Francis Ysidro Edgeworth Research Paper

1. Life

Francis Ysidro Edgeworth was born on February 8, 1845 at the family home Edgeworthstown in County Longford, Ireland, the youngest of five sons. His father Francis Beaufort Edgeworth (1809–1846) was one of 22 children of Richard Lovell Edgeworth (1744–1817), an inventor and educationist whose favorite child was the celebrated novelist Maria Edgeworth (1767–1849). Father and daughter were much interested in political economy and numbered among their acquaintance Bentham Malthus, and Dugald Stewart; after her father’s death Maria also developed a warm friendship with David Ricardo.

Edgeworth’s father, Francis Beaufort Edgeworth, was a favorite of both his elderly father and his half-sister Maria. His mother, 2 years younger than her step-daughter Maria, was Frances Anne Beaufort (1769–1865), of Huguenot stock and elder sister of the admiral who devised the famous Beaufort marine wind scale. Naturally averse to mathematics, Francis left Trinity College, Cambridge without a degree. After a brief courtship at age 22 he married Rosa Florentina Eroles (1815–1864), who was 16, poor, with ‘beautiful eyes,’ and daughter of an exiled Spanish general. After some years in Italy Francis struggled unsuccessfully as owner of a private school in Eltham near London, and in 1840 returned home to Edgeworthstown to run the estate. He died aged 37 after a long illness, in the worst year of the Irish famine.

Despite his disappointing career Edgeworth’s father was not without talent. His writings on poetry and philosophy sufficiently impressed John Stuart Mill to lead him to enquire, in 1840, whether Francis might be available to tutor ‘the eldest son … of a person of very high rank,’ adding mysteriously that ‘there is, probably, no situation of the kind in England in respect to which more important consequences may depend on its being well filled’ (Mineka 1963, p. 439). Moreover, Francis’s friendship was highly valued by the poets Alfred Tennyson and Edward FitzGerald and by the great mathematical physicist Sir William Rowan Hamilton.

Edgeworth himself was actually baptized Ysidro Francis Edgeworth, a form he sometimes used even as an adult. ‘Marshall, remembering his mixed parentage, used to say: ‘‘Francis is a charming fellow, but you must be careful with Ysidro.’’’ (Keynes 1926, p. 152). He was educated by tutors at home, committing to permanent memory whole books of Homer, Virgil, Milton, and Pope. A notable linguist, he was well-read not only in classical Greek and Latin but also in French, German, Italian, and Spanish literature. In 1862 he entered Trinity College, Dublin to read classics and in 1867 transferred to Oxford, where in 1869 he gained a First in the honors examination in classics and philosophy.

He spent the next 22 years in London, living on the heights of Hampstead in two bare rooms that he continued as a pied-a-terre after he moved to Oxford in 1891. Supplementing his slim private means by occasional earnings as a part-time teacher in further education, he studied law systematically (specializing in commercial law) until he was called to the Bar at the Inner Temple in 1877; he never practiced. He also read widely in mathematics and physics, his interests perhaps influenced by those of his father’s friend Hamilton. Certainly, by 1877 he had gained ‘a confident and creative mastery of the calculus of variations’ (Stigler 1986, p. 306).

In 1871 he became an ‘original’ (and lifelong) member of the new Savile Club in central London, whose friendly and distinguished atmosphere apparently suited him well, so much so that during the years 1878–1882 he was quite active in the club’s affairs as either a Committee member or an Honorary Secretary or both. Indeed, it was precisely then and not (as is sometimes alleged) as a reclusive Oxford don that he wrote and published Mathematical Psychics (1881). Two years later he began the many articles on probability and statistics whose quality carried him upwards through a series of appointments at King’s College, London, until in 1891 he was appointed Drummond Professor of Political Economy at Oxford, tenable at All Souls. He retired from this post in 1922 but the College remained his principal home until he died (from pneumonia) in Oxford on February 13, 1926. As the last surviving male Edgeworth in Europe he inherited the family estate from his brother Antonio in 1911 but did not live there.

Although Edgeworth never married it was not for want of trying. He was smitten by the handsome and imperious Beatrice Potter who found him ‘gentlenatured … excessively polite and diffident … pathetic … pedantic’—and boring, even though she glimpsed ‘a hidden fire burning within.’ Insufficiently discouraged, in June 1889 he published a glowing review of her early work and pursued her to a Co-operative Congress in Ipswich where, in spite of his ‘agonized expressions of romantic regard,’ she remained adamantly unimpressed (MacKenzie and MacKenzie 1982, pp. 283–4, 286–7). Three years later she married Sidney Webb.

