Francis Ysidro Edgeworth

Francis Edgeworth was a restless philosophy student at Cambridge on his way to Germany when he decided to elope with a teenage Catalonian refugee he met on the steps of the British Museum. One of the outcomes of their marriage was Ysidro Francis Edgeworth (the name order was reversed later), who was destined to become one of the most brilliant and eccentric economists of the 19th Century.

Edgeworth was born in 1845 in Edgeworthstown, County Longford, Ireland into a large, well-connected and eccentric Anglo-Irish landowning family. The famous novelist Maria Edgeworth, of Castle Rackrent fame, was his (elderly) aunt. Although Edgeworth was the fifth son of a sixth son, all the other heirs eventually died, leaving him to inherit Edgeworthstown in 1911. A lifelong bachelor (for a brief period, he hopelessly attempted to court Beatrice Potter), the Edgeworth line died out with him.

Edgeworth was educated at Edgeworthstown by tutors until 1862, when he went on to study languages and the classics at Trinity College, Dublin. He proceeded in 1867 to Oxford (initially at Exeter, then Magdalen and then, finally, from 1868, Balliol College). He graduated in 1869 with a First in Literae Humaniores, but his degree was only awarded in 1873.

Around 1870, Edgeworth moved to Hempstead, in the environs of London. Very little is known about the next decade of his life. Edgeworth subsisted on private income. He must certainly have studied law, for in 1877, he was called to the bar by the Inner Temple. It is also presumed that he learnt mathematics and statistics on his own. It is likely that his interest in this topic was “inherited” from his father’s friend, William Rowan Hamilton, from his Oxford tutor, Benjamin Jowett, and from his close friendship with his Hempstead neighbor, William Stanley Jevons.

In his first book, New and Old Methods of Ethics (1877), Edgeworth combined his interests, applying mathematics – notably the calculus of variations and the method of Lagrangian multipliers – to problems of utilitarian philosophy. His main concern, following up on Sidgwick, was “exact utilitarianism”, defined loosely as the optimal allocation of resources that maximized happiness of a society. He argued that ultimately it falls upon the “capacity for pleasure” of people in a society. He recognized that, under uncertainty, “equal capacity” ought to be assumed. However, he then went on to argue that certain classes of people “obviously” have a greater capacity for pleasure than others (e.g. men more than women), and thus some amount of inequality is justifiable on utilitarian principles. He struck a Darwinist note when, in an attempt to sound optimistic, he argued that “capacities” would evolve over time in a manner that the egalitarian solution would become justifiable in the future. He resurrected his argument, and gave it a more frighteningly eugenicist tinge, in his 1879 paper.

Although qualified as a barrister, Edgeworth did not practice law but rather fell into the academic underground of Victorian Britain for the next decade. Edgeworth lectured on a wide variety of topics (Greek, English theatre, logic, moral sciences, etc.) in a wide variety of settings, from Bedford College for Women in London to Wren’s private training school for Indian civil servants. The pay was miserable and prestige non-existent. A hopelessly impractical and deferential man, his applications for more permanent and lucrative positions at established academic institutions met with heartbreakingly little success.

He was giving evening lectures on logic at King’s College, London when he published his most famous and original book, Mathematical Psychics (1881). In it, he criticized Jevons’s theory of barter exchange, showing that under a system of “recontracting” there will be, in fact, many solutions, an “indeterminacy of contract”. Edgeworth’s “range of final settlements” was later resurrected by Martin Shubik (1959) as the game-theoretic concept of “the core”. Edgeworth also articulated what eventually became known as “Edgeworth’s conjecture”, namely that as the number of agents in an economy increase, the degree of indeterminacy is reduced. He argued that in the limit case of an infinite number of agents (perfect competition), contract becomes fully determinate and identical to the ‘equilibrium’ of economists. This proposition generated an enormous amount of interest during the 1960s and 1970s. However, as situations of perfect competition are not likely to be met in any society, Edgeworth had argued that the only way of resolving this indeterminacy of contract would be to appeal to the utilitarian principle of maximizing the sum of the utilities of traders over the range of final settlements. Incidentally, it was in this 1881 book that Edgeworth introduced into economics the generalized utility function, U(x, y, z, …), and drew the first ‘indifference curve’.

