Abstract economy
Abstract economy
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,
- Each choice-set <math>X_i</math> is compact, non-empty and convex.
- Each action-correspondence <math>A_i</math> is continuous, and its values are non-empty and convex.
- Each utility function <math>U_i</math> is continuous in <math>x</math> and quasi-concave in <math>x_i</math>.
The continuity conditions on the utility functions can be weakened as follows:
- Each utility function <math>U_i</math> is quasi-concave in <math>x_i</math>, upper semi-continuous in <math>x</math>, and [[graph-continuous-function|graph continuous]].
Another weakening, which does not use graph-continuity, is: For every consumer i, there is: A consumption-set <math>Y_i</math> - as above; An initial endowment vector <math>w_i\in \mathbb{R}^l_+</math> - as above; A preference relation <math>\prec_i</math> that can be equivalently represented by a preference-correspondence <math>P_i: Y_i \twoheadrightarrow Y_i</math>, that depends only on the consumed bundle: <math>P_i(y_i) := \{z_i\in Y_i|z_i \succ_i y_i\}</math>. 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 <math>y_i</math> is at most* the sum of endowments <math>w_i</math>; * For every i: <math>p\cdot y_i = p\cdot w_i</math>; * For every bundle z: if <math>z\succ_i y_i</math> then <math>p\cdot z > p\cdot y_i</math> (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 <math>\prec_i</math> is relatively-open (equivalently, each <math>P_i</math> has an open graph); For every bundle x*, the set <math>P_i(x)</math> is convex and does not contain x (= irreflexivity). Mas-Collel added the condition that the set <math>P_i(x)</math> is non-empty (= non-saturation). * For every i: <math>w_i \gg x_i</math> 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 <math>y_i = (a_i,b_i)</math> of <math>a_i</math> apples and <math>b_i</math> bananas, the set <math>P_i(a_i,b_i) := \{ (a_i',b_i') | (a_i' > a_i) ~or~ (a_i' = a_i ~and~ b_i' > b_i) \}</math>, i.e., each agent wants as many apples as possible, and subject to that, as many bananas as possible. Note that <math>P_i(a_i,b_i)</math> 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 <math>Y_i</math> is non-empty, convex, closed, and bounded below.
- The preference correspondence is non-empty: <math>P_i(y_i)\neq \emptyset</math> (this is a non-saturation condition).
A competitive equilibrium is a price-vector *<math>\mathbf{p}</math> and an allocation <math>\mathbf{y}</math> such that: Feasibility: the sum of all allocations <math>y_i</math> equals the sum of endowments <math>w_i</math> (there is no free-disposal); Budget: for every i, <math>p\cdot y_i \leq p\cdot w_i</math>; Preference: For every i, <math> P_i(y_i)\cap B_i(p, w_i) = \emptyset</math>, where <math> B_i(p, w_i)</math> is the budget-set of i. In other words, for every bundle <math>z\in Y_i</math>: if <math>z\succ_i y_i</math> then <math>p\cdot z > p\cdot y_i</math> (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 <math>z\in Y_i</math>: if <math>z\succ_i y_i</math> then <math>p\cdot z \geq p\cdot y_i</math>.
A *Pareto-optimal* allocation is, as usual, an allocation without a Pareto-improvement. A Pareto-improvement of an allocation <math>\mathbf{y}</math> is defined as another allocation <math>\mathbf{y'}</math> that is strictly better for a subset <math>J</math> of the agents, and remains the same allocation for all other agents. That is:
<math>\sum_{i\in J} y'_i = \sum_{i\in J} y_i.</math> <math>y'_i \in P_i(y_i)</math> for all <math>i\in J</math>. 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.