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In theoretical economics, an abstract economy (also called a generalized N-person game) is a model that generalizes both the standard model of an exchange economy in microeconomics, and the standard model of a game in game theory. An equilibrium in an abstract economy generalizes both a Walrasian equilibrium in microeconomics, and a Nash equilibrium in game-theory.

The concept was introduced by Gérard Debreu in 1952. He named it generalized N-person game, and proved the existence of equilibrium in this game. Later, Debreu and Kenneth Arrow (who renamed the concept to abstract economy) used this existence result to prove the existence of a Walrasian equilibrium (aka competitive equilibrium) in the Arrow–Debreu model. Later, Shafer and Sonnenschein extended both theorems to irrational agents - agents with non-transitive and non-complete preferences.

Abstract economy with utility functions

The general case

Definition

In the model of Debreu, an abstract economy contains a finite number N of agents. For each agent ${\displaystyle i}$, there is:

• A choice-set ${\displaystyle X_{i}}$ (a subset of some Euclidean space ${\displaystyle \mathbb {R} ^{l}}$). This represents the global set of choices that the agent can make.
  • We define the cartesian product of all choice sets as: ${\displaystyle X:=\prod _{j=1}^{N}X_{j}}$.
• An action-correspondence ${\displaystyle A_{i}:X\twoheadrightarrow X_{i}}$. This represents the set of possible actions the agent can take, given the choices of the other agents.
• A utility function: ${\displaystyle U_{i}:X\to \mathbb {R} }$, representing the utility that the agent receives from each combination of choices.

The goal of each agent is to choose an action that maximizes his utility.

Equilibrium

An equilibrium in an abstract economy is a vector of choices, ${\displaystyle x=(x_{1},\ldots ,x_{N})=(x_{i},x_{-i})}$, such that, for each agent ${\displaystyle i}$, the action ${\displaystyle x_{i}}$ maximizes the function ${\displaystyle U_{i}(\cdot ,x_{-i})}$ subject to the constraint ${\displaystyle x_{i}\in A_{i}(x)}$:

${\displaystyle U_{i}(x_{i},x_{-i})=\max _{x_{i}'\in A_{i}(x)}U_{i}(x_{i}',x_{-i})}$

Equivalently, for each agent ${\displaystyle i}$, there is no action ${\displaystyle x_{i}'\in A_{i}(x)}$ such that:

${\displaystyle U_{i}(x_{i}',x_{-i})>U_{i}(x_{i},x_{-i})}$

The following conditions are sufficient for the existence of equilibrium:

• Each choice-set ${\displaystyle X_{i}}$ is compact, non-empty and convex.
• Each action-correspondence ${\displaystyle A_{i}}$ is continuous, and its values are non-empty and convex.
• Each utility function ${\displaystyle U_{i}}$ is continuous in ${\displaystyle x}$ and quasi-concave in ${\displaystyle x_{i}}$.

The continuity conditions on the utility functions can be weakened as follows:

• Each utility function ${\displaystyle U_{i}}$ is quasi-concave in ${\displaystyle x_{i}}$, upper semi-continuous in ${\displaystyle x}$, and graph continuous.

Another weakening, which does not use graph-continuity, is:

• Each utility function ${\displaystyle U_{i}}$ is quasi-concave in ${\displaystyle x_{i}}$, upper semi-continuous in ${\displaystyle x}$, and the function ${\displaystyle W_{i}(x_{-i}):=\max _{x_{i}}U_{i}(x_{i},x_{-i})}$ [which is defined since ${\displaystyle U_{i}}$ is upper semi-continuous] is lower semi-continuous.

The proofs use the Kakutani fixed point theorem.