Edgeworth’s seminal work was given lukewarm reviews by W. Stanley Jevons (1881) and Alfred Marshall (1881). Edgeworth attempted to reach out again to economists and others by restating his theory in a series of journal publications (e.g. 1884, 1889, 1891). Alfred Marshall appropriated Edgeworth’s result in his own Principles (1890) textbook, but his distortion of the idea led to a brief controversial exchange of notes in the Giornale degli Economisti in 1891. Alas, it all to no avail. Marshall had successfully swept the entire matter under the rug, where it was to stay for much of the next eighty years.

Edgeworth’s interests were changing anyway. From 1883 onwards, Edgeworth began making his monumental contributions to probability theory and statistics. In his 1885 book Metretike, Edgeworth presented the application and interpretation of significance tests for the comparisions of means. In a series of 1892 papers, Edgeworth examined methods of estimating correlation coefficients. Among his many results was ‘Edgeworth’s Theorem’ giving the correlation coefficients of the multi-dimensional normal distribution. For his efforts, he was elected President of Section F of the British Association for the Advancement of Science in 1889 and later served as president of the Royal Statistical Society (1912 to 1914).

In 1888, on the strength of testimonials from friends and luminaries such as Jevons and Marshall, Edgeworth finally attained his first professional appointment, to the Tooke Chair in Economic Sciences and Statistics and King’s College, London. But that was only a stepping stone. In 1891, he was elected Drummond Professor and Fellow of All-Soul’s College in Oxford, a much-craved position he would hold until retirement.

In 1891, he was also appointed the first editor of The Economic Journal, the main organ of the fledgling British Economic Association (what later became the Royal Economic Society). This was a task he performed with remarkable diligence until 1911, when the post was assumed by John Maynard Keynes. Edgeworth returned as joint editor in 1919, when Keynes had gotten too busy with other activities. Edgeworth continued actively in this role until his death in 1926.

His interest in economic theory picked up again around this time. In 1894, he published a survey of international trade theory in a series of articles in the Economic Journal. In it, he pioneered the use of offer curves and community indifference curves to illustrate its main propositions, including the “optimum tariff”. In that same year, he engaged Eugen von Böhm-Bawerk in a brief controversy over the opportunity cost doctrine.

In 1897, he published a lengthy survey of taxation. It was here that he articulated his famous “taxation paradox”, i.e. that taxation of a good may actually result in a decrease in price. His paradox was disbelieved by contemporaries, “a slip of Mr Edgeworth”, as E.R.A. Seligman put it. However, many years later, Harold Hotelling (1932) rigorously proved that Edgeworth had been correct. Edgeworth also set the utilitarian foundations for highly progressive taxation, arguing that the optimal distribution of taxes should be such that ‘the marginal disutility incurred by each taxpayer should be the same’ (Edgeworth, 1897).

It was also in 1897 when Edgeworth produced his Giornale article on monopoly pricing, where he criticized Cournot’s exact solution to the duopoly problem with quantity adjustments as well as Bertrand’s “instantly competitive” result in a duopoly model with price adjustment. Instead, Edgeworth showed how price competition between two firms with capacity constraints and/or rising marginal cost curves resulted in indeterminacy. For a modern statement of the “Bertrand-Edgeworth” duopoly model, see Levitan and Shubik (1972).

As a critic of the marginal productivity theory, Edgeworth’s articles (1904, 1911) helped refine the neo-classical theory of distribution on a sounder basis. During the First World War, Edgeworth became particularly interested in questions of war finance. His work in this, although highly original, were a bit too theoretical and did not achieve the practical influence he had hoped.

Finally, as editor of the Economic Journal, Edgeworth often took upon himself the role of reviewing new books in economics. He produced a prodigious amount of reviews, some of which have became classics on their own.

