introduce a few changes: he ceased to refer to the scholastic concepts of ‘just price’ and ‘just wage’, abandoned the intermediate notion of a ‘simply satisfactory’ regime of prices and wages, and (in 1937) assumed the uniformity of the hourly wages per type of labour. He also modified his views on the scope of the Bureau de Calculs. Potron’s knowledge of the economic theory of his times was very poor, and he almost never referred to the economic literature. The lack of understanding was mutual: except in the late 1930s, very few economists took notice of his work. Part of the blame for the lack of recognition must be laid with Potron himself. His notation is cumbersome, the economic meaning of his symbols is rarely transparent, and his economic assumptions are sometimes awkward. No wonder that his economic contributions fell into oblivion until they were rediscovered in 2000 by the French economist Émeric Lendjel (see E1, W4 and also W6). Since then, Potron’s model has inspired scholarly work by Gilbert Abraham-Frois and Emeric Lendjel (see W5, W8, W10, W11), Christian Bidard (see W12), Christian Bidard and Guido Erreygers (see W13, W17), Christian Bidard, Guido Erreygers and Wilfried Parys (see W9, W17), Kenji Mori (see W14) and Wilfried Parys (see W16). In 2004 Gilbert Abraham-Frois and Émeric Lendjel compiled a French edition of his economic works (see E2). Unfortunately, their introduction (see W7) misinterprets some of Potron’s formal results and suggests a Sraffian interpretation of Potron’s views which we have described elsewhere as misleading (see W13). What we try to do here is to present Potron’s model in a matrix algebraic notation using symbols with which present-day economists are familiar. We propose a compact formalization which remains as close as possible to Potron’s original formulation, but wihout exploring all the cases he examined; for instance, we always assume that wages are uniform across firms.

9 The quantity side 9.1 The data Let us begin by listing the economic data from which Potron started. His model involves various goods (which Potron called ‘results of labour’, ‘economic results’ or ‘economic goods’, and which include services), social groups (designated as ‘social categories’ or ‘professions’) and labour types. There are n goods, labelled by the indices i, j = 1, 2, . . . , n, and m social groups, labelled by the indices g , h = 1, 2, . . . , m; each social group is associated with a specific labour type, labelled in the same way. Goods are produced by firms using labour and goods as inputs, over an implicit period of production equal to a year. The methods of production operated by firms

Church, society and economics 35 are of the single-product type and admit constant returns to scale.51 Per product one method of production is available.52 Assuming that one unit of good i requires the input of lih hours of labour type h and aij units of good j, the set of available production methods can be represented by the following matrices:53 • •

L = [lih ], the [n × m] matrix of labour input coefficients; and A = aij , the [n × n] matrix of commodity input coefficients.

Goods are consumed by households. Households belong to different social groups according to the profession of their household heads. Each social group is characterized by a specific consumption profile, expressed as an annual consumption basket, and which is defined independently of its effective income.54 Assuming that a household of social group h consumes bhj units of good j per year, the consumption habits can be represented by the following matrix: •

B = bhj , the [m × n] matrix of consumption baskets.

Within 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 (also designated as ‘simple consumer’, ‘rentier’ or ‘capitalist’), by contrast, is one of which the head does not work. Non-working households are the owners of the firms’ capital and draw their incomes from the firms’ profits.55 Potron assumed that a labourer cannot work more than a well specified number of hours per year. This maximum takes into account the usual periods of rest and holidays:

51 Potron did not refer to the standard notion of returns to scale, but he adopted it when making calculations. When he explicitly mentioned the hypothesis of proportionality between input use and production levels (see section 4 of the first lecture of U3/Chapter 14), it is clear that he had the notion of constant returns in mind. 52 If several methods of production are used for the same good, Potron suggested to consider either the ‘average’ method of production or that of a representative firm (see section 3 of the first lecture of U3/Chapter 14). 53 We denote matrices by capital bold characters and vectors by small bold characters. Scalars are in italics, and can be small as well as capitalized. Matrix M is nonnegative (notation M 0 ) if all its entries are positive or zero, semipositive (notation M ≥ 0 ) if it is nonnegative and nonzero, and positive (notation M > 0 ) if all its entries are positive. The same symbols are used to compare two matrices or two vectors. 54 Potron did not take into account that the consumption level of a household depends upon the number of its members: he reasoned as if the household head was the only member that mattered. 55 Potron envisaged that the levels of savings and profits might induce working household heads to become rentiers, and vice versa (see section 18 of A11/Chapter 5).

36 Church, society and economics •

N , a scalar equal to the maximum number of hours which a labourer can perform during a year.

This number of work-hours is given by law or tradition. Potron defined a state of the economy by the data (L, A, B, N ). He realized that these data are liable to change, but assumed that they could be taken as given ‘during a certain time’.56 Obviously the three matrices L, A and B are nonnegative, and the scalar N positive. Following Potron, we assume that L has no zero column, meaning that each type of labour is used in at least one firm, and that B has no zero row, meaning that each consumption basket contains at least one good. Moreover, we assume that A is an indecomposable matrix (this assumption will be relaxed in section 11). Under the assumption of a given state of the economy, one would expect the data of the economy to include some kind of information on the size of the population. Somewhat surprisingly, Potron’s assumptions about the number of households varied from one publication to the other. We therefore do not treat that number as part of the core data of Potron’s model. We will reconsider the question when we discuss the solutions of his model (section 9.7). 9.2 Quantity variables Whereas the data describe the given state of the economy, the variables are the unknowns of the economic problem, which has two sides: the physical problem which involves the quantity variables, and the value problem which involves the value variables. A set of values of the quantity variables characterizes a specific physical regime; the notion of a value regime is defined in a similar way. Potron was interested in the existence of quantity and value regimes which satisfy a certain number of conditions. In this section we concentrate on the quantity problem. Potron introduced a distinction between principal and secondary unknowns. As far as quantities are concerned, the principal variables are the activity levels of the n available production methods, and the numbers of workers in the m social groups. Let yi be the activity level of the method producing good i, and xh the number of workers of type h. Then we have: • •

y = [ yi ], the [1 × n] vector of activity levels; and x = [xh ], the [1 × m] vector of workers.

All principal variables are required to be strictly positive. The group of secondary quantity unknowns consists of the m numbers of nonworkers (sh ), the m unemployment hours (zh ) per year and the n excess production levels of goods ( fi ). The unemployment hours measure the difference between the number of hours the workers of a given type could work and the hours they 56 Section 18 of A11/Chapter 5.

Church, society and economics 37 actually work. Excess production refers to the difference between production and consumption. In vector terms we have: • • •

s = [sh ], the [1 × m] vector of non-working consumers; z = [zh ], the [1 × m] unemployment vector; and f = [ fi ], the [1 × n] excess production vector.

All secondary unknowns are required to be nonnegative. A limit case is reached when all the secondary unknowns are zero. 9.3 Sufficient production and right to rest A regime of production and labour, defined by adequately chosen levels of the quantity variables, is 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 is written: y

yA + xB

(1)

The second principle, which we call 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: Nx

yL

(2)

Making use of the secondary unknowns, both of these conditions can be expressed in a more precise form as equations rather than inequalities. Since working and non-working households of a given social group have the same consumption habits, the equality which corresponds to (1) is: y = yA + (x + s)B + f

(3)

Likewise, relation (2) leads to the equality: N x = yL + z

(4)

As a matter of fact, Potron always expressed his principles as equations. The main drawback of this procedure is that it involves additional variables. For simplicity, we follow the ‘inequalities’ version, but briefly return to the original ‘equations’ version at the end of this section. 9.4 One problem, three formulations One of the originalities of Potron’s work is that he applied the Perron–Frobenius theorem as soon as 1911, well before any other economist. That theorem, which

38 Church, society and economics constitutes Potron’s mathematical workhorse, was first stated by Perron in 1907 and then improved by Frobenius in 1908 and 1909. By the Perron–Frobenius theorem, we designate a group of related results which can be summarized as follows. Let M be a square, semipositive and indecomposable matrix. Then its eigenvalue with largest modulus, called dominant root and denoted dom(M), is real, positive and simple and is associated with a positive column (or row) eigenvector. For a given scalar λ, there exists a positive column vector c (or a positive row vector r) such that the vector inequality Mc λc (or, alternatively rM λr) holds if and only if dom(M) λ; also, Mc ≤ λc (or, alternatively rM ≤ λr) if and only if dom(M) < λ. Moreover, this last inequality is equivalent to the existence and positivity of the matrix (λI − M)−1 . If the principle of sufficient production holds, inequality (1) with x > 0 and y > 0 and the assumptions about B imply y ≥ yA, therefore the dominant eigenvalue of A must be smaller than 1. In the footsteps of Potron,57 we assume that this condition holds, ensuring the existence and positivity of the inverse matrix (I − A)−1 . A few elementary operations allow us to formulate the previous problem in three different ways. The first approach consists of using (1) to obtain y xB(I − A)−1 , and then to eliminate y in (2) and derive N x xB(I − A)−1 L. Defining: P ≡ B(I − A)−1 L

(5)

the problem is then reduced to finding a vector x > 0 such that: Nx

xP

(6)

The second approach goes the other way around and consists in eliminating x. From (2) we obtain x yL/N , and substituting this into (1) we derive y y(A + LB/N ). Defining: Q ≡ A + LB/N

(7)

the problem can be expressed as that of finding a vector y > 0 such that: y

yQ

(8)

Potron did not notice the existence of a third and even simpler formulation. By introducing the matrix: R≡

0 B L/N A

57 See, e.g., section 17 of A13/Chapter 8.

(9)

Church, society and economics 39 the conditions (1) and (2) are compactly integrated into the problem of finding a vector (x, y) > 0 such that: (x, y)

(x, y)R

(10)

Given the assumptions on L, A and B, the three matrices P, Q and R are semipositive and indecomposable. 9.5 The existence condition Each of these formulations allows us to express the condition for the existence of a satisfactory regime of production and labour in terms of eigenvalues. Potron focused on the properties of the matrices P (of which the order is equal to the number of social groups) and Q (of which the order is equal to the number of goods). Looking at (6), the Perron–Frobenius theorem immediately shows that a satisfactory regime of production and labour exists if and only if: ν ≡ dom(P) ≤ N

(11)

From (8), an equivalent condition is: dom(Q) ≤ 1

(12)

These are, in a nutshell, two fundamental results which Potron demonstrated over and over again. Applying the same technique to matrix R, he could have obtained a third equivalent condition: dom(R) ≤ 1

(13)

When any of these conditions holds as an equality, the inequalities (1) and (2) are transformed into equalities and we are in the limit case mentioned at the end of section 9.2: the only solution is the one in which there are no nonworkers, no unemployment and no excess production. Positive vectors of activity levels and of workers characterizing a satisfactory regime of production and labour are then unique up to a factor of proportionality. By contrast, when any of the inequalities (11), (12) or (13) is strict, there exist infinitely many satisfactory regimes. Potron saw that the distance between N and ν 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.58 9.6 Interpretations of the condition Potron realized that the entries of P can be given an economic interpretation: pgh represents the amount of labour of type h required to obtain a net product 58 See, e.g., A35/Chapter 11.

40 Church, society and economics equal to the consumption basket of social group g.59 Even more interesting is the economic interpretation of condition (11): ν is the minimum number of workhours compatible with the existence of a satisfactory regime of production and labour. Since, in the limit case, the secondary unknowns are zero, it turns out that the dominant eigenvalue of matrix P,60 which Potron called the ‘characteristic number of the given socioeconomic state’, can be taken as the measure of the ‘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’.61 Hence, ν ≤ N 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.62 Simple economic interpretations can also be provided for matrices Q and R and the corresponding inequalities (12) and (13), although Potron never mentioned them. Observe that the labour inputs in matrix L are measured in hours and the scalar N expresses the maximum number of work-hours in a year; this means that the elements of L/N represent the labour inputs measured as fractions of the year. Since the rows of matrix B represent the yearly consumption baskets of the labourers, matrix Q defined by (7) is the matrix of the overall input coefficients when every unit of labour is replaced by its corresponding wage basket. Similarly, matrix R defined by (9) is the matrix of input coefficients where the first m ‘goods’ are the different types of labour measured in yearly units, considered as being produced by the corresponding annual wage baskets, and the last n goods are the usual commodities. It is well known that, for a square matrix C of input coefficients, the associated economy can reproduce itself with a nonnegative surplus if and only if dom(C) is at most equal to 1. Therefore, both conditions (12) and (13) are alternative ways to state that a satisfactory regime exists. 9.7 The secondary unknowns The existence condition indicates when a solution is possible, but it does not determine what the solution will be. The exact values of the principal variables depend upon the specific values of the secondary unknowns. Without much loss of generality, we assume that the existence conditions hold strictly, i.e. we have ν < N and dom(Q) < 1. From equations (3) and (4) we easily derive the relation between the principal and the secondary unknowns. Following the P-approach,

59 60 61 62

See section 16 of A13/Chapter 8. See, e.g., section 4 of A7/Chapter 1. Ibid. Potron adopted various specific values: N = 300 days per year (section 15 of A11/Chapter 5), N = 313 days per year, i.e. 365 days minus 52 Sundays (section 17 of A13/Chapter 8), and N = 3,000 hours per year in the fictitious ‘manna economy’ (section 4 of the appendix to U3/Chapter 14).

Church, society and economics 41 we obtain: x = (sB + f )(I − A)−1 L + z [N I − P]−1

(14)

The Q-approach, by contrast, leads to: y = [(z/N + s)B + f ] [I − Q]−1

(15)

Since P and Q are indecomposable, the inverse matrices on the right-hand sides of these expressions are positive. Hence, whatever nonnegative values the secondary unknowns s, f and z may take, provided they are not all zero, the principal variables x and y are strictly positive. In Potron’s words, ‘there always exists a satisfactory regime of production and labour in which (…) one can assign arbitrary values to the numbers of simple consumers of every social category and to the collective unemployment of each category of labourers’.63 But the reverse does not hold: ‘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’.64 It is this formal property which makes the general distinction between principal and secondary unknowns (we shall return to this question in section 13). An exception to that property of secondary unknowns occurs in the limit case s = f = z = 0. Then ν = N and/or dom(Q) = 1, and there are no activity levels and distributions of workers able to sustain a semipositive combination of the secondary unknowns (s, f , z). A more significant restriction concerns the size of the population. In some publications Potron mentioned that the total number of households is fixed, 65 once he specified that the total number of both working and non-working households is fixed,66 but mostly he remained silent on the issue. His mathematical treatment shows that he had clearly in mind an homothety property: if two economies have the same characteristics (L, A, B, N ) and only differ by their sizes, their satisfactory regimes of production and labour are also homothetic (same relative distribution of workers, same relative activity levels). In that sense, the size of the population is of secondary importance. This does not mean that it does not matter, and Potron expressed himself rather loosely on that topic. Take for instance the above quotation, where Potron asserted that the number of simple consumers can be given arbitrary values. Clearly enough, the ratio between the non-workers and 63 64 65 66

Section 15 of A13/Chapter 8. A35/Chapter 11. See, e.g., section 10 of A11/Chapter 5. See section I of A14/Chapter 9.

42 Church, society and economics the workers cannot be too high in a satisfactory regime of production and labour, and sustaining a great number of non-workers requires a great number of workers. Potron’s statement only holds true only if the size of the population is considered as an adjustment variable, an implicit assumption which shows that Potron did not always pay much attention to ‘real and concrete economic facts ’.67

10 The value side 10.1 The value variables When 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 ( pi ) and the m hourly wages of the various types of labour (wh ). In vector terms this gives: • •

p = [ pi ], the [n × 1] price vector; and w = [wh ], the [m × 1] wage vector.

These variables must all be positive. The secondary value unknowns comprise the n unit profits (πi ), which measure the difference between the selling price and the cost price per unit of good produced, and the m household savings (eh ), which capture the difference between a household’s earnings and its cost of living.68 In vector terms we have: • •

π = [πi ], the [n × 1] vector of unit profits; and e = [eh ], the [m × 1] vector of savings.

The secondary unknowns are required to be nonnegative. Let us consider a worker of type h. His wage bill depends on the number of hours he is effectively employed; we designate the number of hours he actually works by th . Two cases can be distinguished: (i) if th = N (i.e. if he works the maximum number of hours), his income amounts to Nwh ; (ii) if th < N , his income is equal to th wh , and the out of work hours N − th correspond to what Potron called ‘unemployment’. Potron assumed implicitly that if there is collective unemployment in a given profession h, that unemployment is shared equally among all workers of that type (it is therefore partial unemployment). Since the total yearly demand for labour of type h is equal to i yi lih , the number of hours effectively performed by a worker of type h is th = i yi lih /xh . In our

67 Section 8 of A11/Chapter 5. 68 Instead of the standard economic terms ‘profits’ (profits) and ‘savings’ (épargne), Potron systematically used the terms ‘benefits’ (bénéfices) and ‘economies’ (économies). For household savings he also used the inappropriate expression gains, which means earnings (section 6 of the sixth lecture of U3/Chapter 14).

Church, society and economics 43 matrix representation, we list these m scalars as the diagonal entries of a diagonal matrix T: •

T = diag [th ], the [m × m] diagonal matrix of effective working hours.

This diagonal matrix is defined implicitly by the equality: xT = yL

(16)

The principle of the right to rest expressed by inequality (2) can therefore be written alternatively as: T

NI

(17)

10.2 Justice in exchange and the right to life A regime of prices and wages, defined by adequately chosen levels of prices and wages variables, is 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: p

Ap + Lw

(18)

The 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: Nw

Bp

(19)

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

Bp

(20)

Condition (19) is necessary in any case. Potron reflected the difference between the two versions of the right to life in the distinction between a simply and an effectively satisfactory regime of prices and wages, with the first satisfying (18) and (19) and the second (18) and (20). One of the reasons for doing so is that condition (19) can be checked independently of the quantity variables. By contrast, condition (20) 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. Making use of the secondary unknowns π for profits and e for savings, conditions (18) and (20) can also be expressed as equations: p = Ap + Lw + π

(21)

Tw = Bp + e

(22)

44 Church, society and economics 10.3 The weak duality result 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 consists of using (18) and the property that matrix (I − A) has a positive inverse to eliminate the price vector p from (19). The value problem is thus reduced to the existence of a vector of wages w > 0 such that: Nw

Pw

(23)

Alternatively, following the Q-approach the problem can be reduced to the existence of a vector of prices p > 0 such that: p

Qp

(24)

The R-approach, not followed by Potron, amounts to wondering if there exists w > 0 such that: a vector of wages-and-prices p w p

R

w p

(25)

Remarkably enough, the inequalities (23), (24) and (25) are nothing but the transposes of the conditions (6), (8) and (10). We can once again apply the Perron–Frobenius theorem to deduce that any of the three conditions (11), (12) and (13) 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. We refer to it as Potron’s ‘weak’ duality result. To the best of our knowledge, it is the first duality result ever stated and proved for disaggregated economic models. 10.4 The strong duality result As a remarkable extension of this weak duality result Potron demonstrated that the same conditions also guarantee the existence of an effectively satisfactory regime of prices and wages, i.e. a solution p > 0 and w > 0 of (18) and (20). Potron showed the existence of such a solution by considering a slightly modified system. Consider an economy in state (L, A, B, N ), and assume that the condition for the existence of a satisfactory regime of production and labour is met. Let x > 0 and y > 0 define such a satisfactory regime. Suppose now that the labour coefficients are changed in such a way that, at the production levels y, all available labourers x must work the maximum amount of hours, i.e. we change the labour input matrix from L to L such that yL = N x. This can be done by taking L = N LT−1 . For a transformed economy in state (L , A, B, N ) the vectors x and y define a satisfactory

Church, society and economics 45 regime of production and labour, with the peculiarity that expression (2) relative to the right to rest holds as an equality. Knowing this, we can follow two roads. First, we can apply the weak duality result and assert the existence of a price vector p > 0 and a wage vector w > 0 defining a simply satisfactory regime of prices and wages for the transformed economy. If we then change the wages in order that the annual wage bills are the same in the initial and transformed economies (take w = N T−1 w ), it is easy to verify that price vector p > 0 and wage vector w > 0 define an effectively satisfactory regime of prices and wages for the initial economy. Second, we can repeat the proof of the previous section, i.e. apply the Perron–Frobenius theorem to the matrix A + L B/N = A + LT−1 B, show that its dominant eigenvalue is at most equal to one, take p equal to its dominant eigenvector and w equal to T−1 Bp, and finally verify that the price-and-wage vector (p, w) > 0 sustains an effectively satisfactory regime of prices and wages for an economy in state (L, A, B, N ). 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’.69 This property is what we call the ‘strong’ duality result. After 1935, Potron opted for the second way and ceased to refer to the weak duality result as an intermediate step. He dropped the notion of a simply satisfactory regime of prices and wages, and used the notion of a satisfactory regime of prices and wages as a shorthand for an effectively satisfactory regime. 10.5 A remarkable identity Potron was keen to point out that a combination of the quantity and value solutions reveals an interesting result. For this it is necessary to express the quantity and value systems as equations rather than as inequalities. Consider the satisfactory regimes defined by (x, y) and (p, w). Let us left-multiply equality (21) by y, rightmultiply (3) by p and (22) by x, add up these equalities and make use of (16). One obtains:70 xe + yπ = sBp + fp

(26)

The interpretation of the formula is: the sum of the workers’ savings and the firms’ profits represents exactly the sum of the cost of living of the non-workers and the value of excess production. Subtracting f π from both sides we arrive at: xe + (y − f )π = sBp + f (p−π )

(27)

In Potron’s words: ‘This formula means that (…) the sum of the workers’ economies and the firms’ benefits is equal to the non-workers’ total cost of living, 69 A10/Chapter 4. 70 Equation (16) in the second lecture of U3/Chapter 14.

46 Church, society and economics increased by the cost prices of all overproduced commodities’.71 In his lecture series of 1937 he used this result in a rare attempt of economic analysis, thereby referring almost accidentally to ‘finance’ and ‘hoarding’.72

11 Decomposable systems When deriving the results presented above we took the easy road by assuming that matrix A is indecomposable.73 So doing we have made the analysis more restrictive than that of Potron, who from the very beginning of his economic work systematically examined how the results change under the assumption that matrix A is decomposable.74 In this section we briefly consider his research on this topic. In 1911 Potron’s basic mathematical tool, the Perron–Frobenius theorem, was stated only for matrices with positive entries. Since it cannot reasonably be assumed that the input matrix of an economy is always positive, Potron extended the existing results in two directions. First, he identified the case in which the presence of zero entries does not alter the properties established for positive matrices; this happens when matrices are indecomposable. Second, when the zero entries do matter, i.e. in the case of decomposable (or ‘reduced’) matrices, he stated weaker versions of the known properties and interpreted them in economic terms. To evaluate Potron’s contributions, it may be useful to say a few words about the history of the Perron–Frobenius theorem.75 Perron obtained the first version of the theorem in 1907, while he was working on the properties of continued fractions. Frobenius generalized some of Perron’s results in 1908 and 1909, as a by-product of his research into the links between the theory of groups and matrix algebra. There is little doubt that Potron, who had finished his PhD thesis on the theory of groups (see B1) in 1904, became acquainted with this work because he was following Frobenius’s publications on the theory of groups. In extending the results to nonnegative square matrices, Potron made use of a hint in Frobenius’s 1909 paper, suggesting generalizations ‘by continuity considerations’. Summaries published in December 1911 show that Potron had rapidly established the extensions that apply to his economic model.76 But his complete proofs were published only in 1913,77 the year after the publication of Frobenius’s article on nonnegative matrices. Potron was aware that this new paper of Frobenius covered partially the same ground.78 The two articles are, however, 71 U2/Chapter 13. 72 Section 6 of the second lecture of U3/Chapter 14. 73 As an aside it should be noted that this assumption is stronger than strictly required, since the results continue to hold under the weaker hypothesis that matrix Q is indecomposable. 74 See, e.g. sections 3 and 4 of A9/Chapter 3. 75 More details can be found in Thomas Hawkins, ‘Continued fractions and the origins of the Perron–Frobenius theorem’ (Archive for History of Exact Sciences, 2008, 62: 655–717). The exact references to the work of Perron and Frobenius can be found in note 1 of A9/Chapter 3. 76 A9/Chapter 3 and A10/Chapter 4. 77 A13/Chapter 8. 78 Note 1 of A13/Chapter 8.

Church, society and economics 47 quite different. The main lacuna of Potron’s mathematical study is the absence of the distinction between primitive and imprimitive matrices, but that is of little importance for the economic applications. In addition, Potron’s treatment has a few awkward elements; for instance, Potron met difficulties to extend the equivalence between conditions (11) and (12) because the criterion he used, the existence of an associated semipositive eigenvector, no longer characterizes the dominant eigenvalue in the case of decomposability. Potron’s interest in the analysis of decomposable matrices reached a climax when he presented a set of fourteen related theorems in the fourth lecture of his 1937 lecture series. But he left it to the reader to appreciate the value of the possible nuggets; moreover, his proofs involve intricate manipulations of minors, and his notations are artificial (e.g., a cofactor is written as a partial derivative). In his last paper on nonnegative matrices Potron became more personal and made explicit claims on results which he had obtained as opposed to those established by Frobenius.79 It remains to be seen, however, whether Potron’s results are really anything more than corollaries of Frobenius’s results. When deriving the economic consequences of decomposability, Potron concentrated on the matrix Q. After a convenient reordering of the goods and industries, its first rows are represented by a semipositive square submatrix on the left and a zero submatrix on the right. Such a reordering reveals that the principles of sufficient production and of the right to rest can be met even if the goods corresponding to the last columns are not produced. These goods can therefore be deleted from the system, just as the social categories who consume them, but ‘this deletion does not disrupt at all the rest of production’.80 Symmetrically, the principles of justice in exchange and of the right to life can be met even if some produced goods have a zero price and some wages are zero. The free goods can be produced only by means of inputs which are themselves free and by workers who receive a zero wage, whereas the goods involved in the ‘direct or indirect consumption’ of the workers with zero wages must all be free.81 Potron called the regimes in which the principles just mentioned are met with semipositive rather than positive vectors ‘semi-satisfactory’. Curiously enough, he always linked the appearance of zeroes to the limit case dom(Q) = 1, whereas in fact the phenomenon is only due to decomposabilty and may well occur when dom(Q) < 1.82

12 The computational problem The criterion ν ≡ dom(P) < N , or the alternative condition dom(Q) < 1, which ensures the existence of a double satisfactory regime, relies on the comparison of

79 80 81 82

Section 1 of A41/Chapter 16. Note 18 of section 15 of A13/Chapter 8. Ibid. See our paper (W17) for technical details, in particular on the relationships between the decomposability properties of matrices P, Q and R.

48 Church, society and economics the dominant eigenvalue of a matrix with a given number. Before the development of computers and the perfection of specific algorithms, the calculation of the eigenvalues of large matrices was considered as difficult. Potron found a way to avoid it. 12.1 The Potron criterion The difficulty may be by-passed if the condition ν < N or its alternative can be given an equivalent form not requiring the determination of the dominant eigenvalue. Potron stated a first criterion of this type in 1913: he showed that, for a semipositive square matrix M, a given scalar λ is greater than the dominant root of M if and only if all the symmetric minors of matrix λI − M are positive.83 (A symmetric minor of order k, or principal minor in modern language, is obtained by simultaneously deleting n − k rows and columns of the same rank; n is the dimension of the matrix.) Unfortunately, its application requires the calculation of 2n minors, which is practically impossible if the dimension of the matrix is the number of goods (for instance, if n = 100, more than 10 billion determinants have to be calculated). In 1937, Potron greatly improved upon this result by showing that it suffices to check the positivity of the n leading symmetric minors of λI − M, i.e. the minors made of the first k rows and columns (k = 1, . . . , n) of matrix M.84 The number of minors to be calculated is thus reduced from 2n to n. Today, mathematicians usually attribute this result to Ostrowski (1937/1938), and economists to Hawkins and Simon (1949).85 Potron’s papers on the topic86 are slightly anterior to Ostrowski’s. There is little doubt that these parallel publications result from independent findings. Potron’s proof is elegant and, up to a modernization of the wording, coincides with the argument one finds today in textbooks, as an application of the LU factorization of a matrix. Since the ranking of the rows and columns of M is arbitrary (when M is a generalized input–output matrix, it reflects the arbitrary classification of the goods and/or the professions), the criterion can be applied to any matrix which is obtained by permutation of the original one. In 1939 Potron suggested that, in order to reduce the overall calculation time, several calculators might work independently on matrices differing only by the rankings of their rows and columns.87 As soon as one of them finds a negative leading minor, the calculations can stop. On closer look, the idea does not seem efficient. If the dominant eigenvalue of matrix Q is smaller than 1, nobody will ever stumble upon a negative minor; if it is only

83 Section 9 of A13/Chapter 8. 84 See fifth lecture of U3/Chapter 14. 85 See Alexander Ostrowski, ‘Über die Determinanten mit überwiegender Hauptdiagonale’ (Commentarii Mathematici Helvetici, 1937–1938, 10: 69–96) and David Hawkins and Herbert A. Simon, ‘Note: Some conditions of macroeconomic stability’ (Econometrica, 1949, 17: 245–8). 86 See fifth lecture of U3/Chapter 14 and A40/Chapter 15. 87 Section 6 of A41/Chapter 16.

Church, society and economics 49 slightly greater than 1, it is very likely that no soldier in the army of calculators will find a negative minor until the very last stage, when all of them will determine simultaneously the negative sign of the determinant of the full matrix (I − Q). 12.2 The Lanczos connection The LU decomposition of a square matrix consists of writing it as the product of a lower diagonal matrix L with a unit diagonal and an upper diagonal matrix U with nonzero entries on the diagonal. Potron referred to the LDU variant of this procedure, where D is a diagonal matrix and U has a unit diagonal. That decomposition was not new in 1937; as a matter of fact, it is an abstract representation of the well known elimination method for solving linear systems of equations. Its origin can be traced to Gauss in 1801; since then, the method had been repeatedly rediscovered and forgotten. Had the method been common knowledge in 1937, the prestigious Comptes Rendus de l’Académie des Sciences and the Bulletin de la Société Mathématique de France would certainly not have published the part of Potron’s papers relative to its description. Potron attributed the method to Cornelius Lanczos, a specialist in the theory of relativity who also did research on numerical analysis. As a member of the American Mathematical Society since 1931, Potron became aware of Lanczos’s work when he saw a list of abstracts of papers submitted for presentation at meetings of the Society.88 The title of Lanczos’s paper (‘A simple recursion method for solving a set of linear equations’) must have aroused Potron’s curiosity. The first sentence of the abstract (‘In the problem of solving a set of simultaneous linear equations with non-vanishing determinant both the application of determinants and the customary elimination method involve laborious calculations.’) describes exactly the problem that Potron was facing. It is followed by a compact description of the recursion method that Lanczos had found to numerically solve these systems of equations. Although the description is rather cryptic, the method Lanczos referred to is not the LU decomposition, but the one known today as the QR decomposition, where Q is an orthogonal matrix and R an upper triangular matrix. The QR and the LU factorizations can be used to solve the same type of computational problem, but they are clearly distinct from one another. It is therefore curious that Potron proceeded as if Lanczos had described the LU method. The text of Lanczos’s contribution has been lost, so we do not know what was actually in his paper. Potron, however, did have access to the full version since he wrote that Lanczos ‘ha[d] been kind enough to send me his manuscript’.89 88 See ‘Abstracts of papers submitted for presentation to the society’ (Bulletin of the American Mathematical Society, May 1936, 42: 314–348); Lanczos’s abstract has number 173 and is on p. 325. Lanczos presented his paper at the 331st meeting of the Society, held on 10–11 April 1936 at the University of Chicago; see ‘The April meeting in Chicago’ (ibid., July 1936, 42: 456–9); his paper has number 8 and was presented in the Algebra section. 89 Note in section 2 of A41/Chapter 16.

50 Church, society and economics Even if the aim of Lanczos’s method was to find the solution to a system of linear equations, Potron never mentioned it as a way to determine the values of a satisfactory regime. Probably he realized that the size of the matrices constitutes a formidable obstacle.

13 The implementation of a just social order The abstract analysis of the economic problem led Potron to study difficult mathematical questions. He completed his formal analysis by a reflection on the possibilities of implementing satisfactory regimes, and on the organization of society itself. 13.1 Properties of satisfactory regimes It is easy to deal with the case when no satisfactory regime exists (ν > N ): given the number of work-days, the level of productivity is simply insufficient to satisfy demand. There is no possibility to meet all constraints simultaneously, either on the physical or on the value side. As an example, let us assume that some wages are too low. If one tries to remedy the problem by raising them, then, by necessity, the cost prices of some goods will exceed their selling prices. If these prices are increased, again some wages will be too low and ‘all must be started again, and this indefinitely: it is the “race to the rise” ’.90 These manifestations of a deeply rooted economic crisis are unavoidable and independent of the social organization. It is impossible to set the situation straight without a change in the data.91 When ν = N we reach a limit case.92 There exists a unique satisfactory regime of production and labour (characterized by the absence of non-workers, unemployment and excess production) as well as a unique satisfactory regime of prices and wages (characterized by zero profits and savings). But the very uniqueness of these regimes makes their implementation almost impossible. When ν < N infinitely many satisfactory regimes are possible; and the wider the gap, the larger the set of satisfactory regimes becomes. This probably explains why Potron thought it would be inconvenient ‘to assign to the number of workdays a value too close to the characteristic number’.93 In 1937 he voiced some reservations about the (low) value of N ,94 but on the whole he seemed to believe the gap was sufficiently large. That did not guarantee, however, that a satisfactory regime would come about. A defective socioeconomic organization might still

90 91 92 93 94

Section 8 of the sixth lecture of U3/Chapter 14. See, e.g., section 6 of the first lecture of U3/Chapter 14. We already mentioned this at the end of section 9.5. Section 16 of A11/Chapter 5. Section 3 of the sixth lecture of U3/Chapter14. This may be an allusion to the effect of the economic and social reforms initiated by the Front Populaire government. This left-wing governement rose to power in June 1936; it limited the duration of the normal working week to 40 hours and introduced the right to a two-week paid holiday period.

Church, society and economics 51 lead to a bad outcome, or malaise in Potron’s words. This malaise would manifest itself in the same way as in the case mentioned above, viz. by a high level of unemployment and/or increasing prices leading to a high cost of living.95 But a comparison of scalars ν and N would reveal that the state of the economy is basically sound and that the source of the crisis lies in an inadequate choice of the regime. Potron based his explanation of economic crises on the assumption that, in actual economies, firms make decisions relative to production ‘in a haphazard manner’ and choose prices in order to maintain their profits. From a formal point of view, these choices amount to assign values to the principal variables of the equations which are incompatible with those of the secondary unknowns: ‘It is very likely that one does not draw one of the favourable combinations’, that is, characteristic of a satisfactory regime.96 Potron never discussed the possible existence of automatic mechanisms that might steer the economy toward a satisfactory regime. In fact, he never referred to the notions of competition or market, mentioned supply and demand only once,97 and wrote that ‘the prices, the wages, [and] the size of the staff […] result from conventions which, at least theoretically, are free’.98 His ignorance of economics was almost complete: he hardly ever seems to have opened an economics book.99 His scepticism of economic laws transpires in the only passage where he alluded to them: ‘One often speaks about economic laws, rather vague for that matter, which revenge themselves on those who ignore them. I think that the omissions of six systems of equations is much more dangerous’.100 However, he did acknowledge the existence of some economic tendencies. A first is that, on average, production equals consumption, by which he referred to the cyclical or erratic nature of economic activity: ‘the regime of production is made up of periods of full activity and periods of cessation, which alternate with sudden jolts’.101 A second tendency relates to the ‘essentially stable’ nature of satisfactory regimes: as a result of savings or losses households move from one social category

95 In the years around 1910 the ‘high cost of living’ (la vie chère) was a recurrent theme in the press. It also triggered economic studies; see e.g. J. Dessaint, ‘La vie chère, les taxes de consommation et leur incidence’ (Le Mouvement Social, December 1912, 74: 1087–1102 and January 1913, 24–40). Potron referred to that crisis of the ‘high cost of living’ (section 8 of A11/Chapter 5), thus pointing at an application of his construction to concrete facts. 96 A35/Chapter 11. What Potron wanted to say can be illustrated by means of equation (15). Let u = (z/N + s)B + f be the vector of secondary unknowns on the physical side. Any semipositive u leads to a positive vector y of the principal variables, since y = u(I − Q)−1 . But not any positive vector y leads to a semipositive vector u and even if it does, it may be unacceptable for other reasons. 97 This 1935 reference to supply and demand applies to a matrix used by Ragnar Frisch (see A35/Chapter 11). 98 Ibid. 99 In 1913 he briefly referred to the work of Charles Gide and of Victor Brants (see A13/Chapter 9). 100 A35/Chapter 11. 101 U1/Chapter 7.

52 Church, society and economics to another, but these movements tend to cancel out so that ‘the distribution of the population over the various professions and social categories keeps itself constant in an automatic way, so to speak’.102 Potron admitted that if a satisfactory regime were to be reached, the situation would look relatively good. A satisfactory regime is in fact surrounded by other satisfactory regimes, so that a slight change in the data or in the values of some unknowns can be accommodated and does not endanger the satisfactory nature of regime.103 Potron argued that the socioeconomic data change rather slowly; in a given year, technical change affects only a small part of the economy. This implies that satisfactory regimes can change continuously over time, and therefore that it becomes possible to steer the economy onto what might be called an intertemporal satisfactory path. The stability and continuity properties of satisfactory regimes let Potron conclude that ‘the important thing is therefore to construct and implement for a first time a double satisfactory regime’.104 13.2 The search for satisfactory regimes Since Potron discarded the idea of the spontaneous emergence of satisfactory regimes, the question remained how they could be reached. In his letter to Gibrat he suggested to imagine that ‘one had started the calculations around a thousand years ago’ when ‘there were many less equations’.105 It would have been relatively easy to calculate and implement a first satisfactory regime, and from there on it would have been plain sailing. The slow rate of change of the economy would have meant ‘insignificant’ updating work,106 and it would have been possible to fine-tune the satisfactory regimes to the new conditions. Such an initiative had, alas, never been taken, and as a result one had to confront a much more complex and difficult problem from scratch. To solve it, Potron showed much confidence in the power of calculation, at least in his early work. First of all, the data describing the state of the economy (L, A, B, N ) needed to be collected. Potron saw little difficulty in gathering the technical coefficients of production (L, A). Two small-scale surveys, one in a baker’s shop of Paris107 and another in the military factory of Le Bouchet,108 convinced him that these data could be obtained with relative ease. For instance, by studying the book of accounts of the powder factory he was able to calculate the coefficients of the production of B powder in only ten hours’ time. The treatment of fixed capital, taxes and general

102 103 104 105 106 107 108

Section 18 of A11/Chapter 5. See, e.g., section 3 of the first lecture of U3/Chapter 14. Section 10 of the sixth lecture of U3/Chapter 14. A35/Chapter 11. Section III of A14/Chapter 9. Section 6 of A11/Chapter 5 and A12/Chapter 6. Section III of A14/Chapter 9.

Church, society and economics 53 expenses proved to be a bit complicated, but also these nuts could be cracked.109 Comforted by the results of his two experiments, Potron proposed to organize a large-scale survey for the whole economy, involving two steps.110 The first would consist of asking each entrepreneur to draw up his catalogue, i.e. the list of his inputs and outputs. This should allow the identification of the rows and columns of a huge table, starting from the final goods (or first-species units), then those used in the fabrication of the final goods (second-species units), etc. The second step would be quantitative, with the first rows and columns being filled in first. With regard to the data B relative to the standards of living of each profession, he initially assumed that a similar but easier survey would be sufficient.111 In 1937 he suggested that the collective contracts negotiated between employers and employees would provide precise indications on these standards.112 Potron was aware that the size of the table might be a problem. Just before the beginning of the First World War he made an estimate of the total number of coefficients to be calculated, based on the assumption that there are about ten thousand professions and about as many economic goods. Each of the matrices L, A and B would therefore contain roughly 100,000,000 entries. But many of these entries would be zero; from his experiences he estimated that, on average, about two hundred non-zero inputs (goods or labour) would have to be calculated per good produced. Furthermore he assumed there are only six standards of living. All in all, this amounted to 2,000,000 non-zero production coefficients and 60,000 standard of living coefficients; the total number of figures which would correspond to those contained in about ten volumes of logarithm tables.113 He remained confident in the feasibility of the task, although he did express some reservations about the possible poor quality of ordinary industrial accounts: he suspected that these were not kept with the same meticulousness as was customary at military facilities. Yet he believed that the self-interest of the industrial producers would incite them to improve their accounting methods, as this would allow them to determine their production costs with more precision.114 He used a similar argument in the defense of the Taylor system, which starts from a scientific study of the production process involving the cost accounting department.115 Until the end of his life Potron remained interested in industrial statistics: in his very last economic publication he once more referred to the B powder example and criticized Sauvy’s statistical project for ignoring the quantities of labour per profession, which are entries of his data table.116 On the whole, however, he must

109 110 111 112 113 114 115 116

Section 6 of A11/Chapter 5. A12/Chapter 6. Ibid. See, e.g, section 2 of the first lecture of U3/Chapter 14. Section 3 of A14/Chapter 9. A44/Chapter 17. Section VI of A15/Chapter 10. A44/Chapter 17.

54 Church, society and economics have been satisfied with the development of the statistical apparatus in France in the 1930s, since it went into the direction of collecting the empirical data required for his economic model. In his early writings Potron argued that a special institution should be created to collect and update the data, and to determine and adjust satisfactory regimes: the Bureau de Calculs.117 At that time he saw it as an integral part of a Christian design of society inspired by the principles of the encyclical Rerum Novarum. For years Catholic intellectuals such as the Jesuit Gustave Desbuquois had advocated a corporative organization of society, based on the idea that cooperation between social groups rather than class struggle was the appropriate answer to the social question. Potron subscribed to that view, but stressed that ‘par la force des choses’ even a corporative society would be obliged to face the problems he studied in his model.118 Given the complexity of the economic interactions, the ‘tribunal of arbitration’ composed of ‘competent men’ that Desbuquois suggested would need the assistance of a Bureau de Calculs, since only an office of that kind would be capable of handling the information.119 Furthermore, the process of gathering the data and implementing the solutions would be guided by strong professional organizations: ‘one sees the considerable role that the professional organizations would have to play in these surveys. Once the data have been provided, the calculations performed, and the results obtained, an even more important role would be left to these organizations. Having full knowledge of things, they would have to take practical decisions, if necessary with the aid of public authorities, capable of bringing about the gradual implementation of an economic order in conformity with the general plan drawn up by God, but of which He leaves to mankind the task of ensuring its execution’.120 It deserves to be stressed that Potron’s aim was to show that the solution to the economic problem requires the social organization advocated by the Church, an argument rarely heard in circles of those who promoted the views of Rerum Novarum. Apparently, only in the 1930s Potron became fully aware that the scale of the model posed an insurmountable obstacle for the determination of satisfactory regimes: ‘The only rational way to proceed, [viz.] to solve systems (7) and (8), is not implementable in practice, given the huge number of equations, equal to the number of all the things that are produced in the world’.121 As a result, ‘it is only by trial and error that a solution can be found’.122 According to Potron, the use of mathematics gave valuable insights into the nature of the economic problem, but ‘the weakness of our human intelligence’ made the search for an effective solution exceedingly difficult, if not impossible.123 In front of the audience of the Catholic

117 118 119 120 121 122 123

Section 20 of A11/Chapter 5. Ibid. Ibid. Ibid. A35/Chapter 11. The two systems to which Potron referred here correspond to (8) and (24) above. Section 7 of the sixth lecture of U3/Chapter 14. Ibid., section 10.

Church, society and economics 55 Institute of Paris Potron concluded his 1937 lecture series by giving a theological twist to that situation and linking it to the mystery of faith: ‘By letting us assess the theoretical difficulties of the problem, the science of mathematics gives us a new reason to repeat to our Father in Heaven the traditional prayer “Give us today our daily bread” ’.124

14 Technical progress and dynamics In the previous section we already mentioned that Potron touched upon the effects of changes in the data. He was aware that the occurrence of various forms of change requires an adequate treatment of the dynamics of the economic system. In 1912, Potron claimed to have started such a study which led to ‘noteworthy conclusions, especially with regard to the constitution of capital by means of saving’,125 but the only formal piece of dynamic analysis which he published before the Great War dealt with the consequences of a continuous increase in population.126 His long calculations aimed to show that as long as the distribution of population remains the same, the satisfactory regimes could be modified without difficulty because the additional population provides for its own needs, at least if production requires no preliminary equipment. Potron’s analysis became more original when he adopted the more realistic alternative hypothesis of the necessity of fixed capital goods such as machinery.127 He considered the introduction of new machines to be the typical way of generating ‘genuine progress’, defined as a change in the labour and material coefficients which leads to a reduction of the characteristic number of the socioeconomic system.128 While the instalment of new equipment should result in a higher wellbeing, Potron conceded that it was often ‘a cause of miseries and ruins’ in actual economies,129 because it deprived a certain number of labourers of work: ‘In general an equilibrium is reached in the end, but after a period of crisis which will be more or less long, more or less disastrous’.130 This raised the question of whether it would be possible ‘to dampen these dangerous oscillations’, in the same way as hydraulic engineers managed ‘to regulate the course of rivers’.131 Without providing a full formal explanation, Potron claimed that a smooth transition ensuring ‘to everyone the same level of well-being during the transitory 124 125 126 127

128 129 130 131

Ibid. Section 19, note 10, of A11/Chapter 5. See section 19 of A13/Chapter 8. For Potron the necessity of advances (which included wages) provided an economic justification for the legitimacy of interest, a controversial topic in the scholastic tradition, and a social justification for the existence of rentiers, identified with ‘the capitalists who live off their revenues’ (section 18 of A11/Chapter 5). The quoted sentence contains one of the two occurrences of the word ‘capitalist’ in Potron’s works. See section 9 of the sixth lecture of U3/Chapter 14. A35/Chapter 11. Section 19 of A11/Chapter 5. Section VI of A15/Chapter 10.

56 Church, society and economics period, and a greater level of well-being after the establishment of the new state’ could be achieved.132 Starting for simplicity from a double satisfactory regime with no overproduction, Potron distinguished two phases in the transition. During the first period, new machines are manufactured with the help of labourers working with the existing methods of production. The overall effects on the distribution of workers among industries can be determined thanks to the knowledge of the present production coefficients. No change is necessary on the value side. The second period begins when the new machines start to be utilized. On the physical side, the voluntary overproduction of the equipment period is no longer necessary and the new production coefficients apply. These conditions determine the characteristics of the new satisfactory regime of production and labour, again with a redistribution of workers. On the value side, it suffices to switch from the old to a new satisfactory regime of prices and wages. Potron considered that the value dynamics have limited effects and are less complex, but he drew attention to the price of the new machine: he argued that the new equipment should be sold at its cost price on its first sale.133 One specific case he explicitly mentioned coincides with what, in modern language, would be called Harrod-neutral technical progress, and Potron showed that it results in a proportional wage increase.134 The reasoning confirms that Potron firmly believed in the powers of forecasting and calculation: ‘Undoubtedly one will not find useless the effort which must be made to determine in advance these new regimes and to adjust to them, if one thinks of the disasters and misery which any delay inevitably brings along’.135

15 A final portrait Maurice Potron can be characterized as an amateur economist par excellence (see Table 2). Ignorant of the history, the terminology and the methods of mainstream economics, he constructed an economic model all by himself, almost out of nowhere so to speak, as if he was the first to do so. He claimed that this model provided the key to identify the economic problem and to diagnose the causes of society’s economic distress. He developed his views in isolation from economists, using mathematics as the adequate tool to do economic research and to obtain insights which might lead to the improvement of society. Serious questions may be raised about the realism of his proposals for economic policy and the applicability of his suggestions for the organization of society. There is, however, little doubt about the striking originality of his contributions to economic analysis. At least in four cases he showed to be a pioneer ahead of his time.

132 Section 19 of A11/Chapter 5. 133 See A35/Chapter 11. He later modified his views on the subject (see section 6 of the sixth lecture of U3/Chapter 14). 134 See section 9 of the sixth lecture of U3/Chapter 14. 135 Section 19 of A11/Chapter 5.

Church, society and economics 57 Long before Leontief developed his input–output framework, Potron represented the economic system as a giant network of interactions involving industries and social groups. He found that it can be modelled in a simple way with the help of matrices. He described the structure of these matrices in detail and argued that it was a realistic ambition to fill them up with data collected by means of industrial and household surveys. He also discovered that a unique scalar, the maximum eigenvalue of an integrated input–output matrix, can be interpreted as the amount of labour necessary for the reproduction of the economy. By comparing this value to the maximum amount of labour one obtains an elementary check of whether the economy is productive enough to meet consumption. The associated positive row and column eigenvectors admit economic interpretations in terms of activity levels, labour requirements, prices and wages. Here, for the first time, the Perron–Frobenius eigenvalue received an application outside mathematics. Before the Second World War no other economist ever used the Perron–Frobenius theorem. In an attempt to by-pass the effective calculation of the maximum eigenvalue, Potron showed that the criterion he had originally derived can be replaced by an equivalent criterion on the signs of the principal minors of a matrix. This is the earliest formulation of the Hawkins–Simon condition, which Potron obtained years before Hawkins and Simon introduced it to the economics profession. Potron found that the solution of the physical side of his economic model, i.e. the determination of activity levels and labour requirements, is intimately connected to the solution of the value side, i.e. the determination of prices and wages. A generation had to pass before a similar duality property was discovered by John von Neumann. Without any schooling in economics, the Jesuit Maurice Potron single-handedly constructed a complex disaggregated economic model, studied its properties and found a characterization for the existence of a solution. His main motivation was to make a contribution to the implementation of a just social order, showing that an organization of society in accordance with the views of the Church would solve the economic problem. In the process he conceived tools which economists would rediscover and embrace only much later and have now become classical. He may have been an amateur toiling on the outskirts of the economics profession, but he did show flashes of genius.

58

Three-year national service. Centennial of the Revolution (Eiffel tower).

1889

1881 1882 1886

1875 1877 1878 1879 1880

Resignation of Adolphe Thiers. Mac-Mahon elected as President of the Republic to prepare restoration. Divisions amid the royalist tendencies. The Wallon law definitely establishes a Republican regime. Gambetta’s speech on clericalism. Election of Pope Leo XIII. Resignation of Mac-Mahon. Ferry decrees on non-authorized congregations (29 March). Expulsion of the Jesuits (29 June). Ferry law on free primary schooling. Ferry laws on compulsory and lay primary schooling. Creation of the Association Catholique de la Jeunesse Française. Secularization of primary school teachers.

Defeat of Napoléon III in Sedan against the Prussian troops (2 September). Proclamation of the Third Republic (4 September). Monarchist majority at the elections. The ‘bloody week’ (21–28 May): end of the Commune de Paris. The pretender to the throne rejects the tricolour flag. Five-year national service.

1873

1872

1871

1868 1870

French history

Table 2 Maurice Potron and his times

Studies at the École Sainte-Geneviève.

Follows education at home (until 1889).

Maurice Potron is born in Paris (31 May), as the first child of a family of six.

Marriage of Auguste Potron and Cécile Frottin.

Life of Maurice Potron

59

1901

1899

1898

J’accuse, by Émile Zola: beginning of the ‘Dreyfus affair’. Creation of the Ligue des droits de l’homme. Creation of Le Sillon, a Christian social movement, by Marc Sangnier. Waldeck-Rousseau chief of government. Law on associations including provisions against the congregations. Creation of the Action Libérale Populaire (ALP), a Republican political party of Catholic inspiration, by Jacques Piou. Creation of the Ligue des Femmes Françaises (LFF), of royalist inspiration, to defend the right for a religious education.

First congress of the Cercles d’Études Ouvriers in Reims. Creation of the Confédération Générale des Travailleurs (CGT). Christian workers’ congress in Reims. First ecclesiastic congress in Reims. Christian-democratic congress in Lyon.

1893 1895

1896

Encyclical Au Milieu des Sollicitudes asking for the ralliement to the Republic.

‘Algiers Toast’: cardinal Lavigerie asks the Catholics to ‘rally’ to the Republic. Encyclical Rerum Novarum on the condition of the working classes.

1892

1891

1890

Publishes his first scientific paper.

Continued

Attends the Philosophy scholasticate and teaches mathematics at the Maison Saint-Louis in Jersey (1899–1901).

Continues his education as a Jesuit in Canterbury and in Paris. Teaches mathematics at the École Sainte-Geneviève (until 1899).

Enters the Jesuit novitiate in Canterbury (10 November). Takes his First Vows (13 November).

Graduates from the École Polytechnique. Completes his engineering studies at the École d’Application des Poudres et Salpêtres. Decides to opt for a clerical life.

Passes the competitive entrance exam of the École Polytechnique. Makes a spiritual retreat at the Villa Manrèse in Clamart.

60

1908

1907

1906

1905

1904

1903

1902

Émile Combes, chief of government, intends to appoint the bishops without the Pope’s agreement. Creation of the Ligue Patriotique des Françaises (LPDF), close to the ALP, from a split of the LFF. The requests of non-authorized congregations are rejected. Election of Pope Pius X. The members of congregations excluded from teaching. Creation of the Semaines sociales de France by Marius Gonin and Adéodat Boissard. 10-hour work day. Resignation of Combes. Two-year national service. Breaking of the diplomatic relations with the Holy See. Law of separation of the Churches and the State. The Jesuit Gustave Desbuquois takes over the direction of Action Populaire in Reims. Inventories of the Church’s properties. Encyclicals Vehementer Nos and Gravissimo Officii Munere. 1,100 dead at the Courrières coal mine. Strike for the 8-hour work day (1 May). The Amiens chart claims the independence of the CGT from the political parties. Creation of a Ministry of Labour. Encyclical Une Fois Encore and papal decree Lamentabili Sane Exitu. Revolts of wine producers. Strike of postmen. Strike of building workers. Arrests of the CGT leaders (1 May).

French history

Table 2 Cont’d

Is appointed as assistant-counsellor-chaplain of the Union Sociale d’Ingénieurs Catholiques (USIC).

Becomes manager of the Villa Saint-Joseph in Épinay-sur-Seine, where spiritual retreats are organized.

Spends his ‘Third year’ in Canterbury (27 September 1906 – 31 July 1907).

Is ordained deacon (February), then priest (29 June). Participates in a series of astronomic observations at the Paris Observatory (April 1905 – April 1906)

Continues the Philosophy scholasticate at the Institut Catholique de Paris (until 1905). Obtains a PhD in mathematics from the University of Paris.

Attends the Theology scholasticate in Canterbury.

Life of Maurice Potron

61

The Great War. Raymond Poincaré, President of the Republic, calls for the ‘Sacred Union’. 3 September 1915: election of Pope Benedict XV.

Creation of collective conventions between employers’ and employees’ unions. 8-hour working day and 48-hour week. Creation of the Confédération Française des Travailleurs Chrétiens (CFTC). The Senate, afraid of clerical influences, rejects the right to vote for women.

1919

Strikes in car factories.

Strike of postmen. Confirmation of the banning of strikes by public servants. Suspension of the ralliement policy. Law on the workers’ and the peasants’ pensions (65 years). Strike of railway men. Condemnation of Le Sillon by Pius X.

1914–1918

1912

1911

1910

1909

Continued

Teaches mathematics at the École Sainte-Geneviève in Versailles (1919–1930). Teaches mathematics and astronomy at the Facultés Catholiques de Lille (1921–1924).

Publishes for the first time his complete economic model in Échos de l’Union Sociale d’Ingénieurs Catholiques et des Unions-Féderales-Professionnelles de Catholiques (15 October). Publishes several other papers on his model until the outbreak of the war. Professes his Final Vows (2 February). Teaches mathematics and astronomy at the Facultés Catholiques de l’Ouest (until 1914). Is mobilized as an officer in the Powder corps, but obtains a transfer to the artillery. Participates in the battles of Champagne, Verdun, Chemin des Dames, Aisne, and is gassed. Appointed as battery captain and member of the military staff. Attached to the 1st US army (June – November 1918). Receives the Légion d’honneur (25 December 1916). Death of his mother, Cécile Potron (February 1918).

62

1942

1940

1936–1937

1935

1926 1931

1924

The ‘1929 crisis’ hits France, which applies a deflation policy: reduction of public expenses by 10%. The Front Populaire government. Massive strikes in June 1936. Matignon agreements: collective conventions, wage rises, 40-hour week, 2-week paid holidays. The phoney war (Sept. 1939–May 1940) and the rout of the French army (May–June 1940).

After the victory of the Cartel des Gauches at the elections, the government threatens to apply the anti-congregation laws again. High inflation and monetary crisis. Creation of the Jeunesse Ouvrière Chrétienne.

French history

Table 2 Cont’d

Teaches mathematics at the Collège Saint François-Xavier in Vannes (Britanny) Dies from pneumonia (21 January) and is buried in Vannes.

Organizes a series of six lectures on his economic model at the Institut Catholique de Paris (March – April 1937)

Death of his father, Auguste Potron. Is appointed as lecturer, then professeur hors cadre, at the Institut Catholique de Paris (until 1939). Resumes his researches in economics.

Becomes a member of the Versailles section of DRAC (Droits du Religieux Ancien Combattant).

Life of Maurice Potron

Notes on the translations

All translations have been made by the editors, Christian Bidard and Guido Erreygers. Each chapter is preceded by an editorial introduction which provides bibliographic details on the source material and supplementary information on its history and context. Any material inserted by the editors is in square brackets. Insignificant typographical errors are corrected without indication. Editorial footnotes serve to specify the more substantial corrections, to supply missing bibliographic information, to make cross-references, and to clarify obscure points in Potron’s texts. Editorial remarks always end with the abbreviation ‘Eds.’. The footnote numbering restarts from 1 when a new chapter begins. The footnote numbers do not distinguish Potron’s footnotes from those added by the editors. The equation numbers coincide with those given by Potron. Potron sometimes put his equation numbers on the left, and sometimes on the right. In the translations they have been put systematically on the right. Cross-references follow the style adopted in the introduction: ‘Ax/Chapter y’ refers to item Ax in the Potron Bibliography, which is translated as Chapter y in this book.

1

Abstract of a study on just prices and wages

Editors’ note Potron’s first economic publication was an abstract inserted by Gustave Desbuquois as a footnote on pp. 882–3 of his article ‘La loi du juste prix’ (Le Mouvement Social, October 1911, 72: 867–84). It has no explicit title or author, but the information provided by Desbuquois and the fact that parts of it can be found verbatim in Potron’s article in the Échos de l’Union Sociale d’Ingénieurs Catholiques et des Unions-Fédérales-Professionnelles de Catholiques of the same year (A8/Chapter 2) leave no doubt that Maurice Potron is the author of this abstract. Desbuquois’s article is based on the lectures which he gave in August 1911 during the Semaine Sociale de Saint-Étienne. The Semaines Sociales, launched in 1904, were yearly meetings of both laymen and clergymen during which the social doctrine of the Catholic Church was discussed and updated; the session in SaintÉtienne was the eighth of its kind.1 A nearly identical version of Desbuquois’s article in Le Mouvement Social appeared under a different title in the proceedings of that session: Gustave Desbuquois, ‘La justice dans l’échange’, Semaine Sociale de France. Cours de Doctrine et de Pratique Sociales. 8ème Session. SaintÉtienne 1911. Compte-Rendu in-extenso (Lyon, Chronique Sociale de France, 1911: 165–77; Potron’s abstract is on pp. 175–6). Desbuquois pointed out that ‘the problem of the economic organization of exchange according to the principles of justice is infinitely complex’ (p. 881). In his view the problem consisted of reconciling moral imperatives with economic possibilities: on the one hand, moral principles require that those who contribute to production should earn wages high enough to afford them to make a decent living; on the other, the effect of economic laws should be taken into account, since prices depend upon economic factors. Between these two forces an equilibrium must be established. Desbuquois argued that it is the task of economic science to determine 1 More information on the Semaines Sociales can be found in: Jean-Dominique Durand (ed.), Les Semaines Sociales de France. Cent Ans d’Engagement Social des Catholiques Français. 1904–2004 (Paris: Parole et Silence, 2006), and especially in the contribution of Bernard Laurent, ‘Les Semaines Sociales de France et la science économique’ (pp. 267–88). Potron is not mentioned in the book.

A study on just prices and wages 65 the conditions of this equilibrium, and it is then that he referred to the work of Maurice Potron. The note starts with the following introductory remark by Desbuquois: We have not gone further in the study of this problem. In a still unpublished study of which he communicates to us the conclusions, a distinguished mathematician, Father Potron, former Student of the École Polytechnique, has set forth a precise formulation of the mathematical part of the problem and obtained curious results. Here follows the short summary of a study of which the author will soon publish the mathematical part in the Annales de l’École Normale, and will develop the socioeconomic consequences in Le Mouvement Social. (p. 882) The two forthcoming articles mentioned by Desbuquois were published in 1912 in Le Mouvement Social (see A11/Chapter 5) and in 1913 in the Annales Scientifiques de l’École Normale Supérieure (see A13/Chapter 8). * * * 1. For the general state of the world of labour to be satisfactory, the production of everything which supports life (objects of immediate consumption or use: items for food, clothing, housing; next, objects necessary for the production of these, and so forth) must fulfil the following conditions: 1◦ ) For production to be sufficient for consumption, it must be that after a certain lapse of time, for instance one year, the provisions of everything which supports life are back at their initial states; it must also be the case that no labourer is forced to perform a number of normal working days greater than the number of work-days in the period under consideration. 2◦ ) Let us suppose a state of production just sufficient for consumption: for the labourer then to be able to live of his share of labour, it must be that the wage which he effectively earns during a certain period is at least equal to what a decent living costs him in the same period. Two facts complicate the present conditions: on the one hand, consumption and as a result the necessary production increase with the number of labourers; on the other, prices and as a result the cost of living increase with wages. It is therefore uncertain, a priori, whether these two conditions can be fulfilled. 2. In a given state of industry, the production of a given quantity of any item requires the consumption or the use – this comes down to the same – of given quantities of other items, as well as given durations of labour performed by various persons belonging to various social categories. At the same time, to each social category corresponds a type of existence, which can be characterized by the annually consumed quantity of the objects of immediate consumption or use.