The words diffident, excessively polite, and pedantic recur often in friends’ descriptions of Edgeworth’s character, which perhaps accounts for Heilbroner’s remark that: ‘Neurotically shy, he tended to flee from the pleasures of human company’ (1999, p. 173). However, just as frequently those same friends also described Edgeworth as affectionate, humorous, and kind; ‘in my house … few visitors were so beloved as he’ (Bonar 1926, p. 652). Known for his hospitality to foreign visitors, he had ‘the widest personal acquaintance in the world with economists of all nations’ (Keynes 1926, p. 151). The young Joseph Schumpeter, visiting Edgeworth at All Souls, was memorably welcomed to breakfast by his host with ‘rock pheasant’ (presumably, cold roast grouse) and champagne (Samuelson 1951, p. 53n).

Edgeworth was twice President of Section F of the British Association for the Advancement of Science (1889, 1922), President of the Royal Statistical Society in 1912, one of the original Fellows of the British Academy (1903), and an honorary Doctor of Civil Law from the University of Durham (1889). The most important of his few honors was his sole editorship of the Economic Journal (E. J.), which lasted from 1890 until J. M. Keynes succeeded him in 1911, whereupon he became Chairman of the Editorial Board. Keynes, busy at Versailles in 1919, co-opted Edgeworth to serve as Joint Editor and he continued active in this capacity ‘up to the last day of his life’ (Keynes 1926, p. 140).

No complete Bibliography: of Edgeworth’s prolific writings in economics and statistics has yet been published but an incomplete count yields four very short books, about 300 articles and shorter papers, and more than 200 book reviews. Perhaps reflecting his position as editor of the E. J., after 1890 his major papers in economics tended to be surveys, learned and complex and delighting in brilliant paradoxes, rather than articles on original themes. Many of these, especially the surveys of the theories of distribution, index numbers, international trade, monopoly, railway rates, returns to scale, and taxation, he gathered together and reprinted in Edgeworth (1925). All his many papers in probability and statistics (except (1887) and a few others) were reprinted in 1996.

2. Mathematical Psychics

Although subtitled ‘An Essay on the Application of Mathematics to the Moral Sciences,’ Mathematical Psychics actually contains three analytically distinct essays, all rather jumbled together. The first is on methodology and comprises pp. 1–16 (undated page references here are all to (1881)); Appendix I—‘On Unnumerical Mathematics’; and Appendix II. It develops two separate themes, the first being a visionary stress on the importance of maximum problems for economics, at both the individual and social level. It goes on to argue that in many respects problems in economics are analogous to those of classical mechanics, so that ‘Mecanique Sociale may one day take her place along with Mecanique Celeste, throned each upon the double-sided height of one maximum principle’ (p. 12). The second theme is less grandiose and develops in detail the importance of ‘unnumerical mathematics’ for positive economics. The emphasis here on second-order conditions for a maximum sounds very modern: ‘the condition that a required curve shall be, or shall not be, convex … this very relation of concavity, not a whit more indefinite in psychics than in physics … quarried from such data as the law of decreasing utility, of increasing fatigue, of diminished returns to capital and labour’ (p. 92).

Edgeworth wrote most of his second essay, on the Utilitarian Calculus, when he was primarily a mathematically inclined philosopher. It consists of pp. 56–82 (which essentially reprints (1879)); Appendices III and IV; pp. 116–8 and 122–5 of Appendix VI; and pp. 126–34 of Appendix VII. Earlier, pp. 35–79 of his monograph New and Old Methods of Ethics (1877) had applied the calculus of variations and the psychophysics or Fechner and others to Sidgwick’s utilitarianism. The resulting ‘exact utilitarianism’ sought ‘the greatest quantity of happiness of sentients, exclusive of number and distribution,’ where ‘sentients’ explicitly refer not just to mankind but to the whole animal creation (1877, p. 35). After a pioneering and detailed variational analysis, employing Lagrange multipliers for both finitely and infinitely many constraints as well as the idea of a continuum of rational agents, he concluded that for modern humankind ‘(T)here is no indication that … a law of distribution other than equality is to be wished’ (1877, p. 78).

The substance of some comments by Alfred Barratt on this monograph were incorporated in Edgeworth’s article in Mind (1879), which enhanced his earlier analysis by considering the possible negative utility from work as well as positive utility from consumption; but also diminished it disastrously, by treating different individual capacities for enjoyment not, as before, as essentially random variations but instead as systematic differences in the sentient’s position in the ‘order of evolution.’ This drastically modified his previous conclusions concerning the optimality of equal distribution, since now ‘the more capable of pleasure shall ha e more means’ (1879, p. 398, 1881, p. 64), and led him to some startling arguments in favor of inequality.

