Arrow-Debreu equilibrium model

The Arrow-Debreu equilibrium model (also: Arrow-Debreu-McKenzie model ) is a microeconomic model of the entire economy. It is named after Gérard Debreu and Kenneth Arrow as well as Lionel W. McKenzie , represents a further development of the Walrasian equilibrium model developed by Léon Walras and examines an overall economic equilibrium.

The model extends the general equilibrium model to include uncertain expectations and state-dependent quantities and is therefore of great importance for finance theory. It shows that in a market economy, under idealizing conditions, it is not possible to make someone better without making someone else worse off. In short, a market equilibrium is a Pareto optimum .

General

content

General equilibrium models consider market economies in which all consumers and producers act rationally. They describe how consumers and producers simultaneously choose offers and requests, taking into account their budget or technological restrictions. In contrast to partial models, which only analyze individual markets, general equilibrium models characterize macroeconomic allocations in which all markets are cleared at the same time.

history

The first attempt in neoclassical theory to develop a comprehensive model for determining the relative prices in an economy came from Léon Walras , the founder of the Lausanne School . He wanted to turn the classical economics of Adam Smith and David Ricardo into an "exact science". So he tried to describe the economy mathematically . Abraham Wald and later Maurice Allais , Kenneth Arrow and Gérard Debreu described the existence and stability of a general equilibrium for a market economy with private property . Arrow, Allais and Debreu received the Nobel Prize in Economics for their work on general equilibrium theory (AGT) .

Description of the economy

Components

Consider an economy made up of n markets. In this there are I consumers and J companies, whereby the index quantities (the quantity of all consumers) and (the quantity of all producers) are defined for these two groups . Consumers and producers are now considered one after the other, then the initial equipment of the economy: ${\displaystyle {\mathcal {I}}=\{1,\ldots ,I\}}$ ${\displaystyle {\mathcal {J}}=\{1,\ldots ,J\}}$

• The consumption possibility set of a consumer is with , i.e. the set of all consumption bundles possible for i . His preferences are characterized by the order of preferences . (Such includes ordered pairs using , for which it holds that of i weak against is prefers.) The consumer sector thereof can by the sequence on the basis will be described. ${\displaystyle i\in {\mathcal {I}}}$ ${\displaystyle X_{i}\subset \mathbb {R} _{+}^{n}}$ ${\displaystyle X_{i}\neq \emptyset }$ ${\displaystyle \mathbf {x} ^{i}=(x_{1}^{i},\ldots ,x_{n}^{i})}$ ${\displaystyle \succsim _{i}}$ ${\displaystyle ({\hat {\mathbf {x} }}^{i},{\tilde {\mathbf {x} }}^{i})}$ ${\displaystyle {\hat {\mathbf {x} }}^{i},{\tilde {\mathbf {x} }}^{i}\in X_{i}}$ ${\displaystyle {\hat {\mathbf {x} }}^{i}}$ ${\displaystyle {\tilde {\mathbf {x} }}^{i}}$ ${\displaystyle \left\{(X_{i},\succsim _{i})\right\}_{i\in {\mathcal {I}}}}$
• A company's output capacity is . It includes all possible production plans . The sign of each component of is interpreted as follows: ${\displaystyle j\in {\mathcal {J}}}$ ${\displaystyle Y_{j}\subseteq \mathbb {R} ^{n}}$ ${\displaystyle \mathbf {y} ^{j}=(y_{1}^{j},\ldots ,y_{n}^{j})}$ ${\displaystyle y_{k}^{j}}$ ${\displaystyle \mathbf {y} ^{j}}$

• ${\displaystyle y_{k}^{j}<0}$ : Producer j uses product k as input (e.g. work performance, raw materials)
• ${\displaystyle y_{k}^{j}>0}$ : Producer j produces product k as output (e.g. consumer good)

The production sector can therefore be characterized by a sequence . ${\displaystyle \left\{Y_{j}\right\}_{j\in {\mathcal {J}}}}$

• The initial configuration of the economy describes which or how many resources are available to the economy at the beginning of the analysis. It is given by the equipment vector (resource vector) . In addition, you agree to equip one person (with regard to all products). ${\displaystyle \mathbf {e} =(e_{1},\ldots ,e_{n})\in \mathbb {R} _{+}^{n}}$ ${\displaystyle \mathbf {e} ^{i}=(e_{1}^{i},\ldots ,e_{n}^{i})\in \mathbb {R} _{+}^{n}}$ ${\displaystyle i\in {\mathcal {I}}}$

Overall economy

The entire economy can consequently be described as a tuple in the Arrow-Debreu model

${\displaystyle \mathbf {\mathcal {E}} =\left[\left\{(X_{i},\succsim _{i})\right\}_{i\in {\mathcal {I}}},\left\{Y_{j}\right\}_{j\in {\mathcal {J}}},\mathbf {e} \right]}$

describe.