66 A study on just prices and wages What we have there is a set of numerical coefficients which are like experimental data, and which represent certain industrial conditions of fabrication and organization, [and] certain requirements of consumers, in one word, a given socioeconomic state. 3. Between those numerical coefficients, the produced quantities of items, the numbers of labourers of each social category that contribute to the production of every object, the numbers of working hours which are asked from them, the numbers of non-working consumers belonging to each social category, and the surpluses, evaluated for each object, of production over total consumption, there exists a system of relations which is easy to establish. Between those same numerical coefficients, the selling prices of the various objects, the wages of labourers, the benefits of firms, and the annual costs of living corresponding to the various types of existence, there is a second system of relations. All these quantities except the numerical coefficients under consideration can vary under the influence of a variety of causes; but, as long as the socioeconomic state does not change, they will keep on verifying the same equations. One can establish, between the same quantities, two systems of inequalities translating respectively the two desiderata of No. 1. Is each of the systems of equations compatible with the corresponding system of inequalities? That is the question which must be resolved to decide whether, in a given socioeconomic state, it is mathematically possible to realize the two desiderata of No. 1. 4. These are the results obtained: To a given socioeconomic state corresponds for a given period of time, for instance one year, a certain number, which depends upon the numerical coefficients representing that state, and which can be called the characteristic number of the given socioeconomic state, relative to the period under consideration. In order for it to be mathematically possible, in a given socioeconomic state, to carry out the two desiderata of No. 1, a single condition is necessary, and, in general, sufficient: the characteristic number relative to a certain period must be at most 2 equal to the number of work-days of that period. In particular, if the characteristic number is smaller 3 than the number of workdays, no labourer is forced to the maximum labour; there can be non-working consumers; benefits can be left to both firms and wage-earners, since the selling prices exceed the cost prices and the wages the costs of living; but the total sum of these benefits is exactly equal to the total cost of living of the non-workers.

2 [Potron wrote ‘at least’ instead of ‘at most’. – Eds.] 3 [Potron wrote ‘greater’ instead of ‘smaller’. – Eds.]

2

With regard to a mathematical contribution to the study of the problems of production and wages

Editors’ note Almost simultaneously with the previous note, Potron published a short article: ‘A propos d’une contribution mathématique à l’étude des problèmes de la production et des salaires’ (Échos de l’Union Sociale d’Ingénieurs Catholiques et des Unions-Fédérales-Professionnelles de Catholiques, 15 October 1911, 2nd year, No. 7: 4–7). The journal did not mention Potron’s name explicitly, but chose to describe the author as ‘our Assistant-Counsellor-Chaplain’. The Union Sociale d’Ingénieurs Catholiques (USIC) had started to circulate a newsletter entitled Écho de l’Union Sociale in January 1908. In March 1910 this union of engineers affiliated itself with the Unions Fédérales Professionnelles de Catholiques, a federation of unions of Christian employers. On that occasion it was decided that a single newsletter would be published under the name Échos des Unions-Fédérales-Professionnelles de Catholiques, including as a supplement the Échos de l’Union Sociale d’Ingénieurs Catholiques for the members of the USIC. From the middle of 1911, however, two versions of the newsletter were published, one under the heading Échos de l’Union Sociale d’Ingénieurs Catholiques et des Unions-Fédérales-Professionnelles de Catholiques and the other under the heading Échos des Unions-Fédérales-Professionnelles de Catholiques. Potron’s article appeared in the common part of the two journals, not in the supplement for engineers, which means that an identical version of his article was published in both journals.1 Note that none of these journals used to publish theoretical articles. The mention ‘A propos d’une contribution’ in the title is also bizarre, as it suggests that the article is a comment on a pre-existing work. The introductory paragraph, obviously not written by Potron himself, makes it clear that the article is an expanded version of the abstract inserted in Desbuquois’s article: In the Lectures given by Father Desbuquois, head of ‘Action Populaire’ in Reims, during the Semaine Sociale de Saint-Étienne, a communication from 1 The issues of both journals carried the same dates and issue numbers. The front pages of Nos. 7, 8 and 9 of the 1911 volume of the Échos de l’Union Sociale d’Ingénieurs Catholiques et des Unions-Fédérales-Professionnelles de Catholiques erroneously mentioned ‘3rd year’ instead of ‘2nd year’.

68 Mathematical study of production and wages our Assistant-Counsellor-Chaplain was read. It was a brief exposition of the guiding idea and the main results of a study, of which the mathematical principles will soon be published in the Annales de l’École Normale and the socioeconomic consequences developed in Le Mouvement Social. We think it will be of interest to the Fellows of USIC and the Members of UFPC that we reproduce here this abstract and supplement it with a very brief exposition of the mathematical part. (p. 4) The closing paragraph, again not written by Potron, called upon the readers to assist Potron: It would undoubtedly be interesting to be able, in a more extended study, to give numerical examples of some of the equations under consideration. The Assistant-Counsellor-Chaplain relies on the kindness of the Fellows of USIC and the Members of UFPC to obtain the necessary information. (p. 7) An offprint with the more appropriate title ‘Contribution mathématique à l’étude des problèmes de la production et des salaires’ is conserved in the Archives Jésuites de la Province de France in Vanves and does carry Potron’s name as author. Both editorial remarks have been removed. Although the offprint is presented as an ‘extract’ from the Échos de l’Union Sociale d’Ingénieurs Catholiques, Potron modified the mathematical presentation. A notable change concerns the characteristic number of the socioeconomic system: this crucial variable is denoted by ν instead of γ , a notation which Potron maintained in the rest of his work. In the offprint he replaced ωh by ωh and introduced the additional variables ωih . He also deleted equation (7*) and the explanation preceding it. As a result, the variables γih are no longer defined explicitly, and what they stand for can be derived only by taking a close look at the equations. It turns out that Potron modified the meaning of these variables. With one exception all changes occur in section 3 of the paper. Our translation follows the article version; in the appendix we give the modified version of section 3. * * * 1. For the general state of the world of labour to be satisfactory, the production of everything which supports life – objects of immediate consumption or use: items for food, clothing, housing; next, objects necessary for the production of these, and so forth2 – must fulfil the following conditions. 1◦ ) For production to be able to provide for consumption, it must be that after a certain lapse of time, for instance one year, the provisions of everything which supports life are back at their initial states; but that no labourer is forced to perform 2 In the more developed work that will be published in Le Mouvement Social, it will be explained how fictitious objects corresponding to transport costs and taxes can be entered into this list. [See section 4 of A11/Chapter 5. – Eds.]

Mathematical study of production and wages 69 a number of normal working days greater than the number of work-days3 in the period under consideration. This condition expresses simply a physical necessity. 2◦ ) For each labourer to be able to live of his share of labour in the general production, assumed to provide exactly for consumption,4 it must be that the wage which he effectively earns during a certain period, for instance one year, is at least equal to what a decent living costs him in the same period. This condition expresses, for the employee, a physical necessity and, for the employer, a moral duty. Since, on the one hand, consumption, and as a result the necessary production, increases with the number of labourers, and on the other, prices and as a result the cost of living, increase with wages, it is uncertain, a priori, whether these two conditions can be fulfilled. To this preliminary question, of a mathematical order, one should give an answer of the same order. 2. In a given state of industry the production of a given quantity of any item requires the consumption or the use – this comes down to the same – of given quantities of other items, as well as given durations of labour performed by various persons belonging to various social categories. Let A1 , A2 , . . . , As be the quantitative units of all that is useful to life; let in general5 ai1 A1 , ai2 A2 , . . . , ais As

(i = 1, . . . , s)

be the consumption corresponding to the production of Ai , the a being positive or zero scalars; let C1 , C2 , . . ., Cr be the various social categories, and tih (i = 1, . . . , s; h = 1, . . . , r) the number of working days requested from the category Ch for the production of Ai . The a and t are scalars representing the industrial conditions of fabrication and organization. To the social category Ch (h = 1, . . . , r) corresponds a type of existence characterized by an annual consumption of bih Ai (i = 1, . . . , s; h = 1, . . . , r); the b are positive or zero scalars representing the workers’ requirements. At last, N denotes the number of work-days per year. 3. Given these data, between the produced quantities δi Ai (i = 1, . . . , s) equal to the total consumed quantities, the numbers πih (i = 1, . . . , s; h = 1, . . . , r) of 3 This number corresponds, for the period under consideration, to the maximum of labour which can be performed in a normal and continuous way. 4 There must indeed be equilibrium between the average production and consumption; as a matter of fact, periods of overproduction are always followed by periods of restricted production and therefore of unemployment. 5 [In the following equation and in equation (1) below, Potron wrote αi1 instead of ai1 . He corrected the mistake in the offprint. – Eds.]

70 Mathematical study of production and wages workers of categories Ch working for the production of Ai , the numbers ωh of simple consumers of category Ch , we have the relationships δi = a1i δ1 + · · · + asi δs + bi1 (ω1 + π11 + · · · + πs1 ) + · · · + bir (ωr + π1r + · · · + πsr ),

i = 1, . . . , s

(1)

Between the prices αi of Ai , the benefits βi made on Ai , the average daily wages σih of the workers of category Ch producing Ai (i = 1, . . . , s; h = 1, . . . , r), we have the relationships αi = ai1 α1 + · · · + ais αs + ti1 σi1 + · · · + tir σir + βi ,

i = 1, . . . , s

(2)

As the production δi Ai requires δi tih normal working days from the πih labourers of category Ch , the first condition of No. 1 is written N πih ≥ δi tih ,

i = 1, . . . , s; h = 1, . . . , r

(3)

[and] one must obviously have δi > 0, πih > 0, ωh ≥ 0,

i = 1, . . . , s; h = 1, . . . , r

(4)

The effective earnings of the πih labourers of Ch producing δi Ai are then δi tih σih ; their total cost of living is πih (b1h α1 + · · · + bsh αs ); the second condition of No. 1 is therefore written δi tih σih ≥ πih (b1h α1 + . . . + bsh αs ),

i = 1, . . . , s; h = 1, . . . , r

(5)

[and] one must obviously have αi > 0, σih > 0, βi ≥ 0,

i = 1, . . . , s; h = 1, . . . , r

(6)

The first condition of No. 1 can therefore be met always and only if the system (1), (3), (4) admits solutions; and the second, always and only if the system (2), (5), (6) admits solutions. The study of both these systems will be published in the Annales de l’École Normale;6 here are the results. If N is smaller than a certain scalar γ , function of the coefficients a, b, t, none of the two systems admits a solution. If N > γ , both systems admit solutions. If N = γ , one can only assert, in general, that there exist values of π, δ, ω, satisfying (1) with δi ≥ 0, πih ≥ 0, γ πih = δi tih , ωh = 0

6 [See A13/Chapter 8. – Eds.]

(i = 1, . . . , s; h = 1, . . . , r)

Mathematical study of production and wages 71 and that there exist values of α , σ , β satisfying (2) with αi ≥ 0, σih ≥ 0, βi = 0

(i = 1, . . . , s; h = 1, . . . , r)

When solutions exist, there is between them the following relationship: if we call γih the benefit 7 made per working day by any of the πih labourers producing Ai , so that δi tih σih − πih (b1h α1 + · · · + bsh αs ) = δi tih γih

(7)

we find δ1 (β1 + t11 γ11 + · · · + t1r γ1r ) + · · · + δs (βs + ts1 γs1 + · · · + tsr γsr ) = ω1 (b11 α1 + · · · + bs1 αs ) + · · · + ωr (b1r α1 + · · · + bsr αs )

(8)

4. Here is the economic interpretation of these results. Let there be a given socioeconomic system, defined by a given set of industrial conditions regarding fabrication and organization, and by a given set of requirements from consumers of various categories. To this system corresponds, for a given period, one year for instance, a certain number, which can be called the characteristic number of the given socioeconomic system, relative to the period under consideration. If the characteristic number relative to a certain period is greater than the number of work-days of this period, production, as a whole, is smaller than consumption; the considered socioeconomic state is therefore not stable. Moreover, it is mathematically impossible, as long as this state is not modified, to establish everywhere a regime of lucrative prices and wages sufficient for living. If the characteristic number relative to a certain period is smaller than or equal to the number of work-days of this period, production, as a whole, can be equal to consumption during this period, provided there is a suitable distribution of workers and consumers. Moreover, it is always mathematically possible to establish a regime of lucrative prices and wages sufficient for living. If, in particular, the characteristic number relative to a certain period is smaller8 than the number of work-days of that period, production can be made equal to consumption without any labourer having to provide the maximum labour; there can be non-working consumers; if there are, benefits can be left to employers and employees, the total sum of which is exactly equal to the total cost of living of the non-workers.9 7 [In his later work Potron used the term ‘economies’ (économies) instead of ‘benefits’ (bénéfices) to refer to individual savings. – Eds.] 8 [In both versions Potron wrote ‘greater’ instead of ‘smaller’. – Eds.] 9 [In the offprint, Potron inserted here a new paragraph: ‘All these results remain if one takes into account the continuous increase of population, provided that distribution does not change’. This is an announcement of the results presented in section 19 of A13/Chapter 8. – Eds.]

72 Mathematical study of production and wages Let us remark that the equations (1) and (2), of which these results are consequences, translate real and concrete economic facts, and are even written down as such in the book of accounts of the industrials, shopkeepers and consumers. It is therefore not absurd to draw practical conclusions from them, to inquire, for instance, by calculating the characteristic number, whether the often deplored lack of equilibrium between the wage rates and the cost of living is due only to a defective determination of prices and wages; or whether this lack of equilibrium does not stem from the impossibility in which the industry finds itself, on the whole, to provide in a continuous and usual way all that is required by the group of consumers.

Appendix: Section 3 according to the offprint version 3. Given these data, between the produced quantities δi Ai (i = 1, . . . , s) equal to the total consumed quantities, the numbers πih (i = 1, . . . , s; h = 1, . . . , r) of workers of categories Ch working for the production of Ai , the numbers ωh of simple consumers of category Ch , we have the relationships δi = a1i δ1 + · · · + asi δs + bi1 (ω1 + π11 + · · · + πs1 ) + · · · + bir (ωr + π1r + · · · + πsr ),

i = 1, . . . , s

(1*)

Between the prices αi of Ai , the benefits βi made on Ai , the average daily wages σih of the workers of category Ch producing Ai (i = 1, . . . , s; h = 1, . . . , r), we have the relationships αi = ai1 α1 + · · · + ais αs + ti1 σi1 + · · · + tir σir + βi ,

i = 1, . . . , s

(2*)

As the production δi Ai requires δi tih normal working days from the πih labourers of category Ch , the first condition of No. 1 is written N πih = δi (tih + ωih ), ωih ≥ 0,

i = 1, . . . , s; h = 1, . . . , r

(3*)

[and] one must obviously have δi > 0, πih > 0, ωh ≥ 0,

i = 1, . . . , s; h = 1, . . . , r

(4*)

The effective earnings of the πih labourers of Ch producing δi Ai are then δi tih σih ; their total cost of living is πih (b1h α1 + . . . + bsh αs ); the second condition of No. 1 is therefore written10 δi tih σih = πih (b1h α1 + · · · + bsh αs + γih ), γih ≥ 0,

i = 1, . . . , s; h = 1, . . . , r (5*)

10 [In equations (5*) and (7*) the variable γih designates the annual savings of a worker of category Ch active in the production of good Ai ; in the journal version, however, this variable designates his daily savings. – Eds.]

Mathematical study of production and wages 73 [and] one must obviously have αi > 0, σih > 0, βi ≥ 0,

i = 1, . . . , s; h = 1, . . . , r

(6*)

The first condition of No. 1 can therefore be met always and only if the system (1*), (3*), (4*) admits solutions; and the second, always and only if the system (2*), (5*), (6*) admits solutions. The study of both these systems will be published in the Annales de l’École Normale; here are the results. If N is smaller than a certain scalar ν , function of the coefficients a, b, t, none of the two systems admits a solution. If N > ν , both systems admit solutions. If N = ν , one can only assert, in general, that there exist values of π , δ , ω, satisfying (1*) with δi ≥ 0, πih ≥ 0, N πih = δi tih , ωh = 0

(i = 1, . . . , s; h = 1, . . . , r)

and that there exist values of α, σ, β satisfying (2*) with αi ≥ 0, σih ≥ 0, βi = 0

N σih = b1h α1 + · · · + bsh αs

i = 1, . . . , s; h = 1, . . . , r

When solutions exist, there is between them the relationship δ1 β1 + · · · + δs βs + π11 γ11 + · · · + π1r γ1r + · · · + πs1 γs1 + · · · + πsr γsr = ω1 (b11 α1 + · · · + bs1 αs ) + · · · + ωr (b1r α1 + · · · + bsr αs )

(7*)

3

Some properties of linear substitutions with coefficients 0 and their application to the problems of production and wages

Editors’ note In the two previous notes Potron had announced that he would expose his mathematical model in a forthcoming paper in the Annales Scientifiques de l’École Normale Supérieure. That paper was eventually published in 1913 (see A13/Chapter 8). In the meantime, however, Potron found another prestigious outlet for his mathematical research on economics. Already in December 1911 Potron published two dense papers with a rigorous statement of his mathematical results in the Comptes Rendus de l’Académie des Sciences. Whereas his previous notes addressed predominantly Catholic intellectuals, these papers were written for an audience of mathematicians. The first article, presented in the session of 4 December 1911, was ‘Quelques propriétés des substitutions linéaires à coefficients 0 et leur application aux problèmes de la production et des salaires’ (Comptes Rendus de l’Académie des Sciences, 1911, 153: 1129–32; see also the Errata, ibid.: 1541). It deals exclusively with the mathematical aspects. Potron started by referring to results on positive matrices obtained a few years earlier by the German mathematicians Oskar Perron (1880–1975) and Georg Ferdinand Frobenius (1849–1917), and then indicated how his own results can be seen as an extension of these to the case of nonnegative matrices, making use of a continuity argument by Frobenius and a theorem of another German mathematician, Hermann Minkowski (1864–1909). Potron’s note was presented by Paul Appell. Appell (1855–1930) was a famous French mathematician, whose main fields were functional analysis and mechanics. He had been president of Potron’s PhD commission. He was for some time Rector of the University of Paris, and later he represented France at the League of Nations. * * *

Linear substitutions with coefficients

0 75

1. Messrs Perron and Frobenius (the first in M.A., t. LXIV, p. 261; the other in S.A.B., 1908, pp. 471–76; 1909, pp. 514–18)1 have obtained some results concerning matrices with elements > 0. These results amount to the following: if a substitution (a) = xi , k aik xk (i, k = 1, · · · , n) has all its coefficients > 0, the characteristic root of maximum modulus of (a) is real, positive and simple. There exists a function i αi xi , with coefficients > 0, and only one up to a constant factor, that (a) multiplies by this characteristic root. And this function is, up to a constant factor, the only function i αi xi , with coefficients > 0, that (a) multiplies by a constant. 2. Making a generalization partially hinted at by Mr Frobenius, I have shown, by continuity, that, if the coefficients of (a) are assumed only 0, the characteristic root of maximum modulus of (a) is real and nonnegative, and that there exists at least one function i αi xi , with coefficients 0 and not all zero, that (a) multiplies by this characteristic root. 3. Pursuing the study of this more general case, I have shown that, if r denotes this characteristic root of maximum modulus, and n − q the rank of the matrix of the coefficients of the system r αi −

k

aki αk = 0

(i, k = 1, · · · , n)

0 and not all there are at most q distinct functions i αi xi , with coefficients zero, that (a) multiplies by r. I have then shown, by applying a theorem due to Minkowski (Geometrie der Zahlen, pp. 39–45)2 that, in at least one of these functions, at least q − 1 of the coefficients α are zero, the other being 0; and, then, that (a) transforms the variables x corresponding to the coefficients > 0 into linear functions of these variables only. From this I conclude that, if (a) transforms some of these variables x into linear functions of these variables only,3 thus operating a substitution that can be denoted as (a)1 , [and] if r1 is the characteristic root of maximum modulus of (a)1 , there exists at least one linear function of the variables of (a)1 only, with coefficients 0 and not all zero, that (a) multiplies by r1 . Finally, if there exists a linear function of the variables of (a)1 , with coefficients > 0, that (a) multiplies by a constant, this constant cannot be but r1 .

1 [The exact references of these publications are: Oskar Perron, ‘Zur Theorie der Matrices’ (Mathematische Annalen, 1907, 64: 248–63); Georg Ferdinand Frobenius, ‘Über Matrizen aus positiven Elementen’ (Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, 1908: 471–6) and ‘Über Matrizen aus positiven Elementen II’ (Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, 1909: 514–8). – Eds.] 2 [Hermann Minkowski, Geometrie der Zahlen (Leipzig, Teubner, 1910); Potron referred to section 19, ‘Appendix on linear inequalities’, of the first chapter of the book. It should be noted that the bulk of the material of this book had already been published in 1896. – Eds.] 3 In this case, I will say that matrix |aik | is partially reduced.

76 Linear substitutions with coefficients 4.4

0

It follows from these theorems that the system s αi −

k

aki αk =

αi > 0,

bli βl

l

(i, k = 1, . . . , n; l = 1, . . . , p)

0

βl

(1)

(I)

(2)

in which the coefficients a and b are 0, admits solutions only and, if matrix |aik | is not partially reduced, always, if the parameter s is at least equal to the characteristic root r of maximum modulus of |aik |. If matrix |aik | is partially reduced, one can only assert, for s = r, that (1) has, in general, solutions satisfying 0,

αi

i

(i = 1, . . . , n; l = 1, . . . , p)

αi > 0, βl = 0

(3)

Similarly, if s and t denote two independent parameters, and > 0, the system s αi −

k

aki αk =

t βl =

l

bli βl

i cli αi

αi > 0,

(i, k = 1, . . . , n; l = 1, . . . , p)

⎫ (1) ⎪ ⎬

(4) (5)

βl > 0

⎪ ⎭

(II)

in which we have 0, bli

aik

0, cli

0,

i

cli > 0

(i = 1, . . . , n; l = 1, . . . , p)

admits solutions only and, if |aik | is not partially reduced, always if s is the characteristic root σ (t) of maximum modulus of aki + 1t l bli clk . As above, if |aik | is partially reduced,5 one can only assert that (1) and (4) have solutions such that 6 0, βl

αi

0,

i

αi > 0,

l

βl > 0

(i = 1, . . . , n; l = 1, . . . , p)

(6)

Let D = |suik − aik | (uik = 0 for i = k, uii = 1); if one denotes Dik the elements of the adjunct of D, if s is greater than the characteristic root r of maximum modulus of |aik |, and if one sets7 i

cji Dik = Ddjk ,

k

blk djk = Blj

(i, k = 1, . . . , n; j = 1, . . . , p)

4 [Potron erroneously labelled this section 3 instead of 4. This also affects the reference to this section in A10/Chapter 4. – Eds.] 5 [Potron wrote aik instead of |aik |. – Eds.] 6 [In the following expression Potron wrote βl 0 instead of βl 0. – Eds.] 7 [In the expression that follows we have replaced blh by blk . – Eds.]

Linear substitutions with coefficients

0 77

one sees that (II) is equivalent to t βj −

l

t βl = αi > 0,

Blj βl = 0 i cli αi

βl > 0

(i = 1, . . . , n; l , j = 1, . . . , p)

⎫ (7) ⎪ ⎬

(4) (5)

⎪ ⎭

(III)

and that, since the B coefficients are 0, (III) admits solutions only and, if Blj is not partially reduced, always if t is the characteristic root τ (s) of maximum modulus of Blj . Same statement as above if Blj is partially reduced.8 One also sees that the two functions σ and τ are each other’s inverse, and that the conditions: t > 0, s at least equal (or equal) to the characteristic root of maximum modulus of aik + 1t l blk cli , are equivalent to the conditions: s > r , t at least equal (or equal) to the characteristic root of maximum modulus of Bjl . These results apply immediately to the solutions of important economic problems.

8 [System (III) is obtained by substituting (7) for (1) in system (II). The original system (1)–(4) has n + p equations and n + p unknowns, but the new system (7)–(4) has 2p equations and n + p unknowns. In his 1913 paper (see section 8 of A13/Chapter 8) Potron correctly substituted (7) for (4) instead of for (1), yielding a system with n + p equations and n + p unknowns. See also our paper W13. – Eds.]

4

Application to the problems of ‘sufficient production’ and the ‘living wage’ of some properties of linear substitutions with coefficients 0

Editors’ note On 26 December 1911, three weeks after the publication of his previous paper, Potron published another short paper in the same journal: ‘Application aux problèmes de la “production suffisante” et du “salaire vital” de quelques propriétés des substitutions linéaires à coefficients 0’ (Comptes Rendus de l’Académie des Sciences, 1911, 153: 1458–9). It contains a brief explanation of how the mathematical results he had obtained can be applied to an economic problem. This paper too was presented by Paul Appell. * * * In a Note presented at the meeting of 4 December 1911,1 I have given the statements and the principles of proof of several theorems on linear substitutions with coefficients 0. The detailed proofs will be presented in another journal.2 Here I will show how these results apply immediately to the solution of important economic problems. In fact, let Ai (i = 1, . . . , n) be the various results of labour, Ai their units, Pi the establishments that produce them; Ch (h = 1, . . . , p) the various social categories; bih Ai the annual consumption of Ai by a consumer of Ch ; aki Ai that made by Pk (k = 1, . . . , n) to produce Ak ; N the number of work-days in the year; [and] tih the number of working days that the workers of Pi belonging to Ch must provide to produce Ai . N and the numbers a, b, t are data representing the industrial, economic and social state. Then, if one aims at establishing: Either 1◦ a satisfactory regime of production and labour, that is, such that production is equal to consumption and every worker has his normal rest; Or 2◦ a simply satisfactory regime of prices and wages, that is, such that any exchange price is at least equal to the cost price, and any annual maximum wage (corresponding to maximum labour) to the annual cost of living;

1 [See A9/Chapter 3. – Eds.] 2 [See A13/Chapter 8. – Eds.]

Problems of ‘sufficient production’ and the ‘living wage’

79

3◦

Or an effectively satisfactory regime of prices and wages, that is, such that any exchange price is at least equal to the cost price, and any annual effective wage (corresponding to the amount of labour necessary for production to be equal to consumption) to the annual cost of living; one is led in every case to solve a system analogous to system II of No. 43 of the previous Note, with s = 1 and t = N . According to the results of that section, one can state the following theorems: 1◦ A satisfactory regime of production and labour and a simply satisfactory regime of prices and wages are possible only and, in general, always, if the characteristic root of maximum modulus of the matrix aik + N1 h bkh tih is 1; or, when the characteristic root r of maximum modulus of |aik | is < 1, if that ν of the matrix analogous to Blj is N . An exception occurs, as in No. 4, if the matrices are partially reduced. 2◦ 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. In this case one sees, as a consequence of the properties of systems of transposed linear forms, that the annual sum of the firms’ benefits and the workers’ economies represents exactly the annual total cost of living of the non-working consumers.

3 [Here and below, Potron’s reference to ‘No. 3’ is corrected as ‘No. 4’ for the reason explained in note 4 of Chapter 3. – Eds.]

5

Possibility and determination of the just price and the just wage

Editors’ note Until the beginning of 1912, Potron’s economic writings consisted of rather compact notes in which he sketched the contours of his mathematical model and hinted at its economic implications. But he also announced the imminent publication of more elaborate papers, and one of these, dealing with the economic consequences of his model, appeared fairly rapidly: ‘Possibilité et détermination du juste prix et du juste salaire’ (Le Mouvement Social, 1912, 73: 289–316). The journal Le Mouvement Social: Revue Catholique Internationale was the organ of the Action Populaire movement directed by the Jesuit Gustave Desbuquois, who had used it as a vehicle to announce Potron’s work on economics (see A7/Chapter 1). That journal was the successor of two others: at the origin, it was titled L’Association Catholique: Revue des Questions Sociales et Ouvrières, which ran from January 1876 (Vol. 1) to 1891 as the monthly review of L’Œuvre des Cercles Catholiques d’Ouvriers, an organization led by the counts Albert de Mun (1841–1914) and René de la Tour du Pin (1834–1924). It continued under the same title until December 1904 (Vol. 58). From January 1905 (Vol. 59) to December 1908 (Vol. 66), its name was L’Association Catholique: Revue du Mouvement Catholique Social. A second and more significant change occurred when the journal became Le Mouvement Social: according to the editorial notice signed by Gustave Desbuquois and Joseph Zamanski in its first issue, the evolution marks the passage from an ‘aristocratic journal faithful to the defence of the weak and the humble’ to a ‘democratic journal’. Le Mouvement Social itself lasted only a few years, from January 1909 (Vol. 67) to July 1914 (Vol. 78), when the beginning of the Great War interrupted life as it was. Throughout this period Desbuquois and Zamanski served as its editors and contributed frequently to the journal. After the war the publication resumed under the new title Les Dossiers de l’Action Populaire (Revue Bi-mensuelle d’Action Sociale et Religieuse). In this article the influence of Desbuquois is clearly visible in the final section, where Potron purported to show that his model can be seen as an application of the fundamental principles underlying the scholastic doctrine of the just price. The issues of justice in exchange, justice in wage contracts, etc. figured prominently

Determination of the just price and the just wage 81 on the agenda of the Catholic intellectuals who were struggling to come to grips with the ‘social question’. Potron did his best to purge his article of complex mathematical formulas, since even he must have realized that those would surely frighten the readership of Le Mouvement Social. He adopted a literary style and concentrated the mathematics in a single but very long footnote in the middle of the article. (We have transferred this footnote to an appendix.) That said, his way of expressing himself can hardly be called eloquent, and it remains doubtful whether readers who were not familiar with mathematical reasoning managed to grasp the essence of his ideas. * * * 1. In the discussions raised on the law of the just price there seems to be one idea which is not always sufficiently highlighted: it is that the just price, if one defines it as the monetary expression of the true exchange value, is essentially a relative notion. In fact, it seems impossible to appreciate a priori the equivalence between this isolated object and that amount of money considered exclusively as an instrument of exchange. To evaluate the just price of an object one must take account of the purchasing power of the money which is used to evaluate this price, that is of the price of other objects. Moreover, the just price is intimately connected to the just wage; when we say that the product of human labour borrows from this labour a value below which it should not fall, we enunciate the law of the minimum living wage. To evaluate this vital minimum in money, however, we must also take account of the purchasing power of this money in terms of food, clothing [and] housing, in other words, we have to take account of the cost of living, of the price of the objects which are necessary for life. Instead of talking of just prices and just wages in isolated cases, it appears therefore to be clearer and more in conformity with reality to speak of a just regime, of a satisfactory regime of prices and wages, such a regime being characterized by the following two conditions. On the one hand, the reward of the labour which a man can normally perform during a certain time, for instance one year, is at least equal to what a decent living costs for this labourer over the same period; this principle of the minimum living wage is only an application of the principle of the right to life. On the other, for any product the selling price is at least equal to the cost price; this principle of the lucrative price is but an application of the general principle of justice in exchange. Now everybody feels, at least confusedly, that the application of these two principles could well raise some opposition between them. In fact, on the one hand, the cost of living, to which the wage must be at least equal, depends upon the exchange price of the objects necessary for life; on the other, for each product the cost price, to which the exchange price must be at least equal, depends upon the exchange prices of various other products and upon the reward of the labour performed. And one easily realizes that, if the wage is in general below the cost of living, an increase of the wage rates will not necessarily restore the equilibrium, since this increase will inevitably entail a rise of prices, and as a result an augmentation of the cost of living. To all it seems that this difficulty is of a

82 Determination of the just price and the just wage mathematical order; it is indeed to considerations of this order that those who try to solve it turn.1 2. Since this difficulty is of a mathematical order, it seems to me that it calls for researches of the same order, and that, as the saying goes, we first have to put the problem into equations. Now – and it is there that the interest of this work may lie – we do not have to look very far for these equations; they have, so to speak, a concrete existence: they are the equalities between prices, wages, benefits [and] economies which appear in fact in the book of accounts of industrialists, shopkeepers, house mistresses [and] housewives. With regard to the principles of the minimum living wage and of the lucrative price, it is easy to translate them immediately into inequalities which have to be satisfied by the various quantities already linked by the considered equalities. Then the question arises: are the equalities and inequalities compatible? Do there exist quantities that verify both at the same time? That is, in its mathematical form, the problem to be solved. One sees that it presents itself, if we may say so, automatically, and in an absolutely natural and necessary way. By a curious coincidence, to obtain its solution, I had to do nothing more than using and complementing the results, of a purely theoretical and abstract order, obtained recently (in 1908 and 1909) by two mathematicians, one Swedish, Mr Perron, the other German, Mr Frobenius.2 This is what I have done in two Communications to the Academy of Sciences,3 and in a Memorandum which will be published soon by the Annales de l’École Normale.4 I propose here to explain the starting point, to give an exposition of the results, and to derive the conclusions. I 3. I will always conceive the labour contract as coming about between ‘labourers’ and ‘firms’. Exchange takes place either between firms, or between firms and ‘consumers’. This exchange involves a certain amount of ‘money’ on the one hand, and a certain product or ‘result of labour’ on the other. It is useful to explain the precise meaning which I give to these various terms. By ‘Money’ I understand a product which serves as a general means of exchange and payment.

1 See, for instance, J. Zamanski: A propos du Coût de la Vie (Assoc. Cath., 1er sem. 1908, t. LXV, pp. 30–39); and: La Crise Alimentaire (Mouv. Soc., 2e sem. 1911, t. LXXI, pp. 921–934). [Joseph Zamanski, ‘A propos de la coût de la vie’ (L’Association Catholique, 15 January 1908, 65: 30–9) and ‘La crise alimentaire’ (Le Mouvement Social, October 1911, 72: 921–34). – Eds.] 2 [Oskar Perron was German. – Eds.] 3 Comptes Rendus, t. 153, pp. 1129 and 1458, sessions of 4 and 26 December 1911. [See A9/Chapter 3 and A10/Chapter 4. – Eds.] 4 [See A13/Chapter 8. – Eds.]

Determination of the just price and the just wage 83 I call ‘result of labour’ or ‘economic result’ every creation of an economic good, i.e. useful to well-being, or every increase of the utility of a similar good. A ‘result of labour’ may be a new ‘product’, as in industry; or a ‘transference’, as in trade; or a ‘service’, which consists of the realization of certain conditions for the use of material goods, for instance the security provided by the police. By ‘consumer’ I understand not one individual, but one family consisting of father, mother, children or old family members in their care, and eventually even servants. A ‘consumer’ is therefore in reality a consuming family unit. I call ‘labourer’ the head of a family unit who, by performing his activity, contributes to the production of a ‘result of labour’ and, by the reward of his labour, allows the family unit of which he is the head to live. The ‘Firm’, the ‘House’ or the ‘Establishment’, is an abstract entity which makes links between labourers employed by it for the production of the same ‘result of labour’ and furnishes, on the one hand, these labourers with the reward of their labour, and on the other, the consumers with the ‘result of labour’ obtained. When a firm is founded by a financial company, it is natural to consider it as a moral person of which one must distinguish all those who work for it, including Administrators and Directors. I will do the same for firms founded by a ‘boss’, and will always consider him as a labourer distinct from his firm and of whom the labour must be rewarded by it. 4. The various consumers make use5 of the various results of labour to an extent which varies with what one might call their ‘type of existence’ and which can serve to characterize the latter. For the results of labour which are measurable amounts,6 all industrial products for instance, there is no difficulty, once the units have been chosen arbitrarily, to represent by scalars the quantities which a consumer of a given type of existence uses during a certain period, for instance one year. With regard to services, which at first sight do not seem susceptible of a quantitative evaluation, one can nevertheless say that, during a given period, a consumer of a given type of existence uses, in a certain proportion, the complete set of services provided, during that period, by this public administration, for instance by this municipality. And it is precisely in that proportion that he by justice should

5 This use might be: • • •

either a consumption strictly speaking, immediate and complete, for instance in the case of food products; or a use strictly speaking, entailing wear and tear and, as a result, susceptible to being assimilated to a partial immediate consumption, for instance in the case of clothing; or a simple utilization, for instance in the case of transports or of ‘services’.

6 Products strictly speaking are directly measurable, and by means of simple units; transports, by contrast, are indirectly measurable and by means of so-called composite units: the unit of transportation is the transportation of the unit of weight over the unit of distance, the ‘kilometric ton’, to fix ideas.

84 Determination of the just price and the just wage contribute to it by the taxes he pays. One conceives that it is possible to evaluate that proportion to a more or less arbitrary extent. In practice, as far as public administrations are concerned, it is precisely this evaluation which serves as the basis for the distribution of taxes. We can therefore inversely evaluate the ratio between the taxes paid by this consumer to that public administration and the total taxes perceived by that administration, and take this ratio as a measure, at least provisionally, of the proportion in which this consumer uses the services of that administration. The same principle applies to private administrations that provide services of the same order, for instance insurance companies. To sum up, relative to such a period of time, one year for instance, there is, for each type of existence and for each result of labour a numerical coefficient (possibly zero) which indicates to what extent, during the period under consideration, this result of labour is used by a consumer of this type of existence. A type of existence C can therefore be characterized by the set of corresponding coefficients b, relative to one year for instance. If we multiply each of these coefficients b by the price α of the unit of the corresponding result of labour, and if we add up the resulting products bα , we obtain the annual cost of living of the type of existence C. This is the operation performed by the housewife who writes down the annual total of her expenses. 5. Similar considerations apply to the firms which produce the various results of labour, with this difference that each of them uses the results of labour delivered to it by other firms as well as the labour delivered to it by its labourers. Moreover, this use, either of labour, or of the results of labour, can take place in two ways: • •

in an immediate and exclusive way aimed at the general functioning of the establishment, without possible affectation to one result of labour rather than to another; in an immediate and exclusive way aimed especially at the production of a specific result of labour.

With regard to the results of labour used by its general functioning, we can repeat for each firm what has been said about consumers: relative to a given period, for instance one year, there is, for each firm and each result of labour, a numerical coefficient (possibly zero) which indicates to what extent, during the period under consideration, this result of labour is used for the general functioning of the firm in question. In this way, there corresponds to each firm a set of coefficients, relative to one year for instance. If we multiply each of these by the price of the unit of the corresponding result of labour, and if we add up the resulting products, we obtain the annual total of the ‘general-material-costs’ of the firm under consideration. Likewise, there is, for each group of two results of labour, a numerical coefficient (possibly zero) which indicates to what extent the second is used for the specific production of one unit of the first. In this way, there corresponds to each unit of result of labour a set of coefficients. If we multiply each of these by the price of the

Determination of the just price and the just wage 85 unit of the corresponding second result of labour, and if we add up the resulting products, we obtain the ‘material-fabrication-price’ of the unit of the first result of labour under consideration. All these coefficients depend exclusively on the technical industrial conditions of the various firms: fabrication, processes, equipment conditions, etc. They are invariable as long as these conditions do not change. 6. The general functioning of a firm, as well as the specific production of a result of labour, requires a certain amount of labour. In general, this labour is not ‘homogeneous’. Between the high administration of the boss and the mechanical task of the unskilled worker, there are a lot of different types and degrees. The labour of any type can always be evaluated – at least in the case of an average evaluation – in normal days, where for each type of labour the ‘normal day’ represents what can be done, in normal conditions, by a labourer of average dexterity. It is therefore easy to evaluate how many normal days of each type of labour must be performed, either for the annual general functioning of this establishment, or for the specific production of a unit of that result of labour. If we multiply each of these numbers relative to the annual general functioning of an establishment by the average daily wage of the corresponding type of labour, and if we add up the resulting products, we obtain the annual total of the ‘general-labour-costs’ of the establishment under consideration. Doing the same for the specific production of a unit of a result of labour, we have the annual total of the ‘labour-fabrication-price’ of the unit of the result of labour under consideration. The numbers which represent in this way the required amount of labour, either for the annual general functioning of this establishment, or for that specific production, depend exclusively on the technical and administrative industrial conditions: fabrication processes, equipment and organizational conditions, etc. They are invariable as long as these conditions do not change. Now, if we know the proportion according to which the annual total of the general costs of this establishment must be allocated to the various specific productions, and, for this establishment, the annual production of that result of labour, we can easily determine, by means of a simple proportion, which fraction of the general costs must be imputed to the production of a unit of that result of labour. Then the coefficient a – which expresses to what total extent a result of labour B, of which the unit is designated by B, is used for the production of a unit A of the result of labour A – is the sum of two scalars: • •

the first is a coefficient which expresses to what extent B is used for the specific production of A; the second is the product of two coefficients; one expresses to what extent B is used for the annual general functioning of the firm P which produces A; the other indicates which fraction of the total annual of general costs of P can be imputed to the production of A.

86 Determination of the just price and the just wage Likewise, the coefficient t – which expresses, in normal days, the total amount of labour of type T that must be performed for the production of A – is the sum of two scalars: • •

the first indicates the amount of labour T that must be performed for the specific production of A; the second is the product of two coefficients; one expresses the amount of labour T that must be performed for the general functioning of P; the other is the one already used, indicating which fraction of the annual total of general costs of P can be imputed to the production of A.

In this way, to each unit of result of labour A there corresponds a set of coefficients a relative to the various results of labour, and a set of coefficients t relative to the various types of labour. All these coefficients a and t depend on the industrial, technical and administrative conditions of the various establishments. They are invariable as long as these conditions do not change. If we multiply the coefficients a by the prices α of the units of the corresponding results of labour, and the coefficients t by the daily wages σ paid to the corresponding types of labour, and if we add up the resulting products aα and t σ , we obtain the ‘cost price’ of A. As a first numerical example, here are the approximate coefficients of the cost price formula of a two-kilogram loaf of bread, such as they were provided by a brief survey in a baker’s shop of the 1st district of Paris. The cost price of a two-kilogram loaf of bread, in the conditions in which it is made in this shop, is the sum of certain fractions of the prices of the premises, equipment, materials and labour force which are used for its fabrication. Table 5.1 gives the fraction which corresponds to each of these items, in other words the coefficient by which the price of each item is multiplied in the cost price formula of a two-kilogram loaf of bread. The last six coefficients are lacking; to determine each of them, we would need to know the total sum of premia or contributions perceived yearly by the corresponding administration, data which have not yet been obtained. The coefficients 56, … , 61 result directly from experience. To determine the others, we have divided the number of units which are in service of each item by the total number of two-kilogram loaves of bread which represents, during the actual or presumed duration of this item, the total capacity of production of the baker’s shop in question. Thanks to the use of this method, which is sufficient for a first approximation, we did not have to deal with the allocation of the general costs. Moreover, to each type of labour T corresponds a type of decent existence C, which we will define by its coefficients b (No. 4), making, if necessary, the somewhat fluctuating data of observation precise by an arbitrary determination. 7. The coefficients a, b, t, and the number N of work-days per year, represent therefore simultaneously the state of industry and trade, and the needs of the various categories of labourers and consumers. They define with precision what

Determination of the just price and the just wage 87 Table 5.1 Coefficients of the cost price formula of a two-kilogram loaf of bread Item numbers 1

Used items

Coefficients

Premises The yearly renting of a premise including a cellar for 8 m × 4 m ovens, a 4 m × 5 m warehouse, a 3 m × 5 m room, a 6 m × 5 m shop

1/100,000

Equipment 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42

A weighing machine for the flour sacks (150 kg) A goods lift A metre of cable for a goods lift A bolting device A bolting sieve A device to preserve leaven The tinplating of the previous A kneading-trough A wicker basket without canvas A canvas for a wicker basket A set of six shovels to put dough in the oven A scuffle An oven for 60 loaves without tiles or vault The tiles of the previous The vault of the previous The chimney-sweeping of the previous A basket to remove bread from the oven The equipment of a gas lamp The yearly subscription to a working pair of dungarees per week The yearly subscription to a working apron per week The yearly subscription to a working towel per week A store for loaves with three 2.50 m × 3 m floors The washing of a square metre of paint in the shop The renewal of a square metre of paint in the shop A shop store for loaves with three 2.50 m × 3 m floors The equipment of an electric lamp A 25 candles electric bulb A shop counter Scales and accessories A retailing knife A straw chair A shop table A pendulum clock A ream of paper for sacks A small push coach for bread A broom A towel The washing of a towel The washing of an apron An inkpot A litre of ink

1/2,000,000 1/5,000,000 7/50,000 1/5,000,000 3/250,000 1/1,000,000 1/500,000 1/2,000,000 1/500,000 1/100,000 1/30,000 1/50,000 1/2,000,000 1/100,000 1/1,000,000 1/10,000 1/250,000 1/1,000,000 1/5,000 1/50,000 1/25,000 1/5,000,000 3/6,250 1/6,250 1/2,000,000 1/200,000 1/30,000 1/2,000,000 1/1,000,000 1/800,000 1/50,000 1/200,000 1/1,000,000 3/2,000 1/100,000 1/100,000 1/7,500 1/360 1/1,260 1/500,000 1/100,000 Continued

88 Determination of the just price and the just wage Table 5.1 Cont’d Item numbers

Used items

Coefficients

43 44 45 46 47 48 49 50 51 52 53 54 55

A penholder A box of pens A pencil A penknife An envelope A sheet of writing paper The carrying of a letter A ream of printed invoices A Diary of the Employers’ Office A great carrying book A book of accounts A credit report book A shop apron

1/500,000 1/100,000 1/60,000 1/500,000 1/1,000 1/1,000 1/1,000 1/25,000 1/100,000 1/200,000 1/300,000 1/300,000 1/75,000

56 57 58 59 60 61 62 63

Raw material A 150 kilo sack of flour A kilogram of salt A litre of water A kilogram of leaven A kilogram of fine flour 100 kilograms of coal A cubic metre of lighting gas The hectowatt-hour

1/100 1/40 3/5 1/200 1/40 1/360 1/120 1/240

64 65 66 67

Labour force The working day of the employer The working day of the specialized labourer The working day of the assistant The working day of a female worker

1/360 1/360 1/360 1/180

68 69 70 71 72 73

Various services The services of the fire insurance company The services of the glass-breaking insurance company The services of the accident insurance company The services of the State The services of the department The services of the district

one may call a socioeconomic state; and as long as the socioeconomic state does not change, they retain constant values. By contrast, while the socioeconomic state and the coefficients a, b, t , N remain unchanged, we can make changes to: 1◦ The exchange prices α of the units A, the daily wages σ paid by the various firms P for the various works T , and as a result the benefits β of the various firms P, and the economies γ of the labourers of the various categories; 2◦ The annual productions δ A of the various results of labour, the numbers π of labourers of various categories employed by the various firms P, the numbers

Determination of the just price and the just wage 89 ω of the simple consumers of the various categories,7 and as a result the annual excesses ρ A of production over consumption of the various results of labour, and the numbers of days out of work ω of the labourers of the various categories.

But these variables are not independent; in fact, between the variables of each of the two groups there exist two systems of equations of which the coefficients are precisely the coefficients a, b, t , N of the socioeconomic state, assumed to be constant. For the second group of variables (δ, ρ, π, ω, ω), we have a first system of equations by expressing that, for each result of labour A, the annual production δ A is the sum of the annual consumption made of A, by firms as well as by consumers, and of the annual excess ρ A. In this system there are as many equations as there exist distinct results of labour. We have a second system of equations by expressing that, for each category of labourers of each establishment, the scalar N is the sum of the average number of effective working days and the average number of days out of work. In this system there are as many equations as there exist categories of labourers in the various establishments. For the first group of variables (α, β, γ , σ ), we also have two systems of equations. The first is obtained by expressing that, for each result of labour, the exchange price is the sum of the cost price and the benefit. In this system there are as many equations as there exist distinct results of labour. The second system is obtained by expressing that, for each category of labourers of each establishment, the effective annual wage, which depends upon the annual production and upon the number of labourers, is the sum of the annual cost of living and of the annual total of economies. In this system there are as many equations as there exist categories of labourers in the various establishments. 8. As one sees, these four systems of equations express real and concrete economic facts. The equalities of the third system even actually appear in the book of accounts of firms, and those of the fourth system in the book of accounts of the labourers’ households. Moreover, the distinction established between the constants a, b, t , N , and the variables δ, ρ, π, ω, ω, α, β, γ , σ rests also on a real foundation. To modify the a or t coefficients, a change is required in the processes of fabrication, for instance as a result of a new invention, or a change in the conditions of equipment, for instance as a result of the introduction of a new material. To modify the b coefficients, a change is required in the customs of life, for instance as a result of the introduction of a new product. In one word, every change in the a, b, t coefficients, since it is a change of the socioeconomic state, constitutes a

7 [In this article Potron used a special symbol for this variable which looks like a Greek letter, but does not correspond to any letter of the standard Greek alphabet. We have replaced it by ω, the symbol which Potron used for that variable in his 1913 article (see section 13 of A13/ Chapter 8). – Eds.]

90 Determination of the just price and the just wage noteworthy event, always a bit exceptional, and requires some time to take place in a noticeable way. Prices and wages, by contrast, can change often and suddenly, for instance as a result of workers’ claims, legal measures, [or] speculative ploys. Likewise, accidental events such as labour conflicts, [or] stops in industry for whatever reason, can significantly modify production. Therefore – and it is a hypothesis confirmed by the facts – we can assume that, during a certain time, for instance several years, the socioeconomic state, and hence the coefficients a, b, t , N , remain constant, while the other quantities: prices, wages, etc., consumption, production, etc., vary, but always such that the same four systems of equations remain satisfied. It is indeed this hypothesis which one adopts when one asks either for a decrease of prices, or an increase of wages, in order to cope with the crisis of the ‘high cost of living’ without changing anything to the socioeconomic state. But one should not forget that there are in this case two systems of equations which all possible values of prices, wages, economies and benefits must necessarily and always satisfy. 9. The first undoubtedly concrete and actual problem which arises, is then the following: given a certain socioeconomic state, which cannot be changed or for which the time is lacking to change it, to assign to prices and wages such values that the two principles of the right to life and of justice in exchange are met. Now the last principle demands that the exchange prices are at least respectively equal to the cost prices, that is to say, that the benefits, which together with prices and wages verify the third system of equations, are either positive numbers or zero. To give the strictest expression to the right to life, I introduce a fifth system of equations which asserts that for each labourer the annual maximum wage (product N σ of the daily wage by the number of work-days per year) is the sum of the cost of living and of the total annual maximum economies. The principle of the right to life, in its strictest form, is now expressed by saying that the total annual maximum economies, which together with the prices and wages satisfy the fifth system of equations, must be positive scalars or zero. If one remarks that prices and wages are essentially positive scalars, one sees that the problem at hand, the problem of the determination of a simply satisfactory regime of prices and wages, boils down to determining scalars – positive for the prices and wages, positive or zero for the benefits and economies – that satisfy the third and fifth systems of equations. 10. It is nevertheless clear that the principle of the right to life, when correctly interpreted, requires more; it is the wage which can be earned effectively, taking production into account, rather than the theoretical maximum wage, which must be at least equal to the cost of living; in other words, the annual total of economies which can be made effectively, and which together with the prices, wages, annual levels of production [and] numbers of workers of each category satisfy the fourth system of equations, must be positive scalars or zero. However, because of the

Determination of the just price and the just wage 91 supplementary variables which this fourth system contains, we must first deal with the following problem: Supposing first that the total amount of the population remains constant, how do men have to distribute themselves over the various professions of labourers and the various categories of consumers, for the annual production of each result of labour to be exactly equal to the annual consumption, with no labourer having to provide an annual number of working days greater than the number of workdays per year? This is the problem of the determination of a satisfactory regime of production and labour. It is clear that such a regime, once [it is] established, can last as long as the socioeconomic state which serves as its basis. From the mathematical point of view, the first condition requires that the annual excesses – which, together with the annual levels of production, numbers of labourers and of consumers of each category, satisfy the first system of equations – are all zero. The second condition requires that the numbers of days out of work – which, together with the annual levels of production and the numbers of labourers of each category, satisfy the second system [–] are positive or zero. If one remarks that the annual levels of production are essentially positive, and that the numbers of labourers and of consumers of each category are essentially positive or zero, one sees that the problem at hand, the problem of the determination of a satisfactory regime of production and labour, boils down to determining scalars – positive for the annual levels of production, positive or zero for the numbers of labourers and consumers of each category, zero for the annual excesses – satisfying the first and second systems of equations. 11. If one supposes that this second problem is solved, one sees that the determination of a regime of prices and wages for which the principles of the right to life and of justice in exchange are effectively met, boils down to determining scalars – positive for the prices and wages, positive or zero for the benefits and economies – satisfying the third and fourth systems of equations, with the annual levels of production and the numbers of labourers which occur in the last system having values determined by the solutions of the second problem. This third problem is that of the determination of an effectively satisfactory regime of prices and wages. II 12. In the Memorandum which will be published in the ‘Annales de l’École Normale’,8 I have first examined, for each of these three problems, the preliminary question: Is the problem possible? Does it admit solutions? As far as the third problem is concerned, it is clear that the question only arises if the second admits solutions. Incidentally, for the three problems, if there exist solutions,

8 [See A13/Chapter 8. At this point Potron inserted a very long mathematical footnote, which for clarity we have relegated to the Appendix. – Eds.]

92 Determination of the just price and the just wage their determination presents no difficulty at all from the mathematical point of view. So, with regard to the possibility of the three problems, here are the results obtained. To each given socioeconomic state, characterized by specific values of the scalar N and of the coefficients a, b, t, corresponds a certain number, a specific function of N and of the a, b, t, and which I will call the characteristic number of the socioeconomic state; here are its properties. 1◦ If the characteristic number of a socioeconomic state is greater than the number of work-days per year, the first two problems are impossible. Hence, in that case, whatever positive values are given to the prices and wages, the values of the benefits and economies, which together with those of the prices and wages must satisfy the third and fifth systems of equations, cannot all be positive or zero; some of them will unavoidably be negative. Likewise – whatever may be the positive or zero numbers of the labourers and consumers of the various categories, [and] whatever may be the quantities annually produced – the annual excesses and durations of unemployment, which must satisfy the first and second systems of equations, cannot all have positive or zero values; some of them will unavoidably be negative. Hence, in that case, on the one hand some firms will operate with losses, or some workers will not earn enough to support their needs; on the other, for certain products the production will be smaller than the consumption, or for some labourers the number of working days to be performed during the year will be greater than the number of work-days. And this will unavoidably be the case as long as the socioeconomic state is not changed. 2◦ If the characteristic number is equal to the number of work-days per year, the first problem admits in general positive solutions for the prices and wages, [and] zero for the economies and benefits; the second problem admits, in general, positive solutions for the production levels, positive (or zero, if the category does not exist) solutions for the numbers of labourers of the various categories, [and] zero solutions for the numbers of simple consumers, the annual excesses and the durations of unemployment. In an exceptional case one can be forced to assign zero values to certain prices and wages, for the first problem, and to certain production levels, for the second. 3◦ If the characteristic number of a socioeconomic state is smaller than the number of work-days per year, the first two problems are possible. One can even state that the fourth and fifth systems of equations can, in infinitely many ways, be satisfied by values of the prices, wages, economies and benefits which are all positive; and that the first and second systems of equations can, in infinitely many ways, be satisfied by values of the annual production levels, numbers of simple consumers of various categories, [and] durations of unemployment which are all positive, by values of the numbers of labourers of various categories which are all positive (or zero, if the category does not exist, with the corresponding coefficient t being zero), and by values of the annual excesses which are all zero.

Determination of the just price and the just wage 93 13. Let us now deal with the issue of the possibility of the third problem, which, as we have seen, arises only if the second is possible, that is to say if the characteristic number is at most equal to the number of work-days. 4◦ At first, let us assume that the characteristic number is smaller than the number of work-days; [and] let us assign to the production levels and to the numbers of labourers which occur in the fourth system values that satisfy the first and second systems. Then, the third and fourth systems can, in infinitely many ways, be satisfied by values of the prices, wages, economies and benefits which are all positive. But the combination of the equations shows that the annual sum of the firms’ benefits and of the labourers’ wages represents exactly the total annual cost of living of all consumers, whether they are workers or not. 5◦ If the characteristic number is equal to the number of work-days per year, we see first that we can, if applicable, suppress in all equations everything which concerns the results of labour of which production must be zero. We can then state that the third and fourth systems can in general be satisfied by positive values of the prices and wages, and zero values of the economies and benefits. In the exceptional case mentioned above, we can be forced to assign zero values to certain prices and wages. 14. All these results continue to hold if the total number of the population grows, whatever may be the law of its increase, provided that the population distributes itself in a constant proportion over the various professions and the various social categories. One will find in the Annales de l’École Normale the formulas which allow to calculate the most general solutions of these various problems. III 15. I am now going to draw the economic conclusions of these results, recalling that, just as the equations which have served to obtain them, they are intimately connected to the real and concrete domain of the facts. To begin with, if we take into account the weekly periods of rest and the holidays, we can in practice adopt the value of 300 for the scalar N . Therefore, assume that we are given a socioeconomic state (industrial processes, conditions for the installation and organization of the various establishments, needs or requirements of the labourers and consumers of various categories) of which the characteristic number is greater than 300, and that one tries to implement it. Whatever may be the regimes adopted for prices and wages, production and labour, there will always be labourers whose wages will be lower than what is required, or firms that will operate with losses; and if no labourer works more than 300 days per year, there will be products of which the production will be lower than the consumption. If, as a relic from a previous socioeconomic state, the labourers

94 Determination of the just price and the just wage have economies and the firms stocks of money and goods, the new state will be able to persist for some time. But inevitably the economies of the labourers and the stocks of the firms will finally run out, and by the force of things one will be led to the only possible solution: to modify the socioeconomic state. This change will have to consist either in a perfectioning of the industrial processes, which improves the efficiency of labour, or in a restriction of the consumers’ demands, which diminishes consumption. Any other measure, for instance any change of the prices and wages, will only displace the difficulty, change those who suffer the loss, but will not remedy the evil. In fact, in the present case the true and profound cause of the economic malaise is not a defective determination of the prices and wages, but the impossibility of producing all that is needed to satisfy the consumers’ requirements. 16. Assume as given a socioeconomic state of which the characteristic number is 300. As we have seen, in general there does not exist a satisfactory regime of production and labour unless everybody is at work and without any unemployment; and in general there does not exist a satisfactory regime of prices and wages unless the firms renounce to any benefit and the workers to any economies. In the hypothesis that these conditions are fulfilled, it is in general possible to assign positive values to the production levels that are equal to consumption, and positive values to the prices and wages such that the exchange prices are equal to the cost prices. However, in the exceptional case, it may happen that the production levels of some results of labour, some prices and some wages have to be zero. This means that one has to renounce to obtain some results of labour, that some firms must receive their raw materials for free and deliver their products for free, and that some labourers must work for free in the service of these firms whilst also receiving for free all that is needed for their support. This singular case, studied in some detail in the ‘Annales de l’École Normale’, presents only a theoretical interest. Nevertheless one sees that it would in practice be inconvenient to come near to it, that is to say to assign to the number of workdays a value too close to the characteristic number, if this is smaller than 300, or to adopt a socioeconomic state of which the characteristic number is only slightly smaller than 300. 17. Assume, by contrast, that we are given a socioeconomic state of which the characteristic number is smaller than 300. We can, in infinitely many ways, distribute the population over the various professions and social categories so as to obtain a sufficient production, while letting each labourer have his normal rest and some additional leisure, and allowing the existence of non-working consumers. When the choice has been made for one of these distributions, for one of these satisfactory regimes of production and labour, we can, in infinitely many ways, determine the prices and wages so as to effectively ensure a living wage to each labourer, and benefits to each firm, – wages and benefits of which the annual total

Determination of the just price and the just wage 95 is always equal to the total annual cost of living of all consumers, whether they are workers or not. Even when the characteristic number is smaller than 300, however, we can also be confronted with either a defective distribution of the population, or above all a defective regime of prices and wages. Particularly in this last case we will see the occurrence of the same economic phenomena as those mentioned above: there will be labourers whose wages will be lower than what is required, or firms that will operate with losses. But at present the cause of the evil is not anymore inherent to the socioeconomic state itself; to remedy it, nothing needs to be changed to this state; according to the case, it suffices to change either the distribution of the population, or the regime of prices and wages. Calculation cannot indicate which measures should be taken to implement these modifications. These are issues where human liberty comes into play and which belong to the realm of pure economic science. In the presence of a state of economic malaise, calculation can, so to speak, only diagnose with precision, through the determination of the characteristic number, the cause of the evil. By indicating that the evil comes from the socioeconomic state itself, calculation will certainly prevent ineffective and disastrous attempts. By contrast, after having found out that the general socioeconomic state is sound, calculation can indicate which modifications are to be made to the regimes of production and labour, [and] of prices and wages, modifications which will be certainly effective in order to make the malaise go away.9 18. Consider a socioeconomic state of which the characteristic number is smaller than 300, and which is an ideal or a real state according to whether the coefficients b represent the desires of the consumers or their real consumptions; if it is an ideal state, it can be implemented and is stable; if it is a real state, it is stable. I assume, and this hypothesis must be kept in mind, that during a certain time no change occurs in the technical or administrative conditions of trade and industry, no new invention is made, [and] that always the same type of existence corresponds to a given professional or social category, in one word, that the coefficients a, b, t retain constant values. By the force of things, production keeps itself on average equal to consumption. But then we know (No. 13) that the annual total of the firms’ benefits and of the labourers’ wages represents exactly the total cost of living of all consumers, whether they are workers or not. Now it is precisely the firms’ benefits that constitute the revenues of the employed capital. Hence, under the stated hypothesis, the consumers as a whole receive, either in the form of reward for their labour, or in the form of revenues for their capital, only what they have to spend for living. Since no labourer should make a loss, a non-worker can make economies only if another non-worker makes a loss; and a labourer can make economies only

9 [There is an apparent contradiction between the first and the final sentences of the paragraph. On the same topic, see also the passage corresponding to note 83 in A15/Chapter10. – Eds.]