When he wrote those indefensible passages Edgeworth was not yet an economist. But two years later, as virtually an apprentice in economics to his friend, neighbor, and mentor William Stanley Jevons, he wrote his masterpiece, the Economical Calculus. This third essay, entirely on positive economics, consists of pp. 16–56; Appendix V; pp. 118–22 of Appendix VI; and pp. 134–48 of Appendix VII. Its fundamental theorem asserts that: ‘Contract without competition is indeterminate … Contract with perfect competition is perfectly determinate … Contract with more or less perfect competition is less or more indeterminate (p. 20). The wholly normative analysis of the earlier Utilitarian Calculus depended critically on two doubtful assumptions, the cardinal measurability of individual utilities, and their interpersonal comparability, but this theorem needs neither of them. Indeed, it is richly ironical that Edgeworth, the most uninhibited of utilitarians, should have produced in this third essay the most far-reaching (though implicit) ordinalist analysis of any neoclassical economist.

Few works in economics have been so misunderstood as the Economical Calculus. Over the years commentators have discovered there not what it actually contains but only what their own preoccupations have predisposed them to find. For example, those with the deformation professionnelle sometimes induced by Walrasian general equilibrium theory have tended to confuse the aims of Edgeworthian recontract with those of Walrasian tatonnement, while those who believe neoclassical economics to be overly tender towards capitalism have found in Mathematical Psychics ‘a kind of Panglossian Best of All Possible Worlds’ (Heilbroner 1999, p. 173). This is travesty. Edgeworth believed that imperfect competition, with its few competitors, occurs frequently in economic life. Hence, by the theorem, indeterminate contract should occur frequently too, with its ‘characteristic evil’ (p. 29) of ‘deadlock, undecidable opposition of interests, unceasing strife and tumult’ (the last phrase a translation of Edgeworth’s Greek quotation from Demosthenes). This impasse would then lead to ‘a general demand for a principle of arbitration’ (p. 51), a demand which Edgeworth proposes to meet by proffering the following utilitarian rule: the arbitrator should choose that contract which maximizes the total utility of all the parties concerned. ‘Thus the economical leads up to the utilitarian calculus’ (p. 56). In the theorem, articles of contract are whatever is subject to bargain, such as commodities or property rights, and parties may recontract away from any existing contract. The field of competition consists of all those parties (e.g., sellers and buyers) willing and able to enter into contracts for the articles in question. In these general conditions, which include ‘the political struggle for power’ (p. 16) as well as commercial exchange, there may well be neither prices nor markets for the articles. Hence, the theory of supply and demand in perfect markets, as already worked out by Jevons, Marshall, and Walras, does not apply to the general case of Edgeworthian contract and it becomes necessary to devise radically new concepts of equilibrium, appropriate to this general setting.

Edgeworth’s first such concept is a settlement, ‘a contract which cannot be varied with the consent of all the parties to it,’ and the second final settlement, ‘a settlement that cannot be varied by recontract within the field of competition’ (p. 19). A settlement can, therefore, withstand any variation from it proposed by any party to the original contract, while a final settlement can withstand any variation proposed by any interested agent, whether an original contractor or not. A settlement is, thus, precisely an allocation that yields maximum d’ophelimite, a concept first defined 16 years later by Pareto and now known universally as Pareto optimality, its Edgeworthian origins long forgotten. Note that while settlement is a concept of positive economics, Pareto optimality belongs to welfare economics.

Edgeworth explicitly provides for coalitions, whose members may recontract between themselves in order to improve their positions under prevailing contracts. As pointed out first by Shubik (1959), from this perspective the set of final settlements is precisely the core of the relevant cooperative game. (While working with John von Neumann in late 1940, Morgenstern ‘showed him the contract cur e’ (Rellstab 1992, p. 83), but their epoch-making book of 1944 has no reference to Edgeworth.)

When there are only two agents they are the only possible contracting parties, so all settlements are also final settlements. Now for Edgeworth, competition necessarily involves at least three agents (see 1894, p. 378), so two agents represent ‘(α) Contract without competition’ (p. 19). For this case Edgeworth invented and deployed general utility functions and indifference curves to show that the final settlements form a continuous contract curve containing infinitely many points. Contract is, thus, completely indeterminate. (His analysis here remains central to standard microeconomic theory, which uses the ‘exchange box’ invented in 1906 by Pareto to amend Edgeworth’s original exposition; by a delicious irony, economists usually attribute Pareto’s device to Edgeworth!)