A common specification of this economy is a private property economy

${\displaystyle \mathbf {\mathcal {E}} ^{p}=\left[\left\{(X_{i},\succsim _{i})\right\}_{i\in {\mathcal {I}}},\left\{Y_{j}\right\}_{j\in {\mathcal {J}}},\left\{\left(\mathbf {e} ^{i},\mathbf {\theta } ^{i}\right)\right\}_{i\in {\mathcal {I}}}\right]}$

This is a competitive system in which all companies (and their profits) are privately owned, which means that profits are part of the aggregate consumption budget. Since it is a competitive economy, goods are also traded on a decentralized basis on competitive markets, with the market players acting as price takers: Consumers maximize their benefits, producers their profits. The assumption of private property results formally that the consumer's budget is made up of two components: on the one hand, a share of the initial equipment, and on the other hand, a share of the producers' profits. This share amounts to ( would be, for example, the share that person i can claim in the profits of producer 4). According to the requirements is and . ${\displaystyle \mathbf {e} ^{i}\in \mathbb {R} ^{n}}$ ${\displaystyle \mathbf {\theta } ^{i}\in \mathbb {R} ^{J}}$ ${\displaystyle \theta ^{i}=(\theta ^{i1},\ldots ,\theta ^{iJ})}$ ${\displaystyle \theta ^{i4}}$ ${\displaystyle \mathbf {e} =\sum _{i\in {\mathcal {I}}}\mathbf {e} ^{i}}$ ${\displaystyle \sum _{i\in {\mathcal {I}}}\mathbf {\theta } ^{i}=\mathbf {1} }$

The Walrasian balance

Economy with perfect competition

A competitive economy with private property (and perfect competition) has a central price vector that indicates the price of each product. On this basis, every consumer can only consume within the framework of a limited budget ( budget restriction ). In a state of equilibrium, the budget constraint must be respected. ${\displaystyle \mathbf {p} =(p_{1},\ldots ,p_{n})}$

In addition, optimality conditions must be met in equilibrium on both the producer and consumer side . Every consumer must - while maintaining his budget constraints and given the price vector of the economy - choose precisely such a consumption plan for which it is considered to be weakly preferred to any other possible consumption plan. And every producer must follow the maxim of profit maximization, that is, it must apply to every producer that the chosen production plan - given the prices in the economy - is profit-maximizing. (The Arrow-Debreu model does not assume that the optimization problems of consumers and companies always have to have clear solutions.) ${\displaystyle i\in {\mathcal {I}}}$

Finally, the equilibrium allocation must be permissible , in the following sense: If one looks at a competitive economy with private property (and perfect competition), then a concrete “state” of (with specific consumption and production vectors for each consumer or producer) is through one -Allocation vector given. Such an allocation is said to be permissible if it applies to every resource that the total amount consumed corresponds to the initial equipment plus the total amount produced, i.e. if ${\displaystyle \mathbf {\mathcal {E}} ^{p}}$ ${\displaystyle n(I+J)}$ ${\displaystyle \left[\left(\mathbf {x} ^{i}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j}\right)_{j\in {\mathcal {J}}}\right]}$

${\displaystyle \sum _{i\in {\mathcal {I}}}\mathbf {x} ^{i}=\mathbf {e} +\sum _{j\in {\mathcal {J}}}\mathbf {y} ^{j}}$.

Walrasian balance

For the competitive economy with private property , a competitive equilibrium is defined as a tuple ${\displaystyle \mathbf {\mathcal {E}} ^{p}}$

${\displaystyle \left[\left(\mathbf {x} ^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}},\mathbf {p} \right]}$

with the following properties:

1. Each person maximizes their utility, given the equilibrium market prices and their consumption budget. More precisely: Let be the set of all consumption vectors that meet the budget condition: ${\displaystyle {\mathcal {X}}}$ ${\displaystyle \mathbf {x} ^{i}}$

   ${\displaystyle {\mathcal {X}}=\left\{\mathbf {x} ^{i}\left|\mathbf {p} \cdot \mathbf {x} ^{i}\leq \mathbf {p} \cdot \mathbf {e} ^{i}+\sum _{j=1}^{J}\theta ^{ij}\cdot \mathbf {p} \cdot \mathbf {y} ^{j*}\right.\right\}}$

Then it is and it applies: for everyone . ${\displaystyle \mathbf {x} ^{i*}\in {\mathcal {X}}}$ ${\displaystyle \mathbf {x} ^{i*}\succsim _{i}\mathbf {x} ^{i}}$ ${\displaystyle \mathbf {x} ^{i}\in {\mathcal {X}}}$