96 Determination of the just price and the just wage if a non-worker makes a loss. The non-worker who makes economies can move to a higher social category; the labourer who makes economies can move to a category of non-workers; but at the same time, the non-worker who makes a loss must restrain his way of life, move to a lower social category, and can even be forced, in order to live, to move to a category of labourers. Hence, the distribution of the population over the various professions and social categories keeps itself constant in an automatic way, so to speak, as long as the regime of prices and wages is one of the satisfactory regimes which corresponds to that distribution. This way any socioeconomic state of which the characteristic number is smaller than 300 is, by itself, essentially stable. When the characteristic number is smaller than 300, there can be non-working consumers, that is to say who will not exercise their activity in the direct or indirect production of what is required for life in the socioeconomic state under consideration. They are the capitalists, who live off their revenues. Here we touch upon the big question of the legitimacy of interest, and perhaps these considerations will help to shed some light on it. We have to observe that equipment and even wages always represent an advance, since equipment must be constituted before fabrication, [and] equipment and wages must often be paid well before the selling of the product. Without this advance, any production would be impossible; it constitutes therefore a real service delivered. On the other hand, in the cost price formulas elaborated above, only that part of the material enters which is subjected to periodic renewal or restoration. In the case of a tunnel, for instance, we only take into account the maintenance, the necessary periodic restoration of the brickwork, and not the works, done once and for all, of drilling and removing the rubble properly speaking. The part which must be reimbursed annually of the advances of this category could only be fixed in a completely arbitrary way; it is simpler and at the same time more appropriate to treat them as if they had created a permanent object, of an undefined utility, therefore enabling it to be effectively rented. Therefore, the capital employed in each enterprise will in general be rewarded on two grounds. 19. In reality, a socioeconomic state, even it is stable, will not last for ever. New inventions require new installations, and by improving the efficiency of labour, they make a greater well-being possible for all. Let us remark, however, that it is possible to replace one socioeconomic state by another only by going through an intermediate phase, during which one has to produce not only what is required for life in the first socioeconomic state, but also all the new installations that will be used in the second.10 10 It seems to me that a theoretical study of these intermediate periods, which I have already started, leads to noteworthy conclusions, especially with regard to the constitution of capital by means of saving. [This is the only place where Potron used the standard economic term ‘saving’ (épargne) instead of the unusual ‘economies’ (économies). It is unclear to what theoretical study Potron referred here. – Eds.]

Determination of the just price and the just wage 97 But in this second socioeconomic state and even in the intermediary period, the distribution of labourers, the regime of production and labour, [and] the regime of prices and wages will in general differ from those of the first socioeconomic state. The introduction of a new machine, for example, deprives a certain number of labourers of work; by contrast, the construction and maintenance of those machines provides work to other professions. In general an equilibrium is reached in the end, but after a period of crisis which will be more or less long, [and] more or less disastrous, entailing a level of general well-being lower than what it might have been. When, as a result of a progress in industry, a final or transitory socioeconomic state succeeds to a socioeconomic state deemed to be satisfactory, it is always possible to ensure to everyone the same level of well-being during the transitory period, and a greater level of well-being after the establishment of the new state; but this on the condition that one rapidly adopts new regimes of production and labour, [and] of prices and wages, regimes which will in general be different from the old. Undoubtedly one will not find useless the effort which must be made to determine in advance these new regimes and to adjust to them, if one thinks of the disasters and misery which any delay inevitably brings along. To sum up, calculation would therefore have a double role. First it should say whether, from a general point of view, we have reasonable requirements with respect to well-being, whether the normal efficiency of human labour is capable of satisfying them. Once this essential point has been checked, calculation should say how it would be convenient to distribute this labour in order to obtain a sufficient level of efficiency, and next to determine the condition of the normal functioning of that instrument of exchange which is money, in one word to establish a standard regime of prices and wages. 20. It seems to me that there is a great analogy between the theoretical price of each object, when determined as an element of an equilibrium regime, and the ‘just price’ of scholastic doctrine. As a matter of fact, the theoretical price, [as an] element of an effectively satisfactory regime of prices and wages, ensures that in labour and exchange contracts the higher principles of justice and of the right to life are respected. Besides, this theoretical price is only an average, around which the real price, the current price, can oscillate within bounds that could be determined, while the regime continues to be satisfactory. In fact, the systems of equations I have considered admit infinitely many solutions; and by assigning to the variables appropriately chosen values which are positive and close to those, equally positive, which constitute a system of solutions, it is always possible to obtain a second system of solutions, close to the first. This notion of theoretical price, element of an effectively satisfactory regime of prices and wages, contains therefore all the essential elements of the scholastic notion of the ‘just price’; it adds, or at least reveals, the touch of relativity, which distinctly characterizes the notion, and enables its practical determination.

98 Determination of the just price and the just wage In fact, already in his Lectures during the Semaine Sociale of Saint-Étienne on the Law of the Just Price,11 Mr Desbuquois, whilst leaving the task of accomplishing this determination in principle to the æstimatio communis,12 wants it ‘to proceed in practice from the prudent appreciation of competent men, taken in the social environment where the exchange prices are negotiated’.13 I am of his opinion; like him, I also think that it is by ‘the corporative organization, assisted by law where needed’14 that it will be possible to constitute this ‘sovereign tribunal of arbitration, which assembles all rights, … all capabilities’ and ‘to which the right is awarded to appreciate and to fix the expression of justice in exchange’.15 But this tribunal, once it is constituted, will find itself, by the force of things, confronted with the problems which are the subject of this study. It will therefore not be useless for the tribunal to obtain the assistance of a ‘Bureau de Calculs’. This Bureau would keep up to date, so to speak, the representative coefficients of a socioeconomic state which implies for all a sufficient level of well-being and which has a characteristic number sufficiently lower than 300. Next, it would check, by means of the indicated formulas, whether the distribution of the population, [and] the regime of production and labour are convenient for the adopted socioeconomic state; when needed, it would determine and keep up to date a satisfactory regime of production and labour. Finally, it would have to determine and keep up to date an effectively satisfactory regime of prices and wages, a standard regime which would serve as basis for the establishment of current prices and wages. The first operation would therefore consist of determining on the one hand the b coefficients of the formulas of the various types of existence (No. 4), according to reasonable desires, [representing] the needs effectively felt in the various social categories, and on the other the a and t coefficients of the cost price formulas of the various results of labour (No. 6). The determination of the b coefficients is relatively easy, and apparently already accomplished to a large extent. With regard to the determination of the a and t coefficients, we first have to point out the obvious fact that the coefficients of the cost price formula of this result of labour, coming out of that establishment, will all be obtained by addressing only that establishment. To simplify things, we could even do two successive surveys. In the first, strictly qualitative one might say, each industrial or commercial establishment would only be asked to provide: 1◦ The nomenclature of the products which it delivers and the operations it accomplishes; in one word, its ‘catalogue’; 11 Desbuquois: ‘La loi du juste prix’ (Mouvement Social, October 1911, tome LXXI, pp. 867–881). [The reference should read: tome LXXII, pp. 867–884. – Eds.] 12 [On the notion of aestimatio communis (common estimation) in scholastic economics, see for instance: Raymond De Roover, ‘The concept of the just price: Theory and economic policy’ (Journal of Economic History, 1958, 18: 418–34). – Eds.] 13 [Gustave Desbuquois, o.c.: 874. – Eds.] 14 [Ibid.: 880–1. – Eds.] 15 [Ibid.: 874. – Eds.]

Determination of the just price and the just wage 99 2◦

For each article of this catalogue, the nomenclature of everything (raw materials, operations done by foreign establishments, equipment, labour force) which is used for the specific manufacture of that article; 3◦ The nomenclature of everything (as above) which is used, in an indivisible way, for the general functioning of the whole or part of the establishment, and as a result indirectly for the specific fabrication of several articles. Already this first survey would reveal, in a clear and precise way, the interdependency between the various industrial and commercial establishments, [and] between the various results of labour. It would determine the number of formulas to be set up and it would clarify, for each of them, which coefficients have to be calculated, since it would say which results of labour, [and] which types of labour are required for the production of this result of labour. It would also assist industrialists or traders in understanding exactly what the second survey, a quantitative one this time, would ask of them: which quantities of these various results of labour, [and] of these various types of labour, as detailed by the first survey, are required for the production of that result of labour. One sees the considerable role that the professional organizations would have to play in these surveys. Once the data have been provided, the calculations performed, [and] the results obtained, an even more important role would be left to these organizations. Having full knowledge of things, they would have to take practical decisions, if necessary with the aid of the public authorities, capable of bringing about the gradual implementation of an economic order in conformity with the general plan drawn up by God, but of which He leaves to mankind the task of ensuring its execution. M. Potron

[Appendix: The mathematical footnote of section12]16 For those readers who are a bit familiar with mathematical notations – or who simply will not be frightened by them – this is how the three problems in question can be formulated. Let Ch (h = 1, . . . , r) be the categories of the workers of the various establishments and of the non-workers; Ai (i = 1, . . . , s) the various results of labour, Ai their units [and] Pi the establishments that produce them; bih Ai the annual consumption of Ai by a consumer belonging to Ch ; aik Ak (k = 1, . . . , s) the consumption of Ak made by Pi for the production Ai ; δi Ai the annual production of Pi ; πih the number of labourers of Pi belonging to Ch , tih the number of working days demanded from them to produce Ai , ωih their collective unemployment, σih their average individual daily wage, [and] N γih the average annual total of their individual economies; αi the exchange price of Ai ; βi the benefit made by Pi on the sales of Ai ; ωh the

16 [We have corrected several obvious typographical errors. – Eds.]

100 Determination of the just price and the just wage number of non-workers belonging to Ch ; h the total number of consumers of category Ch , [and] NSh the annual cost of living of one of them. The first system comprises the equations δi = a1i δ1 + · · · + asi δs + bi1 h

= π1h + · · · + πsh + ωh ,

1 + · · · + bir

r,

i = 1, . . . , s

(1)

h = 1, . . . , r

(2)

The second system comprises the equations N πih = δi tih + ωih ,

i = 1, . . . , s; h = 1, . . . , r

(3)

The third system comprises the equations αi = ai1 α1 + · · · + ais αs + ti1 σi1 + · · · + tir σir + βi ,

i = 1, . . . , s

(4)

The fourth system comprises the equations δi tih σih = N πih (Sh + γih );

NSh = b1h α1 + · · · + bsh αs ;

(5)

i = 1, . . . , s; h = 1, . . . , r

(6)

The fifth system comprises the equations (6) and σih = Sh + γih ,

i = 1, . . . , s; h = 1, . . . , r

(7)

The first problem then consists of finding α, β, γ , σ, S which satisfy (4), (6), (7) and αi > 0, σih > 0, Sh > 0, βi

0, γih

0,

i = 1, . . . , s; h = 1, . . . , r

(8)

or, which comes down to the same thing, (7) and αi − ai1 α1 − · · · − ais αs = ti1 S1 + · · · + tir Sr + ti1 γi1 + · · · + tir γir + βi , i = 1, . . . , s h = 1, . . . , r NSh = b1h α1 + · · · + bsh αs , αi > 0, Sh > 0, βi

0, γih

0,

The second problem consists of finding δ , π , ω, (3) and δi > 0, πih

> 0, if tih > 0, = 0, if tih = 0,

ωh

0,

h

⎫ ⎪ ⎪ ⎪ ⎬ (9) ⎪

(6) ⎪ ⎪ ⎪ ⎪ (10) ⎭

(I)

, ω which satisfy (1), (2),

0, ωih

0, if tih > 0, = 0, if tih = 0,

Determination of the just price and the just wage 101 or, which comes down to the same thing, (3) and δi − a1i δ1 − · · · − asi δs = bi1

N

h

1 + · · · + bir

r

= δ1 t1h + ω1h + · · · + δs tsh + ωsh + N ωh

δi > 0,

h

> 0, ωh

0, ωih

0, if tih > 0 = 0, if tih = 0

⎫ ⎪ (1) ⎪ ⎪ ⎪ ⎬ (11) ⎪ ⎪ ⎪ (12) ⎪ ⎭

(II)

To tackle the third problem, one must assume that the second problem is solved, that is to say that there exist values of δ, , ω, ω which satisfy (II); or, if one sets ωih = δi τih , that there exist values of δ, , ω, τ which satisfy ⎫ ⎪ ⎪ ⎪ ⎪ N h = δ1 (t1h + τ1h ) + · · · + δs (tsh + τsh ) + N ωh ⎬ δi − a1i δ1 − · · · − asi δs = bi1

δi > 0,

h

> 0, ωh

0, τih

1 + · · · + bir

r

0, if tih > 0 = 0, if tih = 0

(III)

⎪ ⎪ ⎪ ⎪ ⎭

The third problem then consists of finding α, β, γ , σ, S which satisfy (4), (6), (8) and, instead of (5), tih σih = (tih + τih ) (Sh + γih ) that is to say, of finding α, β, γ , S which satisfy ⎫ αi − ai1 α1 − · · · − ais αs = (ti1 + τi1 )S1 + · · · + (tir + τir )Sr ⎪ ⎪ ⎪ ⎪ ⎬ +(ti1 + τi1 )γi1 + · · · + (tir + τir )γir + βi

NSh = b1h α1 + · · · + bsh αs αi > 0, Sh > 0, βi

0, γih

0

⎪ ⎪ ⎪ ⎪ ⎭

(IV)

These four systems are of the same type, studied in the works cited above (No. 2).

6

Mathematical contribution to the study of the problems of production and wages

Editors’ note Potron was convinced that his mathematical model could be applied to the real world. In an attempt to solicit support for the collection of the required empirical data, he wrote a letter to the editors of the Journal de la Société de Statistique de Paris, who decided to publish part of it (May 1912, 53: 247–9; the title ‘Contribution mathématique à l’étude des problèmes de la production et des salaires’ appeared only on the journal’s Table of Contents). It was preceded by the following introductory remark: We have received from Father Potron a very interesting letter with regard to statistical researches to be undertaken. We extract from it the following excerpt; some of our readers may perhaps be able to provide the statistical data which allow Mr Potron to calculate the various coefficients of the formulas he has established; please send them to the secretary-general who will communicate them to the author. The Editors * * * The study of two economic problems, that of the ‘sufficient production’ and that of the ‘living wage’, has led Mr Potron to interesting mathematical considerations which have been the subject of two notes to the Academy of Sciences (Comptes rendus, t. CLIII, p. 1129, 4 December 1911 session, and p. 1458, 26 December 1911 session),1 in which the author has shown how both problems can be written down in equations, and which socioeconomic conclusions can be derived from the results obtained. As for the practical applications, the only statistical data that must be collected are the various coefficients in the formulas of the various cost prices and those of the various types of existence. The first express which quantity of such product or result of labour, – or which quantity of such species of labour, – is necessary

1 [See A9/Chapter 3 and A10/Chapter 4. – Eds.]

Mathematical study of production and wages 103 for the production of such quantity of any other product or result of labour; the second express which quantity of such product or result of labour (things for consumption or direct use) is necessary for the care, say during one year, of the life of an individual of such social rank. The statistical survey concerning the first coefficients will be the most difficult and the longest. It is however facilitated by the obvious fact that all the coefficients in the formula of the cost price of such product or result of labour are to be given by the producers of that product or result of labour themselves. It has seemed however simpler to proceed by two successive enquiries. The first would be, so to say, only qualitative; every industrial or commercial establishment would be asked to give: 1◦ The nomenclature of the products it supplies and the operations it is in charge of; in sum, its ‘catalogue’; 2◦ For each item of that catalogue, the nomenclature of all the items (matters, operations made by other establishments, materials), – and the list of the workforce – used in the special making of this item; 3◦ If necessary, for every service including several special fabrications, the nomenclature of the articles and the labourers used, in an undivided way, for the general working of this service (the service overheads); 4◦ If necessary, the nomenclature of the articles and the labourers used, in an undivided way, for the general working of the establishment (the establishment overheads). This first survey would already show neatly the interdependency of the various industrial and commercial establishments. It would determine what will be the number of formulas to establish, and, in each of these formulas, what will be the coefficients to determine. The second survey, which would be quantitative, would then be concerned by the numerical determination of the coefficients thus precised and distributed in the categories 2◦ , 3◦ and 4◦ . For a product, of which the unit is designated by A, another product, of which the unit is designated by B, can, in general, appear in the three categories. One will usually estimate without difficulty how many B are necessary for the special fabrication of A, let n be that number. But, usually, one will only know that a service comprising several special fabrications, among others that of A, has consumed, during a certain lapse of time, p times the quantity B, and that the establishment, for its general working, has consumed, during a certain lapse of time, q times the quantity B. To determine which fractions of the quantities pB and qB concern the production of A, that is the problem of the distribution of the overhead costs. Its solution contains always a part of arbitrariness, but it is always possible, at least in an approximative way. If k and k are the fractions looked for, the coefficient a, showing how many B are necessary for the production of A, is a = n + kp + k q. It would be the same for a labour coefficient.

104 Mathematical study of production and wages Here is, as a first numerical example, the approximative coefficients of the formula relative to a two-kilogram loaf of bread, as given by the survey taken from the head of a baker’s shop. The cost price of a two-kilogram loaf of bread, under the conditions in which it is fabricated by this first district shop, is the sum of certain fractions of the prices of the premises, the equipment, the materials and the labourers used to make it. Table 6.1 gives the fraction corresponding to each of these items, in other terms the coefficient by which the price of this item is multiplied in the formula of the cost price of a two-kilogram loaf.

Table 6.1 Coefficients of the cost price formula of a two-kilogram loaf of bread Coefficients 1◦ 1/100,000 2◦ 3◦ 4◦ 5◦ 6◦ 7◦ 8◦ 9◦ 10◦ 11◦ 12◦ 13◦ 14◦ 15◦ 16◦ 17◦ 18◦ 19◦ 20◦ 21◦ 22◦ 23◦ 24◦ 25◦ 26◦ 27◦ 28◦ 29◦ 30◦ 31◦ 32◦

1/2,000,000 1/5,000,000 7/50,000 1/5,000,000 3/250,000 1/1,000,000 1/500,000 1/2,000,000 1/500,000 1/100,000 1/30,000 1/50,000 1/2,000,000 1/100,000 1/1,000,000 1/10,000 1/250,000 3/1,000,000 1/5,000 1/50,000 1/25,000 1/5,000,000 3/6,250 1/6,250 1/2,000,000 1/200,000 1/30,000 1/2,000,000 1/1,000,000 1/800,000 1/50,000

Used Premises, Equipment, Raw Material and Labour force The yearly renting of a premise including a cellar for 8 m × 4 m ovens, a 4 m × 5 m warehouse, a 3 m × 5 m room, a 6 m × 5 m shop A weighing machine for the flour sacks (150 kg) A goods lift A metre of cable for a goods lift A bolting device A bolting sieve A device to preserve leaven The tinplating of the previous A kneading-trough A wicker basket without canvas A canvas for a wicker basket A set of six shovels to put dough in the oven A scuffle An oven for 60 loaves without tiles or vault The tiles of the previous The vault of the previous The chimney-sweeping of the previous A basket to remove bread from the oven The equipment of a gas lamp The yearly subscription to a working pair of dungarees per week The yearly subscription to a working apron per week The yearly subscription to a working towel per week A store for loaves with three 2.50 m × 3 m floors1 The washing of a square metre of paint in the shop The renewal of a square metre of paint in the shop A shop store for loaves with three 2.50 m × 3 m floors The equipment of an electric lamp A 25 candles electric bulb A shop counter Scales and accessories A retailing knife A straw chair Continued

Mathematical study of production and wages 105 Table 6.1 Cont’d Coefficients

Used Premises, Equipment, Raw Material and Labour force

33◦ 34◦ 35◦ 36◦ 37◦ 38◦ 39◦ 40◦ 41◦ 42◦ 43◦ 44◦ 45◦ 46◦ 47◦ 48◦ 49◦ 50◦ 51◦ 52◦ 53◦ 54◦ 55◦ 56◦ 57◦ 58◦ 59◦ 60◦ 61◦ 62◦ 63◦ 64◦ 65◦ 66◦ 67◦ 68◦ 69◦ 70◦ 71◦ 72◦ 73◦

A shop table A pendulum clock A ream of paper for sacks A small push coach for bread A broom A towel The washing of a towel The washing of an apron An inkpot A litre of ink A penholder A box of pens A pencil A penknife An envelope A sheet of writing paper The carrying of a letter A ream of printed invoices A diary of the Employers’ Office A great carrying book A book of accounts A credit report book A shop apron A 150 kilo sack of flour A kilogram of salt A litre of water A kilogram of leaven A kilogram of fine flour 100 kilograms of coal A cubic metre of lighting gas The hectowatt-hour The working day of the employer The working day of the specialized labourer The working day of the assistant The working day of a female worker The services of the fire insurance company The services of the glass-breaking insurance company The services of the accident insurance company The services of the State The services of the department The services of the district

1

1/200,000 1/1,000,000 3/2,000 1/100,000 1/100,000 1/7,500 1/360 1/1,260 1/500,000 1/100,000 1/500,000 1/100,000 1/60,000 1/500,000 1/1,000 1/1,000 1/1,000 1/25,000 1/100,000 1/200,000 1/300,000 1/300,000 1/75,000 1/100 1/40 3/5 1/200 1/40 1/360 1/120 1/240 1/360 1/360 1/360 1/180

[We have corrected ‘0.50’ into ‘2.50’. – Eds.]

Each of the coefficients 1 to 55 and 63 to 67 is obtained by dividing the number of units in use of the corresponding item by the total number of loaves made during the effective or presumed life of this item; the coefficients 56 to 62 result directly from experience. The last six coefficients are missing. Each of them is a

106 Mathematical study of production and wages hundred thousandth of the ratio between the sum (premium or contributions) paid yearly by the baker and the total sum (premium or contributions) received yearly by the corresponding administration. We have not yet been able to get this last information. A similar survey, but easier and, apparently, already made in a large part, would give the coefficients of the formulas of the types of existence. We would then have all the elements for a theoretical solution to the two problems and for an a priori hint of their possibility. That hint is obviously all the more interesting if one takes, to estimate the coefficients, the reasonable wants [and] the needs effectively felt in the various social categories. Maurice Potron

7

Relations between the question of unemployment and those of the just price and the just wage

Editors’ note The three-page typed manuscript entitled ‘Relations entre la question du chômage et celles du juste prix et du juste salaire’ is conserved in the Bibliothèque universitaire Lettres, sciences humaines et religieuses of the Université Catholique de l’Ouest in Angers (catalogue number TU908-3-2). It is listed as an off-print – which it is not – under the erroneous title ‘Appelations entre la question du chômage et celles du juste prix et du juste salaire’. The note is not signed or dated, but obviously it must be an early work of Potron. The presence of the manuscript can be explained by the fact that in 1912 Potron was appointed as professor in the Facultés Catholiques de l’Ouest, later renamed as Université Catholique de l’Ouest, in Angers (Maine-et-Loire). The precise references to the equations of the ‘Contribution Mathématique …’ (see A8/Chapter 2) suggest that the note may be seen as a supplement to that publication. What exactly Potron meant when he mentioned ‘the enclosed documents’ is not clear; we suspect that he had joined the note to (drafts of) his early mathematical explorations of his economic model. In fact, the same library holds an off-print of Potron’s ‘Application aux problèmes de la “production suffisante” et du “salaire vital” de quelques propriétés des substitutions linéaires à coefficients 0’ (see A10/Chapter 4) next to the manuscript (catalogue number TU908-3-1). The typescript contains several handwritten corrections, presumably in Potron’s hand, which we have integrated into our translation. * * * On the whole and in general, the average production can hardly exceed the average consumption. Therefore, if the means of production are such that, by utilizing all their capacity, the production per year, for instance, significantly exceeds the annual consumption, it is not possible to utilize constantly this whole capacity of production. Labour will have to be limited to some extent, or even, at certain moments, totally stopped. The serious inconvenience of these stops is that the worker’s cost of living is often just balanced by the wage corresponding to permanent and uninterrupted

108 Unemployment, just price and just wage labour on all work-days. Then any stop in labour, any unemployment, has necessarily disastrous consequences. They are evidently increased by the fact that, since overproduction becomes noticed only when it is exaggerated, the regime of production is made up of periods of full activity and periods of cessation, which alternate with sudden jolts. Is it then impossible to forecast to some extent what the consumption will be during a certain period of time, what the production will have to be during that period, and, next, which amount of work the labourers of various categories can expect? Is it furthermore impossible, when fixing prices and wages, to try and find the equilibrium between the average real wage and the average cost of living of each category? The enclosed documents show, I think, that this is not impossible, and depends only on the determination of three series of numerical coefficients. first series. Each of these coefficients concerns a product of labour A and a category of consumers C. The various coefficients concerning a given category C and the various products A determine the ‘type of existence’ of the category C with precision. second series. Each of these coefficients concerns a group of two products of labour A and B, taken in the order (A, B). It measures the quantity of product B that must be consumed to produce a given quantity of product A. third series. Each of these coefficients concerns a product of labour A and a category of workers-consumers C. It measures the quantity of labour that the production of a given quantity of product A requires from the workers who are consumers of category C. The knowledge of all the coefficients of the second and third series concerning the same product A is necessary for determining the cost price of that product A. These coefficients remain constant as long as the conditions of production of A do not change. The coefficients of the last two series are therefore data resulting from the conditions under which the various commercial and industrial establishments operate; and each of these establishments has to deliver only the coefficients it is concerned with. By contrast, the coefficients of the first series are, to some extent, arbitrary, since they determine the type of existence considered as convenient for every worker, according to his type of labour. Monographs on working families will be able to provide useful pieces of information for their determination. As soon as all these coefficients have been determined, formulas (1) and (3) of the note entitled ‘mathematical contribution …’1 will allow us to determine:

1 [See section 3 of A8/Chapter 2. – Eds.]

Unemployment, just price and just wage 109 • •

a distribution of workers and consumers which ensures that, for each item, production is at least equal to consumption; the average working time demanded from a worker in each category during a given period, for instance one year.

Obviously this average time, measured in hours for instance,2 must be at most equal to the product of the number of work-days in a year by the maximum number of hours in a standard working day. Assuming this condition is met, it is then always possible, as I have shown, to determine by means of formulas (2) and (5) a system of prices and wage rates such that for each item the selling price is greater than the cost price, and that, for each category, the average annual real wage (corresponding to the effective labour) is at least equal to the cost of living. One therefore understands the usefulness that even an approximate determination of the coefficients of the second and third series would have. By starting from the present distribution of workers among the various professions, [and] by estimating, in accordance with experience, the types of existence convenient for these various categories, we could then determine quite easily, by means of the given formulas, how much the industry should produce, the quantity of labour that would be demanded from each category, and a system of lucrative prices and effectively sufficient wages. It is important to notice that, as far as prices and wages are concerned, these results will be very easy to check. Each commercial or industrial establishment, by means of its own particular formula (2), of which it knows all the coefficients with precision, and each labourer, by means of his own particular formula (5), which is nothing but the translation of his annual budget, will be able to evaluate, as far as they are concerned, the exactness of these results.

2 [In previous papers Potron expressed labour time in days rather than hours. – Eds.]

8

Some properties of linear substitutions with coefficients 0 and their application to the problems of production and wages

Editors’ note When Gustave Desbuquois first mentioned Potron’s economic studies in 1911 he alluded to a more formal mathematical exposition soon to be published by the author (see the Editors’ note to Chapter 1). In several of his writings of 1911 and 1912 Potron frequently referred to this forthcoming publication, which finally appeared in 1913: ‘Quelques propriétés des substitutions linéaires à coefficients 0 et leur application aux problèmes de la production et des salaires’ (Annales Scientifiques de l’École Normale Supérieure, 1913, 3rd series, 30: 53–76). The title of this article coincides with that of Potron’s first note published by the Académie des Sciences in 1911 (see A9/Chapter 3). The article contains both a rigorous statement and proofs of Potron’s mathematical propositions (Part I), and an explanation of their economic application (Part II). The journal Annales Scientifiques de l’École Normale Supérieure was founded in 1864 by Louis Pasteur, scientific director of the École Normale Supérieure in Paris. Initially the journal had a broader scope, but since 1900 it published only on mathematics. The 1913 article was Potron’s first publication in this prestigious journal; a second publication, an article on a purely mathematical topic (see A32), appeared in 1934. When Potron first announced his mathematical results in 1911, he presented them as an extension of results obtained by Perron (in 1907) and Frobenius (in 1908 and 1909) on positive matrices. At the beginning of the article Potron pointed out that since then, Frobenius had published an article on nonnegative matrices which covered similar ground. The chronology makes it clear that Potron arrived at his results independently of Frobenius. * * * 1. This work first develops and completes, by extending them to linear substitutions with coefficients 0, various results due to Messrs Perron and Frobenius, that they proved for linear substitutions with coefficients > 0.1 The results thus obtained 1 This work has given rise to two reports to the Academy of Sciences (Comptes rendus, t. 153, p. 1129, session of 4 December 1911; p. 1458, session of 26 December 1911). Mr Frobenius

Linear substitutions with coefficients

0 111

provide the complete solution to the following problem: given a linear substitution with coefficients 0, find a linear function, with coefficients > 0, that this substitution multiplies by a constant factor. As we shall see below, it is precisely to this mathematical problem that the economic problem of Wages can be reduced, problem consisting of finding a regime of prices for the various objects and wages for the various categories of labourers, such that, for every object, the selling price is at least equal to its cost price, and for each labourer assumed to have a standard of living suitable for his category, the wage is at least equal to the cost of living, which itself depends precisely on the prices of various objects.

I. Some properties of linear substitutions with coefficients 0 2. Here are the known results for substitutions with coefficients > 0. If the elements aik (i, k = 1, . . . , n) of a matrix A = |aik | are all > 0, the characteristic root r with maximum modulus of A is real, positive and simple (Perron, M.A., t. LXIV, p. 261; Frobenius, S.A.B., 1908, pp. 471–476; 1909, pp. 514–518).2 Let uik (i, k = 1, . . . , n; uik = 0, i = k; uii = 1) be the elements of the unit matrix of order n, and s a parameter; if s is r, the elements of the adjunct of |suik − aik | are all > 0; consequently, the root of maximum modulus, which is real, positive and simple, of a principal element of the adjunct is < r. It follows that the equations r αi −

k

aki αk = 0 (i, k = 1, . . . , n)

(1)

hold for values all > 0 of α1 , . . . , αn (Frobenius, ibid.). If the equations s αi −

k

aki αk = 0 (i, k = 1, . . . , n)

hold for values all > 0 of α1 , . . . , αn , one has s = r (Frobenius, ibid.). If one considers the substitution (a) = xi , k aik xk , the last two theorems have the following meaning: there exists a function i αi xi with coefficients > 0, and only one up to a constant factor, that (a) multiplies by a constant factor, which can only be r. has published, after these reports, a complete study of matrices with elements 0 (S.A.B., 1912, pp. 456–477). [See A9/Chapter 3 and A10/Chapter 4 for Potron’s Academy notes. The exact reference of the article by Georg Frobenius is: ‘Über Matrizen aus nicht negativen Elementen’ (Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften, 1912, Erster Halbband, Januar bis Juni: 456–77). – Eds.] 2 [For the exact references of the publications by Perron and Frobenius, see note 1 of chapter 3. – Eds.]

112 Linear substitutions with coefficients

0

If the elements of A are only 0, we have the following results (cf. Frobenius, loc. cit.). The characteristic root r of maximum modulus is real and 0. The elements of the adjunct of |ruik − aik | are 0. The root r may be multiple; if p is its order of multiplicity, the principal minors of |ruik − αik | of order > n − p are all zero; a principal minor is > 0. The root r may be zero; then all others are zero simultaneously. 3. One can easily prove these results by means of the following remark: let f (x, t) be a polynomial in x, of which the coefficients are holomorphic functions of a variable t that is finite and 0, and not all zero for t = 0; of which the maximum root α is, for any value > 0 of t, real, positive and simple; which is always > 0 for t > 0 and x > α . For t 0, the root α is a continuous function of t. Let a be its limit for t = 0. One knows that a is a root of f (x, 0). I claim that f (x, 0) is > 0 for x > a, and that the moduli of its roots are a. Let indeed b > a. For t infinitely small > 0, α is close to a, so that b is > α and f (b, t) > 0, therefore f (b, 0) 0. But if b were the modulus of a root b of f (x, 0), b would be the limit of a root β of f (x, t), and the modulus β of that root, having the limit b, would be > α for t infinitely small and positive. 4. Denoting then by t a variable 0, I set aik + t = αik , and I consider the substitution (α ), of which the coefficients αik are all > 0 simultaneously with t, and of which the root ρ of maximum modulus will therefore be real, positive and simple for t > 0. According to the remark in No. 3, the limit r towards which ρ tends when t vanishes is a characteristic root of maximum modulus of A; it is real and 0. It is clear enough that, the elements of the adjunct of |suik − αik | being all > 0 for t > 0 and s ρ , the elements of the adjunct of |suik − aik | will all be 0 for s r. Since a principal element of the adjunct of |suik − αik | has, for t > 0, a real root of maximum modulus, λ > ρ , the corresponding principal element of the adjunct of |suik − aik | admits, as root of maximum modulus, the limit l of λ, which is real and r; this principal element will therefore be > 0 for s > r, and 0 for s = r. For t > 0, there exists a function i μi xi with coefficients > 0 and that (α ) multiplies by ρ ; even, as one sees directly, one can always compel the coefficients μ to satisfy the relation i μi = b, b being an arbitrary scalar > 0 and, consequently, also to satisfy 0 < μi < b (i = 1, . . . , n). Let mi be the limit of μi for t = 0; one has obviously i mi = b, 0 mi b (i = 1, . . . , n); and the substitution (a) multiplies by r the function i mi xi , the coefficients of which are 0 and not all zero. The coefficients m are solutions of rmi −

k

aki mk = 0 (i, k = 1, . . . , n)

Let us look in which case some of them can be zero. Assume for instance that we have m1 > 0, . . . , mp > 0,

mp+1 = · · · = mn = 0

Linear substitutions with coefficients

0 113

The last n − p equations give j

ajh mj = 0, mj > 0, ajh

0 (j = 1, . . . , p; h = p + 1, . . . , n)

This requires ajh = 0 (j = 1, . . . , p; h = p + 1, . . . , n) and the substitution (a) takes the form | xj ,

l

ajl xl ; xh ,

k

ahk xk |

( j , l = 1,..., p; h = p + 1,..., n; k = 1,..., n)

Therefore, if (a) multiplies by a constant factor a linear function u with coefficients 0 not containing all the variables x, (a) transforms each of the variables of u into a linear function of these variables only. I shall express this fact by saying that the matrix |aik | is partially reduced. 5. Conversely, assume that (a) multiplies by a constant factor s a function i mi xi (i = 1, . . . , n) with coefficients > 0. According to No. 4, a transpose of (a) multiplies by r a function i mi yi , the coefficients of which are 0. We therefore have the two systems smi −

k

aki mk = 0,

rmi −

k

aik mk = 0

(i, k = 1, . . . , n)

from which one draws r

i

mi mi =

i

mi

k

aik mk =

k

mk

i

aik mi = s

k

mk mk

hence (s − r)

k

mk mk = 0

Since, according to the hypotheses, k mk mk is > 0, we have s = r. One sees that, if (a) multiplies by a constant factor s a linear function u, with coefficients > 0, of some of the variables only, this factor s is the root of maximum modulus of the substitution (a1 ) performed by (a) on the variables of u. 6. Let us look for all distinct functions, with coefficients 0, that (a) multiplies by r. One knows that, if n − q is the rank of |ruik − aik |, there exist q and only q distinct functions, with any coefficients, that (a) multiplies by r. In fact, the question is to find the distinct solutions of the system r αi −

k

aki αk = 0

(i, k = 1, . . . , n)

(1)

of which all minors of |ruik − aik | of order > n − q are zero, [and] some minor of order n − q is = 0. Let us choose the indices in such a way that the elements of

114 Linear substitutions with coefficients

0

this last minor are the coefficients αq+1 , . . . , αn in n − q distinct equations; then, we can assign to α1 , . . . , αq , q distinct sets of values βj1 , . . . , βjq ( j = 1, . . . , q), and take for instance βjl = 0 ( j , l = 1, . . . , q) for j = l, and βjj = 1. To each set βj1 , . . . , βjq the equations (1) let correspond, for αq+1 , . . . , αn , a unique set βj,q+1 , . . . , βjn . We thus have q distinct sets of solutions, and we know that the most general solution of (1) is given by αi =

j

( j = 1, . . . , q; i = 1, . . . , n)

λj βji

(2)

λ1 , . . . , λq being arbitrary coefficients. Hence, to have the most general expression of the functions, with coefficients 0, that (a) multiplies by r, it suffices to look for the most general solution of the inequalities αi =

j

0

λj βji

( j = 1, . . . , q; i = 1, . . . , n)

(3)

where the λ are taken as unknowns. A solution necessarily exists (No. 4). Therefore (cf. Minkowski, Geometrie der Zahlen, pp. 39–45),3 there exists at least one solution, called extreme solution, given by λj = μj1 ( j = 1, . . . , q), for which q − 1 distinct forms α vanish and the others are 0 and not all zero. Then, if there exist q q distinct extreme solutions given by λj = μjl ( j = 1, . . . , q; l = 1, . . . , q ), we know that the most general solution of (3) is λj = l θl μjl ( j = 1, . . . , q; l = 1, . . . , q ), the θ being arbitrary coefficients 0. Thus, the most general expression of the coefficients of the functions i αi xi , with coefficients 0, that (a) multiplies by r, in other words the most general solution, with numbers 0, of the equations (1) is αi =

j

βji

l

θl μjl =

l

θl

j

μjl βji

i = 1, . . . , n; j = 1, . . . , q; l = 1, . . . , q

(4)

If one sets j

μjl βji = γil

(i = 1, . . . , n; j = 1, . . . , q; l = 1, . . . , q )

(5)

we shall have, as the expression of the most general solution, αi =

l

θl γil

(i = 1, . . . , n; l = 1, . . . , q )

(6)

the θ being arbitrary coefficients 0, and each system γ1l , . . . , γnl of scalars 0, not all zero, and of which at least q − 1 are zero, being what can be called an extreme solution, in scalars 0, of the equations (1). 3 [See note 2 of Chapter 3 for the exact reference. – Eds.]

Linear substitutions with coefficients

0 115

Since the equations (1), when the rank of |ruik − aik | is n − q, admit certainly an extreme solution in which p of the unknowns n − 1 p q − 1 are zero, it follows that (a) multiplies by r a function, with coefficients > 0, of n − p variables only and, as a result, transforms each of these n − p variables into a linear function of these variables only. 7. Consider now, mainly with a view on the application we have in mind, the substitution (a, b) = xi ,

k

aik xk ; yl ,

i

bli xi + syl

(i, k = 1, . . . , n; l = 1, . . . , p)

the b being, like the a, all 0 and, moreover, for any i, b1i , . . . , bpi not all zero. The matrix of (a, b) is sp |aik |.4 Here are its main properties. Assume s at least equal to the characteristic root r of maximum modulus of |aik |; s is then the characteristic root of maximum modulus of (a, b); hence (No. 4) there exists a function i

αi xi +

with coefficients the equations5 s αi −

k

l

βl yl

0 and not all zero, that (a, b) multiplies by s; in other words,

aki αk =

l

bli βl

(i, k = 1, . . . , n; l = 1, . . . , p)

(7)

0 and not all zero. Let us denote by Dik the admit a solution with numbers elements of the adjunct of D = |suik − aik |; if s is greater than the characteristic root of maximum modulus of |aik |, we have (No. 4) D > 0, Dik 0 for i = k, Dii > 0; then one sees directly that, by giving arbitrary values > 0 to β1 , . . . , βp one obtains values > 0 for α1 , . . . , αn . Conversely, if (a, b) multiplies by s a function i αi xi + l βl yl , in which we have αi > 0 (i = 1, . . . , n), βl 0 (l = 1, . . . , p), s is r, and certainly > r, if βl > 0 (l = 1, . . . , p). Let us indeed set Bi =

l

bli βl

(i = 1, . . . , n; l = 1, . . . , p)

and combine the equations (7) with the equations already considered in No. 5, rmi − k aik mk = 0 (i, k = 1, . . . , n),6 which hold for values 0 and not all zero 4 [In modern notation, the matrix (a, b) considered by Potron has dimension (n + p) × (n + p), with the square matrices (a) and (su) as diagonal blocks, and the rectangular matrices (0) and (b) as off-diagonal blocks. Its determinant is sp times that of (a). – Eds.] 5 [In the following expression Potron wrote δl instead of βl . – Eds.] 6 [Potron omitted ‘= 0’. – Eds.]

116 Linear substitutions with coefficients

0

of the m . We obtain r

i

αi mi =

i

k

aik mk =

k

mk Bk

αi

k

mk

i

aik αi =

k

mk (sαk − Bk )

hence (s − r)

mk αk =

k

According to the hypotheses, k mk αk is > 0, we therefore have s − r 0 and, if the β and consequently the B are all > 0, we have s > r. One sees directly, as in No. 4, that if, in a solution with numbers 0 and not all zero of the equations (7), some of the α are zero, the matrix aik is partially reduced. 8. Let us also look, with a view on the application we have in mind, for the solutions with scalars > 0 of the system s αi −

k

aki αk −

t βl −

l

i cli αi

bli βl = 0

=0

(i = 1, . . . , n; l = 1, . . . , p)

(7) (8)

where cl1 , . . . , cln are scalars 0 and, for any l, not all zero. The system under consideration admits solutions > 0 always and only if t is > 0 and if the equations7 s αi −

aki +

k

1 t

l

bli clk αk = 0

(i = 1, . . . , n)

(9)

[admit solutions αi 0 and not all zero,] that is, only if t is > 0 and s characteristic root of maximum modulus of the matrix aik + 1t l bli clk and, if these conditions are met, always, except perhaps if the considered matrix is partially reduced, in which case one can only assert that the system admits solutions 0 and not all zero. This condition can be given another more convenient form for the application we have in mind. Assume indeed that D = |suik − aik | is non zero; denoting by Dik the elements of the adjunct of D, we can replace system (7) by the equivalent system Dαi =

k

Dik

l

blk βl

(10)

from which we draw, by combination with (8), Dt βj =

i

cji

k

Dik

l

blk βl =

l

βl

k

blk

i

cji Dik

7 [We have added αk at the end of the left-hand side of the following expression and inserted the range of index i. – Eds.]

Linear substitutions with coefficients

0 117

Let us set i k

cji Dik = Ddjk

(11) (i, k = 1, . . . , n; l = 1, . . . , p)

blk djk = Blj ,

the B are t βj −

(12)

0, and we obtain l

Blj βl = 0

( j , l = 1, . . . , p)

(13)

The system [(10), (13)] is therefore a consequence of [(8), (9)]. An inverse calculation shows that [(8), (9)] is a consequence of [(10), (13)]. The two systems are therefore equivalent, provided that t D is > 0 and, consequently, their determinants, which are respectively t p suik − aik − 1t l bli clk and Dn tujl − Bjl , vanish for the same pairs (s, t) that do not make t D vanish. As a result, the two equations suik − aik −

1 t

l

blk cli = 0

(14)

and tujl − Bjl = 0

(15)

represent the same curve. Let σ (t) be the root of maximum modulus of suik − aik − 1t l bli clk considered as a polynomial in s, of degree n, and τ (s) the root of maximum modulus of tujl − Bjl considered as a polynomial in t, of degree p. The function σ (t) is finite and continuous for any value > 0 of t. When t is infinitely small, at least one root of equation (14) being infinitely great, σ (t) becomes infinitely great. When t is infinitely great, the n roots of the equation (14) are respectively infinitely close to the n roots of the equation |suik − aik | = 0, therefore σ (t) tends to r.8 Similarly, the B being fractions with polynomials in s of degree n − 1 as numerators, and D as denominator, the function τ (s) is finite and continuous for any value of s > r. When s is infinitely close to r, at least one root of equation (15) being infinitely great, τ (s) is infinitely great.9 When s is infinitely great, the B are infinitely small, therefore the p roots of equation (15) are infinitely small, therefore τ (s) is infinitely small.10 I claim that the two functions σ and τ are each other’s inverse. Indeed, for s1 > r, let t1 = τ (s1 ); I claim that σ (t1 ) = s1 . First, the pair (s1 , t1 ), which 8 [In this sentence Potron wrote ‘small’ instead of ‘great’, and ‘σ (0) = r’ instead of ‘σ (t) tends to r’. – Eds.] 9 [Potron wrote ‘τ (r)’ instead of ‘τ (s)’. – Eds.] 10 [Potron wrote ‘great’ instead of ‘small’. – Eds.]

118 Linear substitutions with coefficients

0

satisfies (15), also satisfies (14); therefore, by definition, σ (t1 ) s1 . If σ (t1 ) were > s1 , since, by increasing indefinitely and continuously t from t1 onwards, σ (t) varies continuously from σ (t1 ) > s1 to r < s1 , there would exist a value t2 > t1 for which one would have σ (t2 ) = s1 ; the pair (s1 , t2 ) which satisfies (14) would also satisfy (15), and t would not be the root of maximum modulus of (15) corresponding to s1 . Similarly, for t1 > 0, let s1 = σ (t1 ); I claim that τ (s1 ) = t1 . First, the pair (s1 , t1 ), which satisfies (14), also satisfies (15), therefore, by definition, τ (s1 ) t1 . If τ (s1 ) were > t1 , since, by increasing indefinitely and continously s from s1 upwards, t varies continuously from τ (s1 ) > t1 to 0 < t1 , there would exist a value s2 > s1 for which one would have τ (s2 ) = t1 ; the pair (s2 , t1 ) which satisfies (15) would also satisfy (14), and s would not be the root of maximum modulus of (14) corresponding to t1 . A very similar reasoning shows that if, for t1 > 0, we have s1 > σ (t1 ), we shall have t1 > τ (s1 ), and if, for s1 > r, we have t1 > τ (s1 ), we shall have s1 > σ (t1 ). Thus, the conditions t > 0, s at least equal – or equal – to the characteristic root of maximum modulus of aik + 1t l blk cli , are equivalent to the conditions s > r, t at least equal – or equal – to the characteristic root of maximum modulus of Bjl . If we consider the curve represented by one or the other of the equations (14) and (15), one or the other of the equations s = σ (t) and t = τ (s) represents a branch of this curve, which is asymptotic to the straight lines s = r and t = 0, and that any parallel to one of the axes, of abscissa > r or of ordinate > 0, meets at a point at finite distance and only one. This curve branch divides the plane into two regions. Considering it allows us to state the following theorem: If, for values all > 0 of the α and the β , the left-hand sides of (7) and (8) are 0, the point (s, t) cannot be located, regarding its position relative to the curve branch considered, in the region of the axes. If it is in the opposite region, there exist values all > 0 of the α and the β making the left-hand sides of (7) and (8) 0 and not all zero. If it is on the curve branch, there exist values all > 0 (exceptionally 0 and not all zero, if the matrices aik + 1t l blk cli and Bjl are partially reduced) of the α and the β making the left-hand sides of (7) and (8) 0. 9. To recognize the position of a quantity s with respect to the root of maximum modulus r of E (s) = |suik − eik |, eik 0 (i, k = 1, . . . , n), one can use the following remarks: For s > r , E (s) is > 0 as well as its minors of any orders; the converse holds because, the pth derivative of E (s) 11 being the sum multiplied by p! of the principal minors, if, for a value s, the polynomial E (s) is > 0 as well as its principal minors, all the derivatives of E (s) are > 0, and we know that s is then greater than the

11 [Potron wrote ‘D (s)’ instead of ‘E (s)’. – Eds.]

Linear substitutions with coefficients

0 119

E (s).12

maximum root of For s = r, if ν is the order of multiplicity of r, E (s) is zero as well as its principal minors of order > n − ν ; the principal minors of order n − ν are 0; some principal minor of order n − ν and the principal minors of order < n − ν which derive from it are > 0; the converse is true, because the polynomial E (s) is then zero as well as its first p − 1 derivatives, its following n − p + 1 derivatives are > 0, and we know that s is then the maximum root of E (s) and that its order of multiplicity is ν .

II. Application of the previous results to the problems of Production and Wages 10. The aim is not at all, as we shall see, to study the action of the very diverse causes that may have an influence on either production or the determination of prices and wages. When I specify, from a mathematical point of view, the conditions in which the problems of Production and Wages are set, my only goal is to find out if these problems may admit satisfactory solutions from an economic point of view. For this, we must first represent mathematically the given relations that exist between the various results of labour. A result of labour will be a transformation or production as such, a transference or transportation, or, more generally, a service (of surveillance, of protection, like those delivered by public administrations, insurance companies, etc.). To obtain this result of labour, one needs, on the one hand, that labour of this or that nature (working force) and, on the other, the use or consumption of these and those results of labour (raw materials, machines, etc.). These are conditions that can be called technical. To the various categories of labour correspond, in general, various social categories of labourers (heads of industry, heads of service, foremen, office and manual workers, unskilled labourers); these are conditions that can be called administrative. To the various social categories correspond, in general, various suitable types of existence, each comprising the use or consumption of these or those results of labour (items of food, housing, clothing, etc.); these are conditions that can be called economic. 11. One can conceive that, these conditions being given, the levels of production, the numbers of labourers assigned to these types of work or belonging to those

12 In fact, if one denotes by Eq (s) the qth derivative, we have E (s) =

q

(s − s0 )q Eq (s0 ) q!

(q = 0, 1, . . . , n)

If Eq (s0 ) > 0 (q = 0, 1, . . . , n), E (s) is > 0 for any value of s s0 . If we have Eq (s0 ) = 0 for q = 0, 1, . . . , p − 1, and Eq (s0 ) > 0 for q = p, p + 1, . . . , n, the formula E (s) =

q

(s − s0 )q Eq (s0 ) q!

(q = p, p + 1, . . . , n)

shows that E (s) is zero for s = s0 but > 0 for s > s0 .

120 Linear substitutions with coefficients

0

social categories, the prices, [and] the wages may vary owing to many causes. One is thus led to consider these various quantities as the unknowns of the question, the data of which are the technical, administrative and economic conditions.13 Assuming given a certain industrial, economic and social state, I shall then have to examine three problems successively: 1◦ Is it possible to determine a distribution of labourers among the various professional and social categories and a distribution of the simple consumers among the various social categories, such that the production can, for each result of labour, be equal to the consumption, without any labourer having to work on other than work-days? I shall call any regime of production and labour which meets both conditions satisfactory; 2◦ Is it possible to determine the set of prices and wage rates in such a way that, for each result of labour, the exchange price is at least equal to the cost price and that, for each labourer, the wage corresponding to the maximum amount of work performed is at least equal to the cost of living? I shall call any regime of prices and wages which meets both conditions simply satisfactory. 3◦ Given a satisfactory regime of production and labour, is it possible to determine the set of prices and wage rates in such a way that the first condition of 2◦ is met and that, for any labourer, the wage corresponding to the amount of work which is effectively required from him, given the production to obtain, is at least equal to the cost of living? I shall call any regime of prices and wages which meets both conditions effectively satisfactory. 12. I shall denote by Ai (i = 1, . . . , s) a result of labour of a certain kind; by Pi the firm producing Ai ; by Ai the quantitative unit of Ai (if Ai designates the services provided by an administration, Ai is the whole of these services provided during a year). The production of Ai requires from Ak (k = 1, . . . , s) a consumption that I shall denote by aik Ak , and demands from the workers of the social and professional category Ch (h = 1, . . . , r) a number of normal working days that I shall denote by tih . The positive or zero scalars a and t represent the industrial conditions of fabrication and organization of the various establishments. In order to live in a convenient way during a year, a consumer of category Ch must consume an amount of Ai that I shall denote bih Ai . The positive or zero scalars b represent the workers’ requirements. At last, N denotes the number of work-days during a year. 13. Given these data, between the annual productions i Ai (i = 1, . . . , s), the annual excesses ρi Ai of the production over the consumption of Ai , the numbers 13 As my only goal here is to expound the way to set the problems in equations, I leave aside all the difficulties that can arise from the determination of the various data taken as starting point. On this topic, see my paper: Possibilité et Détermination du Juste Prix et du Juste Salaire (Mouvement Social, t. LXXIII, avril 1912, pp. 289–316). [See A11/Chapter 5. – Eds.]

Linear substitutions with coefficients

0 121

(h = 1, . . . , r), 14

the numbers πih of the labourers h of consumers of category Ch of Pi belonging to Ch , and the numbers ωh of simple consumers of category Ch , we have the relationships h

= ωh +

i

=

i

aki

k

(i = 1, . . . , s; h = 1, . . . , r)

πih

k

+

h

bih

h + ρi

(16)

(i, k = 1, . . . , s; h = 1, . . . , r)

If δi denotes what i becomes when, all productions being assumed equal to consumptions, we set ρ1 = · · · = ρs = 0, we have δi =

aki δk +

k

h

bih

h

(i, k = 1, . . . , s; h = 1, . . . , r)

(17)

Finally, I introduce a symbol ωih , always zero simultaneously with tih and, if not, defined by N πih = δi tih + ωih

(18)

Between the prices αi of Ai , the benefits βi made by Pi on Ai , [and] the average daily wages σih of the workers of Pi belonging to Ch , we have the relationships αi =

k

aik αk +

h

tih σih + βi

(i = 1, . . . , s; h = 1, . . . , r)

(19)

If N γih denotes the total annual average economies that any of the πih workers of Pi belonging to Ch can make on his effective wage when the annual production is δi Ai , and if NSh denotes the annual cost of living of a consumer of category Ch , we have δi tih σih = N πih (Sh + γih )

NSh =

k

bhk αk

(i, k = 1, . . . , s; h = 1, . . . , r)

(20) (21)

14. This being set, a satisfactory regime of production and labour is possible, as can be seen, always and only if there exist values of δ, π, ω, , ω satisfying (16), (17), (18) with δi > 0,

h

> 0, πih

0, ωh

0, ωih

0

(i = 1, . . . , s; h = 1, . . . , r)

(22)

I shall call (I) the system [(16)–(17)–(18)–(22)].

14 [Here and below, we have corrected minor misprints relative to the ranges of the indices. – Eds.]

122 Linear substitutions with coefficients

0

A simply satisfactory regime of prices and wages is possible always and only if there exist values of α , β , S, σ , γ satisfying (19) and (21) with σih = Sh + γih

(23)

and αi > 0, Sh > 0, σih > 0, βi

0, γih

(i = 1, . . . , s; h = 1, . . . , r)

0

(24)

I shall call (II) the system [(19)–(21)–(23)–(24)]. A satisfactory regime of production and labour being assumed to exist, an effectively satisfactory regime of prices and wages is possible always and only if there exist values of α , S, σ , γ satisfying (19), (20), (21) and (24), where the δ and the π appearing in (20) are solutions of (I). I shall call (III) the system [(19)–(20)–(21)–(24)]. 15. If one takes (18) into account, (16) becomes h

1 N

= ωh +

k

(δk tkh + ωkh )

(25)

Therefore, if a satisfactory regime of production and labour is possible, the relationships δi − h

δi > 0,

h

k

aki δk =

= ωh + N1

> 0, ωh

k

0, ωkh

h

bih

h

(δk tkh + ωkh ) 0,

(i, k = 1, . . . , s; h = 1, . . . , r)

⎫ (17) ⎪ ⎪ ⎪ ⎪ (25) ⎬ ⎪ ⎪ ⎪ (26) ⎪ ⎭

(IV)

are compatible. Conversely, if system (IV) admits solutions, and since the formulas (18) will give values 0 to the π , system (I) admits solutions. Thus, a satisfactory regime of production and labour is possible always and only if system (IV) admits solutions. Similarly, if a simply satisfactory regime of prices and wages is possible, the relationships αi −

k

aik αk = NSh =

αi > 0, Sh > 0, βi

0, γih

h tih (Sh + γih ) + βi k

bkh αk 0,

(i, k = 1, . . . , s; h = 1, . . . , r)

⎫ (27) ⎪ ⎪ ⎪ ⎪ (21) ⎬ ⎪ ⎪ ⎪ (28) ⎪ ⎭

(V)

are compatible. Conversely if system (V) admits solutions, and since the formulas (23) will give values > 0 to the σ , system (II) admits solutions. Thus, a

Linear substitutions with coefficients

0 123

simply satisfactory regime of prices and wages is possible always and only if system (V) admits solutions. Systems (IV) and (V) are precisely of the type studied in Nos. 7 and 8.15 If, as indicated there, we eliminate the between (17) and (25) and the S between (21) and (27), we find the two systems δi −

k

αi −

k

δk aki +

1 N

αk aik +

h

1 N

bih tkh =

h

h

bkh tih =

bih ωh +

h

1 N

k

ωkh

tih γih + βi

(29) (30)

in which the matrices of the left-hand sides are transposes of one another. The application of the results of Nos. 7 and 8 allows us to state the following theorems immediately: A satisfactory regime of production and labour – or a simply satisfactory regime of prices and wages – is possible only if σ (N ), characteristic root of maximum modulus of the matrix aik +

1 N

h

bkh tih

1.16 If σ (N ) < 1, there always exists a satisfactory regime of production and labour in which, production remaining equal to consumption, one can assign arbitrary values to the numbers of simple consumers of every social category and to the collective unemployment of each category of labourers.17 Simultaneously, there always exists a simply satisfactory regime of prices and wages, in which one can assign arbitrary values to the firms’ benefits and the workers’ economies. Assume σ (N ) = 1. If the matrix under consideration is not partially reduced, there still exists a satisfactory regime of production and labour; but, in general, there cannot be either simple consumers in any social category, or unemployment in any category of labourers. Simultaneously, there also exists a simply satisfactory regime of prices and wages; but, in general, there can be neither firms’ benefits nor workers’ economies. If the matrix considered is partially reduced, one can only assert, in general, that (IV) has solutions satisfying, instead of (26), is

δi

0,

i

δi = 1,

h

0,

h

h

> 0, ωh = ωhk = 0

15 [Potron wrote (II) instead of (IV). – Eds.] 16 [If the matrix is partially reduced, the condition σ (N ) 1 is not necessary. – Eds.] 17 [This cannot hold if one assumes that the total population is given. – Eds.]

(31)

124 Linear substitutions with coefficients

0

and that (V) has solutions satisfying, instead of (28), 0,

αi

i

αi = 1, Sh

0,

h

Sh > 0, βi = γih = 0

(32)

I shall say that the corresponding regimes are semi-satisfactory.18

18 The study of these regimes may have some interest, either from a theoretical point of view, or as a limit case, that is, because there may exist satisfactory regimes which hardly differ from a semi-satisfactory regime. Assume therefore that the matrix aik + N1 h bkh tih is partially reduced, that is, we have, by changing the notations where necessary ajl = blh tjh = 0, for j = 1, . . . , p; l = p + 1, . . . , s; h = 1, , r Under the hypothesis σ (N ) = 1 (or N = ν ), one can, in general, only assert that system (IV) admits solutions satisfying, instead of (26), the relationships

δj

0,

j

δj > 0, δl = 0, ωh = ωkh = 0

( j = 1, . . . , p; l = p + 1, . . . , s; k = 1, . . . , s; h = 1, . . . , r);

and that system (V) admits solutions satisfying, instead of (28), the relationships

α j = 0 , αl

0,

l

αl > 0, βi = γih = 0

( j = 1, . . . , p; l = p + 1, . . . , s; i = 1, . . . , s; h = 1, . . . , r);

Aj denotes therefore a result of labour of which the production may be non zero whereas its price is zero, and Al a result of labour of which the production is zero. From the hypothesis that the matrix is partially reduced, it follows that an Al is never used to obtain an Ai ; that a Ch involving the consumption of an Al is not convenient for any worker of a Pj ; that a Ch convenient for a worker in a Pj does not involve the consumption of any Al . Thus neither the firms Pj nor their workers use any Al . The practical implementation of a semi-satisfactory regime of production and labour simply requires deleting some types of existence and the results of labour that these types of existence consume. Thanks to the hypothesis made on the partial reduction of the matrix, it is clear that this deletion does not disrupt at all the rest of production. In a semi-satisfactory regime of prices and wages, there exists a set of types of existence for each of which the cost of living is zero. The workers of these categories do not earn any wage. All the results of labour involved in the direct or indirect consumption of these types of existence have a zero exchange price; moreover, their production only requires the consumption of results of labour with a zero price, and only demands the help of workers for which the cost of living is zero and who do not earn any wage.