Edgeworth’s contributions to economics were stunning in their originality and depth. But he was notoriously poor at expressing his ideas in a way that was understandable to most of his contemporaries. Trained in languages and the classics, he habitually wrote (and spoke) in long, intricate and erudite sentences, sprinkling them with numerous obscure classical and literary references. He was in the habit of inventing words (e.g. brachistopone = the ‘curve of minimal work’) without bothering to define them clearly for readers who could not spot the Greek roots.

If his prose was taxing to read, his use of mathematics was even more negligent of his readers’ abilities. Having taught himself mathematics, Edgeworth must have assumed everyone else had done so as well. He did not bother to provide preliminary explanations of the techniques he was using. Without warning, Edgeworth would glide breathlessly back and forth from his impenetrable prose to no less impenetrable mathematical notation and analysis.

A final point about his personality may be worth mentioning. Many have accused Edgeworth of being excessively “deferential” to authority. He certainly had many ‘heroes’ upon whom he heaped high praise, such as Sidgwick, Jevons and Galton. But he was also quick to wrap his words in velvet when handling opponents such as J. S. Nicholson, Charles Bastable, E.R.A. Seligman and Alfred Marshall.

However, this “deferential” aspect of his character may be due to the simple fact that Edgeworth was not a fighter by nature. Perhaps the painful decades of professional insecurity contributed to this fear. Although he was involved in several critical controversies (as was inevitable for such an original thinker), he would usually withdraw at the first sign of resistance or petulance on the part of his intellectual opponent. His exit would usually be accompanied with a shower of flattery in the hope of avoiding any ill-feeling. And many of his opponents were remarkably vicious. Alfred Marshall, for instance, was a notoriously dirty scrapper and was not averse to intimidation in order to win a debate. If Edgeworth “exhalted (Marshall) to Achilles” (as Joseph Schumpeter (1954: p.831) described it), it might be partly because he wanted to avoid incurring his wrath.

Nevertheless, at Oxford and the Royal Economic Society, Edgeworth was largely regarded as “Marshall’s man” and, indeed, he often solicited Marshall’s opinion on many of his decisions. Whatever his motivations, it is true Edgeworth’s professional activities contributed directly to the ascendancy of the Marshallian neo-classical hegemony and the decline of alternative approaches in Britain around the turn of the century. He blocked Oxford appointments to “undesirable” heterodox scholars such as John A. Hobson. For the Economic Journal, Edgeworth seemed to adopt the editorial policy of rejecting papers which strayed from Marshallian neo-classicism in their analytical approach or threatened to strike up debates on method that might be uncomfortable to the Marshallian orthodoxy. The work of Lausanne School economists, like Barone, were routinely rejected, as were those of the English Historical school. In his own work and book reviews, Edgeworth defended the Marshallian position against the more radical neo-classicals of the day.

Edgeworth himself never established a following and his work had little impact in Britain, with the possible exceptions of Arthur Bowley and W.E. Johnson. Across the water, Edgeworth’s work was respected by Irving Fisher, Knut Wicksell and Vilfredo Pareto, but most of his leads were not followed.

However, as the 20th Century progressed, Edgeworth’s stock grew as Marshall’s faded. In the 1930s, some of his contributions were picked up by Paretians such as Harold Hotelling, John Hicks and Abba Lerner. The 1960s and 1970s were characterized by the flowering of an “Edgeworthian” school, led by Martin Shubik, Herbert Scarf, Gerard Debreu, Robert Aumann, Werner Hildenbrand and other mathematical economists.