1. Given the equilibrium market prices, every company maximizes its profit, that is, it applies to all : to all . ${\displaystyle j\in {\mathcal {J}}}$ ${\displaystyle \mathbf {p} \cdot \mathbf {y} ^{j}\leq \mathbf {p} \cdot \mathbf {y} ^{j*}}$ ${\displaystyle \mathbf {y} ^{j}\in Y_{j}}$
2. The allocation is permitted in. ${\displaystyle \left[\left(\mathbf {x} ^{i*}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j*}\right)_{j\in {\mathcal {J}}}\right]}$ ${\displaystyle \mathbf {\mathcal {E}} ^{p}}$

Such a balance is known as Walrasian balance.

An alternative formulation for the admissibility condition (3.) is common: Obviously, this can be expressed alternatively by means of the individual initial configuration introduced above by means of excess demand . One denotes with

${\displaystyle \mathbf {z} \equiv \sum _{i\in {\mathcal {I}}}\left(\mathbf {x} ^{i}-\mathbf {e} ^{i}\right)}$

the aggregate excess demand of the economy. An allocation is therefore permitted if and only if

${\displaystyle \mathbf {z} =\sum _{j\in {\mathcal {J}}}\mathbf {y} ^{j}}$,

that is, if for each good the aggregated excess demand of all consumers corresponds to the aggregated excess supply of all companies. If this condition is not met, the consumption or production plans of consumers and companies cannot all be implemented at the same time, as the aggregated demand for some goods then deviates from the aggregated supply. Note that household-specific excess demand can have positive or negative components. The sign of the k th component of this vector indicates whether the ( i th) consumer in question buys or sells the product in question: It applies , then i wants to consume more of k than he initially has - and must therefore buy the difference; on the other hand , if he wants to consume less than he initially owns, he will sell the difference. ${\displaystyle \mathbf {x} ^{i}-\mathbf {e} ^{i}}$ ${\displaystyle x_{k}^{i*}>e_{k}^{i*}}$ ${\displaystyle x_{k}^{i*}<e_{k}^{i*}}$

Properties, implications, and existence of Walrasian equilibrium

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

Walras Law

In the equilibrium of an economy with locally unsaturated consumers, Walras law applies to the entire economy. (It refers to an individual preference order on a not saturated locally, if for any and for each environment to one exists, so of i strictly opposite is prefers, so See the article.. Order of preference . ) That is, it is true: ${\displaystyle \succsim _{i}}$ ${\displaystyle X_{i}}$ ${\displaystyle {\hat {\mathbf {x} }}^{i}\in X_{i}}$ ${\displaystyle \epsilon }$ ${\displaystyle N_{\epsilon }({\hat {\mathbf {x} }}^{i})}$ ${\displaystyle {\hat {\mathbf {x} }}^{i}}$ ${\displaystyle {\tilde {\mathbf {x} }}^{i}\in N_{\epsilon }({\hat {\mathbf {x} }}^{i})}$ ${\displaystyle {\tilde {\mathbf {x} }}^{i}}$ ${\displaystyle {\hat {\mathbf {x} }}^{i}}$ ${\displaystyle {\tilde {\mathbf {x} }}^{i}\succ _{i}{\hat {\mathbf {x} }}^{i}}$

${\displaystyle \mathbf {p} \cdot \left(\sum _{j\in {\mathcal {J}}}\mathbf {y} ^{j}\right)-\left[\mathbf {p} \cdot \left(\sum _{i\in {\mathcal {I}}}\mathbf {x} ^{i}\right)-\mathbf {p} \cdot \left(\sum _{i\in {\mathcal {I}}}\mathbf {e} ^{i}\right)\right]=0}$

This means that the value of the aggregated excess demand (across all consumers and companies) must always be zero.

Conditions of existence

There are a number of theorems of existence for the existence of such an equilibrium. In the following an existence theorem based on Arrow and Debreu (1954) is presented.

Existence of an equilibrium: Consider an economy in the sense defined above, and let the following requirements be met: ${\displaystyle \mathbf {\mathcal {E}} }$

(1) The following applies to all consumers : ${\displaystyle i\in {\mathcal {I}}}$

(a) is a compact and convex subset of the ; ${\displaystyle X_{i}}$ ${\displaystyle \mathbb {R} _{+}^{n}}$

(b) is an interior point of ; ${\displaystyle \mathbf {e} ^{i}}$ ${\displaystyle X_{i}}$

(c) is continuous and convex. ${\displaystyle \succeq _{i}}$

(2) The following applies to all companies : ${\displaystyle j\in {\mathcal {J}}}$

(a) is compact and convex; ${\displaystyle Y_{j}}$

(b) . ${\displaystyle \mathbf {0} \in Y_{j}}$

Then have a Walrasian balance. ${\displaystyle \mathbf {\mathcal {E}} }$

Meaning of the conditions of existence

These conditions are by no means all obvious or only purely technical. (1) (b) is particularly problematic, although it can be mitigated; this also applies to the compactness requirement of . The assumptions about the producers tend to appear more natural. ${\displaystyle X_{i}}$

It should be remembered that the above condition is only a sufficient condition for the existence of a general equilibrium. The non-existence cannot therefore be concluded from the violation of some of the points. In addition, some of the conditions of existence can be weakened as mentioned.