Linear substitutions with coefficients 16. To transform these results as in No. 8, I shall k bkh Dki (S) = D (S)dih (S), i tig dih (S) = Tgh (S),

0, if i = k

uik =

1, if i = k

0 125

set19

dih (1) = dih ,

Tgh (1) = Tgh ,

, |Suik − aik | = D(S),D(1) = D,

i, k = 1,..., s g , h = 1,..., r

(33)

∂D(S) = Dik (S), ∂ (Suik − aik )

As in No. 8, I shall denote by ν (S) the characteristic root of maximum modulus of the matrix Tgh (S) , with ν (1) = ν , and by R that of the matrix |aik |; looking at S and N as the coordinates of a point, I shall consider the curve branch S = σ (N ), asymptotic to the straight lines S = R and N = 0; it also represents N = ν (S), and divides the plane into two regions, that of the origin and the axes, where we have S − σ (N ) < 0, and the opposite region, where we have S − R, S − σ (N ) and N − ν (S) > 0. One can then summarize the results of No. 15 as follows: for a satisfactory regime of production and labour – or a simply satisfactory regime of prices and wages – to be possible, it is necessary and, in general, it 19 It is interesting to grasp the concrete meaning of the symbols dih and Tgh and of the root ν . The quantities d1g , . . . , dsg [where g] is one of the scalars 1, . . . , r, are, according to their definition, solution of dig −

k

aki dkg = big

According to (17), dig is therefore what δi becomes when one makes g = 1, h = 0, for h = 1, . . . , r; h = g. Hence, dig Ai (i = 1, . . . , s) is the production of Ai necessary for the subsistence of a consumer of type Cg ; tih dih represents the labour that this subsistence requires from the category of labourers of Pi belonging to the type Ch ; and i tih dig = Thg represents the labour that this subsistence requires from the category of labourers belonging to the type Ch . We know (Frobenius, loc. cit.) that the characteristic root ν of maximum modulus is always located between the smallest and the greatest of the sums Tg =

h

Thg

(g , h = 1, . . . , r)

Tg represents the total labour necessary for the annual subsistence of a consumer of type Cg . Assuming the condition R < 1 to be met, for a satisfactory regime of production and labour, or of prices and wages, to exist, it is therefore necessary that N be greater than the smallest – and it suffices that N be as least equal to the greatest – of the scalars T1 , . . . , Tr . As for the characteristic root ν of maximum modulus, it represents the average number of normal working days that a labourer must perform in order that the annual production obtained represents exactly the exclusive consumption of all workers. Indeed, let θ be that number; we will have, by definition, θπih = δi tih , ωh = 0,

i = 1, . . . , s; h = 1, . . . , r

Then, taking the formulas (16) and (33) into account, we obtain the system θ

The

g

−

h

Thg

h

=0

(h, g = 1, . . . , r).

being all > 0, θ is (No. 5) the characteristic root ν of maximum modulus of Thg .

126 Linear substitutions with coefficients

0

suffices that the point (1, N ) does not belong, with respect to the curve branch considered, to the region of the axes. If the point (1, N ) is on the curve, the possible regimes will in general be strictly satisfactory;20 exceptionally they might be only semi-satisfactory. If one knows that R is < 1, it suffices to know the position of point (1, N ) with respect to the curve branch, that is, the sign of 1 − σ (N ), to compare N with the number ν (1) = ν . I shall call this number ν , function of the coefficients a, b, t, the characteristic number of the socioeconomic system that these coefficients represent. Under the hypothesis R < 1, one can then state the following theorem: for a satisfactory regime of production and labour – or a simply satisfactory regime of prices and wages – to be compatible with a given socioeconomic state, it is necessary and, in general, it suffices that the characteristic number of that socioeconomic state is at most equal to the number of work-days per year. 17. Assume therefore a given socioeconomic state such that σ (N ) is 1, in other words, the condition R < 1 being met, the characteristic number is at most equal to the number 313 of work-days per year; let us assign values satisfying (I) to the δ, π, ω, , ω, and let us delete, where necessary, the Ai for which δi would be zero, by changing where necessary the remaining indices in order that they still follow the natural sequence of integers; let us set 21 (i = 1, . . . , s; h = 1, . . . , r)

ωih = δi τih

(34)

the formulas (18) become N πih = δi (tih + τih )

(35)

the formulas (29) become δi −

k

δk aki +

1 N

h

bih (tkh + τkh ) =

h

bih ωh

(36)

and show that the characteristic root of maximum modulus of the matrix aki + is

1 N

h

bih (tkh + τkh )

1.

20 [Potron here used the concept of a ‘strictly satisfactory’ regime, but without having defined it explicitly. – Eds.] 21 [In expression (34) Potron wrote ‘Tih ’ instead of ‘τih ’. – Eds.]

Linear substitutions with coefficients

0 127

Let us then wonder if an effectively satisfactory regime of prices and wages can exist, that is, values of the α, β, γ , σ, S, satisfying the system ⎫ αi − k aik αk = h tih σih + βi (i, k = 1, . . . , s; (19) ⎪ ⎪ ⎪ ⎪ δi tih σih = N πih (Sh + γih ) h = 1, . . . , r); (20) ⎬ (VI) NSh = k bkh αk , (21) ⎪ ⎪ ⎪ ⎪ ⎭ αk 0, βi 0, γih 0, σih > 0, Sh > 0 (24) or the equivalent system obtained by taking (35) into account, replacing (20) by tih σih = (tih + τih )

1 N

k

1 N

h

bkh αk + γih

(37)

and (19) by αi −

k

αk aik +

bkh (tih + τih ) =

h

γih (tih + τih ) + βi

(38)

The matrix of the left-hand side of (38) is the transpose of that of the left-hand side of (36). Therefore, there exist values > 0 of the α , β , and γ satisfying (38), then values > 0 of the S satisfying (21), then values > 0 of the σ satisfying (20); and these different values are solutions of (VI). Therefore, under the hypothesis stated, an effectively satisfactory regime of prices and wages is always possible, unless the matrix aki + N1 h bih (tkh + τkh ) is partially reduced. In that case, if its maximum characteristic modulus is < 1, in other words if the characteristic number is < 313, an effectively satisfactory regime of prices and wages is still always possible; but if the maximum characteristic modulus is 1, in other words if the characteristic number is 313, the only possibility, in general, is that of an effectively semi-satisfactory regime, that is where the α, σ , and S are only 0 and not all zero. Moreover, if the maximum characteristic modulus is 1, and whether or not the considered matrix is partially reduced, the β and the γ , that is, the firms’ benefits and the workers’ economies can, in general, only have zero values. 18. The comparison of formulas (17) and (19) in No. 13 leads to an interesting remark. One can indeed write them as ⎧ ⎪ ⎨ (i, k = 1, . . . , s; fi = k δk (uki − aki ) = h bih h (17) uik = 0, for i = k ; ⎪ (19) ϕi = k αk (uki − aik ) = h tih σih + βi ⎩ uii = 1). Since the f and the ϕ are linear forms of which the matrices are transposes of one another, we have i

αi f i =

i

δi ϕi

128 Linear substitutions with coefficients

0

from which, taking (21) into account, we derive the formula N

h

Sh

h

=

i

δi (βi +

h

tih σih )

(39)

Here NSh h is the total cost of living of the category Ch , δi βi is the annual benefit of Pi , [and] δi tih σih is the effective total annual wage of the workers of Pi belonging to Ch . Formula (39) therefore expresses that, the socioeconomic state being assumed invariant (which amounts to assuming the constancy of the coefficients a, b, t), the annual sum of the wages of the workers and of the firms’ benefits (or the capitalists’ incomes) is equal to the annual cost of living of all consumers, be they workers or not. 19. Let us examine what these results become if one takes into account the continuous increase of the population, which entails a continuous increase of consumption and consequently of production. We shall make the hypothesis that the increase of population does not change the proportions in which the labourers as well as the simple consumers are distributed among the various categories. Let t be the number which measures, in days, the time elapsed since the beginning of the year; let F(t) be a function which is monodromic, finite, continuous, positive and increasing for 0 t τ = 365, and equal to 1 for t = 0; we shall denote πih F(t) the value, at date t, of the number of workers of Pi – and ωih F(t) that of the number of non-working consumers – belonging to Ch (h = 1, . . . , r). During the time interval (t , t + dt), the total number of consumers of category Ch increases by h dF(t). Until the end of the year, each of the new consumers consumes during a number of days equal to τ − t − ε, 0 ε dt. Because of this increase in all categories, there follows, for the consumption of Ai , an increase which, up to some quantity of order dt 2 , is represented by h bih h (1 − τt )dF(t). And the quantity of Ai consumed during the year by all consumers, [i.e.] those who existed at the beginning and those who have joined during the year, will be h

blh

h

1+

τ

0

1−

t τ

dF(t)

Let then i Ai be the production of Ai , equal to the total consumption of the year; the consumption of Ai by the industrial establishments will be k aik k , and we shall have i

=

k

aik

k

+

h

bih

h

1+

τ

0

1−

t τ

dF(t)

The which appears in the [ ] is the difference of two , the first of which is F(τ ) − 1 and the second, as can be seen by an integration by parts, is F(τ ) − 1 τ 1 τ τ 0 F(t) dt. The [ ] is therefore τ 0 F(t) dt. From the hypotheses made on F(t),

Linear substitutions with coefficients

0 129

δi τ τ 0 F(t) dt,

and after it results that this quantity is > 0. Then, by taking i = deletion of the common factor, we find again the formulas (17). Let us now evaluate the maximum labour on which one can rely. Each of the πih dF(t) labourers of category Ch which join Pi during interval (t , t + dt) will provide, during the remaining of the year, and up to some quantity of order dt, at most N 1 − τt normal working days. The whole of the labourers of Pi belonging to Ch – those who existed at the beginning and those who join during the year – can therefore provide an annual number of normal working days at most equal to N πih 1 +

τ

0

1−

t τ

dF(t) = N πih

1 τ

τ

0

F(t) dt

The number of days that must be performed being one sets N πih

1 τ

τ

0

F(t) dt =

i tih

= δi tih τ1

τ 22 0 F(t) dt,

if

i (tih + ωih )

formula which is nothing but (18), the condition expressing that the regime of production and labour is satisfactory is written ωih 0. Besides, to obtain the annual production i Ai , it suffices that the πih dF(t) labourers of category Ch who join Pi during the interval (t , t + dt) provide collectively, during the rest of the year, a production Ai δi 1 − τt dF(t) and, consequently, a number of working days equal to δi tih 1 − τ1 dF(t). Then, indeed, the total production obtained will be δi 1 +

τ

0

1−

t τ

dF(t) =

δi τ

τ

0

F(t) dt =

i

Their effective total wage will therefore be δi tih σih 1 − τt dF(t); their individual cost of living will be 1 − τt NSh . If therefore one sets δi tih σih 1 −

t τ

dF(t) = N πih 1 −

t τ

(Sh + γih ) dF(t)

formula which is nothing but (20), the condition expressing that the regime of prices and wages is effectively satisfactory is written γih 0. Thus, provided that the distribution of the population does not change, the relations obtained by taking its continuous increase into account do not differ from 22 [Potron designated the upper limit of this integral as ‘τ 0’ instead of ‘τ ’. In the following expression he forgot to write the upper limit. – Eds.]

130 Linear substitutions with coefficients

0

those obtained when assuming it constant. Therefore, all the results of Nos.15 and 16 still hold. τ One sees, similarly, by multiplying by τ1 0 dF(t) both members of (39), that, even when taking into account the population increase, the result of No. 18 still holds.

9

Mathematical contribution to the study of the equilibrium between production and consumption

Editors’ note Potron explored the link between his economic model and the problem of unemployment in his ‘Contribution mathématique à l’étude de l’équilibre entre la production et la consommation’ (Annexe VII of Louis Varlez and Max Lazard (eds), Assemblée Générale de l’Association Internationale pour la Lutte contre le Chômage. Gand, 5–6 Septembre 1913. Procès-verbaux des Réunions et Documents Annexes. Paris, Service des Publications de l’Association Internationale pour la Lutte contre le Chômage, 1914: 163–71). As in his letter to the Statistical Society of Paris, he made a plea for statistical research aimed at determining the numerical values of the coefficients of his model. The Association Internationale pour la Lutte contre le Chômage (International Association on Unemployment) was founded in September 1910 in Paris, at the end of an international conference on unemployment organized at the Sorbonne university. It held general meetings in Ghent (September 1911), Zürich (September 1912) and once again Ghent (September 1913), before the advent of the First World War interrupted its activities. The driving forces of the association were the Belgian lawyer Louis Varlez (1868–1930) and the French sociologist Max Lazard (1875–1953).1 How and to what extent Potron became involved in the association is unknown. In the part of the proceedings in which Potron’s paper appeared, Varlez and Lazard selected a number of documents which had been sent to them by individuals rather than by national delegations.2 Potron’s contribution is noteworthy for two reasons. First, the model he presented is slightly different from the model he had presented before. And second,

1 On the International Association on Unemployment, see: Eric Lecerf, ‘Les Conférences internationales pour la lutte contre le chômage au début du siècle’ (Cahiers Georges Sorel, 1989, 7: 99–126). On Louis Varlez, see: Jasmien Van Daele, Van Gent tot Genève. Louis Varlez. Een Biografie (Gent, Academia Press; AMSAB; Liberaal Archief, 2002). On Max Lazard, see: Raymond Rivet, ‘Max Lazard, 1875–1953’ (Review of the International Statistical Institute, 1953, 21: 210). 2 The association published the proceedings also in different issues of its journal Bulletin Trimestriel de l’Association Internationale pour la Lutte contre le Chômage; Potron’s paper can be found in the issue of April–June 1914, 4th year, No. 2, pp. 509–17.

132 Equilibrium between production and consumption he explicitly referred to two economists, the Frenchman Charles Gide (1847–1932) and the Belgian Victor Brants (1856–1917). * * * ‘An economic organization capable of preventing crises and of maintaining a stable equilibrium between production and consumption’: according to a distinguished economist3 this would be the only really effective remedy against unemployment. Quite a number of practical difficulties impede the implementation of such an organization; but what is perhaps worse is that, in the present state of economics, even the theoretical conditions of a stable equilibrium between production and consumption are known only very imprecisely. The immediate goal of the consumption of an economic good can be either the support of a person’s life – domestic consumption, or the production of another good – industrial consumption. But production and industrial consumption are so entangled with the state of the economy in an interdependency which is at first sight very complex, that it seems impossible to solve even this simple problem: given the demands of domestic consumption, to forecast the effort which will have to be made by the overall production. How then could an economic organism, whatever it may be, act in a different way than automatically in order to maintain an equilibrium of which it does not know the conditions? I would like to outline here a programme of research to be done, from which one will be able to deduce a precise knowledge of these equilibrium conditions. This is the idea which has guided me. In essence, this interdependency between the state of the economy, production and consumption consists of purely quantitative relationships, which have an existence that is in a sense necessary, based on the very nature of things. Therefore, it must be possible to give a precise representation of these relationships in a mathematical form. Such a representation is a necessary instrument for researches of a theoretical nature and cannot but be a useful guide to researches of a practical nature. This is how it can be done. I The labouring population is distributed over a number of ‘establishments’ (or ‘firms’) each of which is devoted to the production of one or more economic goods (products in the proper sense, transportation activities, services). The workforce of an establishment is distributed over a number of jobs. It is easy to give some sort of numerical expression to the interdependency between these establishments. Let us designate by A, B, C, … the various economic goods, and by A, B, C, … the respective sets of the establishments which produce them. To produce a unit

3 Ch. Gide: Principes d’Économie politique, book IV, Chapter 5, p. 360 (4th ed.). [Charles Gide, Principes d’Économie Politique (Paris, L. Larose, 1894 (4th ed.): 560). Gide’s textbook, first published in 1883, went through twenty-six editions in France and was translated into nineteen languages. – Eds.]

Equilibrium between production and consumption 133 A,4

of the establishments A consume or use a certain number of units of A, B, C, …, numbers which I will designate respectively as aA , aB , aC , . . . . Likewise, to produce a unit of B, the establishments B consume or use a certain number of units of A, B, C, … , numbers which I will designate respectively as bA , bB , bC , . . . . And so on. These numbers are the constants of operation of the establishments A, B, C, … . Since they can be changed only by the introduction of new technical processes, they are characteristic for a certain state of the economy. We can call them the ‘constants of production’ with regard to this state. They allow to establish the precise relations which exist between the whole of production and the whole of industrial consumption. As a matter of fact, if I designate as pA , pB , pC , . . . the numbers of units of A, B, C, … annually produced, I have the following relations: entails the consumption of:

the production of: pA units of A pA aA units of A B pB bA pB pC C pC cA ··· ···

pA aB units of B pB bB pC cB ···

pA aC units of C pB bC pC cC ···

… … … ···

The annual consumption in industrial uses is therefore: a number of units of ···

A B C ···

equal to ···

pA aA + pB bA + pC cA + · · · pA aB + pB bB + pC cB + · · · pA aC + pB bC + pC cC + · · · ···

Let us now designate as: TA , TA , TA , . . . TB , TB , TB , . . . TC , TC , TC , . . . ···

the various jobs to be filled in the establishments ···

A B C ···

To produce a unit of A, respectively tA , tA , tA , . . . working days of the jobs TA , TA , TA , . . . have to be performed. To produce a unit of B, respectively tB , tB , tB , . . . working days of the jobs TB , TB , TB , . . . have to be performed. Again, these numbers are the constants of operation of the establishments A, B, C, …, the characteristics of a given state of the economy. We can call them the ‘constants of labour’ with regard to this state. They allow us to establish the precise

4 The definition of these units does not present any difficulties in the case of products in the proper sense or of transportation activities. In the case of ‘services’ one will take as unit, for that kind, the amount of services of that kind delivered annually to a specific number of individuals. It suffices that, in the given technical and administrative conditions, a specific consumption of labour and materials corresponds to the chosen unit. For example, the unit of the service ‘fire insurance’ will be the annual activity of an Insurance Company having for example 1,000 insured customers.

134 Equilibrium between production and consumption relations which exist between the whole of production (characterized by the numbers pA , pB , pC , . . .) and the whole of labour. It is clear that each year must be performed: pA tA , pA tA , pA tA , . . . pB tB , pB tB , pB tB , . . . pC tC , pC tC , pC tC , . . . ···

working days of the jobs working days of the jobs working days of the jobs ···

TA , TA , TA , . . . TB , TB , TB , . . . TC , TC , TC , . . . ···

Let nA , nA , . . . , nB , nB , . . . , nC , nC , . . . be the number of labourers engaged respectively in the jobs TA , TA , . . . , TB , TB , . . . , TC , TC , . . .. The ratios pA tA /nA , pA tA /nA , . . . , pB tB /nB , pB tB /nB , . . . , pC tC /nC , pC tC /nC , . . . will give the average annual numbers of working days of a labourer of each job. Let J be the number of work-days per year; let HA , HA , . . . , HB , HB , . . . , HC , HC , . . . be the average annual numbers of days out of work of a labourer of each job; adding, for each labourer, the number of working days to the number of days out of work one obtains the number of work-days. Hence these equations, equal in number to that of the different kinds of jobs: pA tA /nA + HA = pA tA /nA + HA = · · · = J pB tB /nB + HB = pB tB /nB + HB = · · · = J ···

···

···

These equations express the interdependency of production and labour. If N is the total number of labourers, we have to add: nA + nA + · · · nB + nB + · · · = N Let us now try to calculate the value of the whole of domestic consumption. To each job corresponds a certain standard of living; it can occur that a same standard of living corresponds to different kinds of job, either of the same profession, or of different professions. I suppose that the notations have been chosen in such a way that a same standard of living V corresponds to the jobs TA , TB , TC , . . ., a same standard V to the jobs TA , TB , TC , . . ., and so on. Every standard of living entails per consumer and per year a certain consumption of each economic good, or: at the standard V V ···

a consumption of ⎧ qA , qB , qC , . . . ⎨ qA , qB , qC , . . . units of A, B, C, … ⎩ ···

We have to estimate the number of consumers at each standard of living. Since nA , nA , . . . , nB , nB , . . . , nC , nC , . . . are the numbers of labourers engaged

Equilibrium between production and consumption 135 respectively in the jobs TA , TA , . . . , TB , TB , . . . , TC , TC , . . ., there are: ⎧ ⎧ ⎨V ⎨ nA + nB + nC + . . . at the standard V a number of consumers equal to nA + nB + nC + . . . ⎩ ⎩ ··· ···

To the consumers we have to add the consumers–rentiers, which are distributed in the numbers r , r , . . . over the standards of living U , U , . . ..5 And per consumer and per year we have: at the standard at the standard ···

U U ···

a consumption of a consumption of ···

sA , sB , sC , . . . sA , sB , sC , . . . ···

units of A, B, C, … units of A, B, C, … ···

In total the annual domestic consumption is therefore: a number of units equal to

of the economic goods A, B, C, ···

(nA + nB +··· )qA + (nA + nB +··· )qA +···+ rsA + r sA +··· (nA + nB +··· )qB + (nA + nB +··· )qB +···+ rsB + r sB +··· (nA + nB +··· )qC + (nA + nB +··· )qC +···+ rsC + r sC +··· ···

Bringing together, for each economic good, the annual industrial and domestic consumption, we will obtain the annual total consumption, which has to be balanced by the annual production. Therefore, if there is overproduction of eA , eB , eC , . . . units of A, B, C, …, we will have the following equations obtained by expressing that total production equals total consumption plus overproduction: pA = pA aA + pB bA + pC cA + · · · + (nA + nB + · · · )qA + (nA + nB + · · · )qA + · · · + rsA + r sA + · · · + eA

pB = pA aB + pB bB + pC cB + · · · + (nA + nB + · · · )qB + (nA + nB + · · · )qB + · · · + rsB + r sB + · · · + eB

pC = pA aC + pB bC + pC cC + · · · + (nA + nB + · · · )qC + (nA + nB + · · · )qC + · · · + rsC + r sC + · · · + eC ···

These equations are equal in number to that of specifically distinct economic goods. They express the interdependency of production and consumption.

5 [This is the only instance in Potron’s works where the standards of living attributed to the consumers– rentiers differ from those of workers. – Eds.]

136 Equilibrium between production and consumption Designating by R the total number of rentiers, we have to add: r + r + ··· = R So the state of the economy may change, the distribution of the workforce may modify itself, or the standards of living can ameliorate or worsen: this will produce certain variations in the numbers which occur in these equations; but the equations themselves will always continue to hold, with all the consequences which mathematical reasoning can deduce from them.6 II Let us apply these formulas to the most common problem which occurs when there is a crisis. The state of the economy is given; any change in the technical processes, in the internal organization, in one word in the constants of operation of the productive establishments is out of the question. To reestablish equilibrium, one can only try to cause variations in the pace of production and of consumption, or in the distribution of labourers. In whatever way these variations are made, our formulas continue to be verified, since the constants of production and labour retain their fixed values. All the other quantities remain variable, but what one might call their freedom of variation is clearly diminished: Every socioeconomic regime compatible with the given state of the economy must be such that these formulas are verified; this entails different mathematical consequences, associated with different economic consequences. Here is a way to use these formulas. Suppose the standards of living corresponding to the different jobs are given (one could take, for example, for each job the standard of living of a specific labourer who considers he is more or less satisfied), as well as those corresponding to the different types of rentiers. The formulas will then provide all the possible distributions of the labourers over the various jobs. They show in particular that if the distribution of the rentiers, the overproduction which one wants to obtain of each economic good, and the average unemployment which one allows to subsist in each professional category are given, then the distribution of the labourers is completely determined. One might perhaps say that in this there is much arbitrariness; but, first, this way of proceeding is given only by way of example; and, second, it is better to know one of the satisfactory regimes that one can substitute for the state of crisis than to know none at all. 6 I have developed the study of these formulas in a paper on: Quelques Propriétés des Substitutions Linéaires à Coefficients 0, et leur Application aux Problèmes de la Production et des Salaires (Annales Scientifiques de l’École Normale Supérieure, 3rd series, vol. 30, 1913, February, pp. 53–76). The principal conclusions of this study had been published in two communications to the Academy of Sciences. (Sessions of 4 and 26 December 1911, C.-R., vol. 153, pp. 1129 and 1458.) [The three papers referred to by Potron are A13/Chapter 8, A9/Chapter 3 and A10/ Chapter 4. – Eds.]

Equilibrium between production and consumption 137 The use of these formulas seems also capable of preventing the crises which result from a transformation of the economy that changes a satisfactory state of equilibrium. The previous state of the economy is by hypothesis known. A new invention which has already been tested, but not yet applied, will modify some of the data in a way which can be determined in advance. If we suppose for convenience that nothing is changed with regard to the standards of living and the distribution of the rentiers, the formulas will indicate how one must transitorily change the distribution of labourers to meet the excess of production in the period of transition, and above all what will be the new distribution in the new state of equilibrium. These examples show the interest and the utility of doing research into the actual numerical values of the constants of production and labour, and of updating the data for any changes that might occur. We now have to explain how this research can be done. III Let us first try to form some kind of idea about the number of these constants. We can estimate the number of separate occupations at about ten thousand.7 The number of specifically distinct economic goods must be of the same order of magnitude. On the other hand, a preliminary research permits me to estimate the number of constants of production with regard to a given economic good at about two hundred. Hence about two million constants of production. From the point of view of the standard of living (the only element to be considered in this question), we can classify the workforce of an establishment into six categories: bosses and managers, heads of divisions, foremen, office workers, skilled workmen, unskilled workmen. Hence about sixty thousand constants of labour. The number of these constants is undoubtedly considerable; but a logarithmic table for ordinary use (an octavo volume of some five to six hundred pages) contains always at least 250,000 numbers of six to seven digits each. At most ten volumes of this type will therefore suffice to contain all the constants in question. With respect to the annual modifications which will be induced by the progress of the economy, a small booklet will be enough; as a matter of fact, one octavo page easily contains 500 numbers. The first determination represents therefore only the work required to calculate ten or so logarithmic tables. The updating work is insignificant.

7 In Germany the 1895 survey counted 7,793 denominations of separate professions or occupations. Cf. Victor Brants: Grandes Lignes de l’Économie politique, 5th edition, vol. 1, p. 149, note. [Victor Brants, Grandes Lignes de l’Économie Politique (Louvain, Charles Peeters, 1908: 149). When this book was first published in 1901, it was somewhat confusingly presented as the 3rd edition, since it was compiled from three earlier volumes each of which had gone through two editions. Victor Brants was a Belgian historian and economist at the Catholic University of Louvain. – Eds.]

138 Equilibrium between production and consumption Moreover, the work to determine these constants lends itself to a great amount of division. To calculate with sufficient precision the constants with regard to an economic good, it suffices to address a few of the establishments which produce the good. Next, we may expect to find that, at least in the important establishments, a good deal of the work has already been accomplished by the ‘cost accounting department’. In fact if α, β, γ , . . . designate the exchange prices of the units of the goods A, B, C, …, and sA , sA , . . . the daily wages of the different jobs in the establishment A, then obviously the cost price of a unit of A will be the sum: aA α + aB β + aC γ + · · · + tA sA + tA sA + · · · Every determination of the cost price of a unit of A involves therefore as an essential element the determination, by any kind of process, of the constants of production and labour with regard to A. By their arrangement, the book of accounts of the Production Facilities depending from the Ministry of War allow us to easily find these constants. In here, the expenditures for labour and materials are distributed over different accounts in which not only the amounts spent are recorded (information of no use for the issue at hand) but, and this is essential for us, the corresponding numbers of working hours (or days) and the corresponding quantities of materials or equipment. It is interesting to study in detail what can be learned from these book of accounts, by taking as an example the production of B powder8 in a military gunpowder factory.9 The expenditures which are specifically related to the execution of an order – in this example the production of 196,192 kilograms of B powder – can be found on the specific Special Expenses account for this order, which provides details on the use of manpower and the delivery of raw materials. The common expenditures related to the execution of several orders are, according to their nature (whether they are common to orders of the same or of a different type, whether they depend on the intensity of production or not), distributed over the four general accounts: Workshop Expenses, Minor Production Expenses, Factory Expenses, and Overhead Expenses. Furthermore, the specific Special Expenses account for each order indicates, for each of the four general accounts, what fraction of the Expenses of that account must be assigned to this order. For the example chosen here, these fractions are 8 [B powder (in French: poudre B), also known as white powder (poudre blanche), had been invented in 1886 by the Frenchman Paul Vieille (1854–1934). In comparison to ordinary gunpowder, also known as black powder, it was smokeless and much more powerful. – Eds.] 9 [In 1942 Potron specified that the gunpowder factory in question was the Poudrerie militaire du Bouchet, located in Vert-le-Petit, to the south of Paris (see A44/Chapter 17). As a reserve officer of the Powder corps, Potron spent there some of the military training periods he had to follow at regular intervals (see the archives of the Service Historique de la Défense). – Eds.]

Equilibrium between production and consumption 139 as follows: Workshop Expenses Minor Production Expenses Factory Expenses Overhead Expenses

Manpower 0.244027 0.244045 0.244196 0.244083

Materials 0.244260 0.244078 0.403429 0.244270

Taking as unit 1,000 kilograms of B powder, we will determine everything which is required for their production by multiplying the quantities recorded on the Special Expenses account by 1961.912 = 0.005078, and the quantities recorded on the various general accounts by this number and then by the corresponding coefficient of the table above. In this way, a simple transcript and a few elementary arithmetic operations will provide the constants of production and labour as they result from the annual operation of the establishment in question. The following table indicates how the operations work and how some constants have been determined. The numbers in columns 1, 3, 5, 7 and 9 are the data as found in the accounts; those in columns 2, 4, 6, 8 and 10 are these data multiplied by the appropriate coefficients, so as to reduce these numbers to the chosen unit; those in column 11 are the sums of the numbers given in columns 2, 4, 6, 8, 10; they are the constants we are looking for. The values of the constants obtained in this way can vary from one year to the other, especially with regard to equipment, since the items which are replaced show up in the records only in the year in which they are replaced. Thus, the accountbooks which we have taken as an example show the purchase of a stretching press among the factory Expenses; the corresponding constant is 0.0003. If the machine has a lifetime of ten years, its total use corresponds to 10 times the annual production, which is approximately constant. The true coefficient would therefore be 0.00003. But if one calculates the average over ten years, it is this value that one will find, since the item in question shows up only once. Hence, the average values of the constants calculated over a relatively large number of years of operation will fairly reflect, not only the consumption of materials, but also the effective use of equipment. Not all industrial establishments keep their book of accounts in such a favourable way for our researches. However, since the principle of the determination of the cost price cannot change, it must always be possible to gather from these accounts, at least in an equivalent form, the information we need.10

10 It will always be easy to know exactly, on the one hand the labour force and the materials devoted to a specific fabrication, and on the other the total workforce and the total amount of materials allocated jointly to several specific fabrications. Difficulties can arise only in the distribution of the workforce and the common materials over several specific fabrications. But it has to be remarked that the uncertainty which results from this affects in general only the very small coefficients, and that in each case, whatever distribution is adopted, the total production which must be obtained and the total amount of labour which is to be performed remain more or less constant.

day kilo kilo brush kilo kilo

1. Workforce 2. Sulphuric ether 3. Ethyl alcohol 4. Brushes 5. Firewood 6. Coal 7. 8. Stretching press

5

6

7

8

»

»

»

14,749 74.6885 4,235 121,830 491.4555 » 128,268 651.4000 192 » » 801 » » » » » 9,600 »

5.2482 » 0.2382 0.9936 » 11.9083 »

6,712 291 240 » 6,000 » »

8.3186 0.3607 0.2975 » 7.4372 »

Amount Per unit

Per unit

1

0.0003

10

11

»

»

13.7691 » » » 44.7100 3.0984

Amount Per unit

0.0003

129.1672 491.8162 651.9357 0.9936 157.1267 3,306.9913

Overhead expenses Constants

9

21,887 27.1428 11,108 » » » » » » » » » 51,240 104.9795 36,075 1,606,800 3,291.9846 2,500

Amount

Amount Per unit

4

Amount Per unit

3 Workshop expenses Minor prod. expenses Factory expenses

2

Specific expenses

1

[There seem to be a few inaccuracies in Potron’s table. Taking the ‘per unit’ values as correct, the entry in row 1, column 1 should be ‘14,707’ instead of ‘14,749’, and the one in row 2, column 1 ‘96,773’ instead of ‘121,830’. With regard to the constant of the stretching press (row 8, column 8), it is obvious that Potron applied a different coefficient than the one mentioned in the last column of his previous table. In fact, using the coefficient of that table the constant would have been 1∗ 0.005078∗ 0.403429 ∼ = 0.0021. Furthermore, for the calculation of the per unit values in rows 5 and 6 of column 10, Potron used a conversion coefficient equal to ‘0.244045’ instead of ‘0.244270’. We have corrected two obvious misprints in column 5. – Eds.]

machine

Unit

Type

Table 9.1 Coefficients of the cost price formula of B powder

Equilibrium between production and consumption 141 Moreover, it is worth noting that it is not necessary to be very precise in the determination of these constants. Evidently, starting from these merely rough data one will arrive only at approximate results. But if one knows the degree of approximation of the data, one will be able to determine the degree of approximation of the results and to indicate the bounds which the errors committed in the process will certainly not exceed. On the other hand, as we have seen, the formulas established must serve above all to provide, by way of indications, certain standard regimes; but one should not forget that in the neighbourhood of each standard regime there exists an infinity of regimes which are equally satisfactory. It is therefore not very useful to try and determine the standard regime with a great amount of precision. To sum up, in a given state of the economy there exist absolutely necessary relations between fixed numbers, which depend upon this given state, and variable numbers which represent many variable elements such as production, standard of living and distribution of labourers. Every attempt to organize these variable elements cannot ignore these necessary relations and their consequences. The researches aiming to determine these fixed numbers, these constants of production and labour with regard to the present state of the economy, and to trace their variations caused by the progress of the economy, are therefore not at all at odds with the goal of the International Association on Unemployment. They truly belong to the scientific study of unemployment, to the ‘practical or scientific initiatives capable of having a beneficial effect with respect to the fight against unemployment’.11

11 Statutes of the Association.

10 The scientific organization of labour. The ‘Taylor System’

Editors’ note Potron’s last economic contribution of the period 1911–1914 concerns the theories of scientific management put forward by the American mechanical engineer Frederick Winslow Taylor (1856–1915). The article ‘L’organisation scientifique du travail. Le ‘Système Taylor’ was published in two instalments in Le Mouvement Social, with the first part, ‘A. – Les principes’ (15 June 1914, 77(6): 497–510) covering its basic principles, and the second, ‘B. – Les sanctions de l’expérience’ (15 July 1914, 78(1): 21–33), focusing on its practical effects. There is something odd about the numbering of sections in this article, since part A is split into sections I and II whereas part B is split into sections V and VI. Most probably, however, we are dealing with a simple typographical error. In several respects this article is an outlier in Potron’s work. It is the only publication in which Potron dealt with management issues. Moreover, it is littered with references to the literature on scientific management – in his other writings about economic problems he hardly ever showed that he knew or was interested in what the professional economic literature had to say. Potron was especially interested in the possibilities of applying the Taylor System in France, and in exploring the connections between his own economic model and the system. Potron’s main sources are mentioned in a long footnote appended to the title; in our translation this footnote precedes the text. Potron relied exclusively on French sources, both in the form of translations of Taylor’s publications (most importantly, The Principles of Scientific Management), and of summaries and comments by contemporary French followers of Taylor such as Le Chatelier, de Fréminville and Fremont. Potron borrowed heavily from the work of these authors, especially from that of de Fréminville, but his references are not always accurate. Henry Louis Le Chatelier (1850–1936), an engineer and chemist, was the main propagator of Taylor’s scientific management ideas in France; for this he often used the pages of the Revue de Métallurgie, which he founded and edited.1 Charles de

1 See Léon Guillet, ‘Henry Le Chatelier (1850–1936). Sa vie – son œuvre’ (Revue de Métallurgie, Special issue, January 1937).

The scientific organization of labour 143 la Poix de Fréminville (1856–1936) was also an engineer, who spent the first part of his career in the French automobile sector.2 Charles Fremont (1855–1930), who worked at the École des Mines, published extensively on technology.3 * * * [Note appended to the title] For the redaction of this article the following documents have been used: Taylor: Principes d’Organisation Scientifique des Usines; translated by Mr Jean Royer, with a preface by Mr Henry Le Chatelier (Paris, Dunod and Pinat, 1912). I shall refer to this book as ‘Principes’.4 Le Chatelier: Le Système Taylor, articles published in the Bulletin de la Société des Amis de l’École Polytechnique, No. 12 (January 1914), p. 33; No. 13 (April 1914), p. 15.5 Fremont: A propos du Système Taylor, article published in ‘La Technique Moderne’, vol. VII, No. 9 (1 November 1913), p. 301.6 De Freminville: Conférence de ‘La Technique Moderne’ sur le Système Taylor, supplement to ‘La Technique Moderne’, vol. VIII, No. 6 (15 March 1914). I shall refer to this as C.T.M. – Communication à la Société d’Économie Sociale, in ‘La Réforme Sociale’, 7th series, vol. VII, No. 77 (1 March 1914), p. 317, and No. 78 (16 March 1914), p. 403. I shall refer to it as R.S.7

2 See André Danzin (ed.), Charles De Fréminville: 1856–1936. Pionnier de l’Organisation Scientifique du Travail (Saint Étienne, Aubin, 2000). 3 See George Sarton, ‘Charles Fremont. Historien de la technologie (1855–1930)’ (Isis, 1937, 27: 475–84). 4 [This refers to the French translation of Frederick Winslow Taylor, The Principles of Scientific Management (New York, Harper and Brothers Publishers, 1911), published on behalf of the Revue de Métallurgie. Two different French editions were published by the same editor: a long one (151 pp.) dated 1912, published by H. Dunod and E. Pinat in Paris and printed by A. Burdin in Angers, and a short one (117 pp.) without a date of publication, printed by A. Davy in Paris. Potron always referred to the long version; in this version, Le Chatelier’s introduction, entitled ‘La science économique’, is on pp. 1–25. We complement Potron’s references to this book by the corresponding pages of the original English publication; for this we use the facsimile reprint which is part of the volume: Frederick Winslow Taylor, Scientific Management (New York, Harper and Brothers, 1947). – Eds.] 5 [This refers to Henry Le Chatelier, ‘Le système Taylor, science expérimentale et psychologie ouvrière’ (Bulletin de la Société des Amis de l’École Polytechnique, 1914, 12: 33–48 and 13: 15–34). The same article was published in the Revue de Métallurgie, April 1915, 12: 197–232. – Eds.] 6 [Charles Fremont, ‘A propos du système Taylor’ (La Technique Moderne, 1 November 1913, 7: 301–8). – Eds.] 7 [The first article is: Charles de Fréminville, ‘Les méthodes scientifiques de travail dans l’industrie. Le système Taylor’ (La Technique Moderne, 15 March 1914, 8 (supplement): 16 pp.). The second article is: ‘Le système Taylor et l’organisation scientifique du travail dans les ateliers’ (La Réforme Sociale, 1 March 1914, 7th series, 7: 317–44). This is a report of the meeting of the Sociéte d’Économie Sociale on 12 January 1914. It contains an address by the Chairman

144 The scientific organization of labour ‘Le Génie Civil’, vol. LXII, No. 24 (12 April 1913), p. 474. Une critique du Système Taylor. Observation of Rear-Admiral Edwards on the application of the Taylor system in the shipyards; No. 26 (26 April 1913), p. 514; Réponse à une critique du Système Taylor (by Mr Le Chatelier).8

A – The principles I The ideas of the American engineer F.-W. Taylor on the direction and the organization of factories have been, for about fifteen years, famous in America. To teach them, Harvard University has organized a special Faculty.9 A section of the [American] Society of Mechanical Engineers10 is in charge of studying its applications.11 A bookshop catalogue mentions more than 400 books written on these questions during the last five years.12 Applications of the system have been made in more than sixty different industries.13 The Western Economic Society

8

9

10 11 12 13

of the society, Paul Nourisson (pp. 317–21), followed by the text of de Fréminville’s speech (pp. 321–44). Under the same title, the next issue gives a summary of the discussion which followed de Fréminville’s speech (16 March 1914, 7th series, 7: 403–9). – Eds.] [This refers to: ‘Critique du système Taylor’ (Le Génie Civil, 12 April 1913, 62: 474–5) and Henry Le Chatelier, ‘Réponse à une critique du système Taylor’ (Le Génie Civil, 26 April 1913, 62: 514). The first article summarizes the criticism of Admiral John R. Edwards, ‘The fetishism of scientific management’ (Journal of the American Society of Naval Engineers, 1912, 24: 355–416), and the second gives Le Chatelier’s reply. – Eds.] [This refers most probably to the foundation, in 1908, of the Harvard University Graduate School of Business Administration, better known as the Harvard Business School. It had several scientific management oriented courses in its programme. See Daniel Nelson, A Mental Revolution: Scientific Management since Taylor (Columbus, Ohio State University Press, 1992: 87–9). – Eds.] [Here and elsewhere in the article we have corrected Potron’s spelling of the association’s name. The American Society of Mechanical Engineers was founded in 1880. – Eds.] C.T.M., p. ix, col. 1. C.T.M., p. x, col. 1. [Charles de Fréminville wrote that a catalogue of books on scientific management contained 500 titles, of which 75 per cent had been published in the last five years. – Eds.] Report of the Committee of the [American] Society of Mechanical Engineers on The Present State of the Art of Industrial Management (December 1912), referred to by C.T.M., p. xi, col. 2. – Beyond the classical Midvale and Tabor factories, one can quote, according to C.T.M., the mechanical construction workshop of the Link-Belt Company, the locomotive construction workshops in Shenectady, the wagon factory Pullman, the car construction companies Hudson, Stearns, Franklin, the printing company Plimpton Press, the lock workshop Yale and Towne, the Watertown shipyard, [and] the Work Service of the City of Philadelphia. [We have corrected several misspellings in the names. The report mentioned at the beginning of the note is: ‘The Present State of the Art of Industrial Management’ (Journal of the American Society of Mechanical Engineers, November 1912, 34: ‘Majority report of sub-committee on administration’, 1601–20 and ‘Minority report of sub-committee on administration’, 1621–2; the discussions on the report which took place at the Annual Meeting of the society in December 1912

The scientific organization of labour 145 has devoted its Chicago session to it, on 13 March 1913.14 The general public interest had already been aroused by a deposition at the ‘Interstate Commerce Commission’, on 21 November 1910,15 in which it was told that the application of these principles to the US railways would lead to a five million francs cost reduction per day and could be made with an equal success to all types of activity dealt with in business.16 The implementation of the system has become a speciality. Some distinguished engineers work exclusively on it. Some, like Mr Barth, Mr Gantt, who have long worked with Mr Taylor, belong, so to say, to the same School.17 Others, like Mr Harrington Emerson, are inspired by the same principles but apply them differently.18 Even ladies are involved in this movement: in a very original

were summarized in the issues of March and May 1913, 35: 447–518, 871–7). The firms listed by Potron are: • • • • • • • • • •

14

15

16

17 18

Midvale Steel & Ordnance Company (Nicetown, Pennsylvania); Tabor Manufacturing Company (Philadelphia, Pennsylvania); Link-Belt Company (Chicago, Illinois); American Locomotive Works (Schenectady, New York); Pullman Company (Chicago, Illinois); Hudson Motor Car Company (Detroit, Michigan); F. B. Stearns Company (Cleveland, Ohio); H. H. Franklin Manufacturing Company (Syracuse, New York); Plimpton Press (Norwood, Massachusetts); Yale Lock Manufacturing Company, later the Yale & Towne Manufacturing Company (Stamford, Connecticut). The Watertown Arsenal was a gun manufacturing facility of the United States Army located in Watertown, Massachusetts. Under Mayor Rudolph Blankenburg (1843–1918), the Department of Public Works of the City of Philadelphia, Pennsylvania, was run by Morris L. Cooke, an associate of Frederick W. Taylor; see Donald W. Disbrow, ‘Reform in Philadelphia under Mayor Blankenburg, 1912–1916’ (Pennsylvania History, October 1960, 27: 379–96.) – Eds.] C.T.M., p. ix, col. 2. [The Western Economic Society was founded in Chicago in 1911. It organized a conference on scientific management on 14–15 March 1913. (Cf. the announcement in the American Economic Review, June 1913, 3: 536.) – Eds.] [This is a reference to the testimony given by Louis Brandeis during the Railroad Freight Rate Hearings of the Interstate Commerce Commission, an independent regulatory agency of the railroad industry. Louis Dembitz Brandeis (1856–1941) was a lawyer who was appointed as a Supreme Court Justice in 1916. – Eds.] C.T.M., p. x, col. 1. [Charles de Fréminville quoted both the figure of 1 million US$ and its equivalent of 5 million French francs per day. Brandeis claimed that the railroad companies could save more than 300 million US$ per year by applying scientific management methods. The Western railroad companies immediately invited him to come and work for them (‘Brandeis to teach roads without pay’, The New York Times, 30 November 1910). – Eds.] [Carl Georg Barth (1860–1939), a Norwegian-born mathematician and engineer, and Henry Laurence Gantt (1861–1919), an American mechanical engineer whose name remains attached to the Gantt chart, were associates of Taylor in various scientific management projects. – Eds.] [Harrington Emerson (1853–1931) was an American professional consulting management engineer, and one of Taylor’s rivals on the consulting market. – Eds.]

146 The scientific organization of labour booklet entitled ‘New Housekeeping’, Mrs Christine Frederick tells them how to apply the principles of the Emerson School to housekeeping.19 In France, for one or two years, the attention of engineers and economists has been seeming seriously attracted to these new methods, mainly thanks to the efforts of Mr Le Chatelier, a member of the Institute, Mining General Inspector, Professor at the School of Mining Engineering and at the Faculty of Sciences of Paris. The ‘Revue de Métallurgie’, directed by Mr Le Chatelier, had already published in 1907 a translation of Mr Taylor’s important book, as Études sur l’Organisation du travail dans les Usines.20 It also published in 1912 a translation of another booklet by the same author, as ‘Principes d’Organisation scientifique des Usines’. That last publication and Mr Le Chatelier’s remarkable preface which introduced it drew the attention of many industrial journals and several economic journals. French engineers have gone to the US to study the working of the system from the inside. One of them, Mr de Fréminville, an Arts and Manufactures engineer, and the technical director of the Panhard and Levassor Company,21 gave an account of his impressions in a lecture organized by the journal ‘La Technique Moderne’ on 11 December 1913, [also] in a meeting of the Sociéte d’Économie Sociale on 12 January 1914,22 and in a meeting of ‘Cap’ (Club Action et Pensée) on 12 March 1914.23 As shown by these hints, Taylor’s ideas deserve a serious examination by industrials and economists. Their apparent novelty, [and] their American origin might perhaps arouse some feelings of distrust for a few people. This would be a mistake because, after all, they only constitute a recent state of

19 [Christine Frederick (1883–1970), before her marriage in 1907 known as Christine McGaffey, published widely on household management. The book to which Potron referred is The New Housekeeping. Efficiency Studies in Home Management, Garden City (NY), Doubleday, Page & Company, 1914. – Eds.] 20 [This refers to Frederick Winslow Taylor, Études sur l’Organisation du Travail dans les Usines, published by H. Dunod and E. Pinat in Paris and printed by Burdin in Angers, on behalf of the Revue de Métallurgie, 1907. The book contains French translations of three essays by Taylor which the Revue de Métallurgie had published earlier that year: ‘On the art of cutting metals’ (‘La taille des métaux’, 4: 39–65, 233–336, 401–66), ‘Notes on belting’ (‘Note sur les courroies’, 4: 576–605) and ‘Shop management’ (‘Direction des ateliers’, 4: 633–736). – Eds.] 21 [Panhard et Levassor was a French automobile constructor, taken over by Citroën in the 1960s. – Eds.] 22 [The Société d’Économie Sociale was founded by the French economist and sociologist Frédéric Le Play (1806–1882) in 1856. It published the journal La Réforme Sociale. – Eds.] 23 Bulletin de la Soc. des Amis de l’Éc. Polyt., April 1914, pp. 34–35. [This is a reference to a complementary note inserted at the end of Le Chatelier’s article in the Bulletin de la Société des Amis de l’École Polytechique (cf. footnote 5 above). The note reported on the debate organized by the ‘Club Action et Pensée’. This club seems to have been connected to the Association Générale des Étudiants de Paris; see Paul Desfeuilles, ‘Carl Spitteler’ (La Revue Hebdomaire, September 1915, 2nd series, 25: 671–87) and Philippe Périer, ‘L’évolution de la classification sociologique dans l’école de Le Play (1855–1969). Comment la classification prépare et permet les prévisions’ (Revue Internationale de Sociologie, December 1969, 2nd series, 5: 455–72). – Eds.]

The scientific organization of labour 147 the continuous evolution which, step by step but steadily, improves industrial technique. The problem of the reduction of the labour force necessary to the making of a given task results from the concern, as old as the world, to always seek to improve one’s work with less tiredness. The scientific study of all the operations of mechanical arts is listed, at the end of the XVIIth century, among the works set about by the newly founded Académie Royale des Sciences.24 The precise timing of elementary operations and the systematic study of the ensuing tiredness were not ignored by the ancestors of our modern engineers, like Vauban, Bélidor, Coulomb, Poncelet.25 In France, for several years, Professor Imbert of the Montpellier Faculty of Medecine has been studying photographically the gestures of workers of various jobs and has been measuring the muscular forces with an ergograph.26 And I shall excuse myself to recall a paper published by this Journal in April 1912:27 the basic problems of production, distribution of labourers, just

24 [The Académie Royale des Sciences was founded in 1699, but the origins of the institution can be traced to an initiative launched by Colbert in 1666. It is now one of the five Académies of the Institut de France. – Eds.] 25 Fremont: A propos du Système Taylor, in ‘La Technique Moderne’, vol. VII, No. 9 (1 November 1913, p. 301). – Belidor: Architecture Hydraulique, t. III, pp. 110–111, Paris, 1750; La Science des Ingénieurs, book III, p. 39, 1729. – Coulomb: Mémoire sur la Force des Hommes, p. 280, Paris, 1798. – Description des Arts et Métiers de Messieurs de l’Académie Royale des Sciences, Avertissement, p. 1. – General Poncelet: Introduction à la Mécanique Industrielle, Metz, 1835. [Potron simply copied the references to the works of Bélidor, Coulomb and Poncelet from Fremont’s article. These are the writers and books referred to: • Bernard Forest de Bélidor (1693 (1697?)–1761) was a military and civil engineer, whose main work was Architecture Hydraulique, ou l’Art de Conduire, d’Élever et de Ménager les Eaux pour les Différens Besoins de la Vie (4 Volumes, 1737–1753). His La Science des Ingénieurs dans la Conduite des Travaux de Fortification et d’Architecture Civile (1729) apparently includes extracts from unpublished works by the French engineer Sébastien Le Prestre, marquis de Vauban (1633–1707), famous for his military citadels. • Charles Augustin de Coulomb (1736–1806) was a physicist who published his ‘Résultats de plusieurs expériences destinées à déterminer la quantité d’action que les hommes peuvent produire par leur travail journalier, suivant les différentes manières dont ils emploient leurs forces’, better known as ‘Mémoire sur la force des hommes’, in 1799; it has been included in his Théorie des Machines Simples (Paris, Bachelier, 1821, pp. 255–297). • Jean-François Poncelet (1788–1867) was a mathematician and engineer. The first edition of his Introduction à la Mécanique Industrielle was published in 1829; a second came out in 1839, and a third appeared posthumously in 1870. – Eds.] 26 Dr Imbert: Méthodes de laboratoire et questions ouvrières (Revue générale des Sciences, 30 June 1911). [Armand Imbert, ‘Les méthodes de laboratoire appliquées à l’étude directe et pratique des questions ouvrières’ (Revue Générale des Sciences Pures et Appliquées, 1911, 22: 478–86). Armand Imbert (1850–1922) was a French professor of physiology. – Eds.] 27 Possibilité et Détermination du Juste Prix et du Juste Salaire, in ‘Le Mouvement Social’, vol. LXXIII, No. 4 (April 1912), pp. 289–316. [See A11/Chapter 5. – Eds.]

148 The scientific organization of labour price, just wage, can be solved28 if every producer knows the elements of his cost price, the exact determination of which is precisely required by the Taylor method. However, it would be a serious mistake29 to see, in these scientific studies, the core of the Taylor system. Mr Taylor has developed a method of his own for the industrial application of these laboratory experiments. Some biographical details on Mr Taylor will not be useless to a full understanding of his method. He was the son of a solicitor. An eye illness obliged him to stop the studies he had begun to enter a US university.30 He then served apprenticeship as a pattern maker and became rapidly skilled; but, judging that this profession did not raise interesting questions, he switched to tool-machines work and started in 1878 as an unskilled worker in the workshops of the Midvale Steel Company. Successively accountant, mechanic, foreman, and, after three years, head of the workshop, he understood the inconveniences that a defective labour organization sets between the interests of the workers and those of the management, and decided himself to seek for a remedy to the evil, by improving simultaneously the technical processes and the wage terms.31 The company executive gave him some funds; the first results obtained32 encourage his generosity. Mr Taylor and his first collaborators have been following methodical researches for twenty-five years, which have cost one million francs.33 The first result was the discovery of high cutting speed steel which allowed doubling the efficiency of tool-machines and has ‘revolutionized the mechanical working of metals’.34 Simultaneously, Mr Taylor

28 Annales Scientifiques de l’École Normale Supérieure, 3ème série, t. XXX, 1913, pp. 53–76. [See A13/Chapter 8. – Eds.] 29 This mistake is at the basis of almost all the critiques addressed to the Taylor system, especially by the workers’ unions. It seems to have been shared by A. G. (Le Mouvement Social, 15 March 1913, p. 266), and a bit by Mr Henri du Passage (Etudes, 5 February 1914, p. 387). [A. G., ‘La méthode Taylor ou l’organisation scientifique du travail et la classe ouvrière’ (Le Mouvement Social, 15 March 1913, 75: 266–72); the author could be André Galloo or Alexandre Guillaume. Henri du Passage, ‘Bulletin social’ (Études, 5 February 1914, 138: 374–93); the Taylor system is discussed in section III (pp. 387–93). – Eds.] 30 R.S., 1 March, p. 323. [This reference is inaccurate. Charles de Fréminville alluded to Taylor’s eyesight problems only in the discussion which followed his speech to the Société d’Économie Sociale (‘Le système Taylor et l’organisation scientifique du travail dans les ateliers’, La Réforme Sociale, 16 March 1914, 7th series, 7: 407). – Eds.] 31 Principes, pp. 61–66. [Cf. pp. 48–53 of the original edition. – Eds.] 32 Mr Le Chatelier (Bull. de la Soc. des Amis de l’Éc. Polyt. No. 13, p. 20) mentions that the inspection and cleaning expenses dropped from 310 FF to 55 FF and that the unemployment, consecutive to this operation, was reduced in the same proportion. [See p. 220 of the version published in the Revue de Métallurgie – Eds.] 33 Principes, pp. 114–115. [An inaccurate reference by Potron. As a matter of fact, the statement was not made by Taylor but by Henry Le Chatelier, in the Preface (p. 3) to the French translation. – Eds.] 34 According to Mr Le Chatelier’s testimony (Pref. of Principes, p. 1) and that of Mr de Fréminville (R.S., No. 77, p. 323). [The citation is taken from de Fréminville’s text. In his Preface to the French translation Le Chatelier wrote that Taylor’s ‘discovery has revolutionized the whole mechanical construction’. – Eds.]

The scientific organization of labour 149 conceived the principles of his system of scientific organization of labour (scientific management); these were the topics of two lectures he gave at the [American] Society of Mechanical Engineers, entitled ‘A Piece Rate System’ and ‘Shop Management’.35 Now retired in the neighbourhood of Philadelphia, Mr Taylor is no longer active in implementing himself his system in the factories. He ‘devotes his time to those who have made this instalment their speciality, and to show to the company heads the steps it is necessary to follow during these changes’.36

II Mr Taylor has seen the results of his discovery of high cutting speed steel simultaneously with those of the introduction of his method in many factories. It is to the principles of that method and its results that, by far, he attaches the utmost importance.37 He states that presently 50,000 American workers live under this regime, that their wages have increased by 30 per cent to 100 per cent, that the productivity and prosperity of firms have doubled, that wherever the system has been applied suitably, it has put an end to conflicts and achieved a good relationship between employers and employees.38 The ‘Taylor System’ thus deserves serious study. It seems to me that the best way to proceed is to take the inventor himself as a guide, i.e. to analyse and comment on his ‘The Principles of Scientific Management’.

Basic postulates It is the common interest of employers, employees, [and] consumers that production be plentiful. It is the common interest of employers and employees that the productivity of labour be high. These two statements are opposite to rather widespread ways of seeing. The producers’ associations are prompt to restrict production to maintain prices. The workers’ unions restrict the productivity of their members’ work to avoid unemployment, and also because, due to the defective systems generally adopted (day work or piecework), they have no immediate interest in producing more,

35 [Frederick Winslow Taylor presented his paper ‘A Piece-Rate System, Being a Step toward Partial Solution of the Labor Problem’ at the June 1895 meeting of the American Society of Mechanical Engineers, and ‘Shop Management’ at the June 1903 meeting. – Eds.] 36 Principes, pp. 139–140. [The correct page number is 149. The French translation ends with a ‘Note by the author’, which is not in the original edition. It includes the following statement (our translation): ‘The author invites those who are interested in the issue [of scientific management] to visit him when they will be in the neighbourhood of Philadelphia. He will show them the details of the organization such as it is practised in the various factories of this city. The author, who devotes the greatest part of his time to this cause, considers these visits to be an honour and not an indiscretion.’ – Eds.] 37 Pref. of Principes, p. 2. [This refers to Le Chatelier’s Preface to the French translation. – Eds.] 38 Principes, pp. 46–47. [Cf. p. 28 of the original publication. – Eds.]

150 The scientific organization of labour [and] sometimes it is exactly the contrary. It seems however that Mr Taylor is right in theory. It is in everybody’s interest that the total bulk of goods to share be considerable; in any case, it is not by diminishing it that the sharing, which gives birth to the opposition of interests, can be eased. To really bad solutions which consist in making everybody’s share insufficient or in increasing the shares of some to the detriment of others, one may prefer the solution which consists in increasing the bulk to share. It is a more reasonable, more advantageous, and, it must be said, more Christian solution, after all more in keeping with the obligations of labour and charity imposed upon men.

Basic principles of labour organization In the present organization, that Mr Taylor intends to modify, each worker’s professional knowledge is made of empirical recipes that tradition has taught to him and that he has perhaps improved a bit, if he is clever. Almost never the senior executive intervenes in the details of labour and worries about the question of how to put together and combine these details in the most advantageous way. This is generally an intricate problem, therefore outside the competence of the worker; in any case its solution would require reflections, experiments and trials incompatible with the fulfilment of everyday’s work. As opposed to this approach, the basic principle of the Taylor system is: To share the work and the responsibility between the Management and the workers, the former looking at all which exceeds the competence of the latter. And the first three consequences, which imposes themselves to the Management, are: 1 First to make a scientific study of every elementary operation that the worker must perform, in order to constitute a truly practical science of the job. 2 To communicate that science to the worker by a practical teaching, instead of letting him learn his job as he can. 3 To follow each worker closely to help him to execute the work according to the scientifically determined method, instead of letting him work as he wants; then to give him an immediate participation of the advantages of the new method.39 It is difficult not to admit the rightness of the basic principle. A parallel imposes itself, apparently, with the principles common to various Catholic social schools, specially if, for the slightly administrative, slightly Anglo-Saxon, term ‘Management’, one substitutes the more fatherly, more benevolent, more French,

39 Principes, p. 53 and passim. [These principles are outlined on pp. 52–3 of the French translation, and on pp. 36–7 of the original edition. – Eds.]

The scientific organization of labour 151 patron.40

term One might even believe that there is nothing really new there; but the development of the three consequences shows that Mr Taylor demands much more than what is done today. 1◦ Scientific study of the job. Such a study exceeds by far the worker’s competence. However, only this study can help to estimate the productivity obtained, and decide if it is possible to improve it and how this must be done. Therefore, one will first proceed by analysis, decompose every operation into elementary moves, observing the time necessary to carry out each of them, the corresponding wear and tear of the various tools, specially that special wear and tear which is the worker’s tiredness. Then one will make a synthesis of the elements thus obtained, and one will determine, by making successive variations in the conditions of the experiment, the best way to make the operation, the one which must combine, in the most advantageous way from the point of view of productivity, machine work and human work, the one which must ensure the highest speed and avoid absolutely any overworking, any tiredness that the normal and planned rest could not repair steadily. It is by this aspect of the scientific study of jobs that the Taylor system relates to the previous or contemporary works quoted above. Let me mention its contact point with my own researches.41 I denote A1 , A2 , . . . , An the various economic goods (products strictly speaking, transports, services), A1 , A2 , . . . , An their respective units, C1 , C2 , . . . , Cp the various categories of consumers (bosses, managers, department heads, engineers, foremen, employees and workers of different specialities). I denote Ai or Ak anyone of the economic goods and Ch anyone of the categories of workers. The scientific study of the profession concerns first the time required by each operation. It lets us know exactly how many days of labour, in an establishment producing Ai , the workers of category Ch perform to produce one unit Ai ; I denote that number by tih . The scientific study of the profession must also concern the consumption of the economic goods (destruction of materials, wear and tear of materials, transports, use of ‘services’) implied by each operation. It will therefore let us know exactly, in each establishment producing Ai , how many units Ak of each economic good Ak are consumed to produce the unit Ai . I denote that number by aik .42

40 [The French word patron comes from the Latin words patronus (protector) and pater (father). Depending on the context, we translate it by ‘boss’, ‘employer’ or ‘head’. – Eds.] 41 See Mouvement Social, 15 April 1912, pp. 295–305. [See sections 6–12 of A11/Chapter 5. – Eds.] 42 See, in the Mouvement Social dated 15 April 1912, the table of the approximative values of these coefficients for a two-kilogram loaf of bread. In a study entitled: Contribution mathématique à l’Étude de l’Équilibre entre la Production et la Consommation, published in the Procès Verbaux et Documents Annexes de l’Assemblée générale de l’Association internationale pour la Lutte contre le Chômage (1914, Paris, rue de Babylone, 34), I have given (pp. 169–171) some of the coefficients relative to a ton (1,000 kilograms) of B powder. [The first paper is A11/Chapter 5; the table is in section 6. The second paper is A14/Chapter 9; the coefficients can be found in section III. – Eds.]