Major Works of F.Y.Edgeworth

– Mr. Mathew Arnold on Bishop Butler’s Doctrine of Self-Love, 1876, Mind
– New and Old Methods of Ethics, 1877
– The Hedonical Calculus, 1879, Mind
– Mathematical Psychics: An essay on the application of mathematics to the moral sciences, 1881
– Mr. Leslie Stephen on Utilitarianism, 1882, Mind
– The Law of Error, 1883, Phil Mag
– The Method of Least Squares, 1883, Phil Mag
– The Physical Basis of Probability, 1883, Phil Mag
– On the Method of Ascertaining a Change in the Value of Gold, 1883, JRSS
– Review of Jevons’s Investigations, 1884, Academy
– The Rationale of Exchange, 1884, JRSS
– The Philosophy of Chance, 1884, Mind
– On the Reduction of Observations, 1884, Phil Mag
– A Priori Probabilities, 1884, Phil Mag
– Chance and Law, 1884, Hermathena
– Methods of Statistics, 1885, Jubilee Volume of RSS
– The Calculus of Probabilities Applied to Psychic Research, I & II, 1885, 1886, Proceedings of Society for Psychic Resarch
– On Methods of Ascertaining Variations in the Rate of Births, Deaths and Marriages, 1885, JRSS
– Progressive Means, 1886, JRSS
– The Law of Error and the Elimination of Chance, 1886 Phil Mag
– On the Determination of the Modulus of Errors, 1886, Phil Mag
– Problems in Probabilities, 1886, Phil Mag
– Review of Sidgwick’s Scope and Method, 1886, Academy
– Review of Jevons’s Journals, 1886, Academy
– Four Reports by the committee investigating best method of ascertaining and measuring variations in the monetary standard, 1887, Reports of the BAAS Parts 1 & 3; Part 2
– Observations and Statistics: An essay on the theory of errors of observation and the first principles of statistics, 1887, Transactions of Cambridge Society
– Metretike, or the method of measuring probability and utility, 1887
– On Observations Relating to Several Quantities, 1887, Hermathena
– The Law of Error, 1887, Nature
– The Choice of Means, 1887, Phil Mag
– On Discordant Observations, 1887, Phil Mag
– The Empirical Proof of the Law of Error, 1887, Phil Mag
– On a New Method of Reducing Observations Relating to Several Quantities, 1888, Phil Mag
– New Methods of Measuring Variation in General Prices, 1888, JRSS
– The Statistics of Examinations, 1888, JRSS
– The Value of Authority Tested by Experiment, 1888, Mind
– Mathematical Theory of Banking, 1888, JRSS
– On the Application of Mathematics to Political Economy: Address of the President of Section F of the British Association for the Advancement of Science, 1889, JRSS
– The Mathematical Theory of Political Economy: Review of Walras’s Elements, 1889, Nature
– Review of Wicksteed’s Alphabet, 1889, Academy
– Review of Bohm-Bawerk’s Kapital und Kapitalismus, 1889, Academy
– Review of Bertrand’s Calcul des Probabilites, 1889, J of Education
– Points at which Mathematical Reasoning is Applicable to Political Economy, 1889, Nature
– Appreciation of Gold, 1889, QJE
– The Element of Chance in Competitive Examinations, 1890, JRSS
– Economic Science and Statistics, 1889, Nature
– Review of Marshall’s Principles, 1890, Nature
– Review of Jevons’s Pure Logic, 1890, Academy
– Review of Walras’s Elements, 1890, Academy
– Review of Bohm-Bawerk’s Capital and Interest, 1890, Academy
– La Théorie mathématique de l’offre et de la demande et le côut de production, 1891, Revue d’Economie Politique
– Osservazioni sulla teoria matematica dell’ economica politica, 1891, GdE (trans. ‘On the Determinateness of Economic Equilibrium’)
– Ancora a proposito della teoria del baratto, 1891, GdE
– Review of Ricardo’s Principles, 1891, J of Education
