Francis Ysidro Edgeworth (1845-1926) was a leading figure in the rapid development of economics during the last quarter of the nineteenth century and the first quarter of the twentieth century.

He held the Drummond Chair at Oxford from 1891 and was regarded as second only to the great Cambridge economist Alfred Marshall. He was a prolific and highly original author who, in a cosmopolitan age, had probably the widest correspond­ence with economists all over the world.

He achieved eminence as a statistician as well as an economist, becoming a Guy Medallist (Gold) of the Royal Statistical Society in 1907 and was President of the Society, 1912-14. Indeed, of about 170 papers, roughly three-quarters are concerned with statisti­cal theory. His main contributions to statistics concern work on inference and the “law of error”, the correlation coefficient, transformations (what he called “methods of trans­lation”), and the “Edgeworth expansion”. The latter, a series expansion which provides an alternative to the Pearson family of distributions, has been widely used (particularly since the work of Sargan 1976) to improve on the central limit theorem in approximating sampling distributions. It has also been used to provide support for the bootstrap in pro­viding an Edgeworth correction. His third and final book was Metretike: or the Method of Measuring Probability and Utility (1887). These contributions are not examined here; see Bowley (1928) and Stigler (1978). Edgeworth’s work in probability and statistics has been collected by McCann (ed.) (1996).

His name is familiar to all economists, if only because of the “Edgeworth box”, one of the most widely used analytical devices in the subject.

Mathematical Psychics provides the key to all his later work and his lasting importance to economics. He wrote extensively on a wide range of topics, but the central theme of Edgeworth’s work is clear in his revealing statement, taken from his Presidential address to Section F of the Royal Society, that “It may be said that in pure economics there is only one fundamental theorem, but that is a very difficult one: the theory of bargain in a wide sense” (1925, II: 288).

Taking as his starting point Jevons’s (1871 [1957]) basic analysis of exchange of two goods between two traders, Edgeworth supposed that the objective of each trader is to maximize utility, considered to be a general function of the quantities of the goods held and consumed after trade is concluded. The utility-maximizing approach was immedi­ately congenial to Edgeworth, who was steeped in Utilitarian moral philosophy. He first concentrated on the nature of barter, instead of describing the characteristics of an equi­librium set of prices, that is, one which ensures that the individuals’ responses are mutu­ally consistent.

Edgeworth then defined indifference curves for a trader as showing combinations of amounts consumed for which utility is constant. Using several approaches, he demon­strated that the contract curve is the locus of points of tangency between traders’ indif­ference curves, between limits given by their pre-trade curves (those going through the initial endowment point in the box). The existence of a range of final settlements has important implications. First, without introducing further structure to the barter frame­work, it is not possible to say what the implied rate of exchange is, given only informa­tion about preferences and endowments of individuals. It results in “indeterminacy” whereby all that can be said is that the actual trade depends on the relative bargaining strength of the traders.

On the argument that such higgling is widespread, Edgeworth stated in his unique style that, “The whole creation groans and yearns, desiderating a principle of arbitra­tion, and end of strifes” (1881: 51). His next argument involved two steps. First, he showed that the Utilitarian principle of maximizing total utility places individuals on the contract curve, because the mathematical conditions are equivalent to the tangency of indifference curves.

However, he recognized that this result is not sufficient to justify the use of Utilitarianism as a principle of arbitration; it is only a necessary condition. Edgeworth’s justification for Utilitarianism as a principle of justice, comparing points along the con­tract curve, was as follows:

Now these positions lie in a reverse order of desirability for each party; and it may seem to each that as he cannot have his own way, in the absence of any definite principle of selection, he has about as good a chance of one of the arrangements as another... both parties may agree to commute their chance of any of the arrangements for... the utilitarian arrangement. (1881: 55)

The important point to stress about this statement is that Edgeworth clearly considered willingness to accept the Utilitarian arbitration in terms of choice under uncertainty. His argument is that the contractors, faced with uncertainty about their prospects but viewing alternatives along the contract curve as equally likely, would choose to accept an arrangement along Utilitarian lines. Thus a crucial component of this argument is the use of equal a priori probabilities, something that was later important to Edgeworth in his statistical work. In taking this second step Edgeworth believed that he had provided an answer to an age old question, stating, “by what mechanism the force of self-love can be applied so as to support the structure of utilitarian politics, neither Helvetius, nor Bentham, nor any deductive egoist has made clear” (1881: 128).

The importance to him of this new justification of Utilitarianism cannot be exagger­ated. Indeed, the whole of Mathematical Psychics seems to be imbued with a feeling of excitement generated by his discovery of this justification based on a “social contract”. This provided the crucial link between “impure” and “pure” utilitarianism in a more satisfactory way than his earlier appeal to evolutionary forces, made in his book on New and Old Methods of Ethics, written in 1877, before turning to economics.

The nature of price-taking behaviour - involving an equimarginal principle whereby the ratio of prices must be equal, for both traders, to the ratio of their marginal utilities for each of the relevant goods, had been explored with great originality by Jevons with his “equations of exchange”. Edgeworth made important extensions to this analysis, as well as providing his succinct diagrammatical synthesis (which included showing, in 1881: 113, how Marshall’s “offer curves” can be derived from indifference curves). He showed how his box diagram can be used to illustrate a price-taking equilibrium. This arises where one or more of the mutual tangency positions of indifference curves along the contract curve also corresponds to tangency with a straight line going through the endowment point. This line represents a common budget constraint for the choices of the individuals, whereby the slope represents the exchange ratio and hence the relative price. In equilibrium, individuals acting in isolation and taking prices as given (in contrast to those engaged in barter) have mutually consistent demands and supplies. A price-taking equilibrium, as such a tangency position, must therefore correspond to a point on the contract curve.