At the opposite pole to bilateral exchange lies perfect competition, where indefinitely many traders exchange infinitely divisible articles of contract among themselves. Here Edgeworth shows each resulting market equilibrium allocation to be a settlement (the first proof of the First Fundamental Theorem of Welfare Economics), as well as a final settlement. He conjectures the true result (proved rigorously 91 years later by Debreu and Scarf ) that, in the perfectly competitive replica economy on which his analysis is based, every final settlement is a market equilibrium allocation of some initial endowment. Since the latter allocations are usually unique or at least finite in number, this shows that perfect competition is perfectly determinate.

Edgeworth next displays extraordinary technical virtuosity in dealing with the intermediate case of imperfect competition, where he proves that for a replica economy with 2n agents (where 1 n ), the set of final settlements always shrinks in size as n increases, so that contract becomes less and less indeterminate as the degree of competition increases. The key property needed for this striking result is the strict convexity of individual indifference curves, which Edgeworth actually proves by using assumptions that imply cardinal measurability of utility, but which he could just as well have assumed directly, without need of cardinality.

This completes the proof of the theorem (for details, see Newman 1994). As a coda, Edgeworth joyfully and correctly proves that his arbitration rule always produces a settlement. But this has two grave difficulties. First, the rule itself depends on cardinal measurability of utility. Second, an acceptable rule should produce not just a settlement but a final settlement. Otherwise, for some agents the utilitarian allocation might require their allocated utility levels to be below their initial levels, and so its implementation would require compulsory redistribution.

3. Works In Statistics

Soon after Mathematical Psychics appeared, and possibly influenced by Galton, Edgeworth became attracted intellectually by the ‘law of error,’ i.e., the normal distribution and the central limit theorem. For the next 10 years he devoted his formidable powers and energy to probability, mathematical statistics, and applied statistics, publishing over 30 papers and one brief monograph (1887). Although subsequently he split his creative efforts more evenly between statistics and economics, in those later years he published a further 75 papers in statistics. R. A. Fisher rated Edgeworth of ‘equal eminence’ with Karl Pearson in statistics, while a distinguished historian of the subject has judged him to be ‘the leading theorist of mathematical statistics of the latter half of the nineteenth century’ (Stigler 1987, p. 98).

Edgeworth set himself the task of adapting classical probability theory, developed originally by Pierre Simon Laplace and others for the physical sciences, to the problems of statistical inference in the social sciences. For example, regarding the returns of the Bank of England, ‘to estimate the probability that the differences in the averages for different weeks and months are not accidental … under what circumstances does a difference in figures correspond to a difference in fact’ (quoted in Stigler 1986, p. 308). In 1883 an early result of these efforts was possibly the first derivation of Student’s t distribution (reprinted as McCann 1996, II, No. 1), followed in 1885 by a pioneering version of the analysis variance for a two-way classification (McCann 1996, III, no. 2).

His methodological views on statistical inference in the social sciences were given in his centrally important paper Methods of Statistics of 1885 (McCann 1996, II, no. 3—this reprint has quite illegible diagrams). Generally, Edgeworth took an undoctrinaire Bayesian approach but was quite willing also to use frequentist methods rather than inverse probability, as when in 1905 he used significance tests to compare means (McCann 1996, I, no. 26). In (1887) and elsewhere he stressed the importance of maximizing (expected) utility rather than probability in devising appropriate statistical procedures: ‘what have we to do with probability or frequency, except as a guide to our actions?’ (1887, p. 53). But that monograph has no hint of an axiom system that would unify probability and utility, such as those developed much later by Ramsey, de Finetti, and Savage.

He bequeathed his name to statistical posterity with his development in 1905 of a new series expansion— subsequently the Edgeworth series—as a means of adding correction terms to the central limit theorem in order to deal with skew distributions (McCann 1996, I, no. 26). Less eponymous but more widely known is his term coefficients of correlation and his demonstration in 1892 of their role in expressing the parameters of the multivariate normal distribution (McCann 1996, III, no. 16 and II, no. 21). A less appreciated novelty, anticipating R. A. Fisher, was his proof in 1908–9 of the asymptotic efficiency of maximum likelihood estimators (McCann 1996, I, no. 12).

Finally, as so often with Edgeworth, a pioneering contribution in 1888 to inventory theory (McCann 1996, III, no. 4) was ignored by economists and statisticians alike until very recently, even though it ‘filled a gaping hole in orthodox classical monetary theory’ (Laidler 1991, p. 184). This delightfully written essay on banking demonstrated rigorously and quantitatively what had been known to bankers intuitively and qualitatively for centuries, that a bank’s prudential reserve level rises proportionately, not with its liabilities, but with their square root.

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