Arrow–Debreu model

The Arrow- Debreu equilibrium model ( Arrow- Debreu actually correct and McKenzie model ) is a general equilibrium model. It is named after Gérard Debreu and Kenneth Arrow, which it published in 1954. Only occasionally the financial mathematician Lionel W. McKenzie is mentioned in 1959 some improvements added. It is a further development of Léon Walras developed by Walrasian equilibrium model. This model, as well as any general equilibrium model, forms the national economy as a whole and examined from a macroeconomic equilibrium.

The introduction of state-dependent requirements for the theory of finance is of great importance. It extends the equilibrium analysis for perfectly competitive markets to the case of uncertain expectations and shows that it also depends on product and capital markets, where there is competition to a state in the freight and risk allocation in which it is not possible for an individual better to make without providing at the same time another individual worse ( Pareto optimum).

• 2.1 components
• 2.2 overall economy

• 3.1 economy with perfect competition
• 3.2 Walrasianisches balance

• 4.1 conditions of existence
• 4.2 Meaning of the conditions of existence
• 4.3 Walras ' law
• 4.4 Laws of welfare economics
• 4.5 Uniqueness and stability of the equilibrium

General

Content

General equilibrium models represent a hypothetical economy in which all consumers have complete, reflexive and transitive preferences over their consumption possibility sets, so are rational. The model is the greatest possible micro -depth description of an economy and describes how consumers and producers, given a certain Anfangsausstatung, simultaneously choose consumption and production. The aim of this model is to examine general allocations and the equilibrium of an economy without resorting to the concept of Partialmärkten.

History

The first attempt in neoclassical theory to develop a comprehensive model for the determination of relative prices in an economy comes from Léon Walras, the founder of the Lausanne school. He wanted to do from the classical political economy of Adam Smith and David Ricardo an "exact science." Therefore, he tried to describe mathematically the economy. Abraham Wald and later Maurice Allais, Kenneth Arrow and Gerard Debreu described the existence and stability of a general equilibrium for a market economy with private property. Arrow, Allais and Debreu were awarded the Nobel Prize in Economics for their work on general equilibrium theory ( AGT).

Description of the economy

Components

In the Arrow- Debreu equilibrium model, it is assumed that an economy can be expressed by the following four quantities:

( i) The goods space:

Describes quality, time and place of availability of goods. If there are n different goods which are present in each case up to different locations and different times and in each case on to different grades, the L dimension of the goods chamber is determined by

The consumer sector is described by. Here, the consumption possibility amount of consumers is being, ie the set of all i possible Kosumbündel, I is the set of all users and the preference relation of the consumer i is thus implicitly assume that the Kosumverhalten all consumers described as a homo economicus model about the consumption possibility amount can be.

The production sector is described by. Here, J is the set of all businesses and the sign of a vector of entry is interpreted as follows:

Describes all the resources of the economy are at the beginning of viewing.

Total economy

A total economy is described in the Arrow- Debreu equilibrium model by.

An extension of this description of an economy is an economy with private property.

Herein, the possession of the consumer to the company dar. expresses how much of the company j the consumer i belongs. Since every business is j wholly owned by consumers, is considered. In addition, from suppressed, heard how much of the resources that are available at the beginning, the consumer i. It is therefore

The Walrasian equilibrium

Economy with perfect competition

An economy with private property and perfect competition is characterized in particular by the fact that there is a price vector which specifies a price for all goods and resources and that each consumer has a limited budget with the ability of expressing.

Then denotes the set of all i for the consumer optimal according to his preferences Kosumpläne for a given price vector p and the set of all profit-maximizing production plans for a given price vector p. ( Thus, it is not assumed in the Arrow- Debreu model assumes that the optimization problems of consumers and companies must always have unique solutions. )

Walrasianisches balance

An equilibrium is a state with

Here, the excess demand vector of the i-th consumer. The sign of the k- th component of this vector indicates whether the i-th consumer buys or sells the property in question: Does he want to consume more of good k when he initially has - and therefore the difference must buy; other hand, applies, he wants to consume less than he initially has - and the difference is therefore sell.

The equilibrium condition ( iv ) states so that for each good, the aggregate excess demand of all consumers must be the aggregate excess supply of all companies equally. If it is not met, the consumption or production plans of consumers and businesses can not all be realized simultaneously since then deviates for some goods the aggregate demand from aggregate supply.

Properties of the Walrasian equilibrium

The central point of the Arrow- Debreu general equilibrium model is to study its equilibrium. Hiebei is particularly the existence and efficiency of this state interest.

Conditions of existence

An equilibrium exists in economics, if

( 1) for the consumer applies

(2) the producer applies

Importance of the conditions of existence

These conditions are not all obvious or just technically. Especially (1) (iv) is a problem, even though they can be attenuated. However, all assumptions to consumers are actually relatively problematic, while the assumptions appear more natural to the producers. This shows that the beautiful results that a Partialmarktmodell or a generalized Partialmarktmodell for the existence of an equilibrium allocation provides in a general equilibrium model may no longer be inevitable. This statement is quite severe, but she says that even if all players act rationally and perfectly informed in an economy without transaction costs or other frictions, there may be a situation in which there is no balance.

However, it must be noted that the above conditions is only a sufficient condition for the existence of a general equilibrium. From the violation of some of the points so it can not be concluded that the non-existence. In addition, some of the conditions of existence can be mitigated. However, this list is the most frequently encountered.

Walras ' Law

If consumers are unsaturated, they use their budget completely for consumption purposes, so that must be true:

.

Aggregated you this " budget equation " of all consumers, the result for any price vector p:

This equation requires only the condition ( ii ) and thus applies not only in the Walrasian equilibrium. It says that the value of the aggregate (across all consumers and businesses ) excess demand must always be zero.

Fundamental theorems of welfare economics

If all consumers are unsaturated and there is a Walrasian equilibrium, then this equilibrium is Pareto- efficient.

If an allocation is Pareto- efficient and the conditions of existence are secured there is an income allocation for which is a Walrasian equilibrium.

Uniqueness and stability of equilibrium

The questions of uniqueness and stability of equilibrium have typically not been studied in the Arrow- Debreu model, but under the restrictive assumption that the respective optimization problems of consumers and businesses have a unique solution and can therefore be described by an excess demand function the economy.

Other equilibrium models

• Partialmarktmodell
• Generalized Partialmarktmodell
• General equilibrium model
• Edgeworth box model
• Robinson Crusoe model
• Walrasianisches equilibrium model