Exchange economy as a special case

Definition

An exchange economy is a system with N-1 consumers and ${\displaystyle l}$ homogeneous divisible goods. For each consumer i, there is:

• A consumption-set ${\displaystyle Y_{i}}$ (a subset of ${\displaystyle \mathbb {R} ^{l}}$). This represents the set of bundles that the agent can consume.
  • We define the cartesian product of all consumption sets as: ${\displaystyle Y:=\prod _{j=1}^{N}Y_{j}}$.
• An initial endowment vector ${\displaystyle w_{i}\in \mathbb {R} _{+}^{l}.}$
• A utility function ${\displaystyle V_{i}:Y_{i}\to \mathbb {R} }$. This represents the preferences of the agent. Note that the utility of a consumer depends only on his own consumption, rather than on the entire allocation.

Define the set of possible price-vectors as: ${\displaystyle \Delta :=\{p\in \mathbb {R} _{+}^{l}|\sum _{i=1}^{l}p_{i}=1\}}$.

Equilibrium

A Walrasian equilibrium (aka competitive equilibrium) in an exchange economy is a vector of consumption-bundles and a price-vector, ${\displaystyle (y_{1},\ldots ,y_{N-1},p)}$, such that:

• The total consumption is at most the total endowment: ${\displaystyle \sum y_{i}\leq \sum w_{i}}$.
• The total expense of each agent is at most his budget: ${\displaystyle p\cdot y_{i}\leq p\cdot w_{i}}$.
• For each agent ${\displaystyle i}$, the consumption ${\displaystyle y_{i}}$ maximizes the function ${\displaystyle V_{i}(\cdot )}$ subject to the constraint ${\displaystyle p\cdot y_{i}\leq p\cdot w_{i}}$. I.e, if ${\displaystyle V_{i}(z)>V_{i}(y_{i})}$, then ${\displaystyle p\cdot z>p\cdot w_{i}\geq p\cdot y_{i}}$.

Reduction to abstract economy

Arrow and Debreu presented the following reduction from exchange economy to abstract economy.

Given an (N-1)-agent exchange economy, we define an N-agent abstract economy by adding a special agent called the market maker or market player. The "consumption" of this special player is denoted by p. The components of the abstract economy are defined as follows:

• Each of the first N-1 agents has choice set ${\displaystyle X_{i}=Y_{i}}$, utility function ${\displaystyle U_{i}=V_{i}}$, and action set defined by his budget: ${\displaystyle A_{i}(y,p)=\{y_{i}\in Y_{i}|py_{i}\leq pw_{i}\}}$.
• The market player has a choice set ${\displaystyle X_{N}:=\Delta }$(the set of possible price-vectors), utility function ${\displaystyle U_{N}(y,p):=p\cdot (\sum y_{i}-\sum w_{i})}$, and action set defined by ${\displaystyle A_{N}(y,p)\equiv \Delta }$.

Intuitively, the market player chooses the price in a way that balances supply and demand: for commodities with more supply than demand, the right-hand term in ${\displaystyle U_{N}(y,p)}$ is negative so the market player chooses a low price; for commodities with more demand than supply, the term is positive so the market player chooses a high price.

The following conditions in the exchange economy are sufficient to guarantee that the abstract economy satisfies the conditions for equilibrium:

• Each consumption-set ${\displaystyle Y_{i}}$ is compact and convex, and contains the endowment ${\displaystyle w_{i}}$ in its interior.
• Each utility function ${\displaystyle V_{i}}$ is continuous and quasi-concave.

Moreover, the following additional condition is sufficient to guarantee that the equilibrium ${\displaystyle y}$ in the abstract economy corresponds to a competitive equilibrium in the exchange economy:

• For every agent i, ${\displaystyle y_{i}}$ is not a local (unconstrained) maximum of ${\displaystyle V_{i}}$. For example, it is sufficient to assume that all agents are not satiated.

The definition ${\displaystyle A_{i}(y,p)=\{y_{i}\in Y_{i}|py_{i}\leq pw_{i}\}}$ guarantees that the total expense of each agent is at most his budget. The definition ${\displaystyle U_{i}=V_{i}}$ guarantees that the consumption of each agent maximizes his utility given the budget. And the definition ${\displaystyle U_{N}(y,p):=p\cdot (\sum y_{i}-\sum w_{i})}$ guarantees that the total consumption equals the total endowment.