152 The scientific organization of labour Then if αk is the selling price of the unit Ak and σih the daily wage of a worker of category Ch in an establishment producing Ai , the cost price of the unit Ai is given by ai1 α1 + · · · + ain αn + ti1 σi1 + · · · + tip σip The scientific study of the job leads to the determination of the practical conditions which give the most advantageous cost price. Incidentally it lets us know the values definitely adopted by the coefficients aik , tih . Similarly, if a household of category Ch is run according to the principles of the scientific method, it will know exactly how many units Ak of each economic good Ak it consumes every year. I denote that number by bkh . The yearly cost of living of this household is given by b1h α1 + · · · + bnh αn The scientific study of the household’s habits leads to the determination of the practical conditions which give the most advantageous cost of living. Incidentally it lets us know the values definitely adopted by the coefficients bkh . These remarks will be used later on. This scientific study of the job is an industrial laboratory work, preliminary [and] essential to the introduction of the system. In general, this study will show that it is possible [and] easy to obtain from an ordinary, but educated and trained worker, a productivity much above the productivity obtained up to now. Some striking examples will be given below. A new organization for the preparation, supervision and payment of work is therefore necessary to raise the productivity and maintain it at its normal level, to obtain from each worker the fulfilment of the daily task thus set. It is that organization, and not the scientific study of labour, which is the truly essential characteristic, the true novelty of the Taylor System. It is very important, for a journal devoted to social action, to warn against this serious misinterpretation. In many places of his book,43 Mr Taylor states his main four principles, though following the order of execution rather than the order of logical subordination I have adopted. In a very short section entitled ‘Method[s] of scientific study of labour’,44 he states some general rules to follow in that study. One should be wary, however, not to confuse these rules with the principles of the method of scientific organization of labour.45 Mr Taylor also warns us not to confuse the mechanism 43 Principes, pp. 52, 61, 97, 125. [The pages 52–3, 61, 97 and 125 of the French translation correspond to pp. 36–7, 47–8, 85–6 and 114–5 of the original publication. – Eds.] 44 Principes, pp. 125–129. [The English edition does not have a title for this section, but the text corresponds to pp. 115–9. – Eds.] 45 Cf. Mouvement Social, 15 March 1913, p. 267. [See note 29 above. – Eds.]

The scientific organization of labour 153 of organization with its essential principles, the same mechanism being able to produce happy or disastrous results according to the case.46 As elements of this mechanism, he specifically quotes the study of time and its immediate application, the unification of the movements accomplished by every worker for every species of labour. He shows by means of examples that these elements can be used in a way contrary to the essential principles of the system, and then the results are disastrous. 2◦ Worker’s training. One must start with a psychological training, and convince each worker individually to adopt the method of labour, of which the scientific study has planned the details.47 Explanations, striking examples, the prospect of a less tiring, less long, better paid job must lead to the result. In no case, it must be imposed by force. Only then one can start the technical training, the practical apprenticeship of the labourer, who will have to work under an instructor’s supervision. If the training fails, if the worker does not succeed in reaching the planned productivity, one must conclude that he does not have the skills normally required for the corresponding job. The basic principle concerning the best productivity then stipulates not to dismiss the worker purely and simply, but to employ him in other works. 3◦ Management, control, payment of the trained workers. The goal is no longer only to ascertain and punish their defects, it is first to help them, to the extent they need, in fulfilling the planned task. They are given written and detailed instructions, resulting from a careful preparation of work done by the appropriate service. Next to them, in the workshop itself, are the agents specially in charge of explaining these instructions, of showing them, if necessary, the practical application, of putting promptly at their disposal any necessary equipment, of controlling the work in progress, finally of vouching for the achievement of the stipulated task. As for the payment, the worker must, in any case, receive his wage which is indifferently determined by the day or by the task, and moreover, receive an important daily extra when he accomplishes the normal task. Mr Taylor, as an experienced psychologist, insists a lot on these two points: to make each worker know in advance the normal daily task; – this is a type of yardstick which allows the worker to estimate his progress by himself, to know by himself if he has succeeded; – then, to give an immediate advantage, on which he can count in a certain and definitive way, to whom accomplishes the normal task.

46 Principes, p. 137. [Cf. pp. 128–9 of the original edition. – Eds.] 47 ‘(…) only one workman at a time should be dealt with at the start. Until this single man has been thoroughly convinced that a great gain has come to him from the new method, no further change should be made.’ (Principes, p. 139). [Cf. p. 131 of the original edition. – Eds.]

154 The scientific organization of labour

Mechanism of scientific organization These are the basic principles of the system. Their implementation can be achieved in many different ways. Surely, there will often be an advantage to remain close to the type, resulting from a long experience, described by Mr Taylor and his associates. But it matters a lot, as already told, not to confuse the essential principles of his system and the mechanism of its application. Besides the usual general departments of Management, the general administration of the workshops comprises four special sections: the time, wage and cost price service, also in charge of the scientific studies; the handling and distribution service, which settles the circulation of pieces in the various workshops; the instructions preparation service which, using the results of the scientific studies, writes down the instructions and prepares the tasks; the staff and discipline service. Each of these services, according to the size of the workshop, is ensured by one or several employees. The particular management of each workshop also comprises four services, each of them, according to the size of the workshop, being ensured by one or several agents. The labour distribution service, which carries out the instructions of the handling service and supplies the workers with instructions, materials, tools, models [and] drawings; the instruction service, which ensures the practical implementation of the written instructions given to the workers, trains and helps the yet unskilled workers; the control service, which explains the written instructions if necessary, examines the work done and vouches for it; the maintenance service, which ensures the good working of the machines and all their accessories. The agents of these four services therefore act as foremen; but they do not give any order as such to the workers; they acquaint themselves with the detailed instructions from the management; they teach and help to do the work according to these instructions.48 It is useless to go into the details of the functioning of that organization. But it is worth noticing what the relationships between the workers and the Management become. At first sight, it seems that the worker is about to be condemned to some type of speedy and exhausting automatism. The reality is completely different. The scientific study of the job is made in order to determine experimentally the productivity that a normally skilled worker can obtain without tiredness;49 it precisely prevents the overworking which results from an arbitrary demand for a higher and higher production.50 From a higher point of view, idleness, lazy and slow labour are generally considered as bad for the soul. The Taylorist worker

48 This account of the main features of the mechanism essentially follows the Principes and Mr Le Chatelier’s articles. 49 C.T.M., p. xii, col. 2. 50 At the Tabor factory, which is in a way the model factory of the system, all men are busy but none hurries up (C.T.M., p. iv, col. 1). One does not want at all that the time devoted to some work be shorter than the scheduled time; the control is then more severe (C.T.M., p. xii, col. 1).

The scientific organization of labour 155 finds himself trained to the practise of many natural virtues; and, if he wants to become a good Christian, he will not have to change his work habits at all, contrary to his workmate from the CGT.51 As for the criticism of automatism, one may understand it from the worshippers of freedom for itself, from those who think it is complete only when it includes the possibility of doing wrong. Surely, Taylor does not let the worker free to laze around, to lose his time in useless gropings, to use defective techniques according to his fancy; but the riding instructor does not let the novice horseman take the position he wants, the instructor non-commissioned officer does not let the conscript handle his arm as he wishes, the teacher does not let the pupil work according to his fancy. There is, in any operation, whatever it is, a purely mechanical part. That part, more considerable in manual work, it is certainly advantageous to succeed as soon as possible in doing it mechanically, without thinking about it, in time and with minimum tiredness, saving one’s attention to give it at the exact moment. These advantages certainly offset the inconveniences of subjection, besides restricted to the training period. There is here an unexpected application of the principle that the submission to rule and order ultimately develops and perfects freedom. The labourer, doing his work with much more ease, has the feeling to have become a man of a superior category, which is indeed true.52 On the other hand, his real independence has increased. Sheltered behind written and precise instructions, he does not have to do with ‘a corporal who is always right’; he is not exposed to the risk of ‘being caught without knowing why’; he has the right to let the competent instructor come to his machine, every time he meets a difficulty in the understanding or implementation of the received instructions.53

The payment of labour As it has been said, the system does not advocate any specific type of payment. Its characteristic is to reward the achievement of the daily task by an important, immediate and sure advantage, but which can be given under very diverse forms. Mr Taylor and his associates prefer what they call the ‘bonus’ system: the worker is paid per hour, and his wage is increased by 20 per cent to 30 per cent when he fulfils his task within the time set in advance. The instructor foreman receives a 0.50 fr daily allowance per worker earning the ‘bonus’; that allowance is increased up to 0.75 fr per worker when all his workers earn the ‘bonus’. In other cases, the labour is paid piecework, with a bonus for any ‘piece’ made in the stipulated time. In all cases, the wage rate itself must be progressively increased when the

51 [CGT: Confédération Générale du Travail, the main French workers’ trade union, founded in 1895, initially of a revolutionary and anarchist inspiration and which evolved towards socialism. – Eds.] 52 C.T.M., p. xii, col. 2. 53 C.T.M., p. xii, col. 1.

156 The scientific organization of labour worker fulfils his task usually.54 With workers of a certain mentality, for instance the one that prevails in the Cleveland region,55 it is the only advantage which is practical to give them. So the essential is to let the worker benefit partly from the gains resulting from his improved labour method; the way to give him that part is an incidental detail. And if the prospect of a higher gain is one of the motives that may decide the worker to use the new methods, it would be wrong to think that it is the only or even the main motive to use in all cases. Such are the main lines of the Taylor System, according to his author and his most authorized interpreters. Its principles are undoubtedly attractive and well defensible. It is interesting to see now the lessons that experience has taught them.

B – The lessons of experience V Contrary to what one might think, American industrialists have received Mr Taylor’s ideas with a lot of reservation. Many of them, having gone through long years of experience and success, found it hard to admit that they could improve their business and had an interest to reform. The labour unions, which in America like anywhere else, have unfortunately hardly any other programme besides the class struggle, are understandably irrevocably opposed to a system mainly founded on the strict collaboration of the workers and the Management. The stopwatch timing and overwork which in their view result from it are the objects of their attacks. When the system was introduced in the Watertown Arsenal, they obtained the appointment of a commission of enquiry, of which the reports, submitted in July 1911 and March 1912, have made a lot of noise.56 They have also asked the Senate to make a law prohibiting stopwatch timing and bonus payments in government activities. The [American] Society of Mechanical Engineers57 points out that the persons who are interested in this question are divided into two camps, enthusiastic partisans on the one hand, and energetic opponents on the other. The result of this situation is that, both from the side of the employers and from the side of the workers, the applications of the Taylor system are undoubtedly under very close scrutiny. In these conditions, it is already an assuredly remarkable fact that the objections of the adversaries bear only on

54 C.T.M., p. xiii, col. 2. [What de Fréminville wrote was that the scientific management engineers acknowledged that the increase of the worker’s wage must be consolidated; the fixed part, i.e. the hourly rate, must be increased whereas the variable part, i.e. the bonus, should remain accessory. – Eds.] 55 C.T.M., p. xii, col. 2. 56 C.T.M., p. ix, col. 2; p. x, col. 1; p. xiv, col. 2. – R.S., No. 78, p. 405, Génie Civil, LXII, No. 24, p. 474; No. 26, p. 514. [In the discussion following de Fréminville’s speech to the Société d’Économie Sociale, neither the commission of enquiry nor its reports were mentioned explicitly. – Eds.] 57 ‘The Present State of the Art of Industrial Management’ (C.T.M., p. x, col. 1).

The scientific organization of labour 157 principles, that no other unsuccessful cases are mentioned besides those in which the method has been applied contrary to its spirit, without taking the indispensable precautions, in conditions against which Taylor or his collaborators have strongly advised. Moreover, the specific or general facts brought up as examples by the partisans of the system do not seem to have been contested. Here are some of them. The pig iron loaders of the ‘Bethlehem Steel Company’ each handle 12 1/2 tons per day. One of them agrees to test the method indicated by the scientific study. Under the supervision of an instructor who regulates his pace and his rests, he handles, from the first day and without getting tired, 47 1/2 tons, and earns 9.25 fr. per day instead of 5.75 fr.58 One had no difficulty at all finding men capable of doing this work, be it in the old team (1 out of 8), in the factory, or in the neighbourhood. Most of the old loaders were not dismissed, but immediately assigned, in the same company, to tasks which were more appropriate for them.59 The results obtained, over a period of three years, with regard to the ore shovelers of the same company are indicated by the following table:60

Number of workers Daily tonnage per man Daily wage Cost price per ton

Old system 400 to 600 16 5.75 fr. 0.360 fr.

New system 140 59 9.40 fr. 0.165 fr.

In addition, general sobriety, rise of the standard of living and at the same time economies, [and] excellent relationships with the instructors–foremen.61 By numerous but very simple tiny modifications to the way in which bricks are laid, Mr Frank Gilbreth, one of Mr Taylor’s collaborators,62 obtained a productivity of 350 bricks per man and per hour.63

58 [‘And throughout this time he averaged a little more than $1.85 per day, whereas before he had never received over $1.15 per day, which was the ruling rate of wages at that time in Bethlehem.’ (The Principles of Scientific Management, p. 47) – Eds.] 59 Principes, pp. 55–61, 66–75. [Cf. pp. 40–8 and 53–64 of the original publication. – Eds.] 60 [Potron added ‘fr.’ to the figures for the daily tonnage per man, which is obviously a misprint. For the daily wage Taylor’s figures are $1.15 and $1.88, and for the cost price per ton $0.072 and $0.033 (The Principles of Scientific Management, p. 71). – Eds.] 61 Principes, pp. 75–85. [Cf. pp. 64–77 of the original publication. – Eds.] 62 [Frank Bunker Gilbreth, Sr. (1868–1924) was also a pioneer of scientific management techniques, and more a rival than a collaborator of Taylor. – Eds.] 63 Principes, pp. 85–97. Mr Taylor effectively mentions the number 350. [Cf. pp. 77–85 of the original publication; the figure of 350 bricks per man per hour is mentioned on p. 81. – Eds.]

158 The scientific organization of labour A bicycle ball inspection workshop is scientifically reorganized by Mr Sanford E. Thompson,64 under the general superintendence of Mr Gantt, both collaborators of Mr Taylor. The working day is reduced from 10 1/2 to 8 1/2 hours, and interrupted by periods of 10 minutes of rest after each hour and half of consecutive work, in order to avoid strain. Yet 35 female workers do the work which used to be done by 120, with an accuracy which is 2/3 higher; their wages are doubled; [and] they have paid holidays on Saturday afternoon and on two consecutive days per month.65 The small factory of Tabor, in Philadelphia, performs quite a variety of work: it constructs casting machines and other devices, by the unit or in very small series. It used to have a workforce of 150 labourers supervised by three foremen, and it went bankrupt. Now it has a workforce of 90 labourers only; the management consists of 28 employees; the factory is in full prosperity.66 Above I have quoted several big businesses where the Taylor system functions satisfactorily. The Commission of the [American Society of] Mechanical Engineers counted more than sixty of them in 1912.67 Mr Taylor evaluates at 50,000 the number of American workers living – and living happily – under this regime. It is difficult to verify this figure. According to Mr de Fréminville ‘certain plants have applied the method, but secretly, and under different names; many others, in particular the big factories of machine tools, are in general in a good situation and do not feel the need to change. Without asking the help of specialists, without so far proceeding to important reorganizations, their managers nevertheless gradually benefit from the new principles. Mr Taylor considers them to be followers rather than opponents; he confidently waits the moment when they will be led to apply his methods in a complete way.’68 Here is a very important point, which according to Mr Taylor results from his personal experience of thirty years; no strike, no conflict has been observed anywhere the method has been applied in conformity with his principles, even

64 [Last name misspelled as ‘Thomson’ by Potron. Sanford E. Thompson (1867–1949) was a consultant who collaborated with Taylor. – Eds.] 65 Principes, p. 98. [Cf. pp. 95–6 of the original publication. It must be noted that on the subject of the two days of paid holiday per month, Taylor expressed himself more cautiously than what the French edition suggested: ‘All young women should be given two consecutive days of rest (with pay) each month, to be taken whenever they may choose. It is my impression that these girls were given this privilege, although I am not quite certain on this point.’ (p. 96) – Eds.] 66 C.T.M., p. v, col. 1. On p. xii, col. 2, one will find interesting examples, concerning spinning and winding workshops, cited by Mr Gantt. [The examples from Gantt, illustrated by six graphs, are on pp. xii–xiv. – Eds.] 67 [The majority report of the sub-committee on administration drew up a list of 52 ‘industries in which labor-saving is used’ (‘The present state of the art of industrial management’, Journal of the American Society of Mechanical Engineers, November 1912, 34: 1617). – Eds.] 68 C.T.M., p. xi, col. 2. [Potron presented this as a quotation from de Fréminville’s article, but in fact it is a paraphrase of what de Fréminville wrote on pp. xi–xii. – Eds.]

The scientific organization of labour 159 during the critical period of the change of organization. Mr Le Chatelier appears a bit sceptic with regard to this statement; according to him: ‘It would be worthwhile to go and study on the spot a so surprising fact.’69 Mr de Fréminville, who has followed this advice, and has even been in touch with the American adversaries of the system, does not contest this statement. He does point out the very active opposition of labour unions; but ‘this opposition has appeared in an important way only at the Watertown arsenal’; and there ‘the declarations of the workmen to the Commission of inquiry have shown that they were not overworked and were satisfied with the conditions in which they were employed’.70 In order to obtain these good results it is nevertheless essential that the method always be applied in conformity with its principles, with lots of tact and dexterity. Mr Taylor, in the works cited above, Mr Gantt, in his book Works, Wages, and Profits71 [and] the Commission of [the American Society of] Mechanical Engineers, in its report ‘The Present State of the Art of Industrial Management’, insist a lot on this point. The change of organization must always be decided, with full knowledge of what is at stake, by those who ultimately bear the responsibility: owners, trustees, managers. Before any change, it is necessary that all the preliminary work (scientific study of the job, changes of the equipment, calculation of the wages, of the tasks, of the bonuses, [and] of the cost prices) has been completed and given final results. The role which each has to fulfil in the new organization must be clearly understood and frankly accepted by all, from the managers to the instructors–foremen. The understanding of the method must descend from the top to the bottom of the ladder, [and] everyone be concerned about convincing and instructing his immediate subordinates. In particular, as soon as the required efficiency has been determined, the entire management staff, from the top to the bottom, ‘must consider itself obliged to train the workmen up to the point necessary to obtain this efficiency, and provide them with the means to obtain it’.72 For all these reasons, the introduction of the Taylor system in a factory is a delicate task. If you do not compel yourself to follow the necessary steps, if you start from the doubtful results of an incomplete scientific study, if you neglect the psychological training of the management staff and the individual training of the workmen, in one word if you leave aside this or that essential principle of the system, you expose yourself to a failure. Experience has shown this more than once. It is also necessarily a work of long duration, which in an important establishment requires at least three or four years. Apparently this can explain why

69 Preface of Principes, p. 7. [This refers to Le Chatelier’s Preface to the French translation. – Eds.] 70 C.T.M., p. xiv, col. 2. [The first quotation is a paraphrase. – Eds.] 71 [Henry L. Gantt, Work, Wages, and Profits, New York, The Engineering Magazine Co., 1910. – Eds.] 72 Mr Gantt, C.T.M., p. xiii, col. 1. [Potron slightly changed the wording. – Eds.]

160 The scientific organization of labour the two highly skilled technicians whom we have taken as guides, Mr Le Chatelier and Mr de Fréminville, do not refer to any French or even European example. It seems that Mr Taylor’s ideas were practically unknown in France before 1912. If some establishments have started to apply them, they can hardly have gone beyond the preliminary period of scientific studies. Were the essential principles of the Taylor System at issue during the strike at the Renault automobile factory? That seems doubtful to me. There was talk about stopwatch timing. But it must be repeated that neither stopwatch timing nor even the scientific study of the job are the essence of the Taylor method. These are only indispensable preliminaries, more or less like the laboratory experiments which precede the installation and operation of a chemical plant. Not every attempt to increase efficiency by means of the scientific study of labour is necessarily an application of the Taylor system. And moreover, if a workman of normal ability happens to be overworked, or new work methods are forcibly imposed on an entire workshop, or the management is not concerned about training and educating its workmen, [and] about making the accomplishment of their tasks easier, one can affirm that this has nothing to do with the Taylor system. While the labour unions in America oppose the new methods of work, it seems that a great number of workmen of all categories make the very best of it. That is what the proponents of the system assert very firmly; the opponents do not contradict it by any precise fact. This is the opinion of Mr de Fréminville who has seen the situation on the spot. And according to him, if the Taylor methods could have been applied in America, a fortiori they can be in France.73 There is not a man more independent and more rebellious than the American workman, [who] surely [is] assiduous at work, but ambitious and fond of changing workshop. The French workman, unstable and also rebellious, has nevertheless deep down the habit of respecting the instructions and rules. Although he likes to work in fits and starts, he is clever, ingenious, and attaches himself to a firm where he is treated with justice.74 In France, like anywhere else, the diffusion of the Taylor methods will meet three main difficulties. One must expect a fierce opposition from the side of the CGT-type of unions; because the Taylor system, which brings together the workmen and their managers in a close cooperation beneficial for all, contributes necessarily to better relations between classes. On the side of the employers, there is hesitation in the face of rises in costs, in work, [and] in capabilities asked from the managing staff, [which are] precisely what this cooperation will require. This double difficulty, which partakes of the very essence of the system, appears likely to earn it the sympathetic and active support of all those interested in the Catholic social action. There, it seems, the employer would find the opportunity to display real management, [and] good and sound authority. And the unions with Catholic principles could help him in this task. Professor Bertrand Thompson of

73 C.T.M., p. xv, col. 1. 74 Ibid.

The scientific organization of labour 161 Harvard University, who has been one of Mr Taylor’s collaborators, is himself an advocate of this intervention by the unions.75 On 15 February 1913 in Chicago he said: ‘As before, their role will be to determine and to maintain a minimum level of pay and of standard health conditions in work. The difference is that, in the future, their efforts will be assisted instead of fought against by bosses, and that they will have a new duty: to convince their members to achieve the exactly determined fair working day.’76 Remains the third difficulty: a certain unrest concerning the economic consequences which a relatively generalized application of the system would entail. Some words have to be said about this.

VI It seems reasonable to admit that the Taylor methods will increase the prosperity of the factories that will have adopted them and of the workmen employed in these factories. However, since at constant or even higher production they substantially reduce the workforce, a certain number of workers will find themselves deprived of their jobs. It will most probably be the oldest, because [they are] the least capable of learning the new methods, [and] also the least capable of changing profession.77 Any development of mechanization, which increases production and diminishes the workforce, brings about the same phenomena. Of course one points out that the crises are only transitory, that because of increasing consumption industry develops and hires new labourers, and that in the end general wellbeing advances. But that requires quite some time, and in the meantime many can starve. One also makes the observation that the total amount of things necessary or useful for a decent life is largely insufficient, and that as a result it would not be inconvenient to increase their production. But still those who need them should not find themselves precisely deprived by unemployment of the means to buy them. One begins to grow weary of this progress which cannot come about without collisions, revolts, woes, [and] catastrophes.78 Would it not be possible to use the economic laws themselves to dampen these dangerous oscillations, as one uses the laws of hydraulics to regulate the course of rivers? To sum up, the generalized application of the Taylor method poses again, and in a more pressing way, two important economic problems: the problem of the relations between production, consumption, the distribution of labourers, [and]

75 [The name was misspelled as ‘Thomson’ by Potron. Clarence Bertrand Thompson (1882–1969) was a lecturer of manufacturing at the Harvard Business School. At about the same time as Potron wrote his article, Thompson published an interesting survey on scientific management: ‘The literature of scientific management’ (Quarterly Journal of Economics, May 1914, 28: 506–57). – Eds.] 76 C.T.M., p. xiv, col. 1. [Once again, Potron modified the text of de Fréminville. – Eds.] 77 Nevertheless, in a spinning workshop cited as an example by Mr Gantt, the young female workers have left, and the oldest have stayed (C.T.M., p. xiii, col. 1). 78 Cf. ‘Vers l’Ordre Nouveau’ by Mr Zamanski (Mouvement Social, 15 March 1913, p. 253).

162 The scientific organization of labour the standards of living; [and] the problem of the relations between the standards and costs of living, prices and wages. We have two completely distinct problems there, of which the first moreover conditions the second.79 I have shown it, in this Journal, by means of an exposition of the guiding ideas and the main consequences of my researches into these two problems considered from a mathematical point of view.80 Here I shall briefly summarize them in order to show that a generalized application of the Taylor methods will necessarily provide, for both problems, the elements of at least a theoretical solution. Let A1 , A2 , . . . , An be the various economic goods (products strictly speaking, transports, services), [and] A1 , A2 , . . . , An their respective units: let C1 , C2 , . . . , Cp be the various professional categories of labourers. I will denote by Ai or Ak anyone of the goods A1 , A2 , …, An , and by Ch anyone of the categories C1 , C2 , . . . , Cp . For each unit Ai it produces, the establishment producing Ai must consume of each Ak a well-defined number of units Ak , a number which I will designate as aik . For each unit Ai it must also make its workers of each category Ch work during a well-defined number of days, a number which I will designate as tih . The numbers ai1 , ai2 , . . . , ain , ti1 , ti2 , . . . , tip are in a way the operational constants of the establishment producing Ai ; to determine them, simple operations of industrial accounting are sufficient, provided it is kept with enough order and detail. Now, the scientific study of the job, [which is an] indispensable preliminary to the application of the Taylor method, leads precisely to an exact determination of these numbers. A given state of the economy is then characterized by the set of all numbers aik , tih . Likewise, the standard of living corresponding to the category Ch is characterized by the fact that an individual consumes annually of each Ak a well-defined number of units Ak , a number which I will designate as bkh . The standard of living in question is then defined by the numbers b1h , b2h , . . . , bnh . If now δi represents the number of units Ai annually produced, ηi the excess of the annual production over the consumption of Ai , πih the number of employees of category Ch of the establishment producing Ai , ωih their annual number of days out of work, [and] ωh the number of non-workers of category Ch , there exists a system of relations between all these numbers aik , tih , bkh , δi , ηi , πih , ωih , ωh . They are the equations of the problem of the relations between the levels of production and consumption, the distribution of labourers, [and] the standards of living. One sees that prices and wages do not appear in them.81 79 This is what is right in the ideas of those who are opposed to any limitation of production. 80 Possibilité et détermination du Juste Salaire et du Juste Prix (Mouvement Social, 15 April 1912, pp. 289–316). [See A11/Chapter 5. – Eds.] 81 These relations are obtained by writing down that, for each economic good, the annual production equals the annual consumption plus the excess, hence: δi = a1i δ1 + · · · + ani δn + bi1 h

= π1h + · · · + πnh + ωh ,

i

+ · · · + bip

p + ηi ,

i = 1, . . . , n; h = 1, . . . , p

The scientific organization of labour 163 If αi represents the selling price of the unit Ai , βi the corresponding benefit, σih the daily wage of a worker of category Ch of the establishment producing Ai , [and] γih his annual economies, there exists a system of relations between all the numbers aik , tih , bkh , δi , πih , αi , βi , σih , γih . They are the equations of the problem [of the relations] between the standards and costs of living, prices and wages.82 This second problem is conditioned by the first, since the levels of production (through the δi ) and the distribution of labourers (through the πih ) figure in them. All the quantities that appear in these relations are variables, but the speeds with which they vary are quite different. The aik , tih imposed by the state of the economy can remain constant for several years. The bkh which represent the standards of living can in theory vary in an arbitrary way, [but] in fact they always maintain a certain stability. As far as the other quantities are concerned, nothing stands in the way of sudden and great variations. If you limit yourself to a period of a few years, it is therefore natural to take the present values of the numbers aik , tih , bkh as given, and to let the formulas be used for the determination of the other quantities. That is nevertheless only a particular way of using these formulas. It seems justified by the fact that when a new invention occurs, it is with the old techniques [and] the old equipment that one must prepare the installations necessary for the exploitation of this new invention. It is also justified because the results of experiments [and] trials, even on an industrial scale, can be known well before the beginning of the proper exploitation. This happens in particular with the Taylor system. In order to fix ideas and simplify the question, let us assume that the Taylor system is rapidly generalized, without the occurrence, for some years, of an important new invention. As soon as the scientific studies of the jobs are completed, we will know exactly the numbers aik , tih . We will specify the numbers bkh in such a way that no standard of living would be inferior to what is at present considered to be convenient. The first system of formulas then determines the δi : what the production should be, and the πih : how the labourers should be distributed. The second system then determines the αi and the σih , what the prices and wages should be. In this way one knows in advance what must be for the general state to be satisfactory, that is to say, for the production of each and that, for the labourers of each skill, the available work-days equals the working days plus the days out of work, from which: N πih = δi tih + ωih ,

i = 1, . . . , n; h = 1, . . . , p.

N is the annual number of work-days. 82 These relations are obtained by expressing that, for each economic good, the selling price equals the cost price plus the benefit, hence: αi = ai1 α1 + · · · + ain αn + ti1 σi1 + · · · + tip σip + βi ,

i = 1, . . . , n

and that, for each labourer, the effective wage equals the cost of living plus the economies, hence: δi tih σih = b1h α1 + · · · + bnh αn + γih , πih

i = 1, . . . , n; h = 1, . . . , p.

164 The scientific organization of labour economic good to be at least equal to consumption, for the exchange price of each economic good to be at least equal to its cost price, [and] for the effective wage of each worker to be at least equal to his cost of living. The formulas determine a standard regime; they certainly cannot indicate what must be done to implement it; but by providing a precise knowledge of the result which must be achieved [and] by making it possible to compare at any moment what is to what should be, they can increase the effectiveness of the working of professional associations and corporative organizations, of precautionary institutions, of legal measures, [and] of international agreements.83 The scientific study of the job, an indispensable foundation of the Taylor system, aims to establish, not an immutable method of work, but the best method in relation to the techniques and equipment known at present. This method will be able to serve for a number of years; then the discovery of some new technique will compel [us] to change it. Likewise, the computations which I propose to carry out on the collection of results of the scientific studies of jobs aim to determine, not a final equilibrium state, [which is an] impossible thing, but a temporary equilibrium state, which corresponds to a given state of the economy. The economic regime thus determined will fit reality at least as long as the results of the scientific studies on which it is founded. Like these, it will only need to be updated from time to time. It seems to me that the scientific organization of labour in each establishment needs to be complemented by a scientific organization of the whole economic regime. Both are inspired by the same principles. Gathering and centralizing the data with regard to the numbers aik , tih , bkh , carrying out the computations, [and] communicating the results obtained, boils down to doing for the whole set of industrial and commercial labour, what in the Taylor system the scientific management office does for a single factory. The scientific organization of a factory is most certainly an operation which is complicated, costly, [and] apparently unproductive, but in general of a very high real return. The same will undoubtedly hold for a general organization of the economic regime. To sum up, for a sound appreciation of the Taylor system one must always keep in mind its four essential principles: the fundamental principle: sharing of work and responsibility between management (maximum part) and labourers (minimum part); the first three consequences, which specify the role and the duties of management: scientific study of the job; individual training of the labourer; assistance, control, [and] appropriate remuneration. The scientific study of the job, considered on its own, is an indifferent thing in itself. Its use can be excellent or appalling. It seems inaccurate to see in it the essence of the Taylor system and to present its other parts only as more or less ingenious palliatives. To the contrary: the essence of the system is to allow scientific study to serve the common interest of the boss, the worker, [and] the consumer. The careful examination of the method

83 [The question of whether calculation can indicate the way to implement a satisfactory regime was also examined in section 17 of A11/Chapter 5. – Eds.]

The scientific organization of labour 165 [and] the results already obtained in America definitely seem to show that its application to a specific factory, or even to the factories of a specific country,84 will be beneficial to their bosses, their workers, their customers. In the case of a generalized application, certainly the same can happen, theoretically and a priori. In practice, to avoid crises, an adequate pace will be necessary between the development of the various productions, the rise of the standards of living, and the movement of prices and wages. It seems difficult to obtain this pace by the free play of economic laws. In the absence of precise information on the relations existing between the variable elements, it seems almost impossible to obtain this pace by the functioning of precautionary institutions, by the action of professional associations, by legal measures, [or] by international agreements. However, as I have shown, a general application of the Taylor system will effectively enable us to know precisely the relations of a mathematical order which exist between the variable elements. One can reasonably think that this knowledge, as it has been explained, will allow to maintain between the many variations the pace required for a progressive increase of the general well-being. Let us moreover recall that according to Mr Taylor ‘It is not here claimed that any single panacea exists for all of the troubles (…). As long as some people are born lazy or inefficient, and others are born greedy and brutal, as long as vice and crime are with us, just so long will a certain amount of poverty, misery, and unhappiness be with us also. (…) Prosperity depends upon so many factors entirely beyond the control of any one set of men, any state, or even any one country, that certain periods will inevitably come when both sides must suffer, more or less. It is claimed, however, that under scientific management the intermediate periods will be far more prosperous, far happier (…). And also, that the periods will be fewer, shorter and the suffering less.’85 And if one wants to rise above the sphere of economic interests, it seems definitely the case that the Taylor system, the genuine one, does not contain anything which opposes it to the virtues of justice and charity, quite to the contrary, and that in a Christian work environment its introduction would be relatively easy. It appears also that in an environment where labour would be organized according to the essential principles of the Taylor system, the ground would be fertile for the development of Christian virtues.

84 Cf. Mouvement Social, 15 March 1913, p. 269. [See note 29. – Eds.] 85 Principes, pp. 42–3. [Potron referred in fact to pp. 47–8. His text deviates at several points from the French translation to which he referred, and that in its turn does not correspond exactly to the text of the original publication, which can be found on p. 29. Our text conforms to the English original. – Eds.]

11 On some conditions of economic equilibrium. Letter of M. Potron (90) to R. Gibrat (22)

Editors’ note The first sign of Potron’s renewed interest in economics came in 1935, when he published his note ‘Sur certaines conditions de l’équilibre économique. Lettre de M. Potron (90) à R. Gibrat (22)’ (Centre Polytechnicien d’Études Économiques. X-Crise. Bulletin Mensuel, 1935, Nos. 24–25: 62–5). It was a reaction to comments by Robert Gibrat on the talk ‘Le contenu économique des plans et le planisme’ given by Jacques Branger on 22 February 1935, both of which were published in a previous issue of the journal (Centre Polytechnicien d’Études Économiques. X-Crise. Bulletin Mensuel, 1935, Nos. 20–21: 5–13, 14– 15). Potron saw similarities between a matrix used by Ragnar Frisch in his articles ‘Circulation planning: Proposal for a national organization of a commodity and service exchange’ and ‘Circulation planning: Part III. Mathematical appendix’ (Econometrica, 1934, 2: 258–336; 422–35) referred to by Gibrat and the matrices which he himself had used in his economic publications more than 20 years before. Like so many of those who were active in the X-Crise group, Robert Gibrat (1904–1980) was a Polytechnician and therefore a ‘comrade’ of Potron (the numbers between brackets after Potron’s and Gibrat’s names refer to the year in which they entered the École Polytechnique). X-Crise was the French meeting place for those who were interested in the mathematical and econometric approaches to economic problems, as the reference to Frisch’s work testifies. * * * At the C.P.E.E. meeting of last 22 February, you have talked about a supply and demand matrix,1 considered by Frisch. There is another, very interesting, matrix that I think to have been the first ever to consider, at least for its application to economic problems.

1 [This is the only reference to ‘supply and demand’ in the work of Potron. – Eds.]

On some conditions of economic equilibrium 167 Let us denote by Ai (i = 1, 2, . . .) the various ‘products’ or ‘results of labour’; Ai a given quantity, called ‘unit’ of Ai ; Pi the firm or set of firms producing Ai ; di the number of Ai units produced yearly; di − fi the number of Ai units consumed yearly; ai the price of the Ai unit; bi the benefit made by Pi per Ai unit; cik the number of Ak units consumed to produce an Ai unit. It is of the matrix of the cik that I want to talk. Let us also denote by Ch the various standards of living; bhi the number of Ai units consumed yearly by a consumer with the standard of living Ch ;2 Eih the community of the employees of Pi with the standard of living Ch ; pih the number of these employees; qh the number of consumers who do not work and have the standard of living Ch ; Qh the total number of consumers, workers or not, who have the standard of living Ch ; tih the number of working hours that the community Eih must perform for the production of the Ai unit; wih the yearly number of unemployment hours of the community Eih ; N the maximum (legal) number that any worker can perform every year; sih the hourly wage of an employee belonging to Eih ; NSh the yearly cost of living of a consumer with the standard of living Ch ; Neih the total amount of the yearly economies of a worker in Eih . Between these various numbers, there exist six systems of relations: the first three are the equations of the production–consumption equilibrium; the last three are the equations of the prices–wages equilibrium. If one subtracts from the total production of Ai its total consumption, either industrial or domestic, one obtains the number fi of Ai units. This fact is expressed by the system di =

cki dk + k

bhi Qh + fi

(i = 1, 2, . . .)

h

2 [That magnitude was denoted by bih in Potron’s previous papers. – Eds.]

(1)

168 On some conditions of economic equilibrium By definition of Qh , we have pih + qh

Qh =

(h = 1, 2, . . .)

(2)

i

The community Eih could perform Npih hours of labour every year. It effectively performs tih di hours. The difference gives its number wih of unemployment hours; hence Npih = tih di + wih

(i = 1, 2, . . . ; h = 1, 2, . . .)

(3)

If one writes down that the benefit bi made on the unit Ai is the difference between the selling price and the cost price, one obtains3 ai =

cik ak + k

tih sih + bi

(i = 1, 2, . . .)

(4)

h

The economies represent the difference between the wage and the cost of living; hence, for any Eih taken as a group, di tih sih = Npih (Sh + eih ) (i = 1, 2, . . . ; h = 1, 2, . . .)

(5)

The yearly cost of living of a consumer with the standard of living Ch gives NSh =

bhk ak

(6)

k

Note that (4) and (5)–(6) are only ways of writing down the accounts of the firm Pi and the ‘average budget’ of a member of the community Eih , respectively. Clearly, the objective reality of these six systems cannot be questioned, nor that of the conclusions that a mathematical study will allow to draw from them. Among the various symbols that appear in these equations, none, by the very nature of things, can be negative, except the fi , bk , wih and eih . One can say that the set of values ascribed to these various symbols characterizes a given socioeconomic state. Such a state will be satisfactory and stable if the following conditions are met: The bhi must have ‘convenient’ values; The coefficients of overproduction fi must have positive values in order that the stocks are not exhausted, but not too large; The unemployment hours wih must be positive, but not too large; At last, it is absolutely necessary that the bi and eih never have negative values, which would lead the firms to go bankrupt and the workers to die from starvation. 3 [We have replaced cki by cik . – Eds.]

On some conditions of economic equilibrium 169 The possible action of human will on these diverse coefficients varies a lot with their category. The cik and tih are governed by the processes of fabrication, [and] the organization of the various firms. Acting on them is possible by means of new inventions, by ‘rationalization’; but this action will always be rather slow and, on the whole, rather limited. On the contrary the prices, the wages, [and] the size of the staff are not imposed by any physical necessity. They result from conventions which, at least theoretically, are free. Their modifications can occur very quickly. Whatever may be its practical application, one can, in any case, consider the following problem: assuming the cik , tih , bhi and N are known, how can the values of the other variables be determined in order to have a satisfactory and stable regime? Among the other symbols, one will distinguish the ‘principal’ unknowns, viz.: The di , that determine production; The pih , that determine the workers’ distribution; The ai , that is the prices; The sih , that is the wages; and the ‘secondary’ unknowns, viz.: The qh , numbers of non-workers; The wih , that represent unemployment; The fi , that represent overproduction; The bh , firms’ benefits; The eih , workers’ economies. As the system (1)–(2)–(3) only contains the first two groups of principal unknowns, the best is to start with it. It is the theoretical organization of Production. By means of mathematical combinations,4 one can derive from it a system which contains only the first group of principal unknowns; it has the form di −

Cki dk = Fi ,

(7)

k

where the coefficients Cki only depend upon the data, and the Fi are linear combinations of the secondary unknowns of the first three groups. Once the values of all the unknowns satisfying (7) are known, the system (3) will let us know the second group of principal unknowns, the pih .

4 One can see the detail of these combinations, explained on three equations, in my booklet: Les Équilibres production-consommation et prix-salaires (S.P.E.S. Editions, 17 rue Soufflot, Paris-5e). [This is a reference to an apparently never published preliminary version of the lecture series given by Potron in March–April 1937 (see U3/Chapter 14). – Eds.]

170 On some conditions of economic equilibrium Similarly, from the systems (4)–(5)–(6), one can deduce a system which contains only the principal unknowns of the third group. It has the form ai − k

Cik ak = Bi ;

(8)

the Cik only depend upon the data and the unknowns already determined by the systems (7)–(3), [and] the Bi are linear combinations of the secondary unknowns of the last two groups. Once the values of all the unknowns satisfying (8) are known, the system (5)–(6) will let us know the fourth group of principal unknowns, the sih . Let us notice that the definition of the Cik assumes that the di satisfying (7) are all positive (nonzero). In that case these di satisfy a system of the form di − k

Cki dk = Fi = fi +

bhi qh ,

(9)

h

the coefficients C [being] the same as in (8); they form the same square table, after exchange of rows and columns of the same rank. The coefficients C of (7) and C of (8)–(9) are all nonnegative scalars. It is the same for the Fi , Bi , Fi if the secondary unknowns are given nonnegative values. Besides, these right-hand sides are all zero if all the secondary unknowns are given the value of 0. Here enters a theorem of pure Mathematics.5 Let us call characteristic root of a square table cik the (always real) root of maximum modulus of the equation in s obtained by setting the determinant | uik s − cik | (uii being 1, and uik being 0 if k ≷ i) equal to zero. Then, for the existence of positive values of the xi giving positive values to all the expressions xi − cik xk , it is necessary and sufficient that the characteristic root of the table cik is < 1. If it is equal to 1, one can only assert that there exist positive or zero values (not all zero) of the xi giving values 0 to all the expressions xi − cik xk . If the characteristic root is < 1, the system of equations xi − cik xk = bi , where the bi are arbitrary positive scalars, determines positive values for all the xi . The application of these theorems is immediate. If, in (7), the characteristic root of the table Cki , which like the Cki , only depends upon the cik , bhi , tih and N , is > 1, it is impossible to satisfy (7) with positive values of all the unknowns. If it is < 1, the secondary unknowns qh , wih , fi can be given arbitrary positive values. The Fi will be positive; and (7) will determine positive values for the di . As these values satisfy (9), if the Fi are not all zero (which would occur if the fi and qh had all received arbitrarily the value 0), it is certain that the characteristic root of the table Cki is < 1. Then the other secondary unknowns bi and eih can be 5 These theorems are due to: Perron, Math. Ann., t. 44, p. 261; Frobenius, Sitz. Acad. Berlin, 1908, p. 471; Potron, Ann. de l’Éc. Norm., t. 30, 1913, p. 53. [See note 1 of A9/Chapter 3 for the exact references to the work of Perron and Frobenius. Potron’s article is A13/Chapter 8. – Eds.]

On some conditions of economic equilibrium 171 given arbitrary positive values. All the Bi will be positive, and (8) will determine positive values for all the ai . Therefore, all depends upon the characteristic root of the table Cki . If it is < 1, it is theoretically possible to organize production in a satisfactory way, involving however by necessity some overproduction, some unemployment, [and] some number of non-workers. Assuming this organization implemented, it is theoretically possible to set up a satisfactory regime of prices and wages, with some benefits and, for the workers, some economies resulting from the excess of the effectively earned wage over the cost of living. Such a regime will remain satisfactory as long as the conditions represented by the cik , bhi , tih will not change. The discovery of a new industrial process can considerably modify some of the cik or tih . One must then distinguish two periods. During the first, one manufactures, with the help of what already exists, the new machines, the new equipment. This is a production which, not being intended either for current consumption or for current maintenance, constitutes a genuine overproduction. They are the fi which take high values. The regimes, which were satisfactory before this period, may cease to be so throughout it. The second period is the period of utilization of the new machines, the new equipment. The satisfactory regimes of the first period may cease to be so during the second, for two reasons: first because of the modification of certain cik or tih , then because the fi , huge during the equipment period, have disappeared; there only remains to maintain what exists. This partly explains why the technical improvements, which should provide more well-being to men for less total work, are often on the contrary a cause of miseries and ruins. One often speaks about economic laws, rather vague for that matter, which revenge themselves on those who ignore them. I think that the omission of six systems of equations is much more dangerous. Each firm Pi knows its own cik and tih . But there is nobody to write down and solve the system (7). Everyone takes his di in a haphazard manner, and one is quite surprised to have enormous Fi , which go with a large amount of unemployment or an important overproduction or, on the contrary, negative Fi , which is not less disastrous. If, by miracle, a suitable organization of production has been reached, the difficulties begin again for prices and wages. Pi wants first to salvage its benefit bi . If it increases its price ai , this increases either the cost of living of its employees, or the cost price of other commodities. It is the blind race to the rise between prices and wages. 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 (bhi ) and the state of your industry (cik , tih ), [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

172 On some conditions of economic equilibrium the principal unknowns, because it is easier. It is very likely that one does not draw one of the favourable combinations. Let n be the number of equations (7) or (9). Let us consider in an n-dimension space, the point (d) of coordinates d1 , . . . , dn , and, in another space, the point (F) of coordinates F1 , . . . , Fn . There is a one-toone correspondence between these two points. But when (F) moves in the whole region of its space where the Fi are all positive, it may occur that (d) only moves in a very small part of the analogous region in its space. The thing is clear for n = 2. Consider the two equations x − ax − by = u, y − a x − b y = v The coefficients are positive. The characteristic root is the greatest root of the equation (s − a)(s − b ) − ba = 0, of which the discriminant (a + b )2 − 4(ab − ba ) = (a − b )2 + 4ba is > 0. For this root to be < 1, a necessary and sufficient condition is (a + b ) < 2, (1 − a)(1 − b ) − ba > 0. It is easy to see that these various conditions imply a and b < 1. Any point (x, y) such that u and v are > 0 must be below the half-line y = (1 − a)x/b, and above the half-line y = a x/(1 − b ), both of them in the first quadrant. The angle of these two half-lines has a trigonometric tangent, the expression of which is [(1 − a)(1 − b ) − ba ]/[(1 − a)a + (1 − b )b]

If the characteristic root is close to 1, the numerator is close to 0, [and] the favourable region occupies only a small part of the first quadrant. If the characteristic root is exactly 1, the two half-lines coincide. If the point (x, y) is located on this unique half-line, u and v are zero. If not, they always have opposite signs. One may also build another square table, of which the elements only depend upon the cik , tih and bhi . The characteristic root of this second table has the same position, with regard to the scalar N , as that of the first table with regard to scalar 1. Furthermore, this 2nd characteristic root has an interesting interpretation: it is the average number of hours that each labourer must perform every year in order that the production thus obtained just suffices for what the economy consumes for the needs of the workers alone. Another combination6 of the established formulas gives qh NSh +

fi ai =

pih Neih +

di bi

(10)

If production just balances consumption ( fi = 0), [i.e.] when the whole production di has been absorbed, the last term represents the total of the benefits made by the firms. The formula then shows that the sum of the firms’ benefits and the workers’ economies is exactly equal to the non-workers’ cost of living. 6 See my booklet: Les Equilibres …, or my Mémoire in Ann. de l’Éc. Norm. [See note 4 above and A13/Chapter 8. – Eds.]

On some conditions of economic equilibrium 173 If there is overproduction of some commodities, for instance at the end of an ‘equipment period’, formula (10) can be written pih Neih +

(di − fi )bi −

qh NSh =

fi (ai − bi )

(11)

Since the overproduced commodities are not yet sold, the second term on the left-hand side represents the sum of the effectively made benefits. The right-hand side represents the ‘cost price’ of the overproduced commodities. In that case, the sum of the effective benefits and the workers’ economies exceeds the nonworkers’ cost of living; but the difference represents exactly the cost price of the overproduced commodities. One now sees which difficulties will be met if, for instance, the producers of a new machine absolutely want to make a benefit on its first sale. To avoid perturbations, it would be necessary that the producers of the new equipment sell it at its cost price and that, as soon as the new equipment is working, a new regime of production, prices and wages be established if necessary, fitting with the new conditions of the industry and, in particular, with the maintenance alone of the new equipment. All these considerations are somewhat discouraging. It is a scant consolation to say to oneself: the probability of finding by chance a satisfactory regime of production, prices and wages is so low that the encounter of such a regime is almost impossible. The only rational way to proceed, [viz.] to solve systems (7) and (8), is not implementable in practice, given the huge number of equations, equal to the number of all the things that are produced in the world. Certainly, many cik are zero. If one had started the calculations around a thousand years ago, there were many less equations at that time. All in all, the updatings required by each new invention are quite light. Only some coefficients are modified. One would have enough time to finish the calculations before it is put in operation. In any case, there appears to be no reason to incriminate our present table of the Cik . The economy could surely produce the quantities able to meet all the consumers’ desires, in spite of the decrease of N (the 8-hour work-day). There has undoubtedly been an overproduction of useless objects. They remain in possession of their producers; the money that represents their value (formula 11) is sleeping in the banks or has lost its value, as a result of various manipulations. It has not gone to those who would have needed it to buy objects of vital importance. Please forgive this long letter, my dear Comrade, and trust in my dedicated feelings. Potron (90)

12 On the economic equilibria

Editors’ note In July 1936 Potron attended the International Congress of Mathematicians in Oslo, as a member of the delegation of the Société Mathématique de France. He presented two papers at the conference: ‘Sur l’irréductibilité de certaines intégrales Abéliennes aux transcendantes élémentaires’ in the Section ‘Analysis’, and ‘Sur les équilibres économiques’ in the Section ‘Probability calculus, mathematical statistics, insurance mathematics and econometrics’. Succinct versions of both papers can be found in the conference proceedings (Comptes Rendus du Congrès International des Mathématiciens. Oslo 1936, Oslo, A. W. Brøggers Boktrykkeri, 1937, Vol. II: 89–90, 210–11). Here we give a translation of the second paper only. It seems that Potron originally gave a slightly different title to his contribution. Two reports which appeared shortly after the conference referred to Potron’s paper as ‘Conditions des équilibres production-consommation et prix-salaires’.1 This title resembles that of the apparently never published booklet which Potron mentioned in his letter to Gibrat (see note 4 of A35/Chapter 11). Ragnar Frisch also contributed to the Section in which Potron presented his economic paper: ‘Price index comparisons between structurally different markets’ (Comptes Rendus du Congrès International des Mathématiciens. Oslo 1936, o.c.: 220–1). There is, however, no trace of any contact between Potron and Frisch during or after the conference. * * * Notations: Ai = unit of a ‘result of labour’; Ch = ‘standard of living’;2 qh = number of non-workers in Ch ; bhi = number of Ai consumed, in Ch , per person and per year; cki = number of Ai consumed to produce one Ak ; pih = number of workers 1 See Henri Fehr, ‘Le 10e Congrès International des Mathématiciens. Oslo, 13–18 juillet 1936’ (L’Enseignement Mathématique, 1936, 35: 383) and Marston Morse ‘The International Congress in Oslo’ (Bulletin of the American Mathematical Society, 1936, 42: 780). 2 [In fact, in this and the following paper Ch designates a set of persons who share the same standard of living. – Eds.]

On the economic equilibria 175 in Ch producing the Ai ; tih = total number of hours that the production of one Ai requires from them; sih = their hourly individual wage; ai = price of one Ai ; N = individual maximum number of work-hours per year; NSh = Ch ’s cost of living per person and per year. Obvious relations, and required conditions:3 di −

cki dk −

bhi Qh = fi

Npih − di tih = wih

0,

ai −

tih sih = bi

NSh =

cik ak −

0,

Qh =

di and pih > 0, qh 0,

pih + qh 0

(2)

ai and sih > 0

bhk ak , di tih sih − NSh pih = pih eih

(1)

0

(3) (4)

Algebraic consequences:4 di −

Cki dk = Fi = fi + N −1

Cki = cki + N −1 ai −

Cik = cik + di di −

Cki dk = fi +

bhi qh ,

bhi tkh ,

Cik ak = Bi = bi + di −1 −1

bhi wkh + pih eih ,

pih bhk = Cik + (Ndi )−1 bhi qh .

bhk wih ,

(5) (6) (7)

Thanks to known theorems (Frobenius, S.A.B., 1909, p. 504; Potron, A.E.N., t. 30, 1913, p. 53),5 a system xi − aik xk = bi (aik 0, bi > 0) admits solutions xi , all > 0, always and only if a certain function of the aik is < 1. For (5), the condition refers to the Cki , therefore only on the cik , tik , bhi and N . If it is met, (5) determines a set of positive di for any set of positive fi , wkh , and qh . As these di satisfy (7), the condition is met by the Cki ; therefore, for any set of positive bi and eih , (6) determines a set of positive ai . Positive pih and sih then result from (2) and (4).

3 [We have corrected an error in the second condition of equation (4). It results from a change in the definition of the variable eih : in the previous paper this refers to hourly savings, but in the present (in which its meaning is not defined explicitly) and in the next (in which it is; see equation (7) of U2/Chapter 13) to yearly savings. – Eds.] 4 [We have corrected an error in the second condition of equation (6), caused by a confusion of the indices i and k. – Eds.] 5 [See note 1 of A9/Chapter 3, and A13/Chapter 8. – Eds.]

13 Communication made at the Oslo Congress

Editors’ note An extended version of the economic paper which Potron presented in Oslo (A39/Chapter 12) is conserved in the folder ’Potron’ of the Archives Jésuites de la Province de France. This three-page document, entitled ‘Communication faite au Congrès d’Oslo’, is an unpublished typescript. The title and the footnote are handwritten additions by Potron. The title suggests that this paper has been drafted after the congress. It is also linked to the letter to Gibrat (A35/Chapter 11). The following correspondence table indicates how the numbered mathematical expressions of the three versions are related to one another (if an expression has more than one equation, a superscript distinguishes the different equations). Gibrat letter (Chapter 11)

Oslo version (Chapter 12)

Typescript (Chapter 13)

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10) (11) – – –

(11 ) (12 ) (2) (3) (42 ) (41 ) (51 ) (61 ) (7) – – (52 ) (62 ) (62 )

(1) (3) (2) (6) (7) – (4) (8) (11) – (12) (5) (9) (10)

* * *

Communication made at the Oslo Congress 177 In what follows, Ai (i = 1, . . . , n) will denote a given quantity, called unit, of a result of labour. Ch (h = 1, . . . , m) will denote a standard of living, to which Qh consumers belong, each of whom annually consumes bhi units Ai . I assume that annually dk units Ak are produced, and that the production of one Ak requires the consumption of cki units Ai . At the end of the year, the stock of Ai units has changed by n

n

bhi Qh = fi

cki dk −

di −

(i = 1, . . . , n)

(1)

1

1

Let pih be the number of workers of standard Ch engaged in the production of Ai , tih the total number of hours that the production of one Ai requires from them, and N the maximum number of work-hours, per person and per year. For these workers, the collective unemployment amounts to Npih − di tih = wih

(i = 1, . . . , n; h = 1, . . . , m)

(2)

If qh is the number of non-workers of [standard] Ch , we have n

pih + qh

Qh =

(h = 1, . . . , m)

(3)

1

The problem of the rational organization of production can be set as follows. The cki and tih depend on the state of the economy at a given date. One can consider the bhi as characterizing the standards of living claimed by the workers of various categories. All these numbers being given, the problem is to meet (1), (2) and (3) with values > 0 for the di and pih , 0 for the fi , wih and qh . The sum Qh must moreover be equal to the total size of the population. The elimination of the Qh and pih gives n

di −

Cki dk = Fi = fi + N −1

bhi wkh +

bhi qh

(i = 1,..., n)

(4)

1

Cki = cki + N

−1

m

bhi tkh

(i, k = 1,..., n).

(5)

1

Once satisfactory solutions to (4) are obtained, (2) will give the pih . In order that Qh be equal to the total size of the population, it will suffice to multiply these numbers by the same factor. A purely mathematical theory1 informs us when satisfactory solutions to (4) can be obtained. According to that theory, there exists a function C of the Cki 1 Frobenius 1909. [See note 1 of A9/Chapter 3 for the exact reference. – Eds.]

178 Communication made at the Oslo Congress such that: 1◦ If C > 1, it is impossible that (4) holds with di > 0 and Fi 0. 2◦ If C < 1, the Fi , therefore the fi , wih and qh , can be given arbitrary values 0, not all zero; the system (4) then determines di which are all > 0. 3◦ The function C increases if any of the Cki increases, the other being fixed. According to 3◦ , C will increase simultaneously with the bhi . Therefore, for a given state of the economy, the condition C < 1 imposes an upper limit to the set bih of requirements. Similarly, and other things being equal, it forbids that N decreases below a certain limit. Now, let ai be the price of one Ai [and] sih the hourly wages for the various categories of workers. The benefit per unit Ai is clearly m

n

cik ak −

ai −

tih sih = bi

1

(i = 1, . . . , n)

(6)

1

Every worker’s budget is written down by means of the relation defining his yearly economies, n

di tih sih /pih −

(i = 1, . . . , n; h = 1, . . . , m)

bhk ak = eih

(7)

1

A satisfactory regime of prices and wages will be made of ah and sih > 0, [and] of bi and eih 0, satisfying (6) and (7). The elimination of the sih gives n

ai − 1

Cik ak = Bi = bi + di −1

Cik = cik + di

−1

m

pih eih

(i = 1, . . . , n)

(8)

1

m

pih bhk

(i, k = 1, . . . , n)

(9)

1

Once satisfactory solutions to (8) are obtained, (7) will give the sih . The condition of possibility for the problem C (the analogous of C) < 1. According to (9), (5) and (2), we have2 Cki − Cki = (Ndk )

−1

m

bhi (Npkh − dk tkh ) = (Ndk ) 1

Therefore Cki > Cki and C > C. 2 [Potron forgot bhi in the middle part of the expression. – Eds.]

−1

m

bhi wkh 1

(10)

Communication made at the Oslo Congress 179 Hence, if C > 1, the 2nd problem is impossible as well as the 1st. If C < 1, the di > 0, solutions of (7), satisfy, according to (10), n

di − 1

n

Cki dk = Fi −

1

m

(Cki − Cki ) dk = fi +

bhi qh

(11)

1

It follows that C 1 and, even, if the fi and the qh are not all zero, C < 1. In this case, arbitrary values, ≥ 0 and not all zero, can be given to the Bi , therefore to the bi and eih ; the system (9) determines ai which are all > 0. Let us note that the sum of the 1st members of (8), multiplied respectively by di , is equal to the sum of the 1st members of (11), multiplied respectively by ai . We therefore have ai Fi −

di Bi = 0

or, by subtracting and adding

bi fi ,

(ai − bi ) fi +

bhi ai =

qh

pih eih +

(di − fi )bi

(12)

This formula means that, during a period when the general state (organization of the industry and standards of living) has remained stationary, when the prices and wages have not changed, the sum of the workers’ economies and the firms’ benefits is equal to the non-workers’ total cost of living, increased by the cost prices of all overproduced commodities.

14 The mathematical aspect of some economic problems in relation to recent results of the theory of nonnegative matrices. Lectures given at the Catholic Institute of Paris Editors’ note In March 1937 the following announcement appeared under the heading ‘Works of Comrades’ in the newsletter sent to the members of X-Crise (Centre Polytechnicien d’Études Économiques. X-Crise. Bulletin Mensuel, 1935, No. 36: 57): ‘Our comrade Father Potron (90), PhD in Mathematics, will give six lectures on: The relations which exist between certain economic problems and some fairly recent results of a mathematical theory. These lectures will be held on Thursdays at 5 p.m., in the Institut Catholique, 21 rue d’Assas, Room A (corridor at the end of the atrium, to the left of the entry gate of No. 21). Thursday 11 March A mathematical aspect of the problem of the equilibrium between production and consumption, a given standard of living being assigned to each profession and each social condition. Why and how the possibility question of this problem arises. Thursday 18 March A mathematical aspect of the problem of the adjustment of prices and wages, the standards of living being equal to those of the previous problem. Relation of this problem with the previous. Does a new possibility condition arise? Thursday 8 April The Frobenius theorems (stated and proved in 1909) on matrices or linear substitutions with positive entries. Possibility condition of problems which are somewhat analogous to those of the equilibria production–consumption and prices–wages.