Edgeworth was thus able to clarify the sense in which a price-taking (often called competitive) equilibrium is “optimal”, fully recognizing that it is just one of many Pareto optimal points. This gives rise to what is now referred to as the “First Fundamental Theorem” of welfare economics - that a price-taking equilibrium is Pareto efficient.

Edgeworth then returned to the indeterminacy in barter, asking whether this inde­terminacy results from the absence of competition in the simple two-person market. Edgeworth quickly moved on to examine the implications of introducing further pairs of traders. The analysis of barter with numerous traders again involves Edgeworth’s stylized description of the recontracting process of barter mentioned above. With more traders, the importance of the recontracting process, apart from allowing the dissemina­tion of information, lies in the fact that it makes it possible to analyse the use of collu­sion among some of the traders. Individuals are allowed to form coalitions in order to improve bargaining strength. Recontracting enables the coalitions to be broken up by outsiders who may attract members of a group away with more favourable terms of exchange.

The analysis of many traders, where coalitions can be temporarily formed and broken up by the offer of improved terms from other traders, would appear to present formi­dable difficulties. Yet Edgeworth rapidly demonstrated, again using his famous box diagram, that the introduction of further similar pairs of traders gradually reduces the range of indeterminacy; that is, the length of the contract curve shrinks. With a suffi­ciently large number of traders, the range of indeterminacy shrinks to the finite number of price-taking equilibria. Barter thus replicates price-taking behaviour. Given that coa­litions among traders are allowed in the recontracting process, a price-taking equilibrium cannot be “blocked” by a coalition of traders. In this sense the competitive equilibrium is robust.

The argument that a complex process of bargaining among a large number of indi­viduals produces a result which is identical to a price-taking equilibrium is an impor­tant result that is far from intuitively obvious. The recontracting process can be said to represent a competitive process, and the contract curve shrinks essentially because of the competition between suppliers of the same good, although it is carried out in a barter framework in which explicit prices are not used. The price-taking equilibrium, in contrast, does not actually involve a competitive process. Individuals simply believe that they must take market prices as given and outside their control. They respond to those prices without any reference to other individuals. But the result is that the price-taking equilibrium looks just like a situation in which all activity is perfectly coordinated.

Great stress was placed by Edgeworth on comparison with Lagrange’s “Principle of least action” in examining the overall effects produced by the interactions among many particles. The connection with Edgeworth’s analysis of competition, involving interac­tion among a large number of competitors to produce a determinate rate of exchange, is clear. The fact that in the natural sciences so much could be derived from a single prin­ciple was important for both Jevons and Edgeworth. However, Edgeworth took this to its ultimate limit in arguing that the comparable single principle in social sciences, that of maximum utility, would produce results of comparable value. Referring to Laplace’s massive work, Mecanique Celeste, he suggested that:

Mecanique Sociale may one day take her place along with Mecanique Celeste, throned each upon the double-sided height of one maximum principle, the supreme pinnacle of moral as of physical science... the movements of each soul, whether selfishly isolated or linked sympa­thetically, may continually be realising the maximum energy of pleasure, the Divine love of the universe. (1881: 12)

A strong belief in the value of mathematical analysis in economics, even where the precise numerical form of the relevant relationships cannot be known, imbues all of Edgeworth’s work. When this is combined with his strong adherence to Utilitarianism, it is not difficult to see how Edgeworth was excited to be showing not only why this principle may be accepted in the form of a “social contract”, but how the actions of many utility maximizing individuals in a market can lead to a determinate solution. Thus, while the comparison with Laplace may seem fanciful to some readers, it was far from fanciful to Edgeworth. These elements provide the “plan” with which virtually all his work in economics could be viewed. It is no wonder that Alexander Pope’s state­ment, in his Essay on Man, that it presents “A mighty maze, but not without a plan” was borrowed by Edgeworth to describe the competitive barter process. It also nicely fits Edgeworth’s own muvre. Although he went on to write on a wide range of economic topics, and to make original contributions to mathematical statistics which alone would guarantee a lasting reputation, an appreciation of the preoccupations leading towards, and nature of, this first work is important in placing everything else in perspective: his economic papers are collected in Edgeworth (1925).

It is clear from even a small sample of Edgeworth’s work that the writer brings to it not just a deep and fertile originality, but also a vast range of knowledge covering natural sciences and literature. His writing is highly allusive and contains quotations from Greek and Latin classics as well as a range of English poets. It displays a sharp wit of a kind found in no other writing on the subject, and continues to repay repeated reading.

John Greedy

See also:

British marginalism (II); Competition (III); General equilibrium theory (III); Hermann Heinrich Gossen (I); William Stanley Jevons (I); Alfred Marshall (I); Vilfredo Pareto (I); Arthur Cecil Pigou (I); Value and price (III); Marie-Esprit-Leon Walras (I); Welfare economics (III); Philip Henry Wicksteed (I).