# Arrow–Debreu model

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In mathematical economics, the Arrow–Debreu model suggests that under certain economic assumptions (convex preferences, perfect competition, and demand independence) there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.[1]

The model is central to the theory of general (economic) equilibrium and it is often used as a general reference for other microeconomic models. It is named after Kenneth Arrow, Gérard Debreu,[2] and sometimes also Lionel W. McKenzie for his later improvements in 1959.[3]

The A-D model is one of the most general models of competitive economy and is a crucial part of general equilibrium theory, as it can be used to prove the existence of general equilibrium (or Walrasian equilibrium) of an economy. In general, there may be many equilibria; however, with extra assumptions on consumer preferences, namely that their utility functions be strongly concave and twice continuously differentiable, a unique equilibrium exists. With weaker conditions, uniqueness can fail, according to the Sonnenschein–Mantel–Debreu theorem.

## Convex sets and fixed points

A quarter turn of the convex unit disk leaves the point (0,0) fixed but moves every point on the non–convex unit circle.

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 the Arrow–Debreu approach, convexity is essential, because such fixed-point theorems are inapplicable to non-convex sets. For example, the rotation of the unit circle by 90 degrees lacks fixed points, although this rotation is a continuous transformation of a compact set into itself; although compact, the unit circle is non-convex. In contrast, the same rotation applied to the convex hull of the unit circle leaves the point (0,0) fixed. Notice that the Kakutani theorem does not assert that there exists exactly one fixed point. Rotating the unit disk by 360 degrees leaves the entire unit disk fixed, so that this rotation has an infinite number of fixed points.

### Non-convexity in large economies

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.[4] Ross M. Starr (1969) proved the existence of economic equilibria when some consumer preferences need not be convex.[4] In his paper, Starr proved that a "convexified" economy has general equilibria that are closely approximated by "quasi-equilbria" of the original economy; Starr's proof used the Shapley–Folkman theorem.[5]

## Economics of uncertainty: Insurance and finance

Compared to earlier models, the Arrow–Debreu model radically generalized the notion of a commodity, differentiating commodities by time and place of delivery. So, for example, "apples in New York in September" and "apples in Chicago in June" are regarded as distinct commodities. The Arrow–Debreu model applies to economies with maximally complete markets, in which there exists a market for every time period and forward prices for every commodity at all time periods and in all places.[citation needed]

The Arrow–Debreu model specifies the conditions of perfectly competitive markets.

In financial economics the term Arrow–Debreu is most commonly used with reference to an Arrow–Debreu security. A canonical Arrow–Debreu security is a security that pays one unit of numeraire if a particular state of the world is reached and zero otherwise (a so-called "state price"). As such, any derivatives contract whose settlement value is a function on an underlying whose value is uncertain at contract date can be decomposed as linear combination of Arrow–Debreu securities.

Since the work of Breeden and Lizenberger in 1978,[6] a large number of researchers have used options to extract Arrow–Debreu prices for a variety of applications in financial economics.[7]

## References

1. ^ Arrow, K. J.; Debreu, G. (1954). "Existence of an equilibrium for a competitive economy". Econometrica 22 (3): 265–290. doi:10.2307/1907353.
2. ^ EconomyProfessor.com, Retrieved 2010-05-23
3. ^ McKenzie, Lionel W. (1959). "On the Existence of General Equilibrium for a Competitive Economy". Econometrica 27 (1): 54–71. JSTOR 1907777.
4. ^ a b Starr, Ross M. (1969), "Quasi–equilibria in markets with non–convex preferences (Appendix 2: The Shapley–Folkman theorem, pp. 35–37)", Econometrica 37 (1): 25–38, JSTOR 1909201.
5. ^ Starr, Ross M. (2008). "Shapley–Folkman theorem". In Durlauf, Steven N.; Blume, Lawrence E.,. The New Palgrave Dictionary of Economics 4 (1st ed.). Palgrave Macmillan. pp. 317–318. doi:10.1057/9780230226203.1518.
6. ^ Breeden, Douglas T.; Litzenberger, Robert H. (1978). "Prices of State-Contingent Claims Implicit in Option Prices". Journal of Business 51 (4): 621–651. doi:10.1086/296025. JSTOR 2352653.
7. ^ Almeida, Caio; Vicente, José (2008). "Are interest rate options important for the assessment of interest risk?" (PDF). Working Papers Series n. 179, Central Bank of Brazil.