Lectures given at the Catholic Institute of Paris 181 Thursday 15 April Extension of the Frobenius theorems to matrices with nonnegative entries only. These theorems allow to completely answer the possibility question which arises for the two economic problems considered. Thursday 22 April Is the large number of equations an insuperable obstacle for the calculation of solutions? Exposition of an hitherto unpublished method for the practical solution of a large number of linear equations. Thursday 29 April Some economic conclusions suggested by the formulas obtained. Is the race to the rise between prices and wages inevitable? Rationalization and unemployment. Overproduction and hoarding. The text of the six lectures might be published in a typed booklet of about fifty pages, at the price of 6 francs per copy, due after receipt. Those who would like to receive one or more copies of it are kindly requested to inform Father Potron, 114 rue du Bac, Paris (7th district). The booklet will undoubtedly be published around the end of May.’ Potron affixed posters announcing the lecture series in the Institut Catholique itself but failed to inform its direction. The administration was afraid of the unchecked initiative of a mathematics teacher who intended to intervene on economic and social matters. It finally gave a late authorization with a call to order (poster and letters in the Archives de l’Institut Catholique de Paris). Potron also produced the promised typescript: L’aspect mathématique de certains problèmes économiques en relation avec de récentes acquisitions de la théorie des matrices non négatives. Conférences faites à l’Institut Catholique de Paris, with an appendix entitled Le problème de la manne des Hébreux (s.l., s.d., 34 + 5 pp.). A copy of the booklet is conserved in the folder ‘Potron’ of the Archives Jésuites de la Province de France, and another one is available in the Bibliothèque Universitaire des Sciences et Techniques, Université Bordeaux 1 (catalogue number FR 19069). In 1942 Alfred Barriol reviewed it in the Journal de la Société de Statistiques de Paris (see W2/Appendix II below). Our translation is based upon the typescript conserved in the Archives Jésuites de la Province de France. In the margin of another document conserved in the same folder, Potron listed the estimated number of participants of each session: about 50 for the first two lectures, about 15 for the following three, and about 100 for the last, for which apparently a special publicity effort had been made. Potron also recorded the presence of two prominent mathematical economists,

182 Lectures given at the Catholic Institute of Paris René Roy (1894–1977) and François Divisia (1889–1964), at the second and the sixth lectures. At the end of section 2 of the first lecture, Potron referred to an Appendix entitled ‘Le problème de la manne des Hébreux’. A one-page document with this title containing a summary of the problem and one possible solution was apparently distributed during the first lecture. It announced that ‘These various results will be established during the lecture of Thursday 18 March, 5 p.m., room A’. A copy of this handout is conserved in the Archives Jésuites de la Province de France. Potron annotated it as follows: ‘A letter of Mr Divisia (prof. at the École Polytechnique) underlines the value of this little fantasy.’ The five-page appendix which he inserted at the end of his booklet is a more developed version of this document. The main change with respect to the one-page draft is that Potron switched to biblical units of measurement. In 1953 the Jesuit Michel Vittrant wrote a short but highly critical report on the extended version (see W3/Appendix III below). * * * FIRST LECTURE

The production–consumption equilibrium A mathematical aspect of the equilibrium problem between production and consumption, a given standard of living being assigned to each profession. Why and how does a possibility condition arise? It is rather difficult to speak about economic questions without relying on more or less elementary notions of mathematics. One must in fact state equalities or inequalities between various quantities, combine several quantities between them by arithmetical operations. In theoretical studies, it is necessary to have recourse to the symbols of algebra, which allow us to work on letters without having to know the particular values they represent. Even in numerical calculations, it is often advantageous to work on letters first, and to replace them by their values only in the final formulas. 1. From the point of view adopted in these lectures, we mean by ‘economic good’ something which is produced, which is consumed, used or utilized, which is bought and sold. That thing can in general be measured in a quantitative way. When one speaks about some ‘quantity’ of an economic good, one always states, implicitly or explicitly, the unit used to evaluate it. We say, for instance, so many tons of coal, so many metres of cloth, so many kilowatts of electricity, so many kilometric tons, or so many kilometres-travellers for transports and, even, for less material things, so many hours of lessons of mathematics. Instead of ‘unit of an economic good’, we shall simply say ‘economic unit’. In some cases, the economic unit intended to be used, a machine for instance, constitutes as such an indivisible whole. It is only by a fiction, justified for that

Lectures given at the Catholic Institute of Paris 183 matter, that it can be said: making a kg. of bread requires the use of the nth part of a kneading-trough. That means that the kneading-trough is worn out and must be replaced after the production of n kg. of bread. Among the units of this type, we shall consider: •

•

the set of services delivered during the year by a public administration (municipal, departmental, national), except those services for which the administration plays the role of an ordinary firm (postal transports, fabrication of matches, etc.); the set of services delivered during the year by an insurance company, a bank, etc.

By a fiction analogous to that used in the case of a machine, one can say: under such circumstance, such fraction (to be determined, but determinable) of the unit made of this set of services is used. 2. Some years ago, assigning a given standard of living, considered as ‘convenient’, to every profession might have been viewed as somewhat fanciful, for lack of precise data. Today very neat indications are given by the collective contracts. These contracts enumerate, in a very detailed way, the various professions, and stipulate, for each of them, a determined wage. Knowing what the prices of the various consumption goods were at the time of the contract, it is easy to determine one standard of living corresponding to such a wage. It is the operation made by the provident worker who, at the beginning of the year, draws up his ‘family budget’. Several slightly different standards of living may correspond to the same cost of living. One may either choose the more likely or divide the profession into several possible standards of living. In fact, the main purpose is to estimate a consumption in order to schedule a production that satisfies it, if necessary with some excess. In any case, whatever the economic regime, whatever the state of the economy, whatever the standards of living attributed to the various professions, a problem is set: to construct an ideal production-labour-consumption regime (P–L–C regime) in which both conditions are satisfied: a) production must be at least equal to consumption; b) the duration of work must not, for any worker, exceed a certain maximum (given by custom or law). In an Appendix, under the title ‘The Hebrew Manna Problem’, a very simple example will be given. 3. The economic units can be used in two ways to sustain human lives. Some, called 1st-species units, or rations, are consumed, used or utilized directly; the others, called 2nd-species units, are those that must be consumed, used or utilized

184 Lectures given at the Catholic Institute of Paris to produce the rations. There are mixed units, the ton of coal for instance, which belong to both species. The professions–standards of living will be denoted by the symbols V1 , . . . , Vh , . . . , Vm , the economic units by the symbols A1 , . . . , Ai , . . . , Ak , . . . , An . When we have noted down the budget of a representative of each profession, we shall have drawn up the exhaustive list of the rations, and we shall be able to write down the first m rows of a table. In the row corresponding to profession Vh , we shall write down, in the columns corresponding to the rations A1 , . . . , Ai , . . ., the numbers bh1 , . . . , bhi , . . ., where the symbol bhi represents the number of rations Ai consumed yearly by a household of a labourer with profession Vh . If Aj designates a unit made of a yearly set of services, one may assume that, on the one hand, the yearly sum (taxes, insurance premiums, etc.) paid by a consumer-household of standard Vh to the administration that ‘produces’ Aj and, on the other, the total yearly sum paid to that administration by the community depending on it, are both known. The ratio will give the coefficient bhj . Our table contains m rows corresponding to m professions–standards of living, and as many columns as there are 1st-species units. On its left-side, we shall draw m columns corresponding to the m professions; then, below it, we shall insert one row for each ration, putting them in the same order. Then, in the row of the ration Ak , we shall write down: • •

in the column of each profession Vh , the number tkh of hours performed by this profession for the production of one Ak ; in the column of each unit Aj , the number ckj of units Aj that must be destroyed to produce one Ak . When this becomes necessary, new columns corresponding to 2nd-species units, will be inserted on the right side of the table.

A producer of Ak must know these various numbers to be able to establish his ‘cost price’. If a firm produces economic units of several species simultaneously, one will distinguish between the expenses relative to each particular fabrication and the ‘overall expenses’ relative to the whole firm. These overall expenses are usually shared among the various fabrications, proportionally to specified coefficients. Let us then assume that, during the year, a firm has produced pk units Ak , letting the profession Vh work during fkh hours for the fabrication of the Ak and fh hours for its general working, consuming or using gkj units Aj for the fabrication of the Ak and gj for its general working. If this firm attributes the lth part of its overall expenses to the fabrication of the Ak , the numbers tkh and ckj will be given by pk tkh = fkh + lfh ,

pk ckj = gkj + lgj

If Aj designated a yearly collection of services, one might determine gj like bhj above. Thus the producers know and can provide the numbers to write down in our table. But the numbers provided by several producers of the same ration may be different. One can either take their average or specify that the conditions of

Lectures given at the Catholic Institute of Paris 185 production of such a ration are those of such a factory. Anyway, as soon as one satisfactory regime has been obtained, any regime differing from it slightly, either by the data or by the results, will also be satisfactory. Every time we insert, at the right side of the table, a column for a 2nd-species unit, let us insert simultaneously, at the bottom of the table, a row for that unit. In every row – and inserting new columns if necessary – we shall write down the numbers tkh and ckj provided by the producers. We shall continue these operations until the last row is created without inserting a new column. The table is then finished. It fits with the schema: Professions Standards of living Rations Mixed units 2nd-species units

Rations 0 0 0

Mixed units

2nd-species units 0

It is useful to note that there cannot exist a system of j 2nd-species units, located at the ranks r1 < r2 < · · · < rj , such that, in these columns, all coefficients are zero if they do not belong to one of these rows, i.e. such that we have cik = 0 for any i and k satisfying (i − r1 ) (i − r2 ) (. . .) i − rj = 0,

(k − r1 ) (k − r2 ) (. . .) k − rj = 0

In fact, if the unit of rank r1 (the 1st of the system) has been introduced, it is because it is used for the production of a unit of rank < r1 , not belonging to the system. Therefore, in the column of rank r1 , there appears a coefficient = 0 not belonging to any row of the system. That table, of which the role will be fundamental in all what follows, summarizes the data of the economic problems which are set, in a given state of the economy, by the assignment of a given standard of living to each profession. No doubt the considerable dimensions of such a table make it difficult to assume that it is constructed, especially for its last rows.1 But it is worth noting that each of its entries has a given value, even if we do not know it. Moreover, according to what has been said above, each producer can, by means of simple calculations not requiring much time, construct the row that concerns himself. 4. Now, if we knew the number Qh of consumers–households of each standard Vh and the number pk of Ak units of each species produced during the year, the table would allow us to evaluate the number Th of working hours performed, during the year, by each profession Vh , as well as the number Ci of Ai units of each 1 [Potron wrote ‘the last n rows’, but probably he wanted to refer only to the last 2nd-species goods. – Eds.]

186 Lectures given at the Catholic Institute of Paris species consumed during the year. Assuming indeed that the labour performed is proportional to the production obtained, the column Vh gives n

Th = t1h p1 + · · · + tkh pk + · · · + tnh pn =

tkh pk

(1)

1

Making the same hypothesis on the consumption of raw materials and the wear and tear of equipment, the column Ai gives m

Ci =

n

bhi Qh + 1

cki pk

(i = 1, . . . , n)

(2)

1

If, among the Qh consumers–households of standard Vh , there are Ph of them whose head works and qh whose head does not work, we shall have Qh = Ph + qh

(h = 1, . . . , m)

(3)

If a worker must not ‘normally’ perform more than N hours per year, the differences NPh − Th = Wh

(h = 1, . . . , m)

(4)

give the number of unemployment hours or overtime hours for each profession. On the other hand, the differences pi − Ci = fi

(i = 1, . . . , n)

(5)

give the variation of stocks between the beginning and the end of the year. The professions and numbers of professional workers are positive by nature, and the numbers of non-workers nonnegative; let pi > 0 (i = 1, . . . , n)

(6)

Ph > 0

(7)

qh

(8)

0 (h = 1, . . . , m)

The conditions a) and b) are written fi

0 (i = 1, . . . , n),

Wh

0 (h = 1, . . . , m)

(9)

5. If, in (2), the Qh are replaced by their values (3), then the Ph by their values (4), one obtains Ci =

1 N

m

m

bhi (Th + Wh ) + 1

n

bhi qh + 1

cki pk ; 1

(10)

Lectures given at the Catholic Institute of Paris 187 hence, by (1), n

m

Ci =

Cki pk + 1

bhi 1

Cki = cki +

1 N

Wh + qh , N

(11)

m

tkh bhi

(12)

1

Then (5) becomes n

pi −

Cki pk = Fi

(13)

1 m

F i = fi +

bhi 1

Wh + qh N

(14)

The relations (8)–(9)–(14) imply Fi

0 (i = 1, . . . , n)

(15)

Therefore, if the proposed problem admits a solution, the same for the system (6)–(13)–(15). Conversely, if this happens, one can obviously, and in infinitely many ways, give values satisfying (8)–(9)–(14) to the fi , Wh , qh . Then the Th will be given by (1), then the Ph by (4), finally the Qh by (3). The coefficients Cki are nonnegative scalars depending, according to (12), exclusively upon N and upon the coefficients which characterize the state of the economy and the standards of living. In order that Cki be zero, a necessary and sufficient condition is that cki be so (that is, Ai is not used in the fabrication of Ak ), and that, in every of the m products tkh bhi , at least one factor be zero (that is, either the profession Vh does not work in the production of Ak or, in the standard Vh , no Ai is consumed). 6. We are therefore led to ask the following question: which is the condition on the nonnegative coefficients Cki in order that the system (6)–(13)–(15) admits a solution? In the general case, the condition cannot be found by elementary means. The answer requires higher mathematical theories. Such state of the economy, such standards of living in accordance with the law that fixes N , impose such values of the Cki coefficients. If the conditions of possibility are met, it is possible to construct an ideal regime meeting the conditions a) and b). It will perhaps not be possible to implement such a regime immediately, for lack of industrial equipments or educated professional staff.2 2 [This is a rare reference to the practical difficulties of implementing a theoretically established regime. – Eds.]

188 Lectures given at the Catholic Institute of Paris But if the conditions of possibility are not met, it is impossible to construct such a regime. Therefore it will be impossible to implement one of them, as long as something is not changed in the data, either to the number N , or to the state of the economy, or to some of the scheduled standards of living. SECOND LECTURE

The adjustment of prices and wages A mathematical aspect of the adjustment problem of prices and wages, a given standard of living being assigned to each profession. Relation of this problem with the previous. Does a new possibility condition arise? 1. Our present aim is to study the mechanism according to which the economic goods, once produced, are distributed among the consumers. In our actual economic regime, every firm gives to its workers, in exchange for their labour, and to other enterprises, in exchange for the raw materials and the tools they provide, a well determined quantity of monetary signs. In exchange for each economic good corresponding to his standard of living, every consumer gives to the last producer a well determined quantity of these monetary signs. So, to each profession is attached a certain wage and to each economic unit a certain price. 2. If we denote by sh the hourly wage of the profession Vh , the Ph workers of that profession receive effectively Th sh during the year, Th being given by the formula (1) of the 1st lecture [I, (1)]. If we denote by a1 , . . . , an the prices of the units A1 , . . . , An , the numbers written down in the first m rows of the table (see section 3 of the first lecture) allow us to evaluate the cost of living of a consumer-household of standard Vh . It is n

Sh = bh1 a1 + · · · + bhi ai + · · · + bhn an =

bhk ak

(1)

1

By subtracting that cost of living from the effective wage, one obtains the household’s economies Th sh − Sh = eh Ph

(2)

On the other hand, the scalars written down in the last n rows of the table allow us to evaluate the cost price of the unit Ai : n

ci1 a1 + · · · + cin an + ti1 s1 + · · · + tim sm =

m

cik ak + 1

tih sh 1

Lectures given at the Catholic Institute of Paris 189 By subtracting that cost price from the selling price, one obtains the benefit per unit sold n

ai −

m

cik ak − 1

(i = 1, . . . , n)

tih sh = bi

(3)

1

The prices and wages are positive scalars by nature: ai > 0

(i = 1, . . . , n)

(4)

sh > 0

(h = 1, . . . , m)

(5)

To be effectively satisfactory, a price-wage regime must satisfy two conditions:3 c) each worker’s effective wage must be at least equal to the cost of living corresponding to the standard of living assigned to his profession; d) the selling price of every economic unit must be at least equal to its cost price. These conditions are written eh

0

(h = 1, . . . , m) ,

bi

0

(i = 1, . . . , n)

(6)

Does the system (1) to (6) admit a solution? 3. If, in (3), we replace the sh by their values (2), it follows, by taking (1) into account n

ai − 1

Cik ak = Bi

Cik = cik + Bi = bi +

(i = 1, . . . , n)

tih bhk Ph Th tih Ph eh Th

(7) (8) (9)

From (6), it is required that Bi

0

(i = 1, . . . , n)

(10)

If the system (1) to (6) admits a solution, the same holds for the system (4)–(7)–(10). Conversely, if the latter system admits a solution, one can obviously give arbitrary values satisfying (6)–(9) to the bi and the eh . Then, the Th and Ph being already known, (1) and (2) will give the sh . 3 [Conditions a) and b) have been mentioned in section 2 of the first lecture. – Eds.]

190 Lectures given at the Catholic Institute of Paris 4. The system we are led to consider has the same type as the one already met [I, No. 5]. The coefficients of both systems have moreover interesting relationships. In (8), let us exchange the indices i and k, then subtract in [I, (12)]. Taking [I, (4)] into account, it follows Cki − Cki =

1 N

tkh bhi

Wh Th

(11)

Therefore, in general, we have Cki Cki ; and, by [I, No. 5], these two coefficients are always zero simultaneously. One sees moreover, from [I, (13)], that we have Cki pk = Fi −

pi −

Cki − Cki pk = Fi

(12)

But (11) and [I, (1)] give Cki − Cki pk =

1 N

bhi

Wh Th

tkh pk =

1 N

bhi Wh

From [I, (14), (8), (9)] it then follows Fi = fi +

bhi qh

(13)

hence Fi

0

(14)

Therefore any solution of the system [I, (6)–(13)–(15)] satisfies (12)–(14). The matrices made of the coefficients of (7) and (12)4 are simply transposed from each other: every row is exchanged with the column of the same rank. A noteworthy consequence concerning the possibility conditions will be drawn from this fact. 5. Let us add the equations (7), respectively multiplied by p1 , . . . , pn , then the equations (12), respectively multiplied by a1 , . . . , an ; it follows ai pi −

Cik ak pi =

Bi pi ,

pi ai −

Cki pk ai =

Fi ai

The left-hand sides being clearly equal, one concludes Bi pi =

Fi ai

4 [Potron wrote (13) instead of (12). – Eds.]

(15)

Lectures given at the Catholic Institute of Paris 191 On the one hand, (9) and [I, (1)] give Bi pi =

bi pi +

Ph eh Th

tih pi =

bi pi +

ph eh

On the other hand, (13) and (1) give Fi ai =

fi ai +

qh

bhi ai =

fi ai +

qh Sh

Thus, (15) gives the equality bi pi +

Ph eh =

fi ai +

qh Sh

(16)

The general interpretation of that formula will be given in the last lecture. In the case where there is no excess of production ( f1 = · · · = fn = 0), the formula means that, in the absence of any excess of production, the sum of the firms’ benefits and the workers’ economies represents exactly the total of the non-workers’ costs of living. 6. It is important to note that a P–W satisfactory regime is established on the basis of a well determined P–L–C regime. The implementation of a P–W regime might also entail some difficulties due for instance to the financing of the firms. In this regard, various problems arise which go beyond the frame of these lectures.5 As soon as an ideal regime is constructed, every worker or firm can immediately check that such a regime satisfies the conditions (c) and (d), as far as he/it is concerned. But that P–W regime will be able to work in an effectively satisfactory way only when the P–L–C regime on which it is constructed is implemented. 7. According to the previous results, one sees that, for constructing theoretically a satisfactory economic regime, it suffices to find, if it exists, a solution to each of both systems pi > 0,

pi −

Cki pk

0

(I)

ai > 0,

ai −

Cik ak

0

(II)

If one of these systems admits a solution, it admits infinitely many solutions, except in the case when all these relationships would be equalities; because any set of the values of the unknowns close to a solution gives another solution. To every solution of (I) correspond infinitely many P–L–C regimes, obtained as said above [I, No. 5]. If the pi , fi , Wh , qh are multiplied by the same factor k, all the relations are still satisfied, and one sees that Ph and Qh are also multiplied by k. 5 [A rare reference to problems of finance. – Eds.]

192 Lectures given at the Catholic Institute of Paris This factor will then be determined by the condition that k Qh represents the total population of the community living under the considered economic regime. To every solution of (II) correspond similarly infinitely many P–W regimes, obtained as said in No. 3. One may again introduce a proportionality factor k. Here, there is no theoretical reason allowing us to set this factor. But, in practice, if the monetary unit is a determined weight of gold, the coefficient k will be defined by the condition that the price of this weight of gold be 1. 8. The data of the problem (state of the economy, standards of living and N ) might be such that the systems (I) and (II) have no solution, their various relationships being incompatible. This shows the interest of a mathematical study allowing us to assert if a system of this type admits solutions or not. The existence condition is, we shall see, that the number 1 is greater than the value of a certain continuous function of the coefficients. Therefore if this function is distinctly > or < 1, the inequality will remain if the coefficients are slightly modified. One therefore sees that, to obtain sure conclusions, an approximate knowledge of the coefficients of the table described in section 3 of the first lecture will suffice. Moreover, given the indeterminacy of the problem, an effective economic regime may, without ceasing to be satisfactory, deviate slightly from the conceived theoretical model. One must above all retain that the adjustment of wages to prices is possible only if the prices satisfy the system (II), and does not set any difficulty, even from a practical point of view, if this condition is met.6 If on the contrary (II) has no solution, any adjustment is impossible. THIRD LECTURE

Matrices with positive entries The Frobenius theorems on matrices with positive entries. 1. Let b and c be two matrices with entries bik and cik ; we shall denote by a = pb + qc the matrix of which the entries are aik = pbik + qcik . The unit matrix u is a square matrix of which the entries uik are 0 for i = k and 1 for i = k. If b is a square matrix, we shall denote |b| the determinant with the same entries. If a is a square matrix, the determinant |su − a| is the characteristic determinant, and the equation |su − a| = 0 the characteristic equation of the matrix a. A matrix is said positive when all its entries are positive scalars. 2. Let us first recall two properties of determinants. Let B be a determinant of degree n and entries bik . Let M be one of its minors of degree m, of which the entries belong to the rows of ranks h1 < h2 < · · · < hm and

6 [Here Potron reduced the theoretical determination of satisfactory prices and wages to that of prices, but denied once more any difficulty in implementing such prices and wages. – Eds.]

Lectures given at the Catholic Institute of Paris 193 to the columns of ranks i1 < i2 < · · · < im . Let M be the minor of degree n − m, of which the entries belong to the other rows, of ranks j1 < j2 < · · · < jn−m , and to the other columns, of ranks k1 < k2 < · · · < kn−m . The minors M and M are said complementary. Let us set ⎧ h1 + h2 + · · · + hm = h, ⎪ ⎪ ⎪ ⎨ i + i + · · · + i = i, 1 2 m (1) ⎪ j1 + j2 + · · · + jn−m = j , ⎪ ⎪ ⎩ k1 + k2 + · · · + kn−m = k . It is clear that h + j = i + k = 1 + 2 + · · · + n. Therefore the number h + i + j + k = n (n + 1) is always even. Consequently, the two sums h + i and j + k have always the same parity. THEOREM I. With these notations, we always have B=

(−1)h+i MM

(2) n minors of which the entries belong to the rows of m

where M runs over the ranks h1 , . . . , hm .

First,7 it is clear that, up to its sign, any term of a product MM is a term of B, and any term of B is a term of a product MM . I claim that any term has the same sign in B and in the product (−1)h+i MM to which it belongs. Indeed, if the row of rank h1 is permuted with each of its previous rows, then the row of rank h2 with the previous ones except the first, and so on, and if the some operation is made on the columns, the rows and columns of M are moved to the ranks 1, 2, . . . , m. The minors M and M have not changed their values. The determinant B has changed its sign a number of times equal to h1 + · · · + hm + i1 + · · · + im − 2 (1 + · · · + m) = h + i − m (m + 1) Instead of B, we therefore have B = (−1)h+i B. Let T and T be two terms of M and M (their signs initially ignored), the first indices being the natural sequence 1, . . . , m, m + 1, . . . , n and the second indices 7 [In the following proof Potron referred implicitly to the Leibniz formula for determinants: n

det (b) =

sgn(σ ) σ∈S

bi,σ (i) i =1

where σ runs over the set of permutations of {1, 2, . . . , n}, and sgn(σ ) is the signature of σ . – Eds.]

194 Lectures given at the Catholic Institute of Paris being, in T , a permutation i1 , . . . , im of the numbers 1, . . . , n with I inversions and, in T , a permutation k1 , . . . , kn−m with K inversions. The complete sequence of the second indices obviously admits I + K inversions. From this it follows that the product TT appears in the product MM and the determinant B with the same sign, that of (−1)I +K . Thus any term has the same sign in B and in the product MM to which it belongs, therefore the same sign in B and (−1)h+i MM . [End of proof ] The minor (−1)h+i M is said the algebraic complement of the minor M . In particular, if M is reduced to the element aik , its algebraic complement Aik is the product, by (−1)i+k , of the minor obtained by deletion of the ith row and the kth column. Formula (2) then gives the known expansion formula of a determinant with respect to the entries of a row. The determinant A with the entries Aik is said adjunct of the determinant a with the entries aik . The elements aik and Aik are said homologous. Two minors are homologous when they are made of homologous elements. If, in a determinant, two parallel lines are exchanged, one must make the same operation in the adjunct in order to preserve the homology relationships. Moreover, as the determinant changes its sign, the sign of each element of the adjunct must be changed. Corollary of Theorem I. In a determinant b, let bik be an entry, ik its complementary minor, δjkih the minor of degree 2 the entries of which belong to the rows j and k and the columns i and h, [and] jkih its complementary minor. The algebraic complements are8 ∂b = Bik = (−1)i+k ∂ bik

ik

(3)

∂b = (−1)j+k +i+h ∂δjkih

jkih

(4)

Therefore, by applying formula (2), we have b=

(−1) j+k +i+h δjkih

jkih ,

(5)

the summation being extended to all combinations ( jk) , i and h being given. As the two elements bii and bih only appear in the n minors δikih (k = 1, . . . , n), they only appear in the partial sum (−1)k +h δikih

ikih

8 [Potron made use of partial derivatives to write down the coefficients of a linear or affine form. – Eds.]

Lectures given at the Catholic Institute of Paris 195 But we have δikih = ηk (bii bkh − bih bki ) , ηk denoting ±1 according to the position of the entries. We therefore have ∂b = Bii = ∂ bii ∂b = ∂ bih

(−1)k +h ηk bkh

(−1)k +h+1 ηk bki

ikih

ikih

= Bih

The conclusion is ∂ Bii ∂ Bih =− ∂ bki ∂ bkh

(6)

Therefore, if one expands b with respect to its ith row, then every Bih (h = i) with respect to its ith column, we have b=

bih Bih + bii Bii = bii Bii +

bih bki

∂ Bih ∂ bki

or, according to (6), b = bii Bii −

bih bki

∂ Bii ∂ bkh

(7)

where the summation indices h and k do not take value i. The conclusions are Bih =

∂b =− ∂ bih

bki

∂ Bii ∂ bkh

(8)

Bki =

∂b =− ∂ bki

bih

∂ Bii ∂ bkh

(9)

3. THEOREM II. Let m be a minor of degree j of a determinant a of degree n, and (−1)i+k m be its algebraic complement; let M be the homologous minor of m in the adjunct A. We have M = aj−1 (−1)i+k m .

(10)

Let us first assume that the elements of m are those of the first j rows and columns of a. Let us consider a determinant B, of degree n, of which the entries are, for its first j rows, those of the adjunct A and, for the last n − j rows, those of the unit matrix u. We obviously have B = M . Moreover, the product of B by the transposed determinant of a is a determinant c of which the entries are cik =

Aih akh = uik a

cik =

uih akh = aki

(i = 1, . . . , j) (i > j)

196 Lectures given at the Catholic Institute of Paris We therefore have Ba = Ma = aj m , therefore, in this case, M = a j−1 m

(11)

Let us now assume that the entries of m belong to rows i1 , . . . , in and columns k1 , . . . , kn , in increasing order. Let us make the same operation as in the proof of theorem I. When any two parallel lines are exchanged, the determinant and each element of the adjunct change their signs, so that the minor M is multiplied by (−1) j . At the end of the operation, the determinant a is replaced by b = (−1)i+k a, the minor M by P = (−1)(i+k) j M and, according to (11), we have P = b j−1 m . Hence, the formula (10). [End of proof] In particular, if j = n − 1, the algebraic complement of the entry Aik in the adjunct is equal to an−2 aik ; and, if j = n, we have A = an−1 . 4. THEOREM III. The characteristic equation of a positive matrix has always at least one real root. The greatest root, said ‘maximum root’, is always positive and simple. If r denotes that root, the entries of the adjunct of the characteristic determinant are all positive for s r. a b of degree 2. The c d s − a −b s−d c characteristic determinant is ; the adjunct is . The −c s − d b s−a roots of the characteristic equation (s − a) (s − d) − bc = 0 are separated by the positive numbers a and d. Therefore the maximum root r is positive and simple; and, for s r, the four elements of the adjunct are positive. Let us assume that the theorem holds for any positive matrix of degree < n. When applied to the characteristic determinant b = |su − a|, of which the entries are bik = suik − aik , formula (7) gives The theorem is easily checked for a positive matrix

b = a (s) = (s − aii ) Aii (s) −

aih aki

∂ Aii (s) ∂ bkh

(12)

Let ri be the maximum root of Aii (s), characteristic determinant of a matrix of degree n − 1. We have a (ri ) = −

aih aki

∂ Aii (ri ) 0, the equation a (s) = 0 has, for any i, at least one root > ri > 0. If r is the greatest, the derivative gives a (r) =

Aii (r) > 0

Therefore, this root is simple.

Lectures given at the Catholic Institute of Paris 197 Besides, for any entry of the adjunct, formula (8) gives Aih (s) =

aki

∂ Aii (s) ∂ bkh

(14)

For s r > ri , all the entries of the adjunct of Aii (s), therefore all the terms on the right-hand side are positive. Corollary. If r is the maximum root of a positive matrix a, and therefore also of its transposed a¯ , each of the systems rxi −

aik xk = 0

(i = 1, . . . , n)

(15)

rxi −

aki xk = 0

(i = 1, . . . , n)

(16)

admits a positive solution. 5. THEOREM IV. An entry Aih (s) of the adjunct (i = h) is positive for any value of s greater than the maximum root of the symmetric minor of degree n − 2 Aihih (s) = ∂∂Abiihh(s) . The formula (14), where h is = i, can indeed be written Aih (s) = ahi Aihih (s) +

aki

∂ Aii (s) ∂ bkh

(17)

where the summation index k takes no longer the value h. If one applies formula (9) to the determinant Aii (s), one has ∂ Aii (s) = ∂ bkh

ahj

∂ Aihih (s) ∂ bkj

(18)

where the summation index j does not take the value h. After substitution in (17), one gets Aih (s) = ahi Aihih (s) +

aki ahj

∂ Aihih (s) ∂ bkj

(19)

The theorem results from formula (19). 6. THEOREM V. If the system ρ yi −

aik yk = 0

(i = 1, . . . , n)

admits a nonnegative solution distinct from 0, we have ρ = r.

(20)

198 Lectures given at the Catholic Institute of Paris Let us multiply (16) by yi , (20) by xi ; let us sum over i and subtract; we have (r − ρ )

xi yi =

aki xk yi −

aik xi yk = 0

(21)

for any combination of a solution of (16) with a solution of (20). For the assumed solution of (20) and the positive solution of (16) (coroll. of theor. III), we have xi yi > 0, therefore ρ = r. 7. THEOREM VI. The maximum root r of a positive matrix a is greater than the mod of any root ρ = r of the characteristic equation. Indeed, the system ρ zi −

aik zk = 0

(22)

admits a solution distinct from 0, in which the arg of zj takes a value uj and its mod a value vj , the vj being 0 and not all zero. If all the arg uj have the same value u, the solution considered is zj = vj eiu , and we have ρ vi − aik vk = 0, the vi being 0 and not all zero; then (theor. V), one must have ρ = r, contrary to the hypothesis. Thus the arg uj are not all equal, and the image-points of the products aik zk not on the same half-line coming from 0. We therefore have aik vk > mod

aik zk = mod ρ zi = vi mod ρ

(23)

Let us multiply (16) by vi , (23) by xi (positive solution of (16)), sum up over i, and subtract the equality from the inequality; we have (mod ρ − r) As previously,

xi vi
0), the characteristic determinant becomes b (s) = a (s) − hAik (s). For s = r (previous maximum root), we have b (r) = −hAik (r) < 0; therefore r is smaller than the new maximum root. THEOREM VIII. The maximum root r of a positive matrix a is located between the smallest and the greatest of the sums aj = a1j + · · · + anj or aj = aj1 + · · · + ajn . In |su − a|, let us replace the first row by the sum of the n rows, and let us choose s = r; we have (r − a1 ) A11 (r) + . . . + (r − an ) A1n (r) = 0

Lectures given at the Catholic Institute of Paris 199 The A1i (r) being all > 0, all the differences r − ai cannot have the same signs. THEOREM IX. Let ri be the maximum root of Aii (s) and rk the smallest of the roots r1 , . . . , rn . The characteristic determinant a (s) has a unique root > rk . One sees immediately that the theorem holds for positive matrices of degree 2. Let us admit it for matrices of degree < n. Let rj be the greatest of the roots r1 , . . . , rn . For s > rj , the derivative a (s) = Aii (s) is > 0; therefore a (s), always increasing, vanishes at most once. Let us assume rk < s < rj . Let us consider, in a (s) and its adjunct A (s), the two homologous minors of degree 2 m (s) =

s − ajj

−ajk

−akj

s − akk

,

M (s) =

Ajj (s)

Ajk (s)

Akj (s) Akk (s)

The algebraic complement of m (s) is Ajkjk (s); and theorem II gives a (s) Ajkjk (s) = M (s) = Ajj (s) Akk (s) − Ajk (s) Akj (s)

(25)

Let rjk (< rk ) be the maximum root of Ajkjk (s), and let us assume rk < s < rj . We shall have Ajkjk (s) > 0. According to the theorem admitted for the matrices of degree n − 1, Ajj (s), of which the unique root > rjk is rj , is < 0, whereas Akk (s) is > 0. Ajk (s) and Akj (s) are > 0 (theorem IV). Therefore a (s) is then < 0. 9. THEOREM X. If s is greater than the maximum root r of a positive matrix a, and if b1 , . . . , bn are nonnegative numbers, and not all zero, the system sxi −

aik xk = bi

(i = 1, . . . , n)

(26)

admits a positive solution. The solution is indeed given by a (s) xj =

Aij (s) bi

( j = 1, . . . , n)

We know (theor. III) that a (s) and all the Aij (s) are > 0 for s > r. THEOREM XI. If, the same hypothesis being made on b1 , . . . , bn , the system (26) admits a nonnegative solution (necessarily distinct from the zero solution), we have s > r. According to the corollary of theorem III, the system ryi −

aki yk = 0

(i = 1, . . . , n)

admits a positive solution. By combining (26) and (27), one obtains (s − r) bi yi , and both are > 0.

(27) xi yi =

200 Lectures given at the Catholic Institute of Paris FOURTH LECTURE

Matrices with nonnegative entries Matrices with nonnegative entries. 1. A matrix b of degree n is said nonnegative when all its entries bik are 0. Such a matrix is said decomposable if the set of numbers 1, . . . , n can be decomposed into two distinct groups of m and n − m numbers j1 , . . . , jm and k1 , . . . , kn−m in such a way that all the entries, of which the first index belongs to the first group and the second to the second group are zero. By exchange of parallel lines, the rows and columns of which the rank belong to the group j can occupy the first m ranks. Then, in each of the first m rows, the n − m last entries are zeroes. Hence four submatrices: • •

On the downward diagonal from the left to the right, two square submatrices of degrees m and n − m; On the upward diagonal from the left to the right, two rectangular submatrices, the second being made of zeroes only.

The first two, denoted bm and bm are said component submatrices. It is clear that |b| = |bm | bm . If the submatrix below bm has zero entries only, the matrix b is said completely decomposable. 2. If a matrix a is decomposable, the same holds for su − a, and |su − a| is the product of two complementary symmetric minors, characteristic determinants of the two component submatrices. Any root of |su − a| is a root of one of these determinants, and conversely. Each component submatrix can itself be decomposable. Finally a, like su − a, can be considered as a square table of degree q, the entries of which are matrices Mih (i, h = 1, . . . , q), which are always null (made of zeroes) for h > i. The diagonal matrices Mii , which are square, are the indecomposable component matrices. Then one says that the matrix a is written under a reduced form. A component matrix M (non necessarily indecomposable) is said immediate component if any entry of a belonging to a row of M without belonging to M is null, or if this holds for any entry belonging to a column of M without belonging to M . It is clear that any indecomposable component matrix (i.c.m.) Mii belongs to the immediate component made of the union of the Mjh ( j , h = 1, . . . , i). Let M be an i.c.m. of degree m. Let q1 , . . . , qm be the ranks, on the diagonal of a non reduced form of a, occupied by these entries which, on the diagonal of M , occupy the ranks 1, . . . , m. All the entries of a same line being always moved simultaneously, the entry common to the ith row and kth column of M was, in the non reduced form of a, common to the qi th row and the qk th column. Thus, in the non reduced form of a, the i.c.m. M is completely determined by the ranks of its diagonal entries.

Lectures given at the Catholic Institute of Paris 201 THEOREM I. In the various reduced forms that a decomposable matrix can have, the i.c.m. are always the same up to their order. More precisely, if two i.c.m. M and M , appearing in two normal forms, have a common diagonal entry, they are made of the same entries. The theorem holds necessarily for a decomposable matrix of degree 2. Let us admit it for any matrix of degree < n. Let P and P , of degrees p and p , be two immediate components to which M and M belong respectively. Let k, by hypothesis 1, be the number of the diagonal entries common to P and P . We assume that, if necessary, the exchanges of parallel lines have been made in order that, in a, the diagonal entries of P occupy the ranks 1, . . . , k , k + 1, . . . , p, the first k entries being common to P and P . If the other diagonal entries of a do not all belong to P , P and P are immediate components of a matrix of degree < n which contains M and M , and for which the theorem is admitted. Let us therefore assume that: • • •

the first k diagonal entries, as well as the matrix A1 they define, belong to P and P ; the next p − k, as well as the matrix B2 they define, only belong to P; the last n − p, as well as the matrix C3 they define, only belong to P .

On these conditions, a, P and P are tables of submatrices, as below: A1 B1 C1 a = A2 B2 C2 , A3 B3 C3

P=

A1 B1 , A2 B2

P =

A1 C1 A3 C3

In order that P be an immediate component, it is necessary that either C1 = C2 = 0, or A3 = B3 = 0; in order that P be an immediate component, it is necessary that either B1 = B3 = 0, or A2 = C2 = 0. Therefore a is of one of the four types: A1 0 0 A2 B2 0 A3 0 C3

A1 0 0 0 B2 0 A3 B3 C3

A1 0 C1 A2 B2 C2 0 0 C3

A1 B1 C1 0 B2 0 0 0 C3

In every case A1 is an immediate component of P and P . Assume that the diagonal entry common to M and M is a11 . P and P are of degrees < n. According to the admitted theorem, any i.c.m. of P – or P – coincides with M – or M . And any i.c.m. of A1 is an i.c.m. of P and P . Therefore, any i.c.m. of A1 containing a11 must coincide simultaneously with M and M . Therefore M and M coincide. 3. Let us denote in general by aj and uj (an = a, un = u) the submatrices of a and u, of which the entries belong to the first j rows and columns. Let t be a parameter > 0, aj the positive matrices with entries aik = aik + t (i, k = 1, . . . , j) , rj the maximum root of aj , which is positive, simple and greater than the modulus

202 Lectures given at the Catholic Institute of Paris of any other root of |su − aj |. When t tends to 0, the coefficients and the roots of |su − aj | have limits which are the coefficients and the roots of |su − aj |; all the equalities previously established still hold at the limit; in the inequalities, the sign > must generally be replaced by . The limit rj of rj will be a root of suj − aj , and at least equal to the mod of any other root. Therefore this determinant cannot have a real root > rj ; it is therefore > 0 for s > rj . But, concerning the entries of its adjunct, it can only be said that they are 0 for s rj . The root rj will still be said the maximum root of the matrix (or of its characteristic determinant). 4. The inequalities r1 < r2 < · · · < rn = r

(1)

become at the limit r1

r2

···

rn = r

(2)

Consequently, if ri < r, we have ruj − aj = aj (r) > 0 for any j i; and, if ri = r, we have aj (r) = 0 for any j i. Clearly enough, any sequence of symmetric minors, of degrees 1, . . . , n, each being contained in the next, can be transformed by exchanges of parallel lines into the sequence a1 (s) , a2 (s) , . . . , an (s) = a (s). Hence the following theorem. THEOREM II. If a symmetric minor of a (s) = |su − a| does not vanish for s = r, it is, as well as all its own symmetric minors, > 0 for s r. If a symmetric minor of |ru − a| is zero, any symmetric minor containing it is also zero. Theorem IX of the 3rd lecture still holds for nonnegative matrices. Indeed the proof still applies, with the only difference that, on the hypothesis rj > s > rk rjk , the minors Ajk (s) and Akj (s) are only 0. The formula [III, (25)] which is a (s) Ajkjk (s) = Ajj (s) Akk (s) − Ajk (s) Akj (s)

(3)

still gives us a (s) > 0. THEOREM III. If Aii (r) is > 0 for any i, 1◦ the matrix a is indecomposable; 2◦ the entries of the adjunct of a (r) are all > 0; 3◦ each of the systems rxi −

aik xk = 0,

(i = 1, . . . , n)

(4)

ryi −

aki yk = 0,

(i = 1, . . . , n)

(5)

admits a positive solution.

Lectures given at the Catholic Institute of Paris 203 1◦

If a is decomposable, a (s) is the product of two symmetric minors of which at least one of them, b (s), vanishes at s = r. According to theorem II, the same holds for any Aii (s) containing b (s). 2◦ By making s = r in formula (3), we will have, according to the retained hypotheses, Ajk (r) Akj (r) = Ajj (r) Akk (r) > 0 As Ajk (r) and Akj (r) are 0, this formula shows that they are > 0. 3◦ This solution is made of the entries of any line of the adjunct. THEOREM IV. If the minor Aih (s) vanishes for a value of s greater than the maximum root rih of Aihih (s), it is reduced to 0 . The theorem can be checked for a matrix of degree 3. We have, for instance, A1212 (s) = s − a33 ,

A12 (s) = a21 (s − a33 ) + a31 a23 ,

r12 = a33

For s > a33 , A12 (s) is > 0, except if a21 = a31 a23 = 0, in which case it is reduced to 0. Let us admit the theorem for a matrix of degree < n. By going to the limit, it first results that Aih (s) is 0 for s = s0 > rih . Therefore, if Aih (s0 ) = 0, it is a minimum and the derivative vanishes. According to [III, (14)], we have, if suih − aih = bih , ∂ a (s) = Aih (s) = ∂ bih ∂ Ajj (s) = Ajijh (s) = ∂ bih

aki

∂ Aii (s) ∂ bkh

aki

(k avoiding i)

∂ Ajiji (s) ∂ bkh

(k avoiding i)

(6) (7)

d Ajiji (s). By differentiating (6) at s = s0 , we obtain, by We have ds Aii (s) = taking (7) into account,

0=

aki

∂ Ajiji (s0 ) = ∂ bkh

Ajijh (s0 )

(8)

Every Ajijh (s) is an entry of the adjunct of Ajj (s), a characteristic determinant of degree n − 1. As s0 is > rih , which is maximum root rjih of Ajihjih (s), every Ajhjh (s0 ) is 0; therefore (8) requires Ajijh (s0 ) = 0

( j = 1, . . . , n)

(9)

and, according to the admitted theorem, every Ajijh (s) is reduced to 0. Therefore d ds Aih (s) = 0 for any s. Therefore Aih (s) retains, for any s, the value 0 it takes by hypothesis at s = s0 .

204 Lectures given at the Catholic Institute of Paris Corollary. If Aii (r) = 0 and Ahh (r) > 0, at least one of the two minors Aih (s) or Ahi (s) is reduced to 0. Indeed, one must have Aihih (r) > 0, therefore r > rih , otherwise Ahh (r) would be zero. If, in (3), one makes s = r, one sees that at least one of the two minors here considered vanishes; therefore, it is reduced to 0. 5. THEOREM V. Each of the systems (4) and (5) of theorem III admits, in any case, a nonnegative solution distinct from the solution 0. The theorem can be checked for a matrix of degree 2. For, if we have (r − a) (r − d) − bc = 0, both equations (r − a) x − by = 0,

−cx + (r − d) y = 0

either disappear, or are reduced to one only, which clearly admits such a solution. Let us admit the result for a matrix of degree < n. The determinant of the system (4) being zero, one of the equations, the first for instance, follows from the others, and the system is equivalent to n

rxi −

aik xk = ai1 x1

(i = 2, . . . , n)

(10)

2

The determinant of the system is A11 (r), which is > 0 or = 0 according to whether the maximum root r1 of A11 (r) is < or = r. If r1 < r, the entries of the adjunct of A11 (r) are 0, and the ai1 as well. By setting, for instance, x1 = 1, solving (10) gives values 0 for x2 , . . . , xn . If r1 = r, one can set x1 = 0 [and] the homogeneous system corresponding to a nonnegative matrix of degree n − 1 has, according to the admitted result, a nonnegative solution distinct from the solution 0. THEOREM VI. If a symmetric minor b (s) of a (s) vanishes at s = r, and if all the symmetric minors of b (r) are > 0, the matrix a is decomposable, and the submatrix corresponding to the minor b is an indecomposable component matrix. The theorem is immediately checked for a matrix of degree 2. Let us admit it for a matrix of degree < n. By exchanges of parallel lines, one can move the entries of b (s) to the first m rows and columns. The system (4) can then be written rxi − ai1 x1 − · · · − aim xm − ai,m+1 xm+1 − · · · − ain xn = 0

(11)

rxj − aj1 x1 − · · · − ajm xm − aj,m+1 xm+1 − · · · − ajn xn = 0

(12)

i running from 1 to m and j from m + 1 to n. To the matrix b = am there corresponds the system ryi − a1i y1 − · · · − ami ym = 0

(13)

Lectures given at the Catholic Institute of Paris 205 According to theorem III, 1◦ and 3◦ , the matrix am is indecomposable, thanks to the retained hypotheses, and the system (13) admits a positive solution. According to theorem V, the system (11)–(12) admits a nonnegative solution distinct from the solution 0. In that solution, one may have a) xm+1 > 0, . . . , xn > 0, b) xm+1 = xm+2 = · · · = xn = 0, c) xm+1 = · · · = xk = 0, xk +1 > 0, . . . , xn > 0. [Let us examine the three cases successively:] a) Let us multiply (11) by yi , (13) by xi , sum up over i then subtract; it follows that n

m

yi 1

aih xh = 0

(14)

m+1

The yi and the xh being all > 0, (14) entails aih = 0 (i = 1, . . . , m; h = m + 1, . . . , n). The matrix a is therefore decomposable, am being one of its component matrices. b) By hypothesis, the numbers x, . . . , xm are then 0 and not all zero. They satisfy rxi − ai1 x1 − · · · − aim xm = 0 aj1 x1 + · · · + ajm xm = 0

(i = 1, . . . , m)

( j = m + 1, . . . , n)

(15) (16)

According to the retained hypotheses, the system (15) has rank m − 1 and admits (theorem III, 3◦ ) a positive solution. Therefore x1 , . . . , xm are > 0; and, since the ajk are 0, (16) entails ajk = 0 ( j = m + 1, . . . , n; k = 1, . . . , m). The result is the same. c) The unknowns that are not certainly zero satisfy xk +1 > 0, . . . , xn > 0

(17)

rxi − ai1 x1 − · · · − aim xm − ai,k +1 xk +1 − · · · − ain xn = 0

(18)

aj1 x1 + · · · + ajm xm + aj,k +1 xk +1 + · · · + ajn xn = 0

(19)

rxh − ah1 x1 − · · · − ahm xm − ah ,k +1 xk +1 − · · · − ahn xn = 0

(20)

(i = 1, . . . , m; j = m + 1, . . . , k; h = k + 1, . . . , n). The formulas (17) and (19) require aj1 x1 = · · · = ajm xm = 0

(21)

aj,k +1 = · · · = ajn = 0

(22)

206 Lectures given at the Catholic Institute of Paris By combining (18) and (13), as above (11) and (13), one obtains similarly ai,k +1 = · · · = ain = 0

(23)

According to (22) and (23), matrix a is decomposable and ak is one of its component matrices. But ak is of degree k < m. According to the admitted theorem, ak is decomposable and am is one of its component matrices. [End of proof] It follows from the theorem that, if a is an indecomposable nonnegative matrix, the Aii (r) are all positive, the maximum root is therefore simple, and we have all the conclusions of theorem III.9 THEOREM VII. If the system ρ xi −

aik xk = 0

(i = 1, . . . , n)

(24)

admits a positive solution or, if a is indecomposable, only a nonnegative solution distinct from the solution 0, we have ρ = r. By combination of (5) and (24), one obtains (ρ − r) the hypotheses that xi yi is > 0 in all cases.

xi yi = 0. It results from

6. THEOREM VIII. Let s > r, and n scalars b1 , . . . , bn , all positive or, if a is indecomposable, only nonnegative and not all zero; the system sxi −

aik xk = bi

(i = 1, . . . , n)

(25)

admits a positive solution. In fact, solving the system gives a (s) xj =

Aij (s) bi

(26)

According to the assumptions, a (s) and the Ajj (s) are > 0; the Aij (s) (i = j) are 0 if a is decomposable, > 0 if a is indecomposable. Therefore, by (26), each xj is always > 0. THEOREM IX. Let us set the same assumptions on the bi . If the system (25) admits a positive solution or, when a is indecomposable, only a nonnegative solution distinct from the solution 0, we have s > r. 9 [If some symmetric minor of a vanishes at s = r, it follows from theorem II that a admits a symmetric minor b such that the hypotheses of theorem VI hold, therefore a is decomposable. Consequently, if a is indecomposable, all its symmetric minors are positive by theorem II and we have all the conclusions of theorem III (Potron wrote ‘theorem II’). – Eds.]

Lectures given at the Catholic Institute of Paris 207 By combination of (5) and (25), we have (s − r) to the assumptions, both are positive.

xi yi =

bi yi . And, according

THEOREM X. If the matrix a is indecomposable, the system rxi −

aik xk = bi

(i = 1, . . . , n)

(27)

with bi ≥ 0, has a solution only if the bi are all zero.10 Indeed, the Aik (r) are all > 0 (theorems VI and II).11 The equations (27) are only compatible if one has Ai1 bi = 0, which requires b1 = · · · = bn = 0. 7. Assume the matrix a is decomposable and written under a reduced normal form (No. 2). Let ri be the maximum root of the i.c.m. Mii , of degree mi (i = 1, . . . , q). The equations (25) and the unknowns are split into q systems. Let us denote by Ei the system of the mi equations corresponding to the rows of Mii , by Bi the system of their right-hand sides, by Xi the system of the unknowns corresponding to the columns of Mii . A system Bi or Xi will be said positive (or zero), when its numbers are all positive (or zero). The greatest root ri is obviously the maximum root r. If the matrix a is completely decomposable, the equations and the unknowns are split into entirely independent systems. This case will be excluded. THEOREM XI. If s is > r, for the system (25) to have a positive solution, it suffices that the Bi are nonnegative, and B1 non zero. In fact, the system E1 has (theorem VIII) a positive solution X1 . In E2 , let us move to the right-hand sides the terms containing the unknowns of X1 . Matrix a not being completely decomposable, at least one of the right-hand sides of E2 will be > 0, even if B2 is zero. Therefore E2 has a positive solution X2 , and so forth. THEOREM XII. If s > r1 > ri (i = 2, . . . , n), it suffices to set B1 equal to zero in order that the system (25) admits a positive solution. Indeed, the system E1 then admits (theorems VI and II) a positive solution X1 ; and successive solving is completed as above. THEOREM XIII. If the Bi are all nonnegative, and if the system (25) admits a positive solution, 10 [This last equation is labelled (28) in Potron’s manuscript. We have also corrected two errors in the statement of theorem X. – Eds.] 11 [It follows directly from the remark after the proof of theorem VI that statement 2◦ of theorem III applies. – Eds.]

208 Lectures given at the Catholic Institute of Paris 1◦ if B1 is nonzero, we have s > r1 (theorem IX); 2◦ if B1 is zero, we have s = r1 (theorem VII); 3◦ we always have s > ri ( i = 2, . . . , n). In fact, in the system E2 , at least one of the right-hand sides, which depend on X1 , is > 0; X2 being positive, we have s > r2 , and so forth. THEOREM XIV. If s = rh r1 , if rh > rj ( j = 2, . . . , h − 1), and if the Bi (i = 1, . . . , q) are nonnegative, the system (25) has a solution only if B1 is zero; and, in that solution, X1 is zero. If one takes for X1 the positive solution of E1 (B1 being conveniently chosen), solving successively E2 , . . . , Eh−1 defines positive systems X2 , . . . , Xh−1 . At least one of the right-hand sides of Eh , which depend on X1 , . . . , Xh−1 , is > 0. Therefore (theorem X) Eh has no solution. Thus, it is necessary that X1 be zero, which requires that B1 is also zero. FIFTH LECTURE

Inductive solution of a system of linear equations Unpublished method for solving a system of linear equations and calculating a determinant. This method is due to Mr Cornelius Lanczos, professor at Purdue University, Lafayette, USA. Mr Lanczos has been kind enough to let me know the content of his work, on this topic, addressed to the American Mathematical Society (Bull. Am. Math. Soc., t. 42, 1936, p. 325).12 j

1. Let a be a matrix with m rows and n columns, with entries ai , and b a matrix with n rows and p columns, with entries bkj . Let c be a matrix with m rows and p columns, with entries cik =

n 1

j

ai bkj

(1)

c is said product-matrix of a and b (in this order), and one writes symbolically c = ab.

12 [Cornelius Lanczos, ‘A simple recursion method for solving a set of linear equations’ (Bulletin of the American Mathematical Society, 1936, 42: 325). In fact this is only the abstract of a paper presented at the April Meeting of the American Mathematical Society in 1936. The full paper has never been published and does not seem to have survived. In 1939 Potron claimed that Lanczos had sent him the full manuscript (see note 3 of A41/Chapter 16). – Eds.]

Lectures given at the Catholic Institute of Paris 209 If a and b are two square matrices of degree n, it follows from the multiplication rule of determinants that |c| = |a| × |b|. Let x be a matrix with one column and n rows, with entries xj . The productmatrix ax has m rows and one column. Let b be a matrix with one column and m rows, with entries bi . The convention is that the sign = between two symbols of matrices means that their homologous entries are equal.13 The symbolic equality ax = b

(2)

represents the system of equations n 1

j

ai xj = bi

(i = 1, . . . , m)

(3)

Similarly, the system of equations n

sxi − 1

aki xk = bi

(i = 1, . . . , n)

(4)

is represented by one or the other of the symbolic equalities (su − a) x = b

(5)

sx = ax + b

(6)

Similarly, the symbolic equalities ya = c

(2 )

a¯ y = c

(2 )

represent the systems y j aij = ci

(3 )

aij yj = ci

(3 )

Incidentally, these two systems only differ by the notations. In (2 ) and (3 ), the matrices y and c have but one row; in (2 ) and (3 ), they have but one column. 2. Let a be a square matrix of degree n. We are about to construct three matrices α, β, λ, of degrees n, with the following properties. 13 [Probably under the influence of the work of Lanczos, Potron used subscripts and superscripts to indicate the rows and columns of a matrix, a notation due to Einstein. Note that Potron referred to ‘symbolic equalities’ between matrices and did not mention the notion of vector. – Eds.]

210 Lectures given at the Catholic Institute of Paris Their entries are such that αik = 0

(k > i) ,

αii = 1

(i = 1, . . . , n)

(7)

βik = 0

(k > i) ,

βii = 1

(i = 1, . . . , n)

(8)

λki = 0

(i, k = 1, . . . , n; i = k)

(9)

and we have the symbolic equality αλβ¯ = a

(10)

β¯ denoting the transpose of β . Calculating first αλ = γ , then γ β¯ = a, one obtains n 1

αih λhh βkh = aki

(i, k = 1, . . . , n)

(11)

We shall denote by αm , βm , λm the submatrices of which the entries belong to the first m rows and columns of α, β, λ. Since, in (11), we have zero terms as soon as the summation index h exceeds min (i, k), the first m equations (11) are represented by the symbolic equality αm λm β¯m = am

(12)

We have α11 = β11 = 1 and, by (11), λ11 = a11 . Therefore α1 , β1 and λ1 are known. Let us assume that αm−1 , βm−1 , λm−1 are known. Those of the equations (11) where i = m and k runs from 1 to m − 1 are k −1 k −1 k −1 k k 1 1 1 λk −1 βk + αm λk = akm αm λ1 βk + · · · + αm

(k = 1, . . . , m − 1)

(13)

−1 Provided that the product λ11 λ22 . . . λm m−1 is = 0, these equations define 1 2 m − 1 αm , αm , . . . , αm successively. Those of the equations (11) where k = m and i runs from 1 to m − 1 are

1 i−1 i i m αi1 λ11 βm1 + · · · + αii−1 λii− −1 βm + λi βm = ai

(i = 1, . . . , m − 1)

(14)

Under the same condition, these equations define βm1 , βm2 , . . . , βmm−1 successively. At last, for i = k = m, we have 1 1 1 m−1 m−1 m−1 m αm λ1 βm + · · · + αm λm−1 βm + λm m = am j

(15)

This equation lets us know λm m . As long as a λj is not found equal to zero, the calculation can go on.

Lectures given at the Catholic Institute of Paris 211 3. As αj = βj = 1, the formula (12) gives j

aj = λj = λ11 λ22 . . . λj The method thus provides an easy procedure to calculate |a| when all the aj are = 0. Note that, if the determinant |a| is = 0, it is always possible to rank the columns in such a way that all the aj are = 0. The property is obvious for a determinant of degree 2. Let us admit it for any determinant of order < n. If |a|, of degree n, is = 0, at least one of the Ahn is = 0. One can rank the columns in such an way that Ann = |an−1 | be = 0. Then, according to the admitted property, one can rank the first n − 1 columns in such a way that |a1 | , . . . , |an−2 | be = 0. If a is a nonnegative matrix of maximum root r, the determinants suj − aj are all > 0 for s > r and, if a is indecomposable, even for s = r. Hence, the theorem: THEOREM I. If s is greater than the maximum root r of a nonnegative matrix j a, the λj successively obtained in the calculation of |su − a| are all > 0. If a is indecomposable, the same for s = r and j < n. Then λnn is the first zero coefficient. j If therefore one encounters some zero λj , one can say that s is r, and even, if a is indecomposable, that s is < r. 4. If |a| is = 0, the same method allows us to calculate the solution of the linear system symbolically represented by ax = b

(16)

According to (10), we will have αλβ¯ x = b. Therefore, if one sets β¯ x = y, λy = z, we will have α z = b. Thus the solution of (16) will be obtained by solving successively α z = b,

αi1 z1 + · · · + αii−1 zi−1 + zi = bi

λy = z ,

λii yi = zi

β¯ x = y,

xi + βii+1 xi+1 + · · · + βni xn = yi

(i = 1, . . . , n)

(i = 1, . . . , n)

(17) (18)

(i = n, n − 1, . . . , 1)

(19)

These systems give successively z1 and y1 , …, zn and yn , then xn , xn−1 , . . . , x1 . 5. Let us assume that, a being a nonnegative matrix, these various calculations have been made for solving the system (su − a) x = b j

I claim that, if all the λj have been found > 0, all the non diagonal entries of the matrices α and β are 0.

212 Lectures given at the Catholic Institute of Paris In the previous calculations, one must indeed replace aki by suik − aki . For m = 2, the first equations (13) and (14) give α21 λ11 = −a12

λ11 β21 = −a21

0,

0

(20)

Thus the non-diagonal entries of α2 and β2 are 0. If the same holds for αm−1 and βm−1 , the equations (13) and (14), of which the right-hand sides are now −akm and −am i , show successively that the same will hold for the last rows of αm and βm . Then, if b is a positive matrix, the systems (17), (18), (19) show that z , y, x are positive matrices. Since the system (su − a) x = b, b being positive, gives x positive, one concludes [IV, theor. IX] that s is > r, maximum root of a. Hence the theorem. j

THEOREM II. If the λj relative to the matrix s0 u − a are all > 0, it is sure that s0 is > r, maximum root of a. j

Assume that one finds λj > 0 for j < m, and = 0 for j = m. One concludes |s0 um−1 − am−1 | > 0, |s0 um − am | = 0. Thus s0 is (theorem II) > rm−1 , maximum root of am−1 . But, according to [III, theor. IX] and [IV, No. 2], the only root of |sum − am | that is > rm−1 is the maximum root rm of am . Hence, the theorem. j

THEOREM III. If, relative to the matrix s0 u − a, one finds λj > 0 for j < m and = 0 for j = m, one can assert that s0 is the maximum root of am , therefore r, maximum root of a. 6. Thus, a regular sequence of recursive operations allows us to determine, if it exists, the positive solution of the system (su − a) x = b. If such a solution does not exist, the signal is given with certainty, in the course of the calculation, by the j occurrence of a λj 0. 7. Let a, c, g be three nonnegative matrices such that a = c + g; then b, b , b three positive matrices satisfying b = b + b . If s is greater than the maximum root r(a), the system sx = ax + b = cx + b + gx + b

(21)

defines a positive matrix x. Let us set cx + b = sx ,

gx + b = sx

(22)

The matrices x and x are positive and satisfy x + x = x. One can then rewrite (22) as sx − cx − cx = b ,

sx − gx − gx = b ,

or (su − d) z = p

(23)

Lectures given at the Catholic Institute of Paris 213 by setting dik = dik +n = cik ,

dik+n = dik++nn = gik

zi+n = xi ,

zi = xi ,

pi = bi ,

pi+n = bi

(i, k = 1, . . . , n)

The matrix p is positive. One sees that the existence of a positive matrix x, solution to (21), entails that of a positive matrix z, solution to (23), which requires r (a) r (d); and that, conversely, the existence of a positive matrix z, solution to (23), implies that of a positive matrix x = x + x , solution to (21), which requires r (d) r (a). We therefore have r (a) = r (d). Though the matrix d has degree 2n, it may occur that its component matrices are of a degree less than those of a. Since, after having written the matrix under a normal reduced form, it suffices to make the calculations of Nos. 2, 3, 4 successively for the component matrices, these calculations may finally be simpler. SIXTH LECTURE

Conclusions Some economic conclusions. Rationalization and unemployment. Overproduction and hoarding. The race to the rise. 1. In the first two lectures, we have assumed that the numbers N , cki , tkh and bhi , characterizing the state of the economy and the standards of living assigned to the various professions, are given. We have endeavoured to construct ‘satisfactory’ production–consumption and prices–wages regimes. We have characterized these regimes by: the productions pi , the excess productions fi , the professional workforces Ph , the numbers of non-workers qh , the partial unemployments Wh , the prices ai , the benefits bi , the wages sh , the economies eh .14 These numbers must satisfy the relations: Ci =

cki pk +

bhi (Ph + qh )

(1)

πi =

cik ak +

tih sh

(2)

Th =

tkh pk

(3)

Sh =

bhk ak

(4)

pi > 0,

pi − Ci = fi

0

(5)

ai > 0,

ai − πi = bi

0

(6)

14 [In the following equations Potron introduced the symbol πi to denote the cost price of good i. – Eds.]