– Review of Keynes’s Scope and Method, 1891, EJ
– An Introductory Lecture on Political Economy, 1891, EJ
– Review of Sidgwick’s Elements of Politics, 1891, EJ
– Review of Second Edition of Marshall’s Principles, 1891, EJ – 1925 version
– Correlated Averages, 1892 , Philos Magazine
– The Law of Error and Correlated Averages, 1892, Philos Magazine
– Recent Attempts to Evaluate the Amount of Coin Circulating in a Country,1892, EJ
– Review of Marshall’s Economics of Industry, 1892, Nature
– Review of Cantillon’s Essai, 1892, EJ
– Review of Palgrave’s Dictionary, 1892, EJ
– Review of Bohm-Bawerk’s Positive Theory of Capital, 1892, EJ
– Review of Smart’s Introduction to the Theory of Value, 1892, EJ
– Review of Dusing’s Das Geschlechtverhaltniss, 1892, EJ
– Review of Benson’s Capital, Labor and Trade, 1892, EJ
– Review of Smart’s Women’s Wages, 1893
– Review of Bonar’s Philosophy, 1893, Mind
– Review of Walsh’s Bimetallism, 1893, EJ
– Review of Fisher’s Mathematical Investigations, 1893, EJ
– Professor Böhm-Bawerk on the Ultimate Standard of Value, 1894, EJ
– One More Word on the Ultimate Standard of Value, 1894, EJ
– Review of Wiser’s Natural Value, 1894, EJ
– Theory of International Values: Parts I, II and III, 1894, EJ
– Recent Writings on Index Numbers, 1894, EJ
– The Measurement of Utility by Money, 1894, EJ
– Asymmetric Correlation between Social Phenomenon, 1894, JRSS
– Entries: “Average”, “Census”, “Cournot”, “Curves”, “Demand Curves”, Difficulty of Attainment”, “Distance in Time”, “Error”, 1894, in Palgrave, editor, Dictionary of Political Economy, Vol. 1.
– Review of the Webbs’ History of Trade Unionism, 1894, EJ
– Review of Third Edition of Marshall’s Principles, 1895, EJ – 1925 version
– Mr. Pierson on the Scarcity of Gold, 1895, EJ
– Thoughts on Monetary Reform, 1895, EJ
– The Stationary State in Japan, 1895, EJ
– A Defense of Index-Numbers, 1896, EJ
– Statistics on Unprogressive Communities, 1896, JRSS
– The Asymmetrical Probability Curve (Abstract), 1894, Proceedings of Royal Society
– The Asymmetrical Probability Curve, 1896, PhilMag
– The Compound law of Error, 1896, Phil Mag
– Entries: “Gossen”, “Index Numbers”, “Intrinsic Value”, “Jenkin”, “Jennings”, “Least Squares” and “Mathematical Method”, 1896, in Palgrave, editor, Dictionary of Political Economy, Vol. 2
– Review of Price’s Money, 1896, EJ
– Review of Nicholson’s Strikes and Social Problems, 1896, EJ
– Review of Pierson’s Leerkook, Vol. 1, 1896, EJ
– Review of Pierson’s Leerkook, Vol. 2, 1897, EJ
– Review of Bastable, Theory of International Trade, 1897, EJ
– Review of Grazani’s Istituzioni, 1897, EJ
– La teoria pura del monopolio, 1897, GdE, (trans.’The Pure Theory of Monopoly’)
– The Pure Theory of Taxation: Parts I, II and III, 1897, EJ
– Miscellaneous Applications of the Calculus of Probabilities, Parts 1, 2, 3, 1897, 1898, JRSS
– Review of Cournot’s Recherches, 1898, EJ
– Professor Graziani on the Mathematical Theory of Monopoly, 1898, EJ
– Review of Darwin’s Bimetallism, 1898, EJ
– On the Representation of Statistics by Mathematical Formulae, Part 1 (1898), Parts 2, 3, 4 (1899)
– On a Point in the Theory of International Trade, 1899, EJ
– Review of Davidson, Bargain Theory of Wages, 1899, EJ
– Professor Seligman on the Mathematical Method in Political Economy, 1899, EJ
– Entries: “Pareto”, “Pareto’s Law”, “Probability”, “Supply Curve” and “Utility”, 1899, in Palgrave, editor, Dictionary of Political Economy, Vol. III
– Answers to Questions by Local Taxation Commission, 1899, Reprint of Royal Commission for Local Taxation