Therefore, if the exchange economy satisfies the above three conditions, a competitive equilibrium exists.

In the proof we assumed that ${\displaystyle V_{i}}$ depends only on ${\displaystyle y_{i}}$, but this assumption is not really needed: the proof remains valid even if the utility depends on the consumptions of other agents (externalities), or on the prices.

Abstract economy with preference correspondences

The general case

Definition

In the generalized model of Shafer and Sonnenschein, For each agent ${\displaystyle i}$ there is:

• A choice-set ${\displaystyle X_{i}}$ - as above;
• A constraint correspondence ${\displaystyle A_{i}:X\twoheadrightarrow X_{i}}$ - as above;
• A preference correspondence ${\displaystyle P_{i}:X\twoheadrightarrow X_{i}}$. This represents, for each combination of choices of the other agents, what choices the agent strictly prefers to his current choice.

The model of Debreu is a special case of this model, in which the preference correspondences are defined based on utility functions: ${\displaystyle P_{i}(x):=\{z_{i}\in X_{i}:U_{i}(z_{i},x_{-i})>U_{i}(x_{i},x_{-i})\}}$. However, the generalized model does not require that the preference-correspondence can be represented by a utility function. In particular, it does not have to correspond to a transitive relation.

Equilibrium

An equilibrium in a generalized abstract economy is a vector of choices, ${\displaystyle x=(x_{1},\ldots ,x_{N})=(x_{i},x_{-i})}$, such that, for each agent ${\displaystyle i}$, ${\displaystyle x_{i}\in A_{i}(x)}$ and ${\displaystyle P_{i}(x)\cap A_{i}(x)=\emptyset }$. The equilibrium concept of Debreu is a special case of this equilibrium.

The following conditions are sufficient for the existence of equilibrium in the generalized abstract economy:

• (a) Each choice-set ${\displaystyle X_{i}}$ is compact, non-empty and convex.
• (b') Each action-correspondence ${\displaystyle A_{i}}$ is continuous.
• (b'') The values ${\displaystyle A_{i}(x)}$ are non-empty and convex for every x.
• (c') Each preference-correspondence ${\displaystyle P_{i}}$ has an open graph in ${\displaystyle X\times X_{i}}$ (this is a form of continuity condition).
• (c'') For each ${\displaystyle x\in X}$, the convex hull of ${\displaystyle P_{i}(x)}$ does not contain ${\displaystyle x_{i}}$ (this is a form of non-reflexivity condition: an agent does not strictly prefer a choice to itself).

Exchange economy as a special case

Definition

Mas-Colell generalized the definition of exchange economy in the following way. For every consumer i, there is:

• A consumption-set ${\displaystyle Y_{i}}$ - as above;
• An initial endowment vector ${\displaystyle w_{i}\in \mathbb {R} _{+}^{l}}$ - as above;
• A preference relation ${\displaystyle \prec _{i}}$ that can be equivalently represented by a preference-correspondence ${\displaystyle P_{i}:Y_{i}\twoheadrightarrow Y_{i}}$, that depends only on the consumed bundle: ${\displaystyle P_{i}(y_{i}):=\{z_{i}\in Y_{i}|z_{i}\succ _{i}y_{i}\}}$. Note the preference relation is not required to be complete or transitive.

Equilibrium

A competitive equilibrium in such exchange economy is defined by a price-vector p and an allocation y such that:

• The sum of all prices is 1;
• The sum of all allocations ${\displaystyle y_{i}}$ is at most the sum of endowments ${\displaystyle w_{i}}$;
• For every i: ${\displaystyle p\cdot y_{i}=p\cdot w_{i}}$;
• For every bundle z: if ${\displaystyle z\succ _{i}y_{i}}$ then ${\displaystyle p\cdot z>p\cdot y_{i}}$ (i.e., if the agent strictly prefers z to his share, then the agent cannot afford z).