214 Lectures given at the Catholic Institute of Paris Ph > 0,

NPh − Th = Wh

sh > 0,

Th sh − Sh = eh Ph

0

(7)

0

(8)

We have seen that it suffices to find a positive solution (ai , pi ) of the systems pi −

Cki pk = Fi

0

(9)

ai −

Cik ak = Bi

0

(10)

tkh bhi /N

(11)

Cki = cki + Cik = Cik +

tih bhk Wh /NTh

F i = fi +

bhi qh +

Bi = bi +

tih Ph eh Th

(12)

Wh N

(13) (14)

2. The system (9) admits a solution > 0 only if the maximum root r of the matrix C is < 1 [IV, theor. IX]. If r is < 1, one can give arbitrary values > 0 to the fi , qh and Wh , upon which the Fi depend by (13); (9) always defines a solution > 0 [IV, theor. VIII]. We have moreover seen that any solution to (9) satisfies pi −

Cki pk = Fi = fi +

bhi qh

(15)

If the arbitrary unknowns have been chosen in order that the Fi are > 0, the maximum root r of matrix C will be < 1. It will therefore be possible to give arbitrary values > 0 to the bi and eh , upon which the Bi depend by (14); (9) will always define a solution > 0. Thus, in order that the satisfactory regimes one is looking for can be constructed, a necessary and sufficient condition is that the maximum root r of matrix C is < 1. j This occurs [V, theor. II] always and only if the λj relative to matrix U − C are > 0. The production capacity of the industry seems to show that this condition is presently met, except perhaps for a reservation about N . 3. The calculation of λ11 , λ22 , . . . , λnn is made [V, Nos. 2, 3] by an induction in which the entries of the ith row and column of matrix C intervene only when λii is calculated. Moreover, all the entries of this ith row can be easily calculated by the producer of Ai alone. He knows [I, No. 3] his coefficients cik and tih . It suffices that he also knows the coefficients bhk relative to the only professions he employs. 4. If r is < 1, one can always, provided that some qh are taken > 0, construct satisfactory regimes without overproduction ( f1 = · · · = fn = 0).

Lectures given at the Catholic Institute of Paris 215 Assume first that the matrix C is indecomposable. As, according to (12), Cik and Cik are always zero simultaneously, the matrix C will also be indecomposable. Let us choose one qh > 0; the Fi and Fi corresponding to the rations consumed at this standard of living are > 0. Then the positive solution to (9), which satisfies (15), makes r < 1; and, with the Bi > 0, (10) admits a positive solution. If C, and therefore C , is completely decomposable, each of the systems (9) and (10) is split into completely independent systems. Let us assume that C is not completely decomposable, and let us set U − C (matrix of (9)) under a normal reduced form. We shall have ⎡ ⎤ M 11 0 ... 0 0 ⎢ × M 22 . . . 0 0 ⎥ ⎥ (16) U −C = ⎢ ⎣ ... ... ... ... ... ⎦ × × . . . × M kk ⎡

M11 ⎢ 0 U −C = ⎢ ⎣ ... 0

× M22 ... 0

... ... ... ...

⎤ × × × × ⎥ ⎥ ... ... ⎦ 0 Mkk

(17)

To have a solution > 0 to (9), it suffices [IV, theor. XI] to take > 0 one of the Fi corresponding to M11 ; and, if the corresponding Fi is > 0, it is ensured that r1 is < 1. Among the units corresponding to the rows of M 11 there is at least one 1st-species unit Ai consumed at some standard Vh ; Fi will be > 0 simultaneously with qh . Then the 1st system E 1 of (9) has a solution > 0, which satisfies the 1st system E 1 of (15), one of the right-hand sides of E 1 being > 0; r1 is < 1. If the already determined unknowns are moved to the right-hand sides of E 2 and E 2 , at least one of the right-hand sides will be > 0. As r2 is < 1, E 2 has a solution > 0 satisfying E 2 ; therefore r2 is < 1. And so forth. According to the form (17) of the matrix U − C , the system (10) is split into systems Ek , Ek −1 , …, E1 that will be solved successively. To obtain a solution > 0, it will suffice to take > 0 one of the Bi of system Ek , and 0 all the others. 5. The root r is [III, theor. VII] an increasing function of each of its coefficients Cki , where N appears in the denominator. Decreasing N too much may well make r exceed the critical value 1. Incidentally, it is useless to let N decrease. Formula (13) shows that the Fi only depend upon the νh , νh = qh + Wh /N ; and according to (7), we have Ph + qh = νh + Th /N

(18)

Therefore the productions and the numbers of consumers at the various standards of living only depend upon the νh . These being chosen, and the fi taken zero for

216 Lectures given at the Catholic Institute of Paris instance, (9) defines the pi , then (3) the Th . One can take for qh a convenient fraction of the right-hand side of (18); then Ph follows, finally Wh by (7). A value, even considerable, of Wh does not show any drawback; provided that the prices satisfy (10) and the wages (8), the effective wage is always at least equal to the cost of living. 6. Let us assume that, after a regime with no overproduction, there follows, for instance with a view to equipping a new industry, a regime in which one schedules fi = 0 for i = 1, . . . , l and > 0 for i = l + 1, . . . , n. The number of consumers, for each standard of living, is assumed to remain the same. The productions will increase by δ pi . According to (1) and (3), the increases in consumption and labour will be δ Ci =

cki δ pk

(19)

δ Th =

tkh δ pk

(20)

and, according to (5), we have δ pi − δ Ci = δ pi −

cki δ pk = fi

(21)

If δ Th is smaller than the value of Wh in the first regime, one can maintain the same Ph . One can also assume that nothing is changed in the formulas (2), (4) and (6). Formula (8) shows, for each worker, an increase of earnings15 δ eh = sh δ Th /Ph

(22)

In the first regime, by taking (8) into account, the formula [II, (16)] was written bi pi +

(Ph + qh ) Sh

Th sh =

(23)

In the second regime, the same formula is written bi (pi + δ pi ) +

sh (Th + δ Th ) =

fi ai +

(Ph + qh ) Sh

(24)

Subtracting and taking (22) into account, one obtains bi δ pi +

Ph δ eh =

ai fi

(25)

15 [Here and below, Potron wrote ‘earnings’ (gains) but actually equation (22) measures the changes in the ‘economies’ (épargne). – Eds.]

Lectures given at the Catholic Institute of Paris 217 When the index i runs over the integers 1, . . . , l and the index j over the integers l + 1, . . . , n, (25) can be written16 bi δ pi +

Ph δ eh =

aj fj −

bj δ pj

(26)

On the left-hand side, the two terms represent respectively the increases of the benefits of the firms which are normally working, and of the earnings of all workers. On the right-hand side, by adding and subtracting aj δ pj , one has πj δ pj − aj δ pj − fj

The two terms represent respectively the increases in the expenses and receipts of the firms which have produced the excess units but have not yet sold them. If one subtracts and adds bj fj , the right-hand side of (26) can also be written πj fj − bj δ pj − fj

The two terms represent respectively the cost prices of the excess units, and the increases of the benefits of the firms which have produced them. Assume that the second regime differs from the first only by the production of fn+1 economic units of a new species, which is not yet consumed. We have δ pn+1 = fn+1 and n

m

Ph δ eh = πn+1 fn+1

bi δ pi + 1

(27)

1

One sees that, in all cases, the expenses induced by the production of the excess units, in whatever way they have been financed,17 are found again entirely as increases in the firms’ benefits and the workers’ earnings. To the non-absorption of these excess units will correspond a hoarding of these increases in benefits and earnings. 7. The value of the number n of economic units can be estimated at 10 at least.6 The calculations necessary for a complete solution of the systems (9) and (10) are therefore in practice prohibitive, and it is only by trial and error that a solution can be found.18 However, our previous mathematical study of the problem is not

16 [In equation (26), Potron split the previous sum over i = 1, . . . , n as indicated and made use of the assumption fi = 0 for i = 1, . . . , l. – Eds.] 17 [A second reference to finance. – Eds.] 18 [‘Trial and error’ is the translation of the French word tâtonnement, which has the general meaning of ‘groping’. – Eds.]

218 Lectures given at the Catholic Institute of Paris useless. It shows that, for prices and wages, it suffices in sum to have a positive solution (a) of the system ai −

Cik ak > 0

(i = 1, . . . , n)

(28)

where the row of rank i of matrix C can be easily formed by a producer of the Ai units. We have mentioned (No. 2) why it seems possible to assert a priori the existence of positive solutions to this system, except for a reservation about N . For reasons which are mainly of a psychological rather than a mathematical order, it is easier to find a solution (a) of (28) than a solution (a, s) of ai −

cik ak −

Th sh /Ph −

tih sh > 0

bhk ak > 0

(29) (30)

8. As already noticed, (28) is the necessary and sufficient condition that the prices must meet in order that there exist wages constituting with them a satisfactory regime. Assume that one starts from prices which do not meet (28), and that, Sh being the cost of the standard of living ‘assigned’ to the profession Vh , one starts from wages giving Nsh − Sh < 0

(31)

a − pi − pi > 0

(32)

where pi and pi denote the respective cost prices of raw materials and of labour. Assume moreover that one wants to increase all wages by a same proportion and, if necessary, all prices by a same proportion. Let us multiply all wages by a scalar r (> 1) giving Nrsh − Sh

0

(33)

and necessarily ai − pi − rpi < 0

(34)

at least for some i; otherwise the adjustment would be implemented, with prices satisfying (28).19 We can change the signs of the left-hand sides of (34) by multiplying all prices by lr > 1, l satisfying l ai − pi − pi > 0

(35)

19 [Potron wrote ‘with prices not satisfying (28)’. The initial assumption of section 8 should be read as: Assume that there are no prices which satisfy system (28) and that one starts from prices and wages satisfying conditions (31) and (32). – Eds.]

Lectures given at the Catholic Institute of Paris 219 But, the adjustment being impossible a priori, we shall have, at least for some h Nrsh − lrSh = r (Nsh − lSh ) < 0

(36)

Taking advantage of (33), we multiply the new wages by lr. But then we have, by (34) lr(ai − pi ) − lr 2 pi = lr ai − pi − rpi < 0

(37)

Taking advantage of (35), we shall multiply the new prices by lr. But, then, we shall have simply multiplied the left-hand side of (36) by lr. All must be started again, and this indefinitely: it is the ‘race to the rise’. 9. Up to now, we have considered the coefficients cki and tkh characterizing the state of the economy as constant. In fact they change, but slowly, and the effect of their changes on the coefficients Cki can always be forecast. In any case, a genuine progress must result in a decrease of the maximum root r, which allows for either a reduction of labour or an increase in the standards of living. As far as ‘rationalization’ is concerned, it seems possible to give more precise conclusions. If all the tih are divided by a same factor l (> 1), it will suffice to divide N and Wh , then to multiply sh by l, while preserving all the rest. The entries of matrices C and C , in which only the ratios thk /N and Wh /Th appear, do not change; and all formulas still hold. (Th is also divided by l.) If the tih are not all decreased in the same proportion, one can say nothing on the directions of change of Cik = cik +

tih bhk (Th + Wh ) /NTh = cik +

bhk Ph tih /Th

because we have Th /tih = pi +

pj tjh /tih

( j avoiding i)

and we know nothing about the directions of change of the ratios tjh /tih . To avoid the difficulty, it suffices to assume, for a given profession, different wages sih . If pih denotes the number of workers of profession Vh employed in the production of Ai , we shall have ai −

cik ak −

Npih − pi tih = wih

tih sih = bi 0

pi tih sih /pih − Sh = eih

0

(6 ) (7 )

0

(8 )

One can keep constant the numbers of workers pih , the productions pi , the prices ai , the costs of living Sh , the benefits bi and the economies eih . When the tih decrease, it suffices to increase the sih in order that the products tih sih remain constant. All the formulas will still hold.

220 Lectures given at the Catholic Institute of Paris 10. The important point is therefore to construct and implement for a first time a double satisfactory regime. But, faced with this problem, we must confess, by the mathematical study to which we have proceeded, the weakness of our human intelligence. We have been able to state the possibility condition of the problem, and to point out a theoretical device allowing us to recognize if it is met. We have strong reasons to think it is. But the very length of the calculations stops us and obliges us to proceed by trial and error. Some precise results have nonetheless been obtained. In principle, it is a satisfactory regime of production and labour that must be organized first. A corresponding regime of prices and effective wages will always be possible. To construct such a regime, it is necessary and sufficient to construct first a price system satisfying the system (28) of inequalities, in which any inequality can be set up by a single producer, who has to take into account, in the calculation of the coefficients, the standards of living of the only workers he employs. Then, the determination of effectively satisfactory wages does not present any more difficulty. It is certain, and the author thinks he can well remind those who share his faith of that, that our Father who is in Heaven wants, for all his children, even on earth, a decent life. The implementation of this plan demands the labour and endeavour of men. Often, one focuses too much on its practical, already formidable, difficulties. By letting us assess the theoretical difficulties of the problem, the science of mathematics gives us a new reason to repeat to our Father in Heaven the traditional prayer ‘Give us today our daily bread’. APPENDIX

The Hebrew Manna problem The problem dealt with here presents, under a very simplified form, the essential elements of some current economic problems. The state of the economy being given, the question is to build, on the one hand, a regime of production–labour– consumption (P–L–C) and, on the other hand, a regime of prices and wages (P–W), in order to ensure a given standard of living to each profession. The regimes to be built must satisfy the conditions: a) b) c) d)

production must be at least equal to consumption; the labour provided by each worker must not exceed a certain limit; the selling price of an object must be at least equal to its cost price; the worker’s wage must be at least equal to the cost of living corresponding to his assigned standard of living.

1. A certain number of families are brought together in the desert. Each of them lives in a ‘closed economy’. The only exception concerns food. We assume the existence of three firms: • • •

the SCM which collects the manna; the SCF which collects the fuel; the SBC which bakes the cakes intended for consumption.

Lectures given at the Catholic Institute of Paris 221 None of these firms uses tools. The first two use labour exclusively. The SBC uses labour and two raw materials, manna and fuel, which are provided by the SCM and the SCF. We assume that the workforce has only two categories: 1) those who carry out the work (V1 ), 2) those who prepare, run and control it (V2 ). We know what is required: • • •

to collect one ‘omer’20 of manna: one hour of labour of each type; to collect one ‘homer’21 of fuel: two hours of labour of each type; to bake one cake: one omer of manna, three homers of fuel and three hours of labour of each type.

Every year, each worker’s family must be given a number of cakes ‘considered as convenient’, viz. 100 for a family of category 1, 150 for a family of category 2. By denoting by A1 the omer of manna, by A2 the homer of fuel, by A3 the cake, the data are summarized in the table22 V1 V1 V2 A1 A2 A3

1 2 3

V2

A1

A2

A3

1 2 3

0 0 0 0 1

0 0 0 0 3

100 150 0 0 0

We shall introduce the following symbols which represent, for the time being, unknowns to be determined: Q1 and Q2 , the numbers of families living respectively with 100 and 150 cakes per year; P1 and P2 , the numbers of heads of family who work; q1 and q2 , the numbers of heads of family who do not work; 20 [The ‘omer’ (gomor in Potron’s text) was a Biblical dry measure typically used for grain and 1 of an ‘ephah’. – Eds.] manna. It consisted of 10 21 [The ‘homer’ (chomer in Potron’s text) was another Biblical dry measure. It was equal to 10 ‘ephahs’. See Vittrant’s comment (W3/Appendix III) for an evaluation of both units. – Eds.] 22 [In order to be faithful to the genealogical method he presented in section 3 of his first lecture, Potron should have placed the row relative to the production of cakes (A3 ) above the rows relative to the collection of manna and fuel (A1 and A2 ), since cakes are goods of the first species, and manna and fuel of the second species. The ranking of the goods has, however, no effect on the calculations. – Eds.]

222 Lectures given at the Catholic Institute of Paris d1 , the number of omers of manna collected in the year; d2 , the number of homers of fuel collected in the year; d3 , the number of cakes baked in the year; N , the maximum number of hours that a worker can provide per year. 2. From the table of data, we can immediately say that the annual consumption amounts to d3 omers of manna, 3d3 homers of fuel, and 100Q1 + 150Q2 of cakes. By subtracting consumptions from productions, we have the variations of stocks d1 − d3 = f1 ,

d3 − (100Q1 + 150Q2 ) = f3

d2 − 3d3 = f2 ,

(1)

It is furthermore obvious that Q1 = P1 + q1 ,

Q2 = P2 + q2

(2)

For each type of labourer, the number of working hours provided in the year is T = d1 + 2d2 + 3d3

(3)

Since the maximums allowed are NP1 and NP2 , the differences NP1 − T = W1 ,

NP2 − T = W2

(4)

represent unemployment or overtime hours. 3. The very nature of things requires that d1 > 0,

d2 > 0,

q1

q2

0,

d3 > 0,

P1 > 0,

P2 > 0

0

(5) (6)

Conditions a) and b) can be expressed as f1

0,

f2

0,

f3

0,

W1

0,

W2

0

(7)

Hence we are facing a problem of algebra: to find, for the various symbols linked by the equations (1) to (4), values satisfying the conditions (5) to (7). 4. By relying on (2)–(3)–(4), we can replace the system (1) by23 d1 − d3 = f1 ,

d2 − 3d3 = f2

Nd3 − (250d1 + 500d2 + 750d3 ) = F3

(8) (9)

F3 = 100W1 + 150W2 + N (100q1 + 150q2 ) + Nf3 23 [Potron did not have an equation labelled (8), and his label (9) appears to refer to the whole of the following three equations. We have added (8) to the following equation in order to remain consistent with Potron’s numbering. – Eds.]

Lectures given at the Catholic Institute of Paris 223 Taking into account the first two equations, the third can be replaced by (N − 2,500) d3 = 250f1 + 500f2 + F3

(10)

The conditions (5)–(7) show that the problem would be impossible if N was given < 2,500. Assume N = 3,000; the equation (10) becomes f1 f2 W1 W2 d3 = + + f3 + + + 100q1 + 150q2 6 12 6 30 20

(11)

One sees that the problem admits some indetermination, because the unknown symbols in the right-hand side are submitted only to the conditions (7). In fact, a solution is associated with each system of values given to these symbols. For, once d3 is known, we will obtain d1 and d2 by (9), T by (3), P1 and P2 by (4), Q1 and Q2 by (2). If, for instance, one takes f1 = f2 = f3 = 0,

W1 = 9,000,

W2 = 6,000,

q1 = 1,

q2 = 2

the corresponding solution is d1 = 6,000, P1 = 23,

d2 = 18,000,

P2 = 22,

d3 = 6,000,

T = 60,000

Q1 = Q2 = 24

These figures define a P–L–C regime with the following characteristics: among 48 families, 24 belong to each type, 23 whose head is a worker of the 1st category, 22 whose head is a worker of the 2nd category. During the year, the first 23 are out of work for 9,000 hours, the last 22 for 6,000 hours; 6,000 omers of manna are collected and cooked, 18,000 homers of fuel collected and burnt, and 6,000 cakes produced and consumed. 5. Let us now try to build, on these data, a P–W regime satisfying the conditions c) and d). We assume that a worker of the 1st or 2nd category earns s1 or s2 pennies per hour of labour;24 the SBC pays a1 pennies per omer of manna to the SCM; the SBC pays a2 pennies per homer of fuel to the SCF; the consumers buy at the price of a3 pennies per cake from the SBC. The table of data allows us to evaluate the cost prices immediately: for the omer of manna, s1 + s2 pennies; for the homer of fuel, 2 (s1 + s2 ) pennies; 24 [We have translated the French denier, an old (and small) French monetary unit, as ‘penny’. – Eds.]

224 Lectures given at the Catholic Institute of Paris for the cake, a1 + 3a3 + 3 (s1 + s2 ) pennies. The accounts of the three firms are expressed by the following three equations, where the second members represent the benefit (or the loss) per unit: a1 − (s1 + s2 ) = b1 ,

a2 − 2(s1 + s2 ) = b2 ,

a3 − a1 − 3a2 − 3(s1 + s2 ) = b3 (12)

The effectively earned wages during the year amount to Ts1 /P1 pennies and Ts2 /P2 pennies for a worker of the 1st or 2nd category (T being defined by (1)). These workers’ costs of living amount respectively to 100a3 and 150a3 pennies. Their budgets are therefore expressed by the two equations Ts1 − 100a3 = e1 , P1

Ts2 − 150a3 = e2 P2

(13)

Prices and wages are by nature positive scalars, a1 > 0,

a2 > 0,

s1 > 0,

s2 > 0

a3 > 0

(14) (15)

Conditions c) and d) are fulfilled if b1

0,

b2

0,

b3

0,

e1

0,

e2

0

(16)

The numbers P1 , P2 , T must be assumed to be known, that is, a P–L–C regime must be assumed to be built. The regime built in No. 4 gives25 60,000s1 − 2,300a3 = 23e1 ; 60,000s2 − 3,300a3 = 22e2

(13*)

By inserting these values in (12), one obtains a1 −

7a3 = B1 , 75

B1 = E + b1 ,

a2 −

14a3 = B2 , 75

B2 = 2E + b2 ,

21a3 = B3 75 23e1 + 22e2 E= 60,000

a3 − a1 − 3a2 −

B3 = 3E + b3 ,

(17) (18)

The conditions (16) imply B1

0,

B2

0,

B3

0

(19)

6. If the system (12) to (16) admits a solution, so does the system (14)–(17)–(19). Conversely, if this is the case, one can obviously choose arbitrarily five scalars b1 , b2 , b3 , e1 , e2 satisfying (16) and (18). Then (13*) determines s1 and s2 . 25 [The following equation is labelled (13) by Potron, because it corresponds to his previous equation (13) for the numerical values he has just chosen. To avoid confusion we label it as (13*). – Eds.]

Lectures given at the Catholic Institute of Paris 225 In this particular case, it is possible to finish the calculations. Let us add the equations (17) after respective multiplications by 1, 3, 1; this gives a3 = B1 + 3B2 + B3 = b1 + 3b2 + b3 + 10E 15

(20)

Thus, to a P–L–C regime an infinite number of satisfactory P–W regimes may be associated, since the symbols in the right-hand side of (20) are submitted only to the conditions (16). Let us take, for example, b1 = b2 = b3 = 1, 23e1 = 10, 000, e2 = 0. The solution obtained is a1 = 10.50,

a2 = 20,

a3 = 100,

s1 = 4,

s2 = 5.50

(21)

7. There is room for an interesting remark. The total of the firms’ benefits (d1 b1 + d2 b2 + d3 b3 = 30,000) and of the workers’ economies (23e1 = 10,000) amounts to 40,000 pennies. This is exactly the price of the 4,000 cakes that the three non-workers must consume: one of the 1st category (100), two of the 2nd category (300).

15 On nonnegative matrices

Editors’ note In the fifth lecture of his 1937 lecture series Potron had introduced a new solution method for systems of linear equations inspired by the work of Cornelius Lanczos. In the paper ‘Sur les matrices non négatives’ (Comptes Rendus de l’Académie des Sciences, 1937, 204: 844–6), introduced by Maurice d’Ocagne at the session of 8 March 1937, he gave a short formal presentation of the new method. Cornelius Lanczos (1893–1974) was a Hungarian mathematician and physicist, who worked as Einstein’s assistant at the end of the 1920s; he became a specialist in computational mathematics after moving to the USA because of the political situation in Germany. Maurice d’Ocagne (1862–1938) was a Polytechnic engineer and mathematician, perhaps best known for his research on nomography. * * * 1. One knows that the characteristic equation of a square nonnegative matrix (of which no entry is negative) has always a real root, generally positive (exceptionally zero), the modulus of which is not smaller than that of any other root. That root is called the maximum root of the matrix.1 From the properties of the maximum root one easily deduces, as I have shown it,2 the following theorem: Let a be a nonnegative matrix, with entries aik (i, k = 1, . . . , n). If the linear system cxi −

aik xk = bi

(i = 1, . . . , n)

(1)

where the bi are all positive, defines positive values for the xk , the coefficient c is greater than the maximum root of matrix a. If that condition is met, the system (1) defines values, all positive, for the xk . 1 Frobenius, Sitzungsberichte der Akademie von Berlin, 1908, p. 471; 1909, p. 514; 1912, p. 456. [See note 1 of A9/Chapter 3 and note 1 of A13/Chapter 8 for the complete references to the work of Frobenius. – Eds.] 2 Comptes rendus, 153, 1911, p. 1129 and 1458; Annales de l’Ecole Normale, 30, 1913, p. 53. [See A9/Chapter 3 and A13/Chapter 8. – Eds.]

On nonnegative matrices 227 A new method for solving a linear system3 allows us to simplify the calculations

necessary to check if that condition holds or not. 2. Consider the linear system dik xk = bi

(i = 1, . . . , n)

(2)

and let d be the matrix of the dik . One can determine three matrices α, β, λ, with entries αik , βik , λik , of degrees n, such that αλβ = d , αik = βik = 0

for i < k ,

αii = βii = 1, λik = 0

for i = k (3)

β denoting the transpose of β . If dj , αj , βj , λj denote, in general, the matrices made of the entries in the first j rows and columns of d , α, β, λ, one sees that the equations deduced from (3) give, with |f | denoting the determinant of matrix f , αj λj β j = dj ,

λj = λ11 . . . λjj = dj

(4)

k

dmk =

αmh λhh βkh

(k = 1, . . . , m − 1)

(5)

1 i

dim =

αih λhh βmh

(i = 1, . . . , m − 1)

(6)

1 m−1

dmm =

αmh λhh βmh + λmm

(7)

1

If one knows the matrices αm−1 , βm−1 , λm−1 , the formulas (5), (6), (7) let us know the matrices αm , βm , λm .4 Then it follows from (3) that the system (2) is equivalent to the three systems i

αih uh = bi

(8)

1

λhh zh = uh

(i, h = 1, . . . , n)

(9)

3 The method is due to Mr Lanczos, Bulletin of the American Mathematical Society, 42, 1936, p. 325. [See note 12 of U3/Chapter 14. – Eds.] 4 [An exception occurs if some coefficient λii is zero. Potron was aware of this (see section 2 of the fifth lecture of U3/Chapter 14), but he did not mention it here because the criterion he obtained discards the exceptional case. – Eds.]

228 On nonnegative matrices n

βkh xk = zh

(10)

h

which define successively u1 and z1 , . . . , un and zn , then xn , xn−1 , . . . , x1 . 3. If one takes dik = cuik − aik ,5 where the uik denote the entries of the unit matrix, the system (2) becomes the system (1). One knows that, if c is > r (maximum root of a), the determinants dj are all positive,6 therefore the same holds for λjj = dj / dj−1 . I note that the converse is true, i.e. if the λjj successively calculated by Mr Lanczos’s method are all positive, it is certain that c is > r. In fact, according to (5) and (6), α21 and β21 are 0; and, if the off-diagonal entries of αm−1 and βm−1 are 0, it is the same for αmk and βmi (k , i = 1, . . . , m − 1), therefore for all the off-diagonal entries of αm and βm . Thus the off-diagonal entries of matrices α and β are all 0. Take then n positive scalars bi . The formulas (8), (9), (10) give values, all > 0, for the ui , zh , xk . Thus the system (1), where the bi are all positive, is satisfied by values, all positive, of the xk ; according to the theorem mentioned in No. 1, we therefore have c > r. 4. Thus, if c is < r, the sequence of the λjj , and consequently that of the symmetric minors dj , will contain a term 0.7 The Frobenius theorems only allowed us to assert that a determinant 0 must then be met among all the symmetric minors of any order of |d |.8

5 [The scalar c does not appear in the published Note. – Eds.] 6 See note 1. 7 [In modern terminology this statement concerns the signs of the n leading principal minors of matrix d. – Eds.] 8 See my Memorandum in the Annales de l’École Normale, 30, 1913, p. 53. [See section 9 of A13/Chapter 8. – Eds.]

16 On nonnegative matrices and positive solutions to certain linear systems

Editors’ note In his paper ‘Sur les matrices nonnégatives, et les solutions positives de certains systèmes linéaires’ (Bulletin de la Société Mathématique de France, 1939, 67: 56–61) Potron described in greater detail the Lanczos recursion method and the criterion he derived from it, nowadays known by economists as the Hawkins– Simon condition. In the beginning of the paper he clearly distinguished the results on nonnegative matrices obtained by Frobenius and those he attributed to himself. Stylistically, this paper is unusual because Potron drew up a list of references at the end of it. * * * 1. The nonnegative matrices (of which no entry is negative) have been studied by Frobenius [1]. He has proved the following results: F. I. If a, with entries aik , is a square nonnegative matrix, the characteristic determinant |su − a| has always a real and nonnegative root r, at least equal to the modulus of any other root. That root r is said maximum root of the matrix. F. II. Let aj (an = a) be the matrix made of the entries in the first j rows and columns of a, and rj (rn = r) the maximum root of aj . We have rj rj+1 ( j = 1, . . . , n − 1); in the adjunct of the characteristic determinant, the diagonal entries Aii (s) are all positive, and the other entries Aij (s) nonnegative for s > r; the Aii (r) and Aij (r) are nonnegative; in the determinant |ru − a|, if a symmetric minor of any degree is zero, the same holds for any symmetric minor containing it; if a symmetric minor is positive, the same holds for all those it contains.

230 On nonnegative matrices and positive solutions F. III. If the Aii (r) are all positive: 1◦ the matrix a is indecomposable;1 2◦ the Aij (r) are all positive; 3◦ if x, of entries xi , is a matrix of type (n, 1), and y, of entries yi , a matrix of type (1, n), each of the linear systems (ru − a)x = 0,

y(ru − a) = 0

has a positive solution. F. IV. If one of the Aii (r) is zero, there certainly exists a symmetric minor b(s), identical to Aii (s) or contained in Aii (s), possibly of degree 1, and having both properties: 1◦ b(r) = 0; 2◦ if the degree of b(s) is > 1, any symmetric minor of b(r) is positive. Then the matrix a is decomposable, and the submatrix of which the entries are those of b(0) is an indecomposable component. Corollary. If the matrix a is indecomposable, the Aii (r) are all positive, and we have 2◦ and 3◦ of F. III. I have shown in the past [2] that the following results ensue from these theorems: P. I. Let a be a nonnegative matrix, of type (n, n), indecomposable, and b a nonnegative and nonzero matrix, of type (n, 1): 1◦ for any s > r, the linear system (su − a)x = b has a positive solution; 2◦ if this system has a nonnegative and nonzero solution, s is > r. P. II. Let a be decomposable and b positive. For the considered system to have a positive solution, a necessary and sufficient condition is s > r. 2. These theorems do not provide any simple means to recognize if s is > or < r. The calculation of the successive derivatives of |su − a|, that s > r must make all positive, would require the calculation of all the symmetric minors of degrees n − 1, n − 2, . . ..2 But a new method [3] of calculation of the solution to a linear system, according to which the calculation is made in three stages, allows us to recognize, right from the first stage, if s is > or < r, and consequently, if the system admits a positive solution or not. Here is the method, which has not been published in extenso.3 1 A matrix a of type (n, n) is decomposable if the rows and columns can be put, without changing b 0 the diagonal, in an order such that a has the form , b being of the type (h, h), 0 a zero c d matrix of the type (h, n − h), c and d of types (n − h, h) and (n − h, n − h). The submatrices b and d are called components of a. 2 [Here Potron referred implicitly to section 9 of A13/Chapter 8. – Eds.] 3 Mr Lanczos has published only a very brief abstract in the Bull. Am. Math. Soc. He has been kind enough to send me his manuscript. [See footnote 12 of U3/Chapter 14. – Eds.]

On nonnegative matrices and positive solutions 231 3. Consider any linear system dx = b

(1)

Let us try to determine two triangular matrices α and β (αii = βii = 1, αik = βik = 0 for i < k), and a diagonal matrix λ (λki = 0 for i = k) such that we have αλβ = d

(2)

One sees that (2) gives j j j

αi λj βk = dik

(3)

According to the conditions imposed on the matrices α and β , the only terms = 0 on the left-hand side are obtained for j min[h, k ]. We will therefore have4 k −1 k −1 k −1 k k 1 1 1 λk −1 βk + αm λk = dmk αm λ1 βk + · · · + αm 1 i−1 i i m αi1 λ11 βm1 + · · · + αii−1 λii− −1 βm + λi βm = di m−1 m−1 m−1 m 1 1 1 αm λ1 βm + · · · + αm λm−1 βm + λm m = dm

(k = 1, . . . , m − 1) (i = 1, . . . , m − 1)

(4) (5) (6)

If dm , α m , β m , λm denote the submatrices made of the entries of the first m rows and columns of d, α, β, λ, one deduces from (4)–(5)–(6) the relationship α m λm β m = dm

(7)

Since |α m | = β m = 1, one draws from (7) |dm | = |λm | = λ11 λ22 . . . λm m

(8)

These formulas allow us to calculate the matrices α , β , λ by induction. Assume indeed that α m−1 , β m−1 , λm−1 are known. The formulas (4) will 1 , α 2 , . . ., α m−1 ; similarly the formulas (5) will determine successively αm m m 1 determine successively βm , βm2 , . . ., βmm−1 ; then the formula (6) will give λm m. This calculation can only be stopped if one of the λii is found equal to zero. According to (8), this will happen always and only if |di | is the first zero determinant in the sequence |d1 |, |d2 |, . . . . But, if |d| = 0, it is always possible to put the columns in an order such that all the |di | are = 0. The property is obvious for a determinant of degree 2. Assume it holds for any determinant of degree < n. 4 [In sections 3, 4 and 5 we have proceeded to many typographical corrections, most of them stemming from an inappropriate use of the transposition symbol. – Eds.]

232 On nonnegative matrices and positive solutions If |d| is = 0, at least one of the entries in the last row of its adjunct is = 0; one can always put the columns in an order such that this element = 0 is Dnn = |dn−1 |. Then one can, according to the hypothesis retained, put the first n − 1 columns in an order such that the |di | (i = 1, . . . , n − 2) are all = 0. As it will be proved later, this first stage of the calculation, applied to the matrix d = su − a, suffices to recognize if the number s is smaller or greater than the maximum root r of matrix a. Therefore this first stage of the calculation suffices to foresee if the solution of the system (su − a)x = b is positive or not. It also suffices for the calculation of the determinant |d| = λ11 λ22 . . . λnn . 4. The auxiliary matrices, satisfying (2), having been formed, the system (1) is written αλβ x = b

(9)

It can be solved by solving successively αz = b

(10)

λy = z

(11)

βx = y

(12)

The system (10), which can be written αh1 z1 + · · · + αhh−1 zh−1 + zh = bh

(h = 1, . . . , n)

(13)

gives successively z1 , z2 , . . . , zn . Then (11) gives yi = zi /λii ; then the system (12), which can be written xh + βhh+1 xh+1 + · · · + βnh xn = yh

(14)

gives successively xn , xn−1 , . . . , x1 . 5. Let us now take d = su − a. If s is > r, all the |di | are positive (F. II). According to (8), it is the same for all the λii . Conversely, assume that the λii are found all positive. In (4) and (5), we now have dmk = −akm 0, dim = −am 0. For m = 2, i we have α21 λ11 = −a12 ,

λ11 β21 = −a21

Therefore the nondiagonal entries of the submatrices α 2 and β 2 are 0. If it is the same for the nondiagonal entries of α m−1 and β m−1 , the formulas (4) and (5) i and the β i (i = 1, 2, . . . , m − 1) are show, step by step, that the αm 0. Then, if m one takes for b a positive matrix, the formulas (10) and (11) give positive values

On nonnegative matrices and positive solutions 233 successively to z1 , z2 , . . . , y1 , . . . , yn , and the formulas (12) give positive values successively to xn , xn−1 , . . . , x1 . Thus, if the λii are found all positive, the system (su − a)x = b where b is a positive matrix, has a positive solution; therefore s is greater than the maximum root r. We therefore have the result: in order that s be greater than r, maximum root of the nonnegative matrix a, a necessary and sufficient condition is that the diagonal matrix λ, calculated by induction from the matrix su − a (simultaneously with the triangular matrices α and β ), is a positive matrix. 6. That criterion is advantageous from the point of view of practical calculation. First, λii only depends on the entries of ai . Second, the order in which the λii appear depends on the order adopted for the diagonal entries of a. That order can be modified provided that the same permutations are made simultaneously on the rows and the columns of the same ranks. Several calculators, having adopted different orders, can operate simultaneously. If, in su − a, s is < r, one will thus encounter more rapidly the nonpositive λii which certainly exists. 7. The economic problems relative to the production–consumption and the prices– wages equilibria boil down, as I have shown [2, 4], to the search of two positive one-column matrices, p for the productions, a for the prices, solutions to the systems (u − C)p = f , (u − C)a = b, where C is a given nonnegative square matrix, and f and b two arbitrary positive one-column matrices. The matrices solutions will be positive always and only if the maximum root of C is < 1. The calculation of the successive λii would allow us to recognize if that condition is met, and would prepare the calculation of the solutions.

Bibliography 1. Frobenius, Positive und nicht-negative Matrizen (Sitzungsberichte der Academie von Berlin, 1908, p. 471; 1909, p. 514; 1912, p. 456).5 2. Potron, Les Substitutions linéaires à Coefficients non négatifs (Annales de l’École Normale, t. 30, 1913, p. 53; C. R. Acad. Sc., t. 153, 1911, p. 1129 et 1458; t. 204, 1937, p. 844).6 3. Lanczos, Recursion Method for solving linear Equations (Bulletin of the American Mathematical Society, t. 42, 1936, p. 325).7 4. Potron, L’Aspect mathématique de certains Problèmes économiques (Paris, 1936, at the author’s).8

5 6 7 8

[See note 1 of A9/Chapter 3 and note 1 of A13/Chapter 8 for the complete references. – Eds.] [See A13/Chapter 8, A9/Chapter 3, A10/Chapter 4 and A40/Chapter 15. – Eds.] [See note 9 of U3/Chapter 14 for the complete reference. – Eds.] [See U3/Chapter 14. – Eds.]

17 Letter on industrial statistics

Editors’ note The March 1942 issue of the Journal de la Société de Statistique de Paris contained three items with regard to Potron. In Section VI Alfred Barriol wrote Maurice Potron’s obituary (see W1/Appendix I below), and in Section VII he published a report on the booklet which Potron had compiled at the occasion of his lecture series in 1937 (see W2/Appendix II). Moreover, in Section VIII we find Potron’s very last economic paper, ‘Lettre sur les statistiques industrielles’ (Journal de la Société de Statistique de Paris, 1942, 83: 207–8; the title comes from the Table of Contents). Potron’s paper is a comment on the talk ‘Les statistiques industrielles’ given by Alfred Sauvy (1898–1990), a Polytechnic engineer who worked as a statistician and economist in the French administration and later became famous as a demographer. The talk was given at the Statistical Society of Paris on 18 December 1940, and published by the journal shortly thereafter (Journal de la Société de Statistique de Paris, 1941, 82: 131–8; followed by a discussion: 138–45). Potron drew a parallel between Sauvy’s statistical program and the more precise data on the distributions of labour and costs required for his own table. The two introductory sentences are clearly due to Barriol: ‘We have received from our colleague, Father Potron, a letter in which he gives some observations on Mr Sauvy’s talk (Journal, vol. 82, 1941, p. 131). Our colleagues will read it with interest, as a follow up to the bibliography inserted above.’ * * * Several passages of this talk and of the ensuing discussion confirm, interestingly enough, the views expounded in my lectures on ‘The mathematical aspect of some economic problems’.1

1 [See U3/Chapter 14. – Eds.]

Letter on industrial statistics 235 1)2

I have introduced (p. the notion of quantitative unit of an economic good. I have taken as examples the ton of coal, the metre of cloth, the kilometre-ton of a transported commodity, the hour of lesson in such science. Mr Sauvy mentions correctly (p. 133, lines 12–14) that the choice of a unit is necessary to measure production. Obviously, the unit changes of nature with the product. It will sometimes be a weight, sometimes a length, sometimes a full object. But one hardly sees why this diversity might create difficulties, or why there might be room for much hesitation concerning the unit to adopt. Nobody would have the idea to evaluate the production of the French watchmaking in kilos. The main point is to agree on a precise unit for each good. I ask (p. 2)3 for a numbering of the professions, that I propose to denote by V1 , . . . , Vh , . . . , Vm , and of the units of economic goods, that I propose to denote by A1 , . . . , Ai , . . . , An . According to Mr Sauvy (p. 142),4 there exists an official nomenclature of professions, which allows us to assign a number to each of them. Similarly, there exists an official and complete nomenclature of products. Objections have been raised against the method of classification. That method does not matter for the purpose of my lectures. The one that the construction of the tableau (p. 4)5 might suggest would consist in building some type of genealogical tree. First, all ‘consumption goods’; then, for each of them, those which are used as raw materials, then those used as raw materials for these items, and so forth until exhaustion. Among the pieces of information asked for from the industrialists (p. 133), some would be very useful for the construction of the table: it is, for every product, the produced quantity (in one year, I think), the numbers of workers of various professions employed in its manufacturing, the working hours. It will suffice, for each profession, to divide the annual total of working hours by the produced quantity to know the average time of labour that this profession performs for the production of the unit. These are my coefficients tih . To have the coefficients cik , it would suffice to know, moreover, for the production of the year, the quantity of each raw material consumed. It would surely not be more difficult to obtain this information than the previous. Besides, it is indeed necessary to give these pieces of information to the distributors of raw materials. According to Mr Sauvy (p. 133, bottom), it has been admitted that each industrialist would not be obliged to assign the working hours according to its various products. For my goal, it is necessary that the assignment be made, both for the labour of each profession and for each of the raw materials. The operation is moreover absolutely necessary to determine a somewhat reliable cost price.

2 3 4 5

[See section 1 of the first lecture of U3/Chapter 14. – Eds.] [See section 3 of the first lecture of U3/Chapter 14. – Eds.] [This is a reference to an intervention by Sauvy in the discussion following his talk. – Eds.] [See section 3 of the first lecture of U3/Chapter 14. – Eds.]

236 Letter on industrial statistics The admission made at the top of page 134 is noteworthy.6 One must say it is hardly to the industrialists’ honour! There should not be, apparently, any difficulty in evaluating exactly the professional work performed and the quantities of raw materials entered in a workshop where a certain manufacturing takes place. The head of the workshop must write down all this steadily. As for the work performed and the materials consumed for the general working of a firm with several manufacturings, there are first some items for which the assignment is easy. For instance, it is sound to assign the coal consumed for heating proportionally to the volume of the heated premises, the lighting proportionally to the consumption of bulbs, the driving force proportionally to the consumption of machines. Some coefficients will always remain arbitrary to some extent. But one could always take advantage of the halts in one or the other fabrications to check them. Moreover, these badly determined coefficients will generally multiply rather low quantities. In an establishment depending on the Ministry of Defense, the military Powder Factory of Le Bouchet, to the accounts of which I have had access, the assignment coefficients of the overheads among the various products (B powder, BCNL powder, BS powder, melinite, …) are determined in advance. The workshop books are kept very precisely. The prices, but also the quantities, are always mentioned. Some multiplications and divisions suffice to determine the coefficients cik and tih relative to the various products. The calculations concerning the B powder, which only required me about ten hours, have been published in the proceedings of a colloquium which was held in Belgium during Spring 1913 or 1914. Mr Max Lazard, I think, was its secretary.7 It is therefore a question of organization of book-keeping. To obtain it from the industrialists, one should convince them of the advantage that they could find, not only in giving the statistical data they are asked for, but also in calculating easily the coefficients that will allow them to deduce immediately the exact cost prices from the wage rates and the prices of raw materials. Maurice Potron

6 [Sauvy wrote: ‘In very many cases, the industrialists do not know the cost price of a specific good, precisely because they are unable to say how many hours of labour, and above all which part of the general expenses, of the staff’s wages or of the equipment’s depreciation, are affected to this good.’ (o.c.: 134) – Eds.] 7 [See A14/Chapter 9. The conference was held in Ghent in September 1913. Max Lazard and Louis Varlez were the two main organizers. Lazard chaired the session of the Statistical Society of Paris when Alfred Sauvy gave his talk in 1940. – Eds.]

Appendix I Alfred Barriol: ‘Obituary. Maurice Potron (1872–1942)’

Editors’ note Alfred Barriol (1873–1959), a Polytechnic engineer like Potron (he entered the Polytechnique two years after Potron), was a specialist of finance and actuarial science. He was elected Secretary-General of the Statistical Society of Paris in 1909 and still assumed this function in 1942. The members of the Society, of whom many were Polytechnic engineers, met on a monthly basis. Each issue of the journal reported extensively on the talks and discussions of the meeting held the month before, but Potron’s name never occurred in the reports of these meetings. The Society was interested in the applications of statistical methods to economics and social sciences, especially in view of their use by the State. Potron published twice in the journal (see A12/Chapter 6 and A44/Chapter 17). His name is also mentioned in the minutes of the society on the occasion of: • the announcement of his lectures at the Catholic Institute of Paris (session of 17 March 1937); • his nomination and admission as a member of the Society (sessions of 17 March and 21 April 1937); • the receipt of his paper on ‘The decoration of plane surfaces’ as an application of group theory (see U4); Barriol regretted the lack of illustrations (session of 18 May 1938); and • the publication of his paper ‘Sur les fondements de l’arithmétique’ (see A43) (session of 15 October 1941). Since the journal rarely published obituaries of the members of the Society, Barriol’s ‘Nécrologie’ (Journal de la Société de Statistique de Paris, April 1942, 84: 203–4) suggests the existence of close and warm relationships between Potron and the Secretary-General of the Society. Though the obituary is sometimes imprecise, it is the unique source concerning some details of Potron’s life.1 * * *

1 See the Introduction of this book for a more detailed biography of Potron.

238 Appendix I Father Maurice Potron was born on 31 May 1872 in Paris; his father, an engineer of the École Centrale, personally directed the education of his children (4 sons and 1 daughter)2 up to the baccalauréat level;3 admitted to the Jesuit school of the rue des Postes,4 he prepared for the École Polytechnique and, in 1890, he was admitted at the sixteenth rank after only one year of preparatory school; he graduated at the twelfth rank, in 1892, and entered into the Powder service.5 But he already thought about religious life and his vocation was confirmed during a retreat he made in Clermont.6 As soon as his one-year military service as an artillery second lieutenant was finished, he entered at the novitiate of the (then exiled) Jesuits in Canterbury. His previous school of the rue des Postes asked him as a professor in 1896 and, for three years, he taught the course of preparatory mathematics; he was allowed to follow a philosophy programme and to perfect his [knowledge of] theology, and was ordained priest in 1905: this was the goal he wanted to reach above all. Simultaneously, he defended his PhD thesis in mathematics on the following topic: ‘The groups of order p6 ’.7 During some years he was attached to the preparation of the candidates for X with Father Pupey-Girard and taught some courses at the Catholic University of Lille;8 he was appointed as full professor at the Catholic University of Angers where he was especially in charge of the preparation of the candidates to the licence exam.9

2 [In fact, Maurice had four brothers, Henri (1876–1954), Émile (1878–1955), Robert (1880–1886), who died from diphtheria, and Édouard (1883–1950), and one sister, Marie-Élisabeth (1884–1899), who died from tuberculosis. They were instructed at home by a tutor. – Eds.] 3 [In France, the baccalauréat is the examination at the end of the secondary school. – Eds.] 4 [The rue des Postes (now rue Lhomond) was in the Quartier Latin in Paris, close to the Pantheon. At the end of the nineteenth century the ‘school of the rue des Postes’ was a usual way to refer to the École Sainte-Geneviève, a famous preparatory school for the scientific Grandes Écoles (these preparatory schools are commonly known as Taupes and their students as Taupins). After the Law of Separation and a long legal battle, the Jesuit-run school was finally obliged to close in 1913 but re-opened immediately in Versailles. Its new address was rue de la Vieille Église, but the name of the street was changed into rue des Postes in the 1950s. – Eds.] 5 [The preparation normally took at least two years. According to the records of the École Polytechnique Potron was admitted in 1890 at the 91st rank, and graduated in 1892 at the 15th rank. – Eds.] 6 [Clermont is the name of several towns in France, but Barriol’s reference to it is incorrect. He should have referred to Clamart, a town in the southern suburbs of Paris, where the Jesuits organized spiritual retreats at the Villa Manrèse. Auguste Potron had helped to finance the purchase of the Villa in 1876 and Maurice participated to retreats in 1891 and 1892 (see the folders ‘Clamart, historique’ and ‘Clamart, retraites’ of the Archives Jésuites de la Province de France). – Eds.] 7 Paris, Gauthier-Villars, 1904, in-4, 174 pp. Thesis of the Faculty of Sciences of Paris, No. 1183. [See B1. – Eds.] 8 [Pupey-Girard did not teach the candidates for X (i.e. the École Polytechnique), but was in charge of their spiritual supervision. – Eds.] 9 [The licence corresponds to the bachelor’s degree, after three or four years of studies at the university. – Eds.]

Appendix I 239 Mobilized in 1914, he went first as an artillery lieutenant attached to his previous corps (Powder), but he asked to go to the front; then he was appointed as an artillery instructor-officer for the American troops and he returned in 1918, captain, knight of the Légion d’honneur and bearing the Croix de guerre,10 to take again his position at the Sainte-Geneviève school in Versailles always with his ‘Taupins’. Every week he taught at the Catholic Faculties of Paris and Lille and, from 1931 to 1939, he prepared students for the licence degree in mathematics in both these faculties.11 Too old to be mobilized in 1939, he resumed his task of preparing students for the Grandes Écoles at the Saint François-Xavier college in Vannes;12 but, ill and tired for some years, he fell victim to pneumonia, which carried him off on 21 January 1942. Knowing for a long time the scientific value of Father Potron, and having got in touch with each other by chance thanks to a correspondence relative to statistics, I had introduced him in 1937 to our Society, jointly with our Chairman Mr Huber. Many of us remember to have seen him at our dinners and had the occasion to appreciate the charm of his conversation and the rightness of his views. His beautiful works on the mathematical aspect of some economic problems have been signalled in our Journal; he contributed to several journals of mathematics and was a member of the Mathematical Society of France and of the Political Economy Society. His premature death will be deeply felt in all these circles in which he only had friends who estimated him sincerely for his scientific value and his nice character. We pray his brothers to find here the expression of our grief and our sadness. A. Barriol

10 [Maurice Potron received the Légion d’honneur on 25 December 1916 (publication in the Journal Officiel dated 2 and 3 January 1917, p. 63), but not the Croix de guerre (Cross of War). – Eds.] 11 [Potron taught at the Catholic University of Lille from 1921 to 1924. We have found no trace of his teaching in Lille during the thirties. – Eds.] 12 [In reality, Potron taught secondary school pupils preparing for the baccalauréat. – Eds.]

Appendix II Alfred Barriol: ‘[Report on] L’aspect mathématique de certains problèmes économiques’

Editors’ note Barriol’s accurate report (Journal de la Société de Statistique de Paris, 1942, 84: 205–7) on Potron’s 1937 lecture series booklet shows a good understanding of his model. The lack of clarity of the lectures themselves as well as the final reference to ‘Organization Committees’ indicate that Barriol and Potron had personal discussions on the model. The Secretary-General of the Statistical Society of Paris stressed the need for statisticians to collect and process the required industrial data. * * * The mathematical aspect of some economic problems, by Father Potron, professor at the Catholic Institute of Paris. A booklet of 34 pages +5. At the author’s. Our colleague has been led to give six lectures at the Catholic Institute of Paris, collected in a booklet typewritten by himself, that has been presented to the Society and of which it is necessary to give an account, given the present economic situation, because these lectures give very clear ideas on some problems which have worried and certainly still worry public opinion. They aim, indeed, at studying from a mathematical point of view two problems set, in fact, by collective contracts. The assignment of a determined wage to a professional category amounts, according to the prices at the date of the contract, to assigning a determined standard of living to that professional category. An economic regime will be in practical agreement with these contracts only if, in an effective and constant way, it provides the workers with all the economic goods they must use, according to the standard of living assigned to them. Now, an economic regime can be characterized by four types of elements: (a) the yearly production, and (b) the price of every economic good; (c) the workforce and (d) the hourly wage of each professional category. Besides, such a regime must set itself in a given state of the economy, characterized by the technical and administrative conditions of the various firms.

Appendix II 241 The elements a (productions) and c (workforce) must, in the given state of the economy, satisfy two conditions: A. For any economic good, production must be at least equal to consumption. B. No labourer must be obliged to perform more than a certain number N of work-hours per year. In the first lecture, two groups of formulas are established to express these two conditions; then these two groups are reduced to one, in which only the elements a (productions) intervene; this group of formulas will be denoted by (I). Similarly, the elements b (prices) and d (wages) must, in the given state of the economy, satisfy two conditions: C. The selling price of any economic good must be at least equal to its cost price. D. The effective wage of any labourer must be at least equal to the cost of living corresponding to the standard assigned to his professional category. In the second lecture two groups of formulas are established to express these two conditions; then these two groups are reduced to one, in which only the elements b (prices) intervene; this group of formulas will be denoted by (II). For each of the systems (I) and (II) a mathematical question arises: are the formulas constituting the system compatible? The question is important because, in case of incompatibility, one would be sure that it is absolutely impossible, in the present state of the economy, to implement practically an economic regime in agreement with the collective contracts. The third, fourth and fifth lectures are devoted to the study of this compatibility question, by relying on results obtained in 1911 by the German mathematician Frobenius.1 The result of that study is theoretically very simple: in order that, in each of both systems, the formulas be compatible, a unique condition is necessary and sufficient. It can be given several equivalent forms. One of them is: the legal maximum N must not be smaller than a certain limit which depends both upon the state of the economy and upon the standards of living defined by the collective contracts. In the sixth lecture some conclusions are developed. The formulas (I) are of a nature likely to be of a particular interest for statisticians. The formulas (II) are of a nature likely to be of a particular interest for entrepreneurs. Each of the latter has indeed all information required to write down the formula which concerns each of the products he manufactures. He can check if his selling price and his suppliers’ prices for raw materials and tools satisfy or not that formula. The comparison of

1 [The results of Frobenius were published in 1908, 1909 and 1912; see note 1 of A9/Chapter 3 and note 1 of A13/Chapter 8 for the exact references. – Eds.]

242 Appendix II these formulas relative to firms depending on each other could be used as a basis for a price adjustment. In an appendix entitled The Hebrew Manna Problem, the theory is applied to a very simple example, which however highlights its key features. Concerning that appendix, it is interesting to note that the condition N > 2,500 is given by the Frobenius theorem applied to the system (9).2 The characteristic determinant is s 0

0 s

−1 −3

500 750 − 250 N − N s− N

= s s2 −

1,750 750 s− . N N

The maximum root is the positive root of the second degree equation in s. In order that it be < 1, it is necessary and sufficient that f (s) be > 0 for s = 1, which gives 0 < 1−

750 + 1,750 2,500 = 1− , N N

hence N > 2,500.

The equations are in general not difficult to write down if one has patience and a sufficient knowledge of accounting elements, but there are ‘too many’ equations and, to meet them, one must proceed by trial and error without being put off by the calculations to make. One should proceed from the simple to the composite, naturally, and let every industrialist calculate his own coefficients, which should not be very difficult. The Organization Committees would then be involved, but for that they should have experienced calculators, if not even statisticians that they do not presently have, because none of them has helped to develop the teaching we have organized at the Institute of Statistics and which up to now has benefited foreigners only.3 Father Potron’s work focuses accurately on the questions to deal with, and he must be acknowledged for the great step he has made towards the solution of these intricate economic problems. A. Barriol

2 [Barriol illustrated Potron’s general theory by referring to the notion of dominant eigenvalue and by showing that N must exceed a certain level, when Potron himself proceeded to elementary operations to solve the problem. The matrix considered by Barriol can be derived from equations (8) and (9) in ‘The Hebrew Manna Problem’ (see section 4 of the Appendix of U3/Chapter 14). – Eds.] 3 [Potron himself did not refer explicitly to ‘Organization Committees’ in the lectures. These committees are reminiscent of the Bureau de Calculs put forward by Potron in 1912 (see section 20 of A11/Chapter 5), but Barriol’s sentence suggests that they actually exist. – Eds.]

Appendix III Michel Vittrant: ‘[Report on] Le problème de la manne des Hébreux’

Editors’ note Years after Potron’s death, Michel Vittrant, a fellow Jesuit, wrote a short but extremely negative report on ‘The Hebrew Manna Problem’ (see the appendix to U3/Chapter 14). The report, dated 26 March 1953, is conserved in the folder ‘Potron’ of the Archives Jésuites de la Province de France. It is unclear who commissioned the report or why it was written, but this may be linked to another document from the same folder, a 12-page paper with the title ‘Letters from Father Potron to Father Hoenen’ on the cover (see U5). That document was typewritten by Jean Abelé (1886–1961), a Jesuit and physicist. In a one-page presentation, Abelé wrote: ‘Having found in Father Potron’s papers several copies of a letter followed by studies and observations relative to a series of articles on cosmology published in the “Gregorianum”, we think that it is worthwhile to communicate them.’ The articles (in Latin) to which Abelé referred are written by the Dutch Jesuit Peter Hoenen (1880–1961), professor of philosophical cosmology at the Gregorian University of Rome. In his short introduction Abelé mentioned other mathematical works and courses of Maurice Potron, and he concluded by announcing his intent ‘to write down a notice on the scientific work of Father Potron, and the same for that of Father de Séguier.’ There is no signature, but Abelé’s name has been added by an archivist, the Jesuit Paul Duclos (1917–1993). The same archivist marked the conclusion of Vittrant’s report with a red pencil and added above the signature: ‘Michel V., prof. of physics, ex-“Chinese” ’. Vittrant (1882–1953) spent much of his life in Shanghai. He was at the Zi-Ka-Wei scholasticate from 1915 to 1917, taught at the Catholic French-speaking Université L’Aurore from 1917 to 1940, and was again attached to the scholasticate from 1940 to 1951. Due to the communist revolution, the Jesuits closed the university and left continental China in 1951. After his return to France Vittrant taught physics at the École Sainte-Geneviève. * * *

244 Appendix III 26 March 1953 The Hebrew Manna Problem by Father Potron, professor at the Catholic Institute [Maurice Potron, S.J.] (without date) found in a deposit at the Chantilly library1 This work has no biblical value at all, because the data are contrary to the Bible (Ex. c 16 – Num. c 11)2 and ridiculously distorted. • • • • •

The title is wrong, as the problem is absolutely different from the biblical reality. The preamble is pretentious, since the treated problem is much simpler than the economic reality. The massive collecting of manna and its preservation in ‘stocks’ (p. 2)3 are contrary to the biblical prescripts and even impossible: rot on the next day, except for the Sabbath. One-hour work to collect one omer (3 litres) of manna is absurd. Three homers (3 × 200 or 3 × 300 kg) of fuel to cook a one-omer cake is unheard of.

Useless to continue. This work is good to keep as a remembrance of the defects of its author, a talented mathematician, unable to understand reality and to adapt himself to it. (signed) M. Vittrant Versailles-Ste Geneviève

1 [The Bibliothèque des Jésuites, also called the Bibliothèques des Fontaines, in Gouvieux near Chantilly, was an important Jesuit library, with a large section of China-related items. In 1998–1999 the collection was transferred as a deposit to the Bibliothèque Municipale de Lyon. – Eds.] 2 [In Chapter 16 of Exodus it is told that after the Hebrews had fled Egypt, God supplied them with manna in the morning, at a ratio of one omer per person per day, and with meat in the evening. The manna could not be conserved. On every sixth day God supplied a double ratio of manna, and Moses instructed the Hebrews to bake half of it, so that it could be conserved and consumed on the sabbath. The Hebrews survived for forty years on manna, until they arrived in Canaan. In Chapter 11 of Numbers it is told that the Hebrews had to do with manna only, with which they baked cakes. This led to complaints about a lack of meat, and a conflict with God. – Eds.] 3 [See section 2 of the Appendix to U3/Chapter 14. – Eds.]

The Potron Bibliography

Editors’ note Our bibliography lists both published and unpublished items of Potron, followed by complementary information on his writings. Every item is identified by a combination of a capital letter and a number, where the letter indicates the type of document, and the number the chronological order within each category. In addition we enclose a list of secondary literature on Potron’s economic work, and an overview of archival sources on Potron’s life and work. First, we have Potron’s publications, which we divide into four groups: • • • •

Books [B]; Articles [A]; Book Reviews [R]; Solutions of Mathematical Exercises [S].

The reason for the creation of the last category is that Potron published numerous articles and some small books containing exercises and solutions on differential and integral calculus, drawn from the examination sessions in all Faculties of France. These publications are clearly of a different character than his other writings. Second, we group all of Potron’s unpublished items into a single category: •

Unpublished writings [U].

Third, the complementary information on Potron’s writings consists of two categories: • •

New editions [E]; Publications mentioned by other sources [X].

The X category lists publications by Potron mentioned by other sources but of which we have been unable to locate a copy. At last, the additional material contains three parts: •

Works on Potron’s economic model [W];

246 The Potron Bibliography • •

Interviews relative to Potron [I]; Archival sources on Potron.