– The Incidence of Urban Rates, Parts I, II and III 1900, EJ
– Defence of Mr. Harrison’s Calculation of the Rupee Circulation, 1900, EJ
– Review of J.B. Clark’s Theory of Distribution, 1900, EJ
– Review of Smart’s Taxation of Land-Values, 1900, EJ
– Review of Bastable, Theory of International Trade (3rd edition), 1897, EJ
– Review of Bonar and Hollander, Letters of Ricardo, 1900, EJ
– Mr. Walsh on the Measurement of General Exchange Value, 1901, EJ
– Disputed Points in the Theory of International Trade, 1901, EJ
– Review of Gide’s Cooperation, 1902, EJ
– Review of Wells’s Anticipations, 1902, EJ
– Methods of Representing Statistics of Wages and Other Groups Not Fulfilling the Normal Law of Error, with A.L. Bowley, 1902, JRSS
– The Law of Error, 1902, Encyclopedia Britannica
– Review of Cannan’s History of Theories of Production, 1903, EJ
– Review of Bortkiewicz’s Anwendungen and Pareto’s Anwendungen, 1903, EJ
– Review of Bastable’s Public Finance, 1903, EJ
– Review of Bastable’s Cartels et Trusts, 1903, EJ
– Review of Pigou’s Riddle of the Tarriff, 1904, EJ
– Review of Nicholson’s Elements, 1904, EJ
– Review of Bowley’s National Progress, 1904, EJ
– Review of Plunkett’s Ireland, 1904, EJ
– Review of Northon’s Loan Credit, 1904, EJ
– Review of Graziani’s Istituzione, 1904, EJ
– Review of Dietzel’s Vergeltungzolle, 1904, EJ
– Preface, 1904, in J.R. MacDonald, editor, Women in the Printing Trades
– The Theory of Distribution, 1904, QJE
– The Law of Error, 1905, Transactions of Cambridge Society
– Review of Nicholson’s History of English Corn Laws, 1905, EJ
– Review of Cunynghame’s Geometrical Political Economy, 1905, EJ
– Review of Carver’s Theory of Distribution, 1905, EJ
– Review of Taussig’s Present Position, 1905, EJ
– Review of Henry Sidgwick: A memoir, 1906, EJ
– The Generalised Law of Error, or Law of Great Numbers, 1906, JRSS
– Recent Schemes for Rating Urban Land Values, 1906, EJ
– On the Representation of Statistical Frequency by a Series, 1907, JRSS
– Statistical Observations on Wasps and Bees, 1907, Biometrika
– Review of de Foville’s Monnaie and Guyot’s Science economique, 1907, EJ
– Appreciations of Mathematical Theories, Parts I & II (1907), Parts III & IV (1908), EJ
– On the Probable Errors of Frequency Constants, I, II & III (1908), Add. (1909), JRSS
– Review of Andreades’s Lecture on the Census, 1908, EJ
– Review of Rea’s Free Trade, 1908, EJ
– Review of Withers’s Meaning of Money, 1909, EJ
– Review of Mitchell’s Gold Prices, 1909, EJ
– Review of Jevons’s Investigations, 1909, EJ
– Application du calcul des probabilités à la Statistique, 1909, Bulletin de l’Institut international de statistique
– On the Use of the Differential Calculus in Economics to Determine Conditions of Maximum Advantage, 1909, Scientia
– Applications of Probabilities to Economics, Parts I & II, 1910, EJ
– The Subjective Element in the First Principles of Taxation, 1910, QJE
– Review of John Stuart Mill’s Principles, 1910, EJ
– Review of Colson’s Cours, 1910, EJ
– Review of J. Maurice Clark’s Local Freight Discriminations, 1910, EJ
– Review of Hammond’s Railway Rate Theories, 1911, EJ
– Probability and Expectation, 1911, Encyclopedia Britannica
– Monopoly and Differential Prices, 1911, EJ
– Contributions to the Theory of Railway Rates, Part I & II (1911), Part III (1912), Part IV (1913), EJ
– Review of Moore’s Laws of Wages, 1912, EJ
– Review of Pigou’s Wealth and Welfare, 1913, EJ
– On the Use of the Theory of Probabilities in Statistics Relating to Society, 1913, J of RSS
– A Variant Proof of the Distribution of Velocities in a Molecular Chaos, 1913, PhilMag
