Edgeworth conjecture
Encyclopedia
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 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.
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.
Economics
Economics is the social science that analyzes the production, distribution, and consumption of goods and services. The term economics comes from the Ancient Greek from + , hence "rules of the house"...
, the Edgeworth conjecture is the idea, named after Francis Ysidro Edgeworth
Francis Ysidro Edgeworth
Francis Ysidro Edgeworth FBA was an Irish philosopher and political economist who made significant contributions to the methods of statistics during the 1880s...
, that the core
Core (economics)
The core is the set of feasible allocations that cannot be improved upon by a subset of the economy's consumers. A coalition is said to improve upon or block a feasible allocation if the members of that coalition are better off under another feasible allocation that is identical to the first...
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
Game theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
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
General equilibrium
General equilibrium theory is a branch of theoretical economics. It seeks to explain the behavior of supply, demand and prices in a whole economy with several or many interacting markets, by seeking to prove that a set of prices exists that will result in an overall equilibrium, hence general...
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.
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
Gerard Debreu
Gérard Debreu was a French economist and mathematician, who also came to have United States citizenship. Best known as a professor of economics at the University of California, Berkeley, where he began work in 1962, he won the 1983 Nobel Memorial Prize in Economics.-Biography:His father was the...
and Scarf (1963) and Robert Aumann
Robert Aumann
Robert John Aumann is an Israeli-American mathematician and a member of the United States National Academy of Sciences. He is a professor at the Center for the Study of Rationality in the Hebrew University of Jerusalem in Israel...
(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
Continuum (theory)
Continuum theories or models explain variation as involving a gradual quantitative transition without abrupt changes or discontinuities. It can be contrasted with 'categorical' models which propose qualitatively different states.-In physics:...
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.