Reduction to abstract economy

The "market maker" reduction shown above, from the exchange economy of Arrow-Debreu to the abstract economy of Debreu, can be done from the generalized exchange economy of Mas-Collel to the generalized abstract economy of Shafer-Sonnenschein. This reduction implies that the following conditions are sufficient for existence of competitive equilibrium in the generalized exchange economy:

• Each ${\displaystyle \prec _{i}}$ is relatively-open (equivalently, each ${\displaystyle P_{i}}$ has an open graph);
• For every bundle x, the set ${\displaystyle P_{i}(x)}$ is convex and does not contain x (= irreflexivity). Mas-Collel added the condition that the set ${\displaystyle P_{i}(x)}$ is non-empty (= non-saturation).
• For every i: ${\displaystyle w_{i}\gg x_{i}}$ for some bundle x (this means that the initial endowment is in the interior of the choice-sets).

A negative example

The following example shows that, when the open graph property does not hold, equilibrium may fail to exist.

There is an economy with two goods, say apples and bananas. There are two agents with identical endowments (1,1). They have identical preferences, based on lexicographic ordering: for every vector ${\displaystyle y_{i}=(a_{i},b_{i})}$ of ${\displaystyle a_{i}}$ apples and ${\displaystyle b_{i}}$ bananas, the set ${\displaystyle P_{i}(a_{i},b_{i}):=\{(a_{i}',b_{i}')|(a_{i}'>a_{i})~or~(a_{i}'=a_{i}~and~b_{i}'>b_{i})\}}$, i.e., each agent wants as many apples as possible, and subject to that, as many bananas as possible. Note that ${\displaystyle P_{i}(a_{i},b_{i})}$ represents a complete and transitive relation, but it does not have an open graph.

This economy does not have an equilibrium. Suppose by contradiction that an equilibrium exists. Then the allocation of each agent must be lexicographically at least (1,1). But this means that the allocations of both agents must be exactly (1,1). Now there are two cases: if the price of bananas is 0, then both agents can afford the bundle (1,2) which is strictly better than their allocation. If the price of bananas is some p > 0 (where the price of apples is normalized to 1), then both agents can afford the bundle (1+p, 0), which is strictly better than their allocation. In both cases it cannot be an equilibrium price.

Welfare theorems in abstract economies

Fon and Otani study extensions of welfare theorems to the generalized exchange economy of Mas-Collel. They make the following assumptions:

• Each consumption-set ${\displaystyle Y_{i}}$ is non-empty, convex, closed, and bounded below.
• The preference correspondence is non-empty: ${\displaystyle P_{i}(y_{i})\neq \emptyset }$ (this is a non-saturation condition).

A competitive equilibrium is a price-vector ${\displaystyle \mathbf {p} }$ and an allocation ${\displaystyle \mathbf {y} }$ such that:

• Feasibility: the sum of all allocations ${\displaystyle y_{i}}$ equals the sum of endowments ${\displaystyle w_{i}}$ (there is no free disposal);
• Budget: for every i, ${\displaystyle p\cdot y_{i}\leq p\cdot w_{i}}$;
• Preference: For every i, ${\displaystyle P_{i}(y_{i})\cap B_{i}(p,w_{i})=\emptyset }$, where ${\displaystyle B_{i}(p,w_{i})}$ is the budget-set of i. In other words, for every bundle ${\displaystyle z\in Y_{i}}$: if ${\displaystyle z\succ _{i}y_{i}}$ then ${\displaystyle p\cdot z>p\cdot y_{i}}$ (if the agent strictly prefers z to his share, then the agent cannot afford z).

A compensated equilibrium has the same feasibility and budget conditions, but instead of the preference condition, it satisfies:

• Compensated Preference: For every i and for every bundle ${\displaystyle z\in Y_{i}}$: if ${\displaystyle z\succ _{i}y_{i}}$ then ${\displaystyle p\cdot z\geq p\cdot y_{i}}$.