– On the Use of Analytical Geometry to Represent Certain Kinds of Statistics, Parts I-V, 1914, J of RSS
– Recent Contributions to Mathematical Economics, I & II, 1915, EJ
– On the Relations of Political Economy to War, 1915
– The Cost of War and ways of reducing it suggested by economic theory, 1915
– Economists on War: Review of Sombart, etc., 1915, EJ
– Review of Pigou’s Economy and Finance of War, 1916, EJ
– Review of Preziosi’s La Germania alla Conquista dell’ Italia, 1916, EJ
– British Incomes and Property, 1916, EJ
– On the Mathematical Representation of Statistical Data, Part I (1916), Parts II-IV (1917), J of RSS
– Review of Gill’s National Power and Prosperity, 1917, EJ
– Review of Lehfeldt’s Economics in Light of War, 1917, EJ
– Some German Economic Writings about the War, 1917, EJ
– After-War Problems: Review of Dawson at al., 1917, EJ
– Review of Westergaard’s Scope and Methods of Statistics, 1917, JRSS
– Review of Anderson’s Value of Money, 1918, EJ
– Review of Moulton and Phillips on Money and Banking, 1918, EJ
– Review of Loria’s Economic Causes of War, 1918, EJ
– Review of Arias’s Principle, 1918, EJ
– Review of Smith-Gordon, Rural Reconstruction of Ireland and Russell’s National Being, 1918, EJ
– On the Value of a Mean as Calculated from a Sample, 1918, EJ
– An Astronomer on the Law of Error, 1918, PhilMag
– Currency and Finance in Time of War, 1918
– The Doctrine of Index-Numbers According to Prof. Wesley Mitchell, 1918, EJ
– Psychical Research and Statistical Method, 1919, JRSS
– Methods of Graduating Taxes on Income and Capital, 1919, EJ
– Review of Cannan’s Money, 1919, EJ
– Review of Andreades’s Historia, 1919, EJ
– Review of Lehfeldt’s Gold Prices, 1919, EJ
– A Levy on Capital for the Discharge of the Debt, 1919
– Mathematical Formulae and the National Commission on Income Tax, 1920, EJ
– On the Application of Probabilities to the Movement of Gas Molecules, Part I (1920), Part II (1922), Phil Mag
– Entomological Statistics, 1920, Metron
– Review of Gustav Cassel’s Theory of Social Economy, 1920, EJ
– Review of Bowley’s Change in Distribution of National Income, 1920, JRSS
– Review of the Webbs’ History of Trade Unionism, 1920, EJ
– Molecular Statistics, Part I (1921), Part II (1922), JRSS
– On the Genesis of the Law of Error, 1921, PhilMag
– The Philosophy of Chance, 1922, Mind
– The Mathematical Economics of Professor Amoroso, 1922, EJ
– Equal Pay to Men and Women for Equal Work, 1922, EJ
– Review of Keynes’s Treatise on Probability, 1922, JRSS
– Review of Pigou’s Political Economy of War, 1922, EJ
– Statistics of Examinations, 1923, JRSS
– On the Use of Medians for Reducing Observations Relating to Several Quantities, 1923, Phil Mag.
– Mr. Correa Walsh on the Calculation of Index Numbers, 1923, JRSS
– Index Numbers According to Mr. Walsh, 1923, EJ
– Women’s Wages in Relation to Economic Welfare, 1923, EJ
– Review of Marshall’s Money, Credit and Commerce, 1923, EJ
– Review of The Labour Party’s Aim, 1923, EJ
– Review of Bowley’s Mathematical Groundwork, 1924, EJ
– Review of Fisher’s Economic Position of the Married Woman, 1924, EJ
– Untried Methods of Representing Frequency, 1924, JRSS
– Papers Relating to Political Economy, 3 volumes, 1925
– The Plurality of Index-Numbers, 1925, EJ
– The Element of Probability in Index-Numbers, 1925, JRSS
– The Revised Doctrine of Marginal Social Product, 1925, EJ
– Review of J.M. Clark’s Overhead Costs, 1925, EJ
– Mr Rhode’s Curve and the Method of Adjusrtment, 1926, JRSS

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