Principles of Welfare Economy

→ Main article : Welfare theorems

1. Law of Welfare Economy

If the individual orders of preference of all consumers are locally unsaturated and there is a Walras equilibrium, then this equilibrium is also Pareto-efficient.

2. Law of Welfare Economy

When an allocation

${\displaystyle \left[\left(\mathbf {x} ^{i}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j}\right)_{j\in {\mathcal {J}}}\right]}$

Pareto efficient and some other conditions are met, then there is a price vector and a transfer scheme such that there is a Walras equilibrium (with transfers). ${\displaystyle \mathbf {p} \neq \mathbf {0} }$ ${\displaystyle \left[\left(\mathbf {x} ^{i}\right)_{i\in {\mathcal {I}}},\left(\mathbf {y} ^{j}\right)_{j\in {\mathcal {J}}},\mathbf {p} \right]}$

Uniqueness and stability of equilibrium

The questions about the uniqueness and stability of the equilibrium have typically not been examined in the Arrow-Debreu model, but under the limiting assumption that the respective optimization problems of consumers and companies have a clear solution and that the economy can therefore be described by a surplus demand function. The Sonnenschein-Mantel-Debreu theorem states that these functions have certain general properties, but otherwise no concrete statements about their shape are possible. If there is a heterogeneity in factor endowments and preferences, no clear balance is guaranteed.

Other equilibrium models

• Partial market model
• Generalized partial market model
• General equilibrium model
• Edgeworth box model
• Robinson Crusoe model
• Walrasian equilibrium model

literature

• Kenneth J. Arrow, Frank Hahn: General Competitive Analysis. North Holland, 1971, ISBN 0-444-85497-5 .
• William DA Bryant: General equilibrium. Theory and evidence. World Scientific, Hackensack 2010, ISBN 978-981-281-834-8 (e-book: ISBN 978-981-281-835-5 ).
• Gerard Debreu : Theory of Value. An Axiomatic Analysis of Economic Equilibrium. Yale University Press, New Haven and London 1959.
• Gerard Debreu: Existence of Competitive Equilibrium. In: Kenneth J. Arrow, Michael D. Intrilligator (Eds.): Handbook of Mathematical Economics. Volume 2. North Holland, Amsterdam 1982, ISBN 978-0-444-86127-6 , pp. 697-743, doi: 10.1016 / S1573-4382 (82) 02010-4 .
• David M. Kreps: Microeconomic Foundations I. Choice and Competitive Markets. Princeton University Press, Princeton 2012, ISBN 978-0-691-15583-8 .
• Andreu Mas-Colell, Michael Whinston, Jerry Green: Microeconomic Theory. Oxford University Press, Oxford 1995, ISBN 0-19-507340-1 .
• James C. Moore: General equilibrium and welfare economics. An introduction. Springer, Berlin a. a. 2007, ISBN 978-3-540-31407-3 , doi: 10.1007 / 978-3-540-32223-8 .

Individual evidence

1. ↑ On this, for example, Debreu 1982; in detail Bryant 2010, Chapter 2.
2. ↑ Kenneth J. Arrow, Gerard Debreu: Existence of an equilibrium for a competitive economy. In: Econometrica. 22, No. 3, 1954, pp. 265-290, JSTOR 1907353 .
3. ↑ Cf., also on evidence, Kreps 2012, p. 342 ff.
4. ↑ refers to a binary relation B to X as continuous if the quantities (upper contour quantity) and (lower contour amount) for all completed with respect to X are. ${\displaystyle \{\mathbf {x} _{a}\in X|\mathbf {x} _{a}B\mathbf {x} _{b}\}}$ ${\displaystyle \{\mathbf {x} _{a}\in X|\mathbf {x} _{b}B\mathbf {x} _{a}\}}$ ${\displaystyle \mathbf {x} _{b}\in X}$
5. ↑ See Arrow and Hahn 1971.
6. ^ Wolfram Elsner, Torsten Heinrich, Henning Schwardt: The Microeconomics of Complex Economies . Academic Press, 2015, ISBN 978-0-12-411585-9 , pp. 115-117.