Arrow-Debreu model
Encyclopedia
In mathematical economics
, the Arrow–Debreu model (also referred to as the Arrow–Debreu–McKenzie 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.
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
, Gerard Debreu
, and sometimes also Lionel W. McKenzie
for his later improvements in 1959.
The AD 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. (See also the Sonnenschein–Mantel–Debreu theorem.)
and Debreu
proved the existence of general equilibria by invoking Kakutani's
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.
from 1959 to 1961 by F. M. Bator, M. J. Farrell, T. C. Koopmans
, and T. J. Rothenberg. proved the existence of economic equilibria when some consumer preferences
need not be convex
. 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
.
, 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 market
s, in which there exists a market for every time period and forward prices for every commodity at all time periods and in all places.
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 (1978), a large number of researchers have used options to extract Arrow–Debreu prices for a variety of applications in financial economics.
Mathematical economics
Mathematical economics is the application of mathematical methods to represent economic theories and analyze problems posed in economics. It allows formulation and derivation of key relationships in a theory with clarity, generality, rigor, and simplicity...
, the Arrow–Debreu model (also referred to as the Arrow–Debreu–McKenzie model) suggests that under certain economic assumptions (convex preferences
Convex preferences
In economics, convex preferences refer to a property of an individual's ordering of various outcomes which roughly corresponds to the idea that "averages are better than the extremes"...
, perfect competition
Perfect competition
In economic theory, perfect competition describes markets such that no participants are large enough to have the market power to set the price of a homogeneous product. Because the conditions for perfect competition are strict, there are few if any perfectly competitive markets...
and demand independence) there must be a set of prices such that aggregate supplies will equal aggregate demands for every commodity in the economy.
The model is central to the theory of general (economic) 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...
and it is often used as a general reference for other microeconomic models. It is named after Kenneth Arrow
Kenneth Arrow
Kenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....
, Gerard 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 sometimes also Lionel W. McKenzie
Lionel W. McKenzie
Lionel Wilfred McKenzie was the Wilson Professor Emeritus of Economics at the University of Rochester. He was born in Montezuma, Georgia. He completed undergraduate studies at Duke University in 1939 and subsequently moved to Oxford that year as a Rhodes Scholar...
for his later improvements in 1959.
The AD 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
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...
(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. (See also the Sonnenschein–Mantel–Debreu theorem.)
Convex sets and fixed points
In 1954, ArrowKenneth Arrow
Kenneth Joseph Arrow is an American economist and joint winner of the Nobel Memorial Prize in Economics with John Hicks in 1972. To date, he is the youngest person to have received this award, at 51....
and 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...
proved the existence of general equilibria by invoking Kakutani's
Shizuo Kakutani
was a Japanese-born American mathematician, best known for his eponymous fixed-point theorem.Kakutani attended Tohoku University in Sendai, where his advisor was Tatsujirō Shimizu. Early in his career he spent two years at the Institute for Advanced Study in Princeton at the invitation of the...
theorem
Kakutani fixed point theorem
In mathematical analysis, the Kakutani fixed-point theorem is a fixed-point theorem for set-valued functions. It provides sufficient conditions for a set-valued function defined on a convex, compact subset of a Euclidean space to have a fixed point, i.e. a point which is mapped to a set containing...
on the fixed points of a continuous
Hemicontinuity
In mathematics, the notion of the continuity of functions is not immediately extendible to multi-valued mappings or correspondences. The dual concepts of upper hemicontinuity and lower hemicontinuity facilitate such an extention...
function
Multivalued function
In mathematics, a multivalued function is a left-total relation; i.e. every input is associated with one or more outputs...
from a compact
Compact space
In mathematics, specifically general topology and metric topology, a compact space is an abstract mathematical space whose topology has the compactness property, which has many important implications not valid in general spaces...
