## Arrow-Debreu Model

## Arrow-Debreu Model

# Arrow-Debreu Model

The Arrow-Debreu Model is named after the Nobel laureates Kenneth Arrow (b. 1921) and Gerard Debreu (1921-2004). It is a formalized Walrasian economic equilibrium system, and the existence of its competitive equilibrium was proven by Arrow and Debreu in their joint work in 1954. Solving the long-standing problem of proving the existence of equilibrium in a Walrasian system, the Arrow-Debreu Model has been the central piece of the general equilibrium theory of economics since the 1950s.

At around the same time, the economist Lionel McKenzie (b. 1919) proved the existence of a competitive equilibrium of a general equilibrium model using a similar set of techniques, so the Arrow-Debreu model is sometimes also referred to as the Arrow-Debreu-McKenzie model.

The Arrow-Debreu model specifies a competitive economy in which there are finite numbers of consumers, commodities (some being used as production inputs), and production units. Consumers have a set of well-defined preferences (continuous, nonsatiated, and convex), and each consumer holds an initial endowment of the commodities, with a positive quantity of at least one commodity. The technology that converts inputs into outputs is either nonincreasing returns to scale or constant returns to scale. In this economy, every producer maximizes profit and every consumer maximizes utility over their budget sets. The equilibrium of the economy is characterized by a set of prices at which the excess demand is zero for every commodity, and producers make zero profit. These market-clearing prices are reached through a *tâtonnement* process, in which “a fictitious price-setter” facilitates the price adjustment following a set of rules that resembles the way in which prices are reached in the real competitive economy.

Formulated in a purely mathematical form, the Arrow-Debreu model can be easily modified into spatial or intertemporal models with proper definition of the commodities based on the commodity’s location or time of delivery. When commodities are specified to be conditional on various states of the world, the Arrow-Debreu model can easily incorporate expectation and uncertainty into the analysis. Theoretical extensions and applications have been made to analyze financial and monetary markets and international trade, as well as other subjects. With a general equilibrium structure, the model is applicable in assessing the overall impact on resource allocation of policy changes in areas such as taxation, tariff, and price control.

The model has been subject to the criticism that many of the assumptions it makes do not fit the workings of the real economy. However, this criticism is not unique to the Arrow-Debreu model; it also applies to all general equilibrium models that are heavily dependent upon rigorous mathematical proofs. In the case of the Arrow-Debreu model, the assumption that each consumer has to have in the initial endowment at least a positive quantity of all commodities (strong survival assumption) or of at least one commodity (weak survival assumption) has drawn substantial criticism. The *tâtonnement* process, which requires that all purchases be made when the competitive equilibrium is reached, is also claimed to be incompatible with the workings of a real economy, where purchasing at non-market-clearing price is often observed.

**SEE ALSO** *Arrow, Kenneth J.; Debreu, Gerard; General Equilibrium; Market Clearing; Prices;* Tâtonnement

## BIBLIOGRAPHY

Arrow, Kenneth J., and Gerard Debreu. 1954. Existence of an Equilibrium for a Competitive Economy. *Econometrica* 22 (3): 265-290.

Arrow, Kenneth J., and F. H. Hahn. 1971. *General Competitive Analysis*. San Francisco: Holden Day.

Debreu, Gerard. 1959. *Theory of Value*. New York: Wiley.

*Chung-Ping A. Loh*