Edgeworth conjecture

In economics, the Edgeworth conjecture is the idea, named after Francis Ysidro Edgeworth, that the core of an economy shrinks to the set of Walrasian equilibria as the number of agents increases to infinity.

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 which 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 those which 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.

Edgeworth himself did not quite prove this result — hence the term "conjecture" rather than "theorem" — although he did provide most of the necessary intuition and went some way towards it. In the 1960s formal proofs were presented under different assumptions by Debreu and Scarf (1963) and Robert Aumann (1964). Both of these results however showed that the conditions sufficient for this result to hold were a bit more stringent then Edgeworth anticipated. 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. Robert Aumann's result relied on an existence of a continuum of agents.

These proofs of the Edgeworth conjecture are seen as providing some qualified support for the idea that a large economy functions approximately as a price taking competitive economy of General equilibrium theory, even though agents have the power to set prices.

References

*Aumann, Robert (1964). "Markets with a Continuum of Traders", "Econometrica"
*Debreu, Gerard and Scarf, Herbert (1963). "A Limit Theorem on the Core of An Economy", "International Economic Review"
*cite book
last =Edgeworth
first =F. Y.
authorlink =Francis Ysidro Edgeworth
title =Mathemathical Psychics: An essay on the mathematics to the moral sciences
publisher =Kegan Paul
date =1881
location =London
note= Reprinted in Diamond, M.A.