We add an asterisk to the identification numbers of the items which are relevant for Potron’s economics; these are all concentrated in the periods 1911–1914 and 1935–1942. The chapter numbers of the items which have been translated in this volume are enclosed in square brackets at the end of the corresponding bibliographic entries. The Editors’ notes at the beginning of each chapter contain additional information on the texts, including variant editions. Acknowledgements When compiling this bibliography we benefited from help by Robert Bonfils, Christian de Borchgrave, Chantal Dufour, Jocelyne Dufour, Anne-Catherine Putz, the staff of the Archives Nationales in Paris, the Archives Jésuites de la Province de France in Vanves, the Archives Municipales in Vannes, and the archives of the Université Catholique de Lille and the Université Catholique de l’Ouest in Angers. We were fortunate to have access to Wilfried Parys’s Annotated Potron Bibliography (see W16). His annotations provide details on Potron’s writings and on the references to Potron in the scientific literature. We very much appreciate the dedication with which he has traced Potron’s publications and continues to do so.

1 Publications 1.1 Books [B1] Les Groupes d’Ordre p6 . Paris, Gauthier-Villars, 1904, 177 p. [B2] (with François Michel) La Composition de Mathématiques dans l’Examen d’Admission à l’École Polytechnique, de 1901 à 1921: Exercices d’Application du Cours de Mathématiques Spéciales. Paris, Gauthier-Villars, 1922, 452 p. [B3] Exercices de Calcul Différentiel et Intégral. Volume I: Résumé Théorique et Énoncés d’Exercices. Volume II: Solutions des Exercices. Paris, Hermann, 1926–7, 332 + 258 p. [B4] Les Groupes de Lie (Mémorial des Sciences Mathématiques, 81). Paris, GauthierVillars, 1936, 64 p. [B5] (with Jean Armand de Séguier) Théorie des Groupes Abstraits (Mémorial des Sciences Mathématiques, 91). Paris, Gauthier-Villars, 1938, 41 p.

1.2 Articles [A1] ‘Sur la génération de quelques courbes remarquables par le campylographe du P. Marc Dechevrens’, Annales de la Société Scientifique de Bruxelles, 1901–2, 26 (Seconde Partie – Mémoires): 41–56. [A2] ‘Sur quelques groupes d’ordre p6 ’, Bulletin de la Société Mathématique de France, 1904, 32: 296–300.

The Potron Bibliography 247 [A3] [A4] [A5] [A6] [A7] * [A8] *

[A9] *

[A10]*

[A11]* [A12]* [A13]* [A14]*

[A15]* [A16] [A17] [A18]

‘Les Gpm ( p premier) dont tous les Gpm−2 sont abéliens’, Bulletin de la Société Mathématique de France, 1904, 32: 300–14. ‘Sur les groupes d’ordre pm ( p premier) dont tous les sous-groupes d’ordre pm−2 sont abéliens’, Comptes Rendus de l’Académie des Sciences, 1904, 139: 396–9 (Session of 8 August 1904). ‘Sur les groupes d’ordre pm (p premier, m > 4) dont tous les diviseurs d’ordre pm−2 sont abéliens’, Comptes Rendus de l’Académie des Sciences, 1904, 139: 963–4 (Session of 5 December 1904). ‘Sur une formule générale d’interpolation’, Bulletin de la Société Mathématique de France, 1906, 34: 52–60. ‘[Abstract of a study on just prices and wages]’, pp. 882–3 in G. Desbuquois, ‘La loi du juste prix ’, Le Mouvement Social, October 1911, 72: 867–84. [Chapter 1] ‘A propos d’une contribution mathématique à l’étude des problèmes de la production et des salaires’, Échos de l’Union Sociale d’Ingénieurs Catholiques et des Unions-Fédérales-Professionnelles de Catholiques, 15 October 1911, 2nd year, No. 7: 4–7. [Chapter 2] 0 et leur ‘Quelques propriétés des substitutions linéaires à coefficients application aux problèmes de la production et des salaires’, Comptes Rendus de l’Académie des Sciences, 1911, 153: 1129–32 (Session of 4 December 1911). [Chapter 3] ‘Application aux problèmes de la ‘production suffisante’ et du ‘salaire vital’ de quelques propriétés des substitutions linéaires à coefficients 0’, Comptes Rendus de l’Académie des Sciences, 1911, 153: 1458–9 (Session of 26 December 1911); see also ‘Errata’, 1541. [Chapter 4] ‘Possibilité et détermination du juste prix et du juste salaire’, Le Mouvement Social, 15 April 1912, 73: 289–316. [Chapter 5] ‘[Contribution mathématique à l’étude des problèmes de la production et des salaires]’, Journal de la Société de Statistique de Paris, May 1912, 53: 247–9. [Chapter 6] ‘Quelques propriétés des substitutions linéaires à coefficients 0 et leur application aux problèmes de la production et des salaires’, Annales Scientifiques de l’École Normale Supérieure, 1913, 3rd series, 30: 53–76. [Chapter 8] ‘Contribution mathématique à l’étude de l’équilibre entre la production et la consommation’, in: L. Varlez and M. Lazard (eds), Assemblée Générale de l’Association Internationale pour la Lutte contre le Chômage. Gand, 5–6 Septembre 1913. Procès-verbaux des Réunions et Documents Annexes. Paris, Service des Publications de l’Association Internationale pour la Lutte contre le Chômage, 1914, Appendix VII: 163–71. [Chapter 9] ‘L’organisation scientifique du travail. Le système Taylor’, Le Mouvement Social, 1914, 77: 497–510 (‘A. Les principes’), 15 June 1914; 78: 21–33 (‘B. Les sanctions de l’expérience’), 15 July 1914. [Chapter 10] ‘Sur une représentation du groupe des 27 droites en groupe de collinéations quaternaires’, Comptes Rendus de l’Académie des Sciences, 1921, 173: 346–8 (Session of 8 August 1921). ‘Sur le groupe quaternaire primitif d’ordre 25920’, Journal de l’École Polytechnique, 1922, 2nd series, 22: 69–89. ‘Monde [Le système du]’ in: Adhémar d’Alès (ed.), Dictionnaire Apologétique de la Foi Catholique Contenant les Preuves de la Vérité de la Religion et les Réponses

248 The Potron Bibliography

[A19] [A20] [A21] [A22] [A23] [A24] [A25] [A26] [A27] [A28] [A29] [A30] [A31] [A32] [A33] [A34] [A35]* [A36]

aux Objections Tirées des Sciences Humaines, Quatrième édition entièrement refondue, Paris, Gabriel Beauchesne, 1926, Vol. III: 867–78. ‘Sur les théorèmes fondamentaux de la théorie des groupes continus finis de transformations’, Comptes Rendus de l’Académie des Sciences, 1926, 183: 841–2 (Session of 8 November 1926). ‘Sur les partages d’un système d’entiers en groupes de sommes données’, Comptes Rendus de l’Académie des Sciences, 1927, 184: 572 (Session of 7 March 1927). ‘Sur les partages d’un système d’entiers en groupes de sommes données’, Journal de l’École Polytechnique, 1927, 2nd series, 26: 39–43. ‘Quelques remarques sur les équations aux dérivées partielles et les intégrales singulières des équations différentielles’, Nouvelles Annales de Mathématiques, 1927, 6th series, 2: 78–82. ‘Sur les théorèmes fondamentaux de la théorie des groupes continus finis de transformations’, Bulletin des Sciences Mathématiques, 1927, 2nd series, 51: 91–6 (I), 101–14 (II). ‘Sur un théorème fondamental de la théorie des groupes continus finis de transformations’, Comptes Rendus de l’Académie des Sciences, 1931, 192: 1302–4 (Session of 18 May 1931). ‘Sur l’irrationalité du nombre π ’, Revue de Mathématiques Spéciales, 1931, 41: 473–5. ‘Sur l’irréductibilité des polynomes à plusieurs variables’, Bulletin de la Société Mathématique de France, 1932, 60: 127–8. ‘Sur un type de problèmes de cinématique résolus par l’intégrale singulière d’une équation différentielle’, Revue de Mathématiques Spéciales, 1932, 42: 217–8. ‘Sur certaines transformations conformes dans un espace de Riemann’, Comptes Rendus de l’Académie des Sciences, 1932, 195: 747–9 (Session of 2 November 1932). ‘Sur les espaces de Riemann admettant un groupe de transformations isométriques à n(n + 1)/2 paramètres’, Comptes Rendus de l’Académie des Sciences, 1932, 195: 850–2 (Session of 14 November 1932). ‘Sur les espaces de Riemann admettant un groupe isométrique à n(n + 1)/2 paramètres’, Journal de Mathématiques Pures et Appliquées, 1934, 9th series, 13: 197–216. ‘Sur la différentielle binôme’, Bulletin Mathématique des Facultés des Sciences et des Grandes Écoles, June 1934, 1: 161–9; see also ‘Complément’, September– October 1934, 1: 247. ‘Sur les normalisants des s2 dans les groupes gauche et quadratique’, Annales Scientifiques de l’École Normale Supérieure, 1934, 3rd series, 51: 141–51. ‘Sur l’intégrale de différentielle binôme’, Bulletin de la Société Mathématique de France, 1934, 62, Special supplement: 36–9 (Session of 11 April 1934). ‘Sur l’intégrale de différentielle binôme’, Journal de l’École Polytechnique, 1935, 2nd series, 33: 161–74. ‘Sur certaines conditions de l’équilibre économique. Lettre de M. Potron (90) à R. Gibrat (22)’, Centre Polytechnicien d’Études Économiques. X-Crise. Bulletin Mensuel, July-August 1935, Nos. 24–25: 62–5. [Chapter 11] ‘Sur une expression de la courbure tangentielle d’une courbe tracée sur une surface’, Bulletin Mathématique des Facultés des Sciences et des Grandes Écoles, April 1935, 2: 97–101.

The Potron Bibliography 249 [A37] ‘Sur l’irréductibilité des polynômes à plusieurs variables’, Bulletin de la Société Mathématique de France, 1935, 63: 226–30. [A38] ‘Sur l’irréductibilité de certaines intégrales abéliennes aux transcendantes élémentaires’, Comptes Rendus du Congrès International des Mathématiciens. Oslo 1936. Oslo, A. W. Brøggers Boktrykkeri A/S, 1937, Vol. II: 89–90. [A39]* ‘Sur les équilibres économiques ’, Comptes Rendus du Congrès International des Mathématiciens. Oslo 1936. Oslo, A. W. Brøggers Boktrykkeri A/S, 1937, Vol. II: 210–1. [Chapter 12] [A40]* ‘Sur les matrices non négatives’, Comptes Rendus de l’Académie des Sciences, 1937, 204: 844–6 (Session of 8 March 1937). [Chapter 15] [A41]* ‘Sur les matrices non négatives et les solutions positives de certains systèmes linéaires’, Bulletin de la Société Mathématique de France, 1939, 67: 56–61. [Chapter 16] [A42] ‘Sur la décomposition d’un groupe continu fini’, Journal de Mathématiques Pures et Appliquées, 1940, 9th series, 19: 143–61. [A43] ‘Sur les fondements de l’arithmétique’, Revue Générale des Sciences Pures et Appliquées, 1940, 51: 141–4. [A44]* ‘[Lettre sur les statistiques industrielles]’, Journal de la Société de Statistique de Paris, 1942, 83: 207–8. [Chapter 17] [A45] ‘Orientations nouvelles de l’enseignement des mathématiques’, Xavier (Journal of the Collège Saint François-Xavier), Vannes, Easter 1942, 52: 16–18.

1.3 Book reviews [R1] ‘J.-A. de Séguier, Éléments de la théorie des groupes abstraits’, Études, 5 April 1905, 103: 140–1. [R2] ‘B. Baillaud and H. Bourget (eds), Correspondance d’Hermite et de Stieltjes’, Études, 20 May 1906, 107: 533. [R3] ‘É. Borel, Leçons sur les fonctions de variables réelles et les développements en séries de polynômes/R. Baire, Leçons sur les fonctions discontinues’, Études, 20 May 1906, 107: 538–9. [R4] ‘G. Vivanti, Leçons élémentaires sur la théorie des groupes de transformation’, Études, 20 June 1906, 107: 856–7. [R5] ‘G. Darboux, Étude sur le développement des méthodes géométriques’, Études, 5 January 1907, 110: 139–40. [R6] ‘G. Papelier, Formulaire de mathématiques spéciales’, Études, 5 February 1907, 110: 428. [R7] ‘M. Gandillot, Abrégé sur l’hélice et la résistance de l’air’, Études, 5 April 1914, 139: 141–2. [R8] ‘Colonel Compaing de la Tour-Girard, Les outils. Leur étude géométrique’, Études, 20 November 1926, 189: 511–12.

1.4 Solutions of mathematical exercises [S1] ‘Rapports des correcteurs sur les compositions du Concours général (10 juin 1913) – Mathématiques’, Bulletin des Facultés Catholiques de l’Ouest, September 1913, 20th year, No. 4: 3–5.

250 The Potron Bibliography [S2] ‘Calcul différentiel et intégral. Session de juin 1933 avec indications sur les solutions par M. l’abbé Potron’, Bulletin Mathématique des Facultés des Sciences et des Grandes Écoles, 1934: 49–64, 82–4. [S3] ‘Calcul différentiel et intégral. Session de novembre 1933’, Bulletin Mathématique des Facultés des Sciences et des Grandes Écoles, 1934: 85–96, 109–29. [S4] ‘Calcul différentiel et intégral ( juin–juillet 1934) avec indication des solutions par M. l’Abbé Potron’, Bulletin Mathématique des Facultés des Sciences et des Grandes Écoles, 1934: 254–6, 271–88, 305–18; 1935: 19–23. [S5] ‘Calcul différentiel et intégral. Session de novembre 1934 avec indication des solutions par M. l’Abbé Potron’, Bulletin Mathématique des Facultés des Sciences et des Grandes Écoles, 1935: 24–32, 46–64, 79–86. [S6] ‘Calcul différentiel et intégral. Énoncés et indications sur les solutions par M. l’abbé Potron. Session de juin 1935’, Bulletin Mathématique des Facultés des Sciences et des Grandes Écoles, 1936: 47–64, 77–92. [S7] ‘Calcul différentiel et intégral. Énoncés et indications sur les solutions par M. l’abbé Potron. Session de novembre 1935’, Bulletin Mathématique des Facultés des Sciences et des Grandes Écoles, 1936: 93–6, 110–28, 141–55. [S8] ‘Agrégation des sciences mathématiques. Concours de 1937. Problème de calcul différentiel et intégral. Solution par l’abbé Potron, professeur à l’Institut Catholique de Paris’, Bulletin Mathématique des Facultés des Sciences et des Grandes Écoles, 1938: 97–110. [S9] ‘Calcul différentiel et intégral. Première session de 1937. Énoncés et indications sur les solutions par M. l’abbé Potron, professeur à l’Institut catholique’, Bulletin Mathématique des Facultés des Sciences et des Grandes Écoles, 1938: 211–24, 235–56, 281–8, 314–18; 1939: 18–23; see also ‘Rectification’, 1939: 160. [S10] ‘Calcul différentiel et intégral. Deuxième session de 1937’, Bulletin Mathématique des Facultés des Sciences et des Grandes Écoles, 1939: 24–32, 45–64, 77–84.

2 Unpublished writings [U1]* ‘Relations entre la question du chômage et celles du juste prix et du juste salaire’, typescript conserved in the library of the Université Catholique de l’Ouest (Angers), Catalogue number TU908-3-2, date uncertain (around 1913), 3 p. [Chapter 7] [U2]* ‘Communication faite au Congrès d’Oslo’, typescript conserved in the folder ’Potron’ of the Archives Jésuites de la Province de France (Vanves), date uncertain (around 1936), 3 p. [Chapter 13] [U3]* ‘L’aspect mathématique de certains problèmes économiques en relation avec de récentes acquisitions de la théorie des matrices non négatives’, typescript with the text of six lectures presented at the Institut Catholique de Paris in 1937, with an appendix entitled ‘Le problème de la manne des Hébreux’, conserved in the folder ‘Potron’ of the Archives Jésuites de la Province de France (Vanves), undated (around 1936), 34 + 5 p. [Chapter 14] [U4] ‘La décoration des surfaces planes’, typescript conserved in the folder ‘Potron’ of the Archives Jésuites de la Province de France (Vanves), 12 p. [U5] ‘Sur la philosophie des sciences mathématiques’, copy of a typescript attached to a letter to Peter Hoenen, S. J., dated 18 December 1939, conserved in the folder ‘Potron’ of the Archives Jésuites de la Province de France under the general name ‘Lettres du P. Potron au P. Hoenen’, 12 p.

The Potron Bibliography 251

3 Complementary information 3.1 New editions [E1] ‘Sur certaines conditions de l’équilibre économique. Lettre de M. Potron (90) à R. Gibrat (22)’, Cahiers d’Économie Politique, 2000, 36: 153–60. [New edition of A35, with an introduction by Émeric Lendjel (W4).] [E2] Gilbert Abraham-Frois and Émeric Lendjel (eds), Les Œuvres Économiques de l’Abbé Potron, Paris, L’Harmattan, 2004, pp. 57–207. [New edition of A8, A9, A10, A11, A12, A13, A35, A39, A44, U2 and U3, with an introduction (W7), notes, bibliography and appendix by the editors.]

3.2 Publications mentioned by other sources [X1] Exercices de Calcul Différentiel et Intégral. Solutions des Problèmes Posés au C.D.I. dans les Facultés de France en Juin-Juillet 1933. Paris, Hermann, 1933, 44 p. [X2] Solutions des Problèmes Donnés au Certificat de Calcul Différentiel et Intégral, à la Session de Juin–Juillet 1933, dans toutes les Facultés de France. Paris, Hermann, date unknown, 50 p. [X3] Certificat de Calcul Différentiel et Intégral. Solutions des Problèmes Posés à la Session de Juin 1933. Paris, Hermann, date unknown. [X4] Exercices de Calcul Différentiel et Intégral. Paris, Hermann, 1934, 43 p. [X5] Certificat de Calcul Différentiel et Intégral. Solutions des Problèmes Posés à la Session de Novembre 1933. Paris, Croville-Morant, date unknown. [X6] Calcul Différentiel et Intégral. Certificat d’Études Supérieures. Sessions de 1934. Énoncés et Indications sur les Solutions. Paris, Hermann, 1936, 74 p.

4 Additional material 4.1 Works on Potron’s economic model [W1] Barriol, Alfred, ‘Nécrologie. Maurice Potron (1872–1942)’, Journal de la Société de Statistiques de Paris, April 1942, 84: 203–4. [Appendix I] [W2] Barriol, Alfred, ‘[Report on] L’aspect mathématique de certains problèmes économiques’, Journal de la Société de Statistiques de Paris, April 1942, 84: 205–7. [Appendix II] [W3] Vittrant, Michel, ‘[Report on] Le problème de la manne des Hébreux’, manuscript dated 26 March 1953, conserved in the folder ‘Potron’ of the Archives Jésuites de la Province de France. [Appendix III] [W4] Lendjel, Émeric, ‘Une contribution méconnue dans l’histoire de la pensée économique: le modèle de l’abbé M. Potron (1935)’, Cahiers d’Économie Politique, 2000, 36: 145–51. [W5] Abraham-Frois, Gilbert and Émeric Lendjel, ‘Une première application du théorème de Perron–Frobenius à l’économie: l’abbé Potron comme précurseur’, Revue d’Économie Politique, July–August 2002, 111: 639–66. [W6] Lendjel, Émeric, X-Crise (1931–1939): Entre l’Atelier de Modèles et le Bureau des Méthodes, Mémoire pour l’habilitation à diriger les recherches, Vol. II, Université de Marne-la-Vallée, 2002.

252 The Potron Bibliography [W7] Abraham-Frois, Gilbert and Émeric Lendjel, ‘Introduction’, in: Gilbert AbrahamFrois and Émeric Lendjel (eds), Les Œuvres Économiques de l’Abbé Potron, Paris, L’Harmattan, 2004, pp. 8–55. [W8] Abraham-Frois, Gilbert and Émeric Lendjel, ‘Predecessors to Leontief: “Father” Potron’s early contributions to input–output analysis’, International Journal of Applied Economics and Econometrics, January–March 2006, 14: 15–34. [W9] Bidard, Christian, Guido Erreygers and Wilfried Parys, ‘[Book review of Gilbert Abraham-Frois and Émeric Lendjel (eds), Les Œuvres Économiques de l’Abbé Potron]’, 2006, 13: 163–7. [W10] Abraham-Frois, Gilbert and Émeric Lendjel, ‘Father Potron’s early contributions to input–output analysis’, Economic Systems Research, December 2006, 18: 357–72. [W11] Abraham-Frois, Gilbert and Émeric Lendjel, ‘Potron, Maurice (1872–1942)’, in: William A. Darity (ed.), International Encyclopedia of the Social Sciences, 2nd edition, Farmington Hills (Mich.), Macmillan, 2007, Vol. 6: 403–4. [W12] Bidard, Christian, ‘The weak Hawkins-Simon condition’, Electronic Journal of Linear Algebra, January 2007, 16: 44–59. [W13] Bidard, Christian and Guido Erreygers, ‘Potron and the Perron–Frobenius theorem’, Economic Systems Research, December 2007, 19: 439–52. [W14] Mori, Kenji, ‘Maurice Potron’s linear economic model: a de facto proof of Fundamental Marxian Theorem’, Metroeconomica, July 2008, 59: 511–29. [W15] Bidard, Christian, Guido Erreygers and Wilfried Parys, ‘ “Our daily bread”: Maurice Potron, from Catholicism to mathematical economics’, European Journal of the History of Economic Thought, March 2009, 16: 123–54. [W16] Parys, Wilfried, ‘Annotated Potron Bibliography’, Department of Economics, University of Antwerp, Research paper 2010-003. [W17] Bidard, Christian and Guido Erreygers, ‘The Potron matrices’, EconomiX, Université Paris Ouest-Nanterre-La Défense and Department of Economics, University of Antwerp, mimeo, 2010.

4.2 Interviews [I1] ‘Maurice Potron et sa famille’, mimeo, 90 p., 2005. (Souvenirs of Denise SalmonLegagneur, niece of Maurice Potron: interview conducted by Christian Bidard on 3 December 2005). [I2] ‘Maurice Potron à Vannes, d’après les souvenirs d’anciens élèves’, mimeo, 5 p., 2006. (Summary of a survey of Potron’s ancient pupils in the Collège Saint François-Xavier, conducted by Christian Bidard in 2006).

4.3 Archival material 4.3.1 Archives Jésuites de la Province de France (Vanves) The folder ‘Potron’ contains: • a file with personal papers and a few letters from or to Potron concerning his career;

The Potron Bibliography 253 •

offprints of A8, A12, and copies of U2, U3, U4 and U5, as well as Vittrant’s report W3; • a number of letters sent to Raymond Alezais, including a detailed plan for a book on astronomy; most of these letters concern difficulties in the solutions of the exercises collected in B3; • other documents and advertisements for publications, solutions of exercises, short extracts from courses or books. Information on Potron and his family can also be found in the folders ‘PupeyGirard’. These contain information on Pupey’s life (H. Pu 50), spiritual retreats (H. Pu 52), interventions, court cases and affairs (H. Pu 54), correspondence (H. Pu 56), the Action Libérale Populaire (H. Pu 58), the Ligue Patriotique des Françaises (H. Pu 60), and the First World War (H. Pu 62). The last folder has a letter from Potron written during the war. The folder ‘Sainte-Geneviève’ (E.Ve 90/1) contains a file entitled ‘Affaire Potron’, with letters and documents concerning student complaints about his teaching at the École Sainte-Geneviève in Versailles. Moreover, the office of the Province de France in Paris holds a book offered in 1992 to André Bouler, with a reproduction of most of his paintings and caricatures. One of them is a pencil sketch of Maurice Potron’s caricature which served as a preparatory work for the Vannes aquarelle (see section 4.3.6 below). 4.3.2 Archives de l’Institut Catholique de Paris The folder ‘Potron’ contains 33 documents. Six letters, dating from October 1924 to March 1925, concern Potron’s application as a teacher at the Institute. A letter of 24 February 1928 is about Potron’s appointment as chargé de conférence. Several letters dating from 1929 to 1936 deal with his wish to be promoted to a full professorship. There are also documents relating to the courses and public lectures given by Potron, including a poster announcing his 1937 lectures on his economic model (cf. U3; see Photo 5 in the introduction). The folder ‘Baudrillart’ contains three letters from Potron to Mgr Baudrillart, director of the Institute, concerning Potron’s position. 4.3.3 Archives de l’Université Catholique de Lille The archives possess five letters from Potron dating from July 1921 to October 1924 with regard to his teachings at the university. In the last letter he asked for a recommendation concerning his application for a position at the Institut Catholique de Paris. 4.3.4 Archives de l’Université Catholique de l’Ouest (Angers) University posters from the period 1913–1919 mention Potron’s courses in mathematics. The library of the university holds U1.

254 The Potron Bibliography 4.3.5 Archives de l’École Sainte-Geneviève The archives possess two photographs of Maurice Potron with students of his preparatory class for the École Polytechnique during the academic year 1927–1928 (see Photo 2 in the introduction). 4.3.6 Archives municipales de Vannes The archives of the Collège Saint François-Xavier have been deposited here. These include the school’s newsletter Xavier, which published Potron’s last paper (A45) as well as the text of the memorial address by the school’s Rector at Potron’s burial mass. There are also two photographs of Potron with pupils and colleagues. The school itself is in possession of an aquarelle painted in 1940 by André Bouler (then a pupil of the college, who became a Jesuit later), caricaturing the teaching staff, including Potron (see Photo 6 in the introduction). 4.3.7 Bibliothèque de l’Observatoire de Paris In the Complément à l’inventaire Bigourdan, folder ‘A-F 14’, there are five Carnets d’observations (observation notebooks) containing observations and calculations made by Potron between April 1905 and May 1906. 4.3.8 Archives Nationales de France The jury reports on the thesis (by É. Picard) and on the oral defense (by P. Appell) are in the file ‘AJ16 5538’, conserved at the Paris site of the archives. 4.3.9 Archives de l’École Polytechnique The archives conserve files of admission results, physical description and examination results. There is also an official photograph (see Photo 1 in the introduction), and another more informal photograph representing Potron and some of his fellow students. 4.3.10 Archives of the Sisters of the Sainte-Famille de Bordeaux The Rome archives of the congregation hold an eight-page report written by the Mother Superior of the Mours orphanage describing Potron’s priesthood ceremony. 4.3.11 Service Historique de la Défense Folder ‘Potron’ (5YE.139069) summarizes Potron’s military career: Polytechnique, les Poudres, regular training sessions as officer, the Great War and a 1928 request for a veteran certificate, sent by DRAC.

The Potron Bibliography 255 4.3.12 DRAC The association Droit du Religieux Ancien Combattant (recently renamed Défense et Renouveau de l’Action Civique) has a few archives relative to its first years, including the mention of Potron’s membership in the Versailles-Sainte Geneviève section of DRAC. 4.3.13 Archives of the Mouvement des Cadres Chrétiens The Mouvement des Cadres Chrétiens (MCC) in Paris holds the archives of the USIC, an organization which still exists formally, and has a more or less complete collection of the newsletters (Échos) circulated among the members of the USIC and the UFPC (more details in the editors’ note of Chapter 2). 4.3.14 Archives of the Action Catholique des Femmes The Action Catholique des Femmes in Paris holds the archives of the Ligue Patriotique des Françaises (LPDF), including the collection of the Petit Écho of the league. 4.3.15 Potron family archives Papers conserved by descendants of the Potron family include some letters to and from Maurice Potron, photographs (see Photos 3 and 4 of the introduction) and other family-related documents. The family also holds a dedicated souvenir of a pilgrimage of Maurice Potron and his parents to Rome, a vermeil crucifix offered by Élise Leemans-Potron on the occasion of his communion and a chalice consecrated by Mgr Potron for his diaconate.

Name index

NOTE: Page number followed by n refer to footnotes and page numbers in bold refer to tables and figures. Abelé, Jean 243 Abraham-Frois, Gilbert 2, 34, 251–2 Alezais, Raymond 27, 253 Amette, Léon-Adolphe, cardinal 15, 20 Anderson, Robert D. 3n4 Appell, Paul 16, 74, 78, 254 Bagnera, Giuseppe 16 Baldé, Marthe 9, 10 Balzac, Honoré de 7 Barriol, Alfred 15, 31n47, 181, 234, 237–42, 251 Barth, Carl Georg 145 Baudrillart, Alfred, cardinal 25, 253 Belanger, Auguste 18n21 Bélidor, Bernard Forest de 147 Benedict XV, pope 61 Beylard, Hugues 17n19, 19n22 Bidard, Christian 34, 47, 63, 252 Blankenburg, Rudolph 145n13 Blum, Léon 31 Boissart, Adéodat 7, 60 Bonfils, Robert 246 Bouler, André 31, 253–4 Brandeis, Louis Dembitz 145n15–16 Branger, Jacques 166 Brants, Victor 51n99, 132, 137n7 Brunel, Claudie 19n22 Chambord, count of 3 Clement XIV, pope 4 Colbert, Jean-Baptiste 147n24 Combes, Émile 4, 60 Cooke, Morris L. 145n13 Coulomb, Charles Augustin de 147

Courtault, Jean-Michel 16n18 Cova, Anne 19n22 d’Alès, Adhémar 247 d’Ocagne, Maurice 27, 226 Danzin, André 143n2 de Baudicour, Joseph 20 de Bélidor, Bernard Forest 147 de Borchgrave, Christian 246 de Brigode, baroness 19 de Fréminville, Charles de la Poix 29, 142–3, 144n7, 144n12, 145n16, 146, 148n30, 148n34, 156n54, 158–60, 161n76 de la Tour du Pin, René 6–7, 80 de Lamennais, Félicité Robert 6 de Mun, Albert 6–7, 23, 80 De Roover, Raymond 98n12 de Saint-Laurent, Octavie-Thomas, countess 19 de Séguier, Jean Armand 16–17, 27, 243, 246 Dechevrens, Marc 15 Della Sudda, Magali 19n24 Deponcin, Bernard 21n28 Desbuquois, Gustave 7, 28, 30n56, 54, 60, 64–5, 67, 80, 98, 110, 247 Desfeuilles, Paul 146n23 Dessaint, J. 51n95 Disbrow, Donald W. 145n13 Divisia, François 182 Droulers, Paul 7n6 du Passage, Henri 17n19, 18n21, 148n29 Duclos, Paul 243 Dufour, Chantal 8n8, 246 Dufour, Jocelyne 8n8, 246

Name index 257 Duhem, Pierre 28 Dumons, Bruno 19n22 Durand, Jean-Dominique 7n7, 64n1 Edwards, John R., admiral 144 Einstein, Albert 209n13, 226 Emerson, Harrington 145–6 Erreygers, Guido 34, 47, 63, 252 Eymieu, Antonin 19 Fehr, Henri 174n1 Ferry, Jules 3, 58 Frederick, Christine 146 Fremont, Charles 29, 142–3, 147n25 Frisch, Ragnar 30, 51n97, 166, 174 Frobenius, Georg Ferdinand 17, 38, 46–7, 74–5, 82, 110–11, 125n19, 170n5, 175, 177n1, 226n1, 229, 233, 241 Frossard, Marie 19 Frottin, Cécile see Potron, Cécile Frottin, Édouard Jean-Pierre 1, 7, 9 Galloo, André 148n29 Galois, Évariste 15 Gambetta, Léon 3, 58 Gantt, Henry Laurence 145, 158–9, 161n77 Gauss, Carl Friedrich 15, 49 Gibrat, Robert 30, 52, 166, 176 Gide, Charles 51n99, 132 Gilbreth, Frank Bunker 157 Gonin, Marius 7, 60 Gregory XVI, pope 5 Grelon, André 17n19 Guillaume, Alexandre 148n29 Guillet, Léon 142n1 Harmel, Léon 6, 18 Hawkins, David 48, 57 Hawkins, Thomas 46n75 Hilaire, Yves-Marie 17n19, 19n22 Hoenen, Peter 243, 250 Huber, Michel 239 Imbert, Armand 147 Kabanov, Youri 16n18 Lanczos, Cornelius 31, 49–50, 208, 209n13, 226, 227n3, 228–9, 230n3, 233 Laurent, Bernard 64n1 Lavigerie, Charles Martial, cardinal 59 Lazard, Max 131, 236, 247 Le Blancq, Frank 15n14

Le Chatelier, Henry 29, 142–4, 146, 148–9, 154n48, 159–60 Le Play, Frédéric 6, 146n22 Lecerf, Eric 131n1 Leemans, Élise 1, 8, 10, 14, 20, 255 Leemans, Émile 1, 8, 10 Lendjel, Émeric 2, 30n45, 34, 251–2 Leo XIII, pope 3, 6, 17, 28, 58 Leontief, Wassily 56 Leroy, Henri-Joseph 7 Lestra, Jeanne 19 Louis-Napoléon Bonaparte 4; see also Napoléon III, emperor Louis-Philippe, king 3 Mac-Mahon, Marshal Patrice de, president 3, 58 Mariès, Louis 26n37 Mayeur, Jean-Marie 3n4, 17n19, 19n22 McGaffey, Christine see Frederick, Christine Michel, François 26–7, 246 Minkowski, Hermann 74–5, 114 Mori, Kenji 34, 252 Morse, Marston 174n1 Napoléon III, emperor 3, 58 Nelson, Daniel 144n9 Nourrisson, Paul 144n7 Ostrowski, Alexander 48 Ozanam, Frédéric 6 Parys, Wilfried 34, 246, 252 Pasteur, Louis 110 Périer, Philippe 146n23 Perron, Oskar 38, 46, 74–5, 82, 110–11, 170n5 Picard, Émile 15–16, 254 Pinte, Élie 27 Piou, Jacques 59 Pius VII, pope 4 Pius IX, pope 6 Pius X, pope 4, 60–1 Poincaré, Henri 16 Poincaré, Raymond, president 61 Poncelet, Jean-François 147 Ponson, Christian 19n22 Potron, Auguste 1, 7–9, 10, 18, 20–1, 23–4, 58, 62, 238n6 Potron, Cécile 1, 8–9, 10, 19, 23, 58, 61 Potron, Denise see Salmon-Legagneur, Denise Potron, Édouard 9, 10, 24, 32, 238n2

258 Name index Potron, Élisabeth 9, 10 Potron, Élise see Leemans, Élise Potron, Émile 9, 10, 23–4, 238n2 Potron, Étienne-Marie, bishop 14, 255 Potron, Henri 9, 10, 22–4, 238n2 Potron, Marie-Élisabeth (or Marie) 9, 10, 22, 238n2 Potron, Marthe see Baldé, Marthe Potron, Robert 9, 10, 23, 238n2 Pupey (-Girard), Henri (-Régis) 17–21, 23–4, 28, 238, 253 Putz, Anne-Catherine 246 Rebérioux, Madeleine 3n4 Richard, François, cardinal 19 Rivet, Raymond 131n1 Roy, René 182 Saint Ignatius of Loyola 4, 11, 21 Saint Jean-François Régis 21 Salmon-Legagneur, Denise 8, 10, 22n30–1, 24, 252 Salmon-Legagneur, Isabelle 8n8 Sangnier, Marc 59 Sarti, Odile 19n24 Sarton, George 143n3 Sauvy, Alfred 31, 53, 234–6

Simon, Herbert A. 48, 57 Subileau, Françoise 17n19 Sylow, Ludwig 16 Taqqu, Murad S. 16n18 Taylor, Frederick W. 29, 142–65 Thiers, Adolphe 3, 58 Thompson, Clarence Bertrand 160, 161n75 Thompson, Sanford E. 158 Van Daele, Jasmien 131n1 Varé, Louis-Sulpice 7 Varlez, Louis 131, 236n7, 247 Vauban, Sébastien Le Prestre, marquis de 147 Vieille, Paul 138n8 Vittrant, Michel 32, 182, 221n21, 243–4, 251, 253 von Neumann, John 57 Waldeck-Rousseau, Pierre 59 Wallon, Henri 58 Watrigant, Henri 18 Whitaker, Robert J. 15n15 Zamanski, Joseph 28, 80, 82n1, 161n78 Zola, Émile 59

Subject index

NOTE: Page numbers followed by n refer to footnotes and page numbers in bold refer to tables and figures. Académie (Royale) des Sciences 15, 82, 102, 110n1, 136n6, 147 Academy of Sciences see Académie (Royale) des Sciences accounting/accounts 52–3, 72, 82, 88, 89, 105, 138–9, 140, 162, 168, 224, 236, 242 accounting identities 45, 66, 73, 79, 93, 127–8, 172–3, 179, 191, 217, 225 Action Française 6 Action Libérale Populaire 19, 59–60, 253 Action Populaire 7, 28, 30n46, 60, 67, 80 administration 83–6, 95, 106, 119–20; see also administrative conditions administrative conditions 133n4, 144n9, 144n13, 151, 154, 158n67, 183–4; see also administration ALP see Action Libérale Populaire American Locomotive Works 144–5n13 American Mathematical Society 49, 208n12 American Society of Mechanical Engineers 144, 149, 156, 158–9 Association Internationale pour la Lutte contre le Chômage 131, 141 astronomy see cosmology benefits of firms 42n68, 45, 66, 70–2, 79, 82, 88–95, 99, 121, 123, 127–8, 163, 167–9, 171–2, 177–9, 189, 191, 213, 217, 219, 224–5; see also profits benefits of wage-earners see economies of households Bethlehem Steel Company 157 book-keeping see accounting B powder 22, 52–3, 138–40, 152n42, 236

bread: ‘our daily bread’ 55, 220; production coefficients 86–8, 104–6, 183 Bureau de Calculs 33–4, 54, 98–9, 242n3 calculation 16, 28, 35n51, 47–9, 52, 54–7, 95, 97, 99, 117, 140, 159, 164, 173, 181–2, 185, 210–4, 217, 220–1, 225, 227, 230–3, 236 campylograph 15 capital 35, 52, 55, 95–6 capitalist 35, 55n127, 96, 128; see also non-worker; non-working consumer; rentier; simple consumer catalogue of inputs and outputs 53, 98–9, 103 Catalogus Universalis Nostrorum 11, 13n12 category (of consumers; of households; of labour; etc.) see social category Catholic, Catholicism 2–8, 17–9, 21, 24, 29, 54, 59, 64, 74, 81, 150, 160 Catholic Institute of Paris see Institut Catholique de Paris Catholic University of Angers see Université Catholique de l’Ouest Catholic University of Lille see Université Catholique de Lille Catholic University of Louvain 137n7 Centre Polytechnicien d’Études Économiques see X-Crise Cercles Catholiques d’Ouvriers 6, 80 CFTC see Confédération Française des Travailleurs Chrétiens CGT see Confédération Générale du Travail

260 Subject index characteristic number of a socioeconomic state/system 39, 50, 55, 66, 68, 71–2, 92–6, 98, 126–7 characteristic root of maximum modulus see dominant eigenvalue/root Christian 6, 29, 33, 54, 59, 67, 150, 155, 165 Church 1–8, 15, 17, 24, 33, 54, 57, 60, 64 Club Action et Pensée 146 coefficients of consumption 35–6, 52–3, 65–6, 69, 78, 83–4, 108, 120, 134–5, 152, 162, 167, 174, 177, 184, 221 coefficients of production 22, 35–6, 40, 44, 52–3, 56, 65, 69, 78, 84–6, 87–8, 103, 104–5, 108, 120, 132–4, 140, 151, 162, 167, 174–5, 177, 184, 221, 235, 244 collective contracts/conventions 53, 61–2, 183, 240–1 Collège Saint François-Xavier 26, 31, 32, 62, 239, 252, 254 commodity input coefficients see coefficients of production Commune de Paris 5–6, 58 competition 33, 51 computational issues 47–9, 164 concrete facts 41, 50, 72, 93 Confédération Française des Travailleurs Chrétiens 61 Confédération Générale du Travail 5, 59–60, 155, 160 constant returns to scale 35, 186 constants of fabrication/production see coefficients of production consumer 66, 71–2, 78, 82–4, 86, 89, 91–5, 99–100, 109, 120–1, 125, 128, 134–5, 149, 151, 164, 167–8, 173, 177, 184–6, 188, 215–6, 223 consumption basket/habits 35–7, 39–40 consumption of a household 78, 99, 120; see also standard of living; type of existence corporatism 6, 33, 54, 98–9, 164 cosmology 25, 27–8, 60–1, 243 cost of living: 42–3, 45, 65–6, 69–73, 78–9, 81–2, 84, 89–90, 93–5, 100, 107–9, 111, 120–1, 128–9, 152, 162–4, 167–8, 171–2, 175, 179, 183, 188–9, 216, 219–20, 224, 241; high 5, 51, 90; zero 94, 124n18 cost price 42–3, 45, 50, 56, 66, 79, 81, 86–90, 94, 96, 98, 102–6, 108–9, 111, 120, 138–9, 148, 152, 154, 157, 159,

163–4, 168, 171, 173, 179, 184, 188–9, 213n14, 217–8, 220, 223, 235–6, 241 data: empirical/statistical 31, 33, 52–4, 56, 86, 99, 102, 108, 120n13, 137, 139, 141, 164, 183, 185, 221–3, 234, 236, 240, 244; socioeconomic/theoretical 34–6, 50–2, 69, 72, 78, 120, 169–70, 185, 188, 192 decent existence/life 64–5, 69, 81, 86, 161, 220 decomposable matrix 32, 46–7, 77, 113, 116, 118, 123, 124n18, 127, 200–8, 215, 230 determinant 17, 48–9, 115n4, 117, 170, 192–200, 202, 204, 208–9, 211, 227–9, 231–2, 242 dominant eigenvalue/root 37–41, 45–8, 57, 75–7, 79, 111–9, 123–7, 170–2, 196–9, 201–8, 211–5, 219, 226, 228–30, 232–3, 242 duality property 32, 57, 66, 70, 73, 79, 92, 109, 123–4, 170–1, 175, 178–9, 214; strong 44–5; weak 44–5 dynamics 33, 55–6, 96–7, 128–30, 171, 219 earnings of a household 42, 70, 72, 216–7; see also economies of a household École Centrale des Arts et Manufactures 1, 9, 18, École d’Application des Poudres et Salpêtres 11, 22, 59, 238, 254 École des Mines 143, 146 École Normale Supérieure 110 École Polytechnique 1, 11, 12, 15, 18, 21–3, 26, 29, 59, 65, 166, 182, 237–8, 253–4; see also X-Crise École Sainte-Geneviève 11, 13, 17, 21, 25, 26n36, 27, 58–9, 61, 238n4, 239, 243–4, 253–4 Econometric Society 31 econometrics 29, 166, 174 economic crisis 29, 51, 55, 62, 90, 97, 132, 136–7, 161, 165; see also malaise economic good 34, 53, 83, 132, 134–8, 151, 162–4, 182, 188, 235, 240–1; see also product; result of labour economic laws 51, 64, 161, 165, 171 economic model: physical/quantity side 32, 34–41, 43, 45, 50, 54, 56–7; value side 32, 36, 42–5, 50, 55–7 economic result see economic good

Subject index 261 economics: literature of 31, 34, 142; present state of 132 economies of households 42n68, 45, 71n7, 79, 82, 88–96, 99, 121, 123, 127, 157, 163, 167–9, 171–2, 177–9, 188, 191, 213, 216n15, 219, 225; see also savings encyclicals and papal decrees: Au Milieu des Sollicitudes 3, 59; Gravissimo Officii Munere 4, 60; Lamentabili Sane Exitu 60; Rerum Novarum 6–7, 17–18, 28, 33, 54, 59; Une Fois Encore 60; Vehementer Nos 4, 60 equilibrium: economic 97, 164, 166–75; after the introduction of new equipment 55, 97, 137; of production and consumption 51, 69n4, 107, 131–6, 167, 180, 182, 213, 233; of wages and cost of living 72, 81, 108 equipment 55–6, 85–6, 87–8, 89, 96, 99, 104–5, 138–9, 153, 159, 163–4, 171, 173, 186–7, 236n6; see also machine/machinery excess production see overproduction exchange prices see prices existence conditions of satisfactory regimes 32–3, 36, 39–40, 43–4, 46–7, 66, 71, 79, 92–3, 109, 123–7, 170–1, 175, 177–9, 187–8, 191–2, 214, 223, 233, 241–2; see also duality property experiments and trials 52, 148, 150–1, 160, 163 Facultés Catholiques de l’Ouest see Université Catholique de l’Ouest Facultés Catholiques de Lille see Université Catholique de Lille Facultés Catholiques de Lyon 27 family see household F.B. Stearns Company 144–5n13 finance 46, 83, 191, 217, 237 French education system: Combes’s anticlerical policy 4, 60; Falloux law 4; Ferry laws 3, 58; role of the Jesuits 4–5 Frobenius theorems: on nonnegative matrices 46, 110n1, 125n19, 181, 226n1, 229–30, 233, 241; on positive matrices 37, 46, 74–5, 82, 110–2, 170n5, 175, 177n1, 180, 192–9; 226n1, 233, 241; see also Perron-Frobenius theorem Front Populaire 50n94, 62 genealogical tree of goods 183–5, 235; see also table of coefficients general costs see overhead costs

gold as monetary unit 192 Grandes Écoles 11, 25, 27, 238n4, 239 Gregorian University of Rome 243 group theory 15–18, 26–7, 46, 237–8 Harrod-neutral technical progress 56 Harvard Business School 144n9, 161n75 Harvard University 144, 161 Hawkins-Simon condition 33, 48, 57, 229 Hebrew Manna Problem 30–1, 183, 220–5, 242–4 H.H. Franklin Manufacturing Company 144–5n13 hoarding 46, 181, 213, 217 household: 35, 83, 220; budget/cost of living/earnings 42, 51, 152, 183, 188; (see also economies of households; savings); head 35, 83, 186, 221, 223; management 146n19; non-working see non-worker; number of 36, 41, 185, 220–1; social category/standard of living 35, 37, 51, 108, 152, 184–6, 188, 221; survey 56, 108; working 35, 37, 41, 89, 108, 221, 223 Hudson Motor Car Company 144–5n13 implementation: of satisfactory regimes 33, 50–4, 57, 93–9, 124, 132, 164, 171, 173, 187–8, 191–2, 220, 241; of the Taylor system 145–6, 149, 154–5, 164–5 indecomposable matrix 36, 38–9, 41, 46, 200–8, 215, 230 indetermination of satisfactory regimes see infinitely many satisfactory regimes infinitely many satisfactory regimes 39, 50–2, 92–4, 97, 123, 141, 169–72, 187, 189, 191–2, 223–5 input coefficients see coefficients of production input-output matrix/table 29, 35, 36, 40, 46, 48, 57; see also table of coefficients, of the economy instability see stability Institut Catholique de Paris 14, 25–7, 30, 54, 60, 62, 180–1, 237, 240, 244, 253 interest on capital 55n127, 96 International Association on Unemployment see Association Internationale pour la Lutte contre le Chômage International Congress of Mathematicians (Oslo, 1936) 27, 30, 174, 176 Interstate Commerce Commission 145

262 Subject index Jesuit archives 11n9, 11n11, 17n19, 18n20–1, 20n26–7, 23n32, 24n33–4, 26n36, 27n39, 28n43, 68, 176, 181–2, 238n6, 243, 244n1, 246, 250–3 Jesuit missions: Canterbury (novitiate) 13–15, 59–60, 238; Jersey (Maison Saint-Louis) 4, 9, 15, 18n21, 59; Xujiahui, Shanghai (Zi-Ka-Wei) 15, 243 Jesuit order: 1, 3–6, 9, 11–15, 17–21, 23–4, 25, 28, 31, 58–9; members 7, 14–21, 24, 25, 26–8, 54, 60, 68, 176, 181–2, 238, 243–4, 246, 250–3 just price 28, 33–4, 64, 80–101, 107–9, 147–8; see also lucrative price just regime see satisfactory regime just wage 33–4, 80–101, 107–9, 148; see also living wage justice in exchange 43, 47, 80–1, 90–1, 98 L’Abeille 18 la vie chère see cost of living, high labour input coefficients see coefficients of production Lanczos solution method 30, 48–9, 208–13, 226–8, 229–33 Le Sillon 59 Leibniz formula 193n7 LFF see Ligue des Femmes Françaises Lie group 26 Ligue des droits de l’homme 59 Ligue des Femmes Françaises 19, 59–60 Ligue Féminine d’Action Catholique Française 19n23 Ligue Patriotique des Françaises 19–20, 60, 253, 255 Link-Belt Company 144–5n13 Lissajous curves 15 living wage 78–9, 81–2, 94, 102; see also just wage livret de travail 5 LPDF see Ligue Patriotique des Françaises lucrative price 71, 81–2, 109; see also just price machine/machinery 55–6, 97, 119, 139, 148, 151, 154–5, 158, 171, 173, 182–3, 236; see also equipment malaise 49, 94–5; see also economic crisis Manna see Hebrew Manna Problem market 33, 51; see also supply and demand Mathematical Society of France see Société Mathématique de France

mathematics as an appropriate tool 33, 54, 56–7, 81–2, 98–9, 111, 119, 132, 136–7, 173, 182, 217–8, 220, 240 matrix of commodity input coefficients: 35–6, 40, 52–3, 76, 78–9, 92, 120, 125–6, 162–3, 167, 170, 174–5, 177, 184–5, 187, 214, 219; 235; decomposable 46, 79; dominant eigenvalue of 38, 79, 125–6; indecomposable 36, 45 matrix of consumption coefficients 35–6, 40, 52–3, 78, 92, 120, 126, 162–3, 167, 170, 174–5, 177–8, 184, 214 matrix of labour input coefficients 35–6, 40, 52–3, 78, 92, 120, 126, 162–3, 167, 170, 174–5, 177, 184, 214, 219, 235 maximum number of work-days see work-days maximum number of work-hours see work-hours maximum root see dominant eigenvalue/root Midvale Steel & Ordnance Company 144–5n13, 148 minimum number of working hours 39 minor 47–9, 57, 112–4, 118–9, 121, 192–7, 199–200, 202–4, 206, 228–30 misery: and disasters 56, 97; and poverty and unhappiness 165; and ruins 33, 55, 171; of the working class 5–6 monetary aspects 81, 97, 188, 192, 223n4 money 81–2, 93, 97, 173 Montpellier Faculty of Medicine 147 non-worker 35–7, 39, 41, 45, 50, 66, 71, 95–6, 99–100, 162, 171–4, 177, 179, 186, 191, 213, 221, 225; see also capitalist; non-working consumer; rentier; simple consumer non-working consumer 36, 66, 71, 79, 94, 128; see also capitalist; non-worker; rentier; simple consumer nonnegative square matrix 30, 35n53, 36–7, 46–7, 74–5, 110, 180–1, 187, 200–8, 211–2, 226–33 normal rest 78, 94 number of effective/normal working days see working days number of effective/normal working hours see working hours number of equations, formulas, goods, see size of the economic problem

Subject index 263 number of work-days of a period see work-days number of work-hours of a period see work-hours Observatoire de Paris 27–8, 60, 254 organization: Catholic 2; economic 64, 132, 164, 166, 169, 171, 177; industrial 66, 69, 71, 85, 90–1, 93, 120, 136, 144, 169, 177, 179, 236; professional 54; rational 177; scientific see Taylor system; social 6, 33, 50, 54, 56–7; see also corporatism Organization Committees 240, 242 out of work days/hours 42, 89, 91, 134, 162–3, 223; see also unemployment overhead costs 52, 84–6, 103, 138–40, 184, 236 overproduction 36–7, 45–6, 50, 56, 66, 69n4, 89, 91–2, 108, 120, 135–7, 162, 168–9, 181, 183, 191, 213–4, 216–7 P-L-C regime see production-labour-consumption regime P-W regime see price-wage regime Panhard et Levassor 146 Paris Observatory see Observatoire de Paris Parti Ouvrier Français 5 partially reduced matrix see decomposable matrix Perron theorems on positive matrices 37, 46, 74, 82, 110–1, 170n5; see also Perron-Frobenius theorem Perron-Frobenius theorem 28, 37–9, 44–6, 57 physical side see economic model Plimpton Press 144–5n13 population: distribution 51–2, 56, 71n9, 91, 93–6, 98, 109, 128–9, 132, 147; growth 55, 71n9, 93; size 36, 41–2, 123n17, 128–30, 177, 192; working 5, 21 positive square matrix 35n53, 38, 46, 74–5, 110–1, 180, 192–9, 201, 212–3 powder factory of Le Bouchet 22, 52, 138n9, 236 Powder school see École d’Application des Poudres et Salpêtres price 32–4, 42–5, 47, 50–1, 56–7, 64–6, 69–72, 78–101, 104, 107–9, 111, 119–30, 138, 150, 152, 162–5, 167–71, 173–5, 178–81, 183, 188–92, 213–4, 216–220, 223–5, 236, 240–2; current/real 97–8; theoretical 97;

zero 47, 94, 124n18; see also cost price, just price; lucrative price; race to the rise price-wage regime 189, 191–2, 220, 223, 225 prices-wages equilibrium 167, 180, 213, 233; see also price-wage regime; satisfactory regime of prices and wages principal unknowns/variables 36, 40–1, 51, 169–72; see also quantity unknowns/variables; value unknowns/variables product 34, 81–3, 89, 93, 98, 102–3, 108, 132–3, 151, 162, 167, 235–6, 241; see also economic good; result of labour production-consumption equilibrium see equilibrium of production and consumption production-labour-consumption regime 183, 191, 220, 223–5; see also satisfactory regime of production and labour profession 21, 33–5, 42, 48, 52–3, 91, 93–7, 109, 134, 137, 151, 161, 180, 182–9, 213–4, 218–20, 235; see also social category; social group professional organizations 18, 54, 99, 164–5 profits 35, 42–3, 45, 50–1; see also benefits of firms Pullman Company 144–5n13 Purdue University 208 quantity side see economic model quantity unknowns/variables 36–7, 43; see also principal unknowns/variables; secondary unknowns race to the rise (between prices and wages) 33, 50, 171, 181, 213, 219 ralliement policy 3, 6, 59, 61 ration see unit, first species rationalization 169, 181, 213, 219 reduced matrix see decomposable matrix regime 36–7, 50–1, 56; standard 97–8, 141, 164; see also satisfactory regime of prices and wages; satisfactory regime of production and labour Renault automobile factory 160 rentier 35, 55n127, 135–7; see also capitalist; non-worker; non-working consumer; simple consumer result of labour 34, 78, 82–6, 88–9, 93–4, 98–9, 102–3, 119–20, 124n18, 167, 174, 177; see also economic good; product

264 Subject index retreats see spiritual retreats right to life 43, 47, 81, 90–1, 97 right to rest 37, 43, 45, 47 satisfactory regime of prices and wages: 32–3, 45, 50, 56, 81, 94, 171, 178 (see also price-wage regime; prices-wages equilibrium); effectively 43–5, 79, 91, 97–8, 120, 122, 127, 129, 189, 191, 220; semi- 47, 124, 126; simply 34, 43–5, 78–9, 90, 120, 122–3, 125–6; strictly 44, 126; see also infinitely many satisfactory regimes satisfactory regime of production and labour 32–3, 39–41, 44–5, 50–3, 55–6, 78–9, 91, 94, 98, 120–6, 220; see also production-labour-consumption regime; infinitely many satisfactory regimes savings 35n55, 42–3, 45, 50–1, 55, 71n7, 72n10, 96n10, 175n3; see also economies of households scholastic doctrine 33–4, 55n127, 80, 97–8; see also just price; just wage School of Mining Engineering see École des Mines scientific management see Taylor system scientific organization of labour see Taylor system secondary unknowns 36–7, 39–43, 51, 169–72; see also quantity unknowns/variables; value unknowns/variables selling price see price Semaines Sociales de France 7, 60, 64 services 34, 83–4, 88, 103, 105, 119–20, 132, 133n4, 151, 162, 183–4 simple consumer 35, 41, 70, 72, 89, 92, 120–1, 122, 128; see also capitalist; non-worker; non-working consumer; rentier size of the economic problem 52–4, 86–8, 99, 104–6, 137–41, 173, 217, 220, 242; see also calculation; implementation Société d’Économie Politique 31n47, 239 Société d’Économie Sociale 144n7, 146, 148n30, 156n56 Société de Statistique de Paris 102, 131, 234, 236n7, 237, 239–40 Société Mathématique de France 27, 174, 239 Society of Jesus see Jesuit order social category 41, 47, 51, 65–6, 69–72, 78, 86, 89–96, 98–9, 106, 108–9, 111, 119–21, 123–5, 128–9, 136–7, 151–2,

160, 162–3; 177–8, 221, 223–5, 240–1; see also profession; social group social Catholicism 6–7 social group 34–7, 39, 53, 57; see also profession; social category social question 2, 5–7, 18, 53, 81 socioeconomic state 39, 55, 66, 68, 71, 88–98, 126, 128, 136, 168; see also data; state of the economy socioeconomic system see socioeconomic state Sorbonne university see University of Paris species see unit spiritual retreats: at the castle of Courcelles 22; for engineering students 18, 20–1; for engineers 18, 20; organized by Pupey-Girard 17–18, 20–1, 253; by the sisters of Notre-Dame du Cénacle 8; at the Villa Manrèse (Clamart) 11, 18, 59, 238n6; at the Villa Saint-Joseph (Épinay) 21, 60; at the Villa Saint-Régis (Mours) 20–1, 23–4; for workers 17–18, 21 Sraffian interpretation 34 stability: of equilibrium 132; of coefficients 163; of regimes 51–2, 71, 95–6, 168–9 standard of living 53, 111, 134–7, 141, 157, 162–3, 165, 167–8, 171, 174, 177, 179–80, 182–9, 192, 213, 215–6, 218–20, 240–1; see also consumption basket/habits; type of existence standard regime 97–8, 141, 164 state of industry, of industry and trade, etc. see state of the economy state of the economy 36, 44–5, 50, 52, 55, 65, 69, 86, 132–3, 136–7, 141, 162–4, 171, 177–8, 183, 185, 187–8, 192, 213, 219–20, 240–1; see also socioeconomic state Statistical Society of Paris see Société de Statistique de Paris statistics 29, 31, 53, 102–3, 131, 234–6, 241–2; see also survey strictly satisfactory regime of prices and wages 44, 126 substitution 74–8, 107, 110–3, 115, 136, 180, 233, 247; see also decomposable matrix; indecomposable matrix sufficient production 37–8, 47, 78–9, 94, 102 supply and demand 51, 166

Subject index 265 surplus production see excess production survey 53, 57, 86–8, 98–9, 102–6, 108, 137n7; see also statistics table of coefficients: of the economy 52–4, 137, 170–1, 184–5, 188, 192, 235 (see also input-output matrix/table); of the Manna economy 221–3; of the production of B powder 140, 151–2n42, 236; of the production of bread 87–8, 104–5, 151–2n42 Tabor Manufacturing Company 144–5n13, 154n50, 158 taxation 52, 68n2, 84, 184 Taylor system 29, 53, 142–65 technical coefficients of fabrication/production see coefficients of production technical progress 52, 55–6, 96–7, 171, 219 transportation 68n2, 83n5–6, 119, 132, 133n4, 151, 162, 182–3, 235; see also services trial and error 54, 217, 220, 242 tribunal of arbitration 54, 98 type of existence 62–3, 69, 83–4, 86, 95, 98, 102, 106, 108–9, 119, 124; see also standard of living UFPC see Unions Fédérales Professionelles de Catholiques UIC see Union des Ingénieurs Catholiques unemployment 29, 36–7, 39, 41–2, 50, 69n4, 92, 94, 99, 108, 123, 131–2, 136, 141, 148n32, 150, 161, 167–9, 171, 177, 181, 186, 213, 222; see also out of work days/hours Union des Ingénieurs Catholiques 18 Union des Retraites Régionales 20–1 Union du Sacré Coeur 20 Union par le Secrétariat Central 20 Union par le Service Central 20n25 Union par le Syndicat Central 20n25 Unions Fédérales Professionnelles de Catholiques 18, 20, 23, 64, 67–8, 255 Union Sociale d’Ingénieurs Catholiques 17–18, 20, 24, 60, 64, 67–8, 255 unit: first species 53, 183–5, 215; second species 53, 183–5; mixed species 184–5 Université Bordeaux 1 181

Université Catholique de l’Ouest 24, 26, 61, 107, 238–9, 246, 253 Université Catholique de Lille 25, 61, 238–9, 246, 253 Université L’Aurore 243 University of Chicago 49n88 University of Paris 1, 15, 60, 74, 131, 146 USIC see Union Sociale d’Ingénieurs Catholiques utility of a good 83, 96 value side see economic model value unknowns/variables 36, 41–2; see also principal unknowns/variables; secondary unknowns wage 32–5, 40, 42–5, 47, 50–1, 55–7, 64–6, 69–72, 78–130, 138, 148–9, 152–9, 162–5, 167–9, 171, 173, 175, 178–81, 183, 188–9, 192, 213–4, 216–220, 224–5, 236, 240–2; decent 6; differentiated 219; effective 65, 69–70, 72, 78–9, 89–90, 121, 128–9, 163n82, 164, 171, 188–9, 216, 220, 241; increase/rise 56, 62; maximum 78, 90; nominal 5; real 5, 108–9; uniform 33–4; zero 47, 94, 124n18; see also just wage; living wage; race to the rise Watertown Arsenal 144–5n13, 156, 159 well-being 55, 83, 96–8, 165, 171 Western Economic Association 144–5 work-days: (maximum) number 50, 65–6, 69, 71, 78, 86, 90, 92–3, 109, 120, 126, 134, 163n81 work-hours: (maximum) number 36, 40, 42, 44, 167, 175, 177, 186, 222, 241 working days 65, 69–70, 72, 78, 91–2, 99, 120, 129, 133–4, 138, 163n81; minimum/average number 89, 125n19; see also characteristic number of a socioeconomic state/system working hours 39, 43, 66, 138, 167, 185, 222, 235; minimum/average number 39, 172; see also characteristic number of a socioeconomic state/system X-Crise 29–30, 166, 180 Yale & Towne Manufacturing Company 144–5n13