Edgeworth's limit theorem is an economic theorem, named after Francis Ysidro Edgeworth, stating that the core of an economy shrinks to the set of Walrasian equilibria as the number of agents increases to infinity.
That is, among all possible outcomes which may result from free market exchange or barter between groups of people, while the precise location of the final settlement (the ultimate division of goods) between the parties is not uniquely determined, as the number of traders increases, the set of all possible final settlements converges to the set of Walrasian equilibria.
Intuitively, it may be interpreted as stating that as an economy grows larger, agents increasingly behave as if they are price-taking agents, even if they have the power to bargain.
Edgeworth (1881) conjectured the theorem, and provided most of the necessary intuition and went some way towards its proof. Formal proofs were presented under different assumptions by Debreu and Scarf (1963) as well as Aumann (1964), both proved under conditions stricter than what Edgeworth conjectured. Debreu and Scarf considered the case of a "replica economy" where there is a finite number of agent types and the agents added to the economy to make it "large" are of the same type and in the same proportion as those already in it. Aumann's result relied on an existence of a continuum of agents.
The core of an economy
The core of an economy is a concept from cooperative game theory defined as the set of feasible allocations in an economy that cannot be improved upon by subset of the set of the economy's consumers (a coalition). For general equilibrium economies typically the core is non-empty (there is at least one feasible allocation) but also "large" in the sense that there may be a continuum of feasible allocations that satisfy the requirements. The conjecture basically states that if the number of agents is also "large" then the only allocations in the core are precisely what a competitive market would produce. As such, the conjecture is seen as providing some game-theoretic foundations for the usual assumption in general equilibrium theory of price taking agents. In particular, it means that in a "large" economy people act as if they were price takers, even though theoretically they have all the power to set prices and renegotiate their trades. Hence, the fictitious Walrasian auctioneer of general equilibrium, while strictly speaking completely unrealistic, can be 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
thumb|Figure 1 - An Edgeworth box showing exchange between two people.
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.
Trade with less than perfect competition
[[Image:Edgeworthprocess2.svg|thumb|Figure 2 - Trade between two pairs of people.