Exchange, Contract and Indeterminacy

In modern economic analysis, the analytical tools invented by Edgeworth in 1881, such as the indifference map and the contract curve, are now used in a vast range of contexts. They were introduced by Edgeworth to examine the nature of barter among individuals.

The PROBLEM to which attention is specially directed in this introductory summary is: How far contract is indeterminate—an inquiry of more than theo­retical importance, if it show not only that indeterminateness tends to [be pres­ent] widely, but also in what direction an escape from its evils is to be sought (Edgeworth 1881: 20; upper case in original).

Edgeworth began his analysis by taking the case of two individuals, A and B, exchanging quantities, x and y, of two goods. The framework is that described by Jevons, where the first individual holds all of the initial stocks of the first good, and the second individual holds all the stocks of the second good. Edgeworth wrote the utility functions of each individual in terms of the amounts exchanged, rather than consumed. He then immediately defined the general (rather than additive) utility function, the contract curve and indiffer­ence curves.

Following Edgeworth’s introduction of the general utility function, he raised the question of the equilibrium which may be reached with, ‘one or both refusing to move further’. In barter the conditions of exchange must be reached by voluntary agreement, or contract, between the two parties, and of course it is fundamental that egoists would not agree to a contract which would make them worse off than before the exchange.

The problem of obtaining the equilibrium values of x and y which, ‘cannot be varied without the consent of the parties to it’ was stated as follows: ‘It is required to find a point (x, y) such that, in whatever direction we take an infi­nitely small step, [utilities] do not increase together, but that, while one increases, the other decreases’ (ibid.: 21). The locus of such points, ‘it is here proposed to call the contract-curve’. Edgeworth’s alternative derivations of the contract curve involved the movement, from an arbitrary position, along one person’s indifference curve. He stated, ‘motion is possible so long as, one party not losing, the other gains’ (ibid.: 23). Here, Edgeworth used the Lagrange multiplier method of maximising one person’s utility subject to the condition that the other person’s utility remains constant. After presenting the results for the two-person two-good case, Edgeworth (ibid.: 26) examined the contract curve in the case where three individuals exchange three goods. This involved an early use of determinants in economics.

The concept of the contract curve helps to specify a range of “efficient exchanges”. The essential feature of the analysis from Edgeworth’s point of view is that there is a range, rather than a unique point, so that ‘the settle­ments are represented by an indefinite number of points’ along the contract curve (ibid.: 29; italics in original). At any particular settlement, the rate of exchange is expressed in terms of the amount of one good which is given up in order to obtain a specified amount of the other good. Hence, the existence of a range of efficient contracts means that the rate of exchange (or effective price ratio) is “indeterminate”.

Edgeworth argued that his analysis of indeterminacy in contract between two traders can be applied to a wide variety of contexts, including trade unions and employers’ associations. Having shown the possibilities of indeterminacy, Edgeworth went on to show how ‘the escape from its evils’ requires either competition or arbitration. He quickly moved on to the introduction of fur­ther traders.

In Edgeworth’s problem of two traders exchanging two goods, the defini­tion of a range of efficient exchanges along the contract curve is analytically separate from the question of whether or not two isolated traders would actu­ally reach a settlement on the contract curve, through barter. However, these two aspects were not clearly separated by Edgeworth because at the beginning of his analysis he introduced his stylised description of the process of barter: this is the “recontracting” process. Edgeworth did not wish to assume that individuals initially have perfect knowledge. Instead, he supposed that ‘there is free communication throughout a normal competitive field’ (ibid.: 18). Knowledge of the other traders’ dispositions and resources is obtained by the formation of tentative contracts, which are not assumed to involve actual transfers and can be broken when further information is obtained. Edgeworth introduced this in typical style, alluding to Alfred Tennyson’s poem “Maud; A Monodrama”: ‘“Is it peace or war?”, asks the lover of Maud, of economic competition, and answers hastily: it is both, pax or pact between contractors during contract, war, when some of the contractors without the consent of others recontract’ (ibid.: 17).

The recontracting process thus enables the dissemination of information among traders.

Edgeworth’s analysis was extremely terse. He introduced a second person A and a second person B, assumed to be exact replicas of the initial pair, with identical tastes and endowments. This simplification allows the same diagram to be used as in the case when only two traders are considered in isolation. Two basic points can be stated immediately. First, in the final settlement all individuals will be at a common point in the Edgeworth box. Second, the settlement must be on the contract curve. The first property arises because if two individuals have identical tastes, their total utility is maximised by shar­ing resources equally.

The question at issue is whether the range of indeterminacy along the con­tract curve is reduced by the addition of these traders. Suppose with just one pair, the type-5 trader has all the bargaining power and pushes the A trader to the limit of the contract curve where B obtains all the gains from trade. With the two pairs of traders no longer in isolation, the ability of a type-A trader to turn to someone else (or form a coalition), rather than deal with a single trader, means that the Bs now compete against each other. The stylised process of recontracting with the two Bs competing against each other will produce a final settlement with all traders at a common point on the contract curve, where the limit has moved inwards along the old contract curve.

With many pairs of such traders, Edgeworth showed that a final settlement is on the contract curve, and looks just like a price-taking equilibrium. If there are multiple equilibria, the recontracting process causes the number of final settlements to shrink to the number of price-taking equilibria. For a discus­sion of utility functions involving multiple equilibria, and a comparison of bargaining, competitive and utilitarian solutions, see Creedy (1994a).

This argument relating to the shrinking contract curve, first established by Edgeworth, is often referred to as the Edgeworth limit theorem; for a more detailed exposition, see Creedy (1986). The fact that the price-taking solution is necessarily on the contract curve gives rise to what is now referred to as the “First Fundamental Theorem” of welfare economics, that a price-taking equi­librium is Pareto efficient. Furthermore, the use of price-taking, compared with recontracting, provides a considerable reduction in the amount of infor­mation required by traders. Given an equilibrium set, individuals only need to know the prices of goods, whereas in the recontracting process they have to learn a considerable amount of information about other individuals’ prefer­ences and endowments. However, Edgeworth placed most stress on the equiv­alence of the competitive price-taking solution with a barter process involving large numbers.

Given that coalitions among traders are allowed in the recontracting pro­cess, a price-taking equilibrium cannot be blocked by a coalition of traders, and the competitive equilibrium is robust. The argument that a process of bargaining among a large number of individuals produces a result which replicates a price-taking equilibrium, allowing for the free flow of information using recontracting and enabling coalitions of traders to form and break up, is an important result that is far from intuitively obvious.

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. However, the result is that the price-taking equilibrium looks just like a situation in which all activity is perfectly co-ordinated.

Edgeworth (1881: 28) also derived, from his indifference curves, the recip­rocal demand curve, or offer curve, of each individual, although such curves (introduced by Marshall as diagrammatic representations of Mill's model of international trade) were then called ‘demand-and-supply curves'. Edgeworth's contribution was to define offer curves in terms of indifference curves, ‘the locus of the point where lines from the origin touch curves of indifference' (ibid.: 113). He mentioned them only briefly in the text (ibid.: 39), but the lack of emphasis is understandable, since in imperfect competition they are not relevant. When there is a lack of competition, giving rise to indetermi­nacy, there is nothing to ensure that individuals will trade on their offer curves and, as Edgeworth argued, ‘the conceptions of demand and supply at a price are no longer appropriate' (ibid.: 31). It is this general preference, in favour of the analysis of barter in non-competitive situations, to which Marshall later objected.

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