A Pareto-optimal allocation is, as usual, an allocation without a Pareto-improvement. A Pareto-improvement of an allocation ${\displaystyle \mathbf {y} }$ is defined as another allocation ${\displaystyle \mathbf {y'} }$ that is strictly better for a subset ${\displaystyle J}$ of the agents, and remains the same allocation for all other agents. That is:

• ${\displaystyle \sum _{i\in J}y'_{i}=\sum _{i\in J}y_{i}.}$
• ${\displaystyle y'_{i}\in P_{i}(y_{i})}$ for all ${\displaystyle i\in J}$.

Note that this definition is weaker than the usual definition of Pareto-optimality (the usual definition does not require that the bundles of other agents remain the same - only that their utility remains the same).

Fon and Otani prove the following theorems.

• Every competitive equilibrium is Pareto-optimal.
• Under certain conditions on the preferences, for every Pareto-optimal allocation, there exists a price-vector with which it is a compensated equilibrium.

See also

A further generalization of these equilibrium concepts for a general model without ordered preferences can be found in Barabolla (1985).

References

1. ^ Debreu, Gerard (1952-10-01). "A Social Equilibrium Existence Theorem". Proceedings of the National Academy of Sciences. 38 (10): 886–893. Bibcode:1952PNAS...38..886D. doi:10.1073/pnas.38.10.886. ISSN 0027-8424. PMC 1063675. PMID 16589195.
2. ^ Arrow, Kenneth J.; Debreu, Gerard (1954). "Existence of an Equilibrium for a Competitive Economy". Econometrica. 22 (3): 265–290. doi:10.2307/1907353. JSTOR 1907353.
3. ^ Shafer, Wayne; Sonnenschein, Hugo (1975-12-01). "Equilibrium in abstract economies without ordered preferences" (PDF). Journal of Mathematical Economics. 2 (3): 345–348. doi:10.1016/0304-4068(75)90002-6. hdl:10419/220454. ISSN 0304-4068.
4. https://arxived.org/pdfs/files/425be847d31d3cc9665d00b8f0585038.pdf
5. Glicksberg, I. L. (1952-02-01). "A Further Generalization of the Kakutani Fixed Point Theorem, with Application to Nash Equilibrium Points". Proceedings of the American Mathematical Society. 3 (1): 170–174. doi:10.2307/2032478. ISSN 0002-9939. JSTOR 2032478.
6. Fan, K. (1952-02-01). "Fixed-point and Minimax Theorems in Locally Convex Topological Linear Spaces". Proceedings of the National Academy of Sciences. 38 (2): 121–126. Bibcode:1952PNAS...38..121F. doi:10.1073/pnas.38.2.121. ISSN 0027-8424. PMC 1063516. PMID 16589065.
7. ^ Dasgupta, Partha; Maskin, Eric (1986-01-01). "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory". The Review of Economic Studies. 53 (1): 1–26. doi:10.2307/2297588. ISSN 0034-6527. JSTOR 2297588.
8. Mas-Collel, Andrew (1974-12-01). "An equilibrium existence theorem without complete or transitive preferences". Journal of Mathematical Economics. 1 (3): 237–246. doi:10.1016/0304-4068(74)90015-9. ISSN 0304-4068.
9. ^ Vincy Fon and Yoshihiko Otani (1979-06-01). "Classical welfare theorems with non-transitive and non-complete preferences". Journal of Economic Theory. 20 (3): 409–418. doi:10.1016/0022-0531(79)90044-9. ISSN 0022-0531.
10. Barbolla, Rosa (1985-01-01). "Equilibrium and quasi-equilibrium existence theorems for a general model without ordered preferences". Economics Letters. 19 (3): 205–210. doi:10.1016/0165-1765(85)90022-9. ISSN 0165-1765.

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• General equilibrium theory
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