Arrow–Debreu model

It was proposed by Kenneth Arrow, Gérard Debreu in 1954,[1] and Lionel W. McKenzie independently in 1954,[2] with later improvements in 1959.

Arrow (1972) and Debreu (1983) were separately awarded the Nobel Prize in Economics for their development of the model.

Arrow and Debreu have recently treated this question with techniques permitting proofs.This statement is precisely correct; once there were beliefs, now there was knowledge.

In rather short order, it was no longer "as it is" in Marshall, Hicks, and Samuelson; rather, it became "as it is" in Theory of Value.This section follows the presentation in,[6] which is based on.

The households possess proportional ownerships of producers, which can be thought of as joint-stock companies.

The households receive a budget, income from selling endowments, and dividend from producer profits.

The households possess preferences over bundles of commodities, which, under the assumptions given, makes them utility maximizers.

The households choose the consumption plan with the highest utility they can afford using their budget.

is chosen to be "large enough" for the economy so that the restriction is not in effect under equilibrium conditions (see next section).

Note that the above proof does not give an iterative algorithm for finding any equilibrium, as there is no guarantee that the function

In 1954, McKenzie and the pair Arrow and Debreu independently proved the existence of general equilibria by invoking the Kakutani fixed-point theorem on the fixed points of a continuous function from a compact, convex set into itself.

In contrast, the same rotation applied to the convex hull of the unit circle leaves the point (0,0) fixed.

The assumption of convexity precluded many applications, which were discussed in the Journal of Political Economy from 1959 to 1961 by Francis M. Bator, M. J. Farrell, Tjalling Koopmans, and Thomas J. Rothenberg.

[9] Ross M. Starr (1969) proved the existence of economic equilibria when some consumer preferences need not be convex.

[10] (Uzawa, 1962)[11] showed that the existence of general equilibrium in an economy characterized by a continuous excess demand function fulfilling Walras's Law is equivalent to Brouwer fixed-Point theorem.

Thus, the use of Brouwer's fixed-point theorem is essential for showing that the equilibrium exists in general.

[12] In welfare economics, one possible concern is finding a Pareto-optimal plan for the economy.

Verify that Walras's law holds, and so the expenditures match income plus profit, and so it is possible to provide each household with exactly the necessary budget.

This modification is similar to the generalization of the minimax theorem to the existence of Nash equilibria.

The definition of market equilibrium assumes that every household performs utility maximization, subject to budget constraints.

In detail, we continue with the economic model on the households and producers, but we consider a different method to design production and distribution of commodities than the market economy.

In Debreu and Scarf's paper, they defined a particular way to approach an infinitely large economy, by "replicating households".

Now define the "underdog coalition" consisting of the worst-treated household of each type, and they propose to distribute according to

The converse is true under minor additional assumption:[16] (Debreu and Scarf, 1963) — If

The assumption that production possibility sets are convex is a strong constraint, as it implies that there is no economy of scale.

Since the work of Breeden and Lizenberger in 1978,[18] a large number of researchers have used options to extract Arrow–Debreu prices for a variety of applications in financial economics.

[19] No theory of money is offered here, and it is assumed that the economy works without the help of a good serving as medium of exchange.To the pure theorist, at the present juncture the most interesting and challenging aspect of money is that it can find no place in an Arrow–Debreu economy.

This circumstance should also be of considerable significance to macroeconomists, but it rarely is.Typically, economists consider the functions of money to be as a unit of account, store of value, medium of exchange, and standard of deferred payment.

After that, households and producers merely execute their planned productions, consumptions, and deliveries of commodities until the end of time.

Here, "generically" means "on all points, except a closed set of Lebesgue measure zero", as in Sard's theorem.