Notes on General Equilibrium
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General Equilibrium Theory
“The two central problems of the theory that this monograph presents are (1) the explanation of the prices of commodities resulting from the interaction of the agents of a private ownership economy through markets, (2) the explanation of the role of prices in an optimal state of an economy. The analysis is therefore organized around the concept of a price system or, more generally, of a value function defined on the commodity space.”
— Gérard Debreu, The Theory of Value
Now, let us view a competitive economy from the general equilibrium perspective. As Debreu (1959) emphasizes, general equilibrium theory seeks to explain how the prices of commodities emerge from the interactions of agents in a market economy, and how those prices, in turn, sustain an optimal allocation of resources.
Consider an economy with \(H\) commodities indexed by \(h \in \mathcal{H} = {1, \dots, H}\), \(I\) consumers indexed by \(i \in \mathcal{I} = {1, \dots, I}\), and \(J\) producers or firms indexed by \(j \in \mathcal{J} = {1, \dots, J}\).
Consumers derive utility from consuming goods. Let \(X\) denote the consumption correspondence, which maps each consumer \(i \in \mathcal{I}\) to a subset \(X(i) = X_i \subseteq \mathbb{R}^{\mathcal{H}}\) of the commodity space, representing the set of consumption bundles available to that consumer. Each element \(x_i \in X_i\) is a consumption bundle (or consumption vector), describing quantities of all commodities that the consumer may feasibly consume.
Let \(R\) be a preference correspondence that maps each \(i \in \mathcal{I}\) to a reflexive binary relation \(R(i) = R_i \subseteq X_i \times X_i\) over \(X_i\), representing the consumer’s preference ordering such that \((x_i, x_i) \in R_i\) for all \(x_i \in X_i\).
Each producer \(j \in \mathcal{J}\) operates a production set \(Y(j) = Y_j \subseteq \mathbb{R}^{\mathcal{H}}\), where each element \(y_j \in Y_j\) is a production bundle (or production vector) describing quantities of inputs and outputs across commodities.
The total resources available in the economy are represented by an aggregate endowment vector \(\bar{\omega} \in \mathbb{R}^{\mathcal{H}}\), where \(\bar{\omega} = (\bar{\omega}(h))_{h \in \mathcal{H}} = (\bar{\omega}h)_{h \in \mathcal{H}}\) specifies the total amount of each commodity in the economy, not necessarily individualized among agents. These commodities are assumed to be pre-existing and dated. It means that each good has a defined origin in time.
Definition 1 (Economy) The list \((\mathcal{H}, \mathcal{I}, \mathcal{J}, X, R, Y, \bar{\omega})\) is an economy if
- \(\mathcal{H}\) is a nonempty finite set of commodities;
- \(\mathcal{I}\) is a nonempty finite set of consumers;
- \(\mathcal{J}\) is a nonempty finite set of producers or firms;
- \(X\) is a correspondence from \(\mathcal{I}\) to the commodity space \(\mathbb{R}^{\mathcal{H}}\);
- \(R\) is a correspondence from \(\mathcal{I}\) to \(\mathbb{R}^{\mathcal{H}} \times \mathbb{R}^{\mathcal{H}}\) such that \(R(i)\) is a reflexive binary relation over \(X_{i}\) s.t. \((x_{i},x_{i}) \in R_{i}\) for each \(x_{i} \in X_{i}\);
- \(Y\) is a correspondence from \(\mathcal{J}\) to \(\mathbb{R}^{\mathcal{H}}\); and
- \(\bar{\omega}\) is a point of \(\mathbb{R}^{\mathcal{H}}\).
Definition 2 (Feasible Allocation) Let \((\mathcal{H}, \mathcal{I}, \mathcal{J}, X, R, Y, \bar{\omega})\) be an economy. Then \((x, y) \in (\mathbb{R}^{\mathcal{H}})^{\mathcal{I}} \times (\mathbb{R}^{\mathcal{H}})^{\mathcal{J}}\) is a feasible allocation for \((\mathcal{H}, \mathcal{I}, \mathcal{J}, X, R, Y, \bar{\omega})\) if \((x, y) \in X \times Y\) and \(\sum_{i \in \mathcal{I}}x_{i}=\bar{\omega}+\sum_{j \in \mathcal{J}}y_{i}\).