seen as a "short-cut" to getting the right answer. Illustration when there are only two commodities Francis Ysidro Edgeworth first described what later became known as the limit theorem in his book Mathematical Psychics (1881). He used a variant of what is now known as the Edgeworth box (with quantities traded, rather than quantities possessed, on the relevant axes) to analyse trade between groups of traders of various sizes. In general he found that 'Contract without competition is indeterminate, contract with perfect competition is perfectly determinate, [and] contract with more or less perfect competition is less or more indeterminate.' Trade without competition If trade in two goods, X and Y, occurs between a single pair of traders, A and B, the potential outcomes of this trade can be shown in an Edgeworth box (Figure 1). In this diagram A and B initially possess the entire stock of X and Y respectively (point E). The lines U(a) and U(b) are the indifference curves of A and B which run through points representing combinations of goods which give utility equal to their initial holdings. As trade here is assumed to be non-coercive, neither of the traders will agree to a final settlement which leaves them worse off than they started off and thus U(a) and U(b) represent the outer boundaries of possible settlements. Edgeworth demonstrated that traders will ultimately reach a point on the contract curve (between C and C') through a stylized bargaining process which is termed the recontracting process. As neither person can be made better off without the other being made worse off at points on the contract curve, once the traders agree to settle at a point on it, this is a final settlement. Exactly where the final settlement will be on the contract curve cannot be determined. It will depend on the bargaining process between the two people; the party who is able to obtain an advantage while bargaining will be able to obtain a better price for his or her goods and thus receive the higher gains from trade. This was Edgeworth's key finding - the result of trade between two people can be predicted within a certain range but the exact outcome is indeterminate. This finding was (erroneously) disputed by Alfred Marshall and the discussions between the two on this point is known as the barter controversy. Trade with less than perfect competition Suppose a