, 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
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
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 EconomyJournal of Political Economy
The Journal of Political Economy is an academic journal run by economists at the University of Chicago and published every two months by the University of Chicago Press. The journal publishes articles in both theoretical economics and empirical economics...
from 1959 to 1961 by F. M. Bator, M. J. Farrell, T. C. Koopmans
Tjalling Koopmans
Tjalling Charles Koopmans was the joint winner, with Leonid Kantorovich, of the 1975 Nobel Memorial Prize in Economic Sciences....
, and T. J. Rothenberg. proved the existence of economic equilibria when some consumer preferences
Convex preferences
In economics, convex preferences refer to a property of an individual's ordering of various outcomes which roughly corresponds to the idea that "averages are better than the extremes"...
need not be convex
Convex set
In Euclidean space, an object is convex if for every pair of points within the object, every point on the straight line segment that joins them is also within the object...
. 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
Shapley–Folkman lemma
In geometry and economics, the Shapley–Folkman lemma describes the Minkowski addition of sets in a vector space. Minkowski addition is defined as the addition of the sets' members: for example, adding the set consisting of the integers zero and one to itself yields the set consisting of...
.
Economics of uncertainty: Insurance and finance
Compared to earlier models, the Arrow–Debreu model radically generalized the notion of a commodityCommodity
In economics, a commodity is the generic term for any marketable item produced to satisfy wants or needs. Economic commodities comprise goods and services....
, 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 market
Complete market
In economics, a complete market is one in which the complete set of possible gambles on future states-of-the-world can be constructed with existing assets without friction. Every agent is able to exchange every good, directly or indirectly, with every other agent without transaction costs...
s, in which there exists a market for every time period and forward prices for every commodity at all time periods and in all places.
The Arrow–Debreu model specifies the conditions of perfectly competitive markets.
In financial economics
Financial economics
Financial Economics is the branch of economics concerned with "the allocation and deployment of economic resources, both spatially and across time, in an uncertain environment"....
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
Numéraire
Numéraire is a basic standard by which values are measured. Acting as the numéraire is one of the functions of money, to serve as a unit of account: to measure the worth of different goods and services relative to one another, i.e. in same units...
if a particular state of the world is reached and zero otherwise (a so called "state price
State prices
In financial economics, a state-price security, also called an Arrow-Debreu security , is a contract that agrees to pay one unit of a numeraire if a particular state occurs at a particular time in the future and pay zero numeraire in all other states...
"). 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 (1978), a large number of researchers have used options to extract Arrow–Debreu prices for a variety of applications in financial economics.
See also
- Model (economics)Model (economics)In economics, a model is a theoretical construct that represents economic processes by a set of variables and a set of logical and/or quantitative relationships between them. The economic model is a simplified framework designed to illustrate complex processes, often but not always using...
- John Geanakoplos (1987). "Arrow–Debreu model of general equilibrium," The New Palgrave: A Dictionary of Economics, v. 1, pp. 116–24.
- Incomplete marketsIncomplete marketsIn economics, incomplete markets refers to markets in which the number of Arrow–Debreu securities is less than the number of states of nature...
External links
- Notes on the Arrow–Debreu–McKenzie Model of an Economy, Prof. Kim C. Border California Institute of TechnologyCalifornia Institute of TechnologyThe California Institute of Technology is a private research university located in Pasadena, California, United States. Caltech has six academic divisions with strong emphases on science and engineering...
- "The Fundamental Theorem" of Finance; part II. Prof. Mark RubinsteinMark RubinsteinMark Edward Rubinstein is a leading financial economist and financial engineer. He is currently Professor of Finance at the Haas School of Business of the University of California, Berkeley, where he is involved in teaching courses in the , an academic program that is focused on equipping...
, Haas School of BusinessHaas School of BusinessThe Walter A. Haas School of Business, also known as the Haas School of Business or simply Haas, is one of 14 schools and colleges at the University of California, Berkeley.... - Option Prices and State Prices, Prof. Don Chance Louisiana State UniversityLouisiana State UniversityLouisiana State University and Agricultural and Mechanical College, most often referred to as Louisiana State University, or LSU, is a public coeducational university located in Baton Rouge, Louisiana. The University was founded in 1853 in what is now known as Pineville, Louisiana, under the name...