2 10.B Pareto Optimality and Competitive Equilibria

セッテイングと定義・定理のみ簡単にまとめておく。

\(i = 1, ..., I\) consumers
\(j = 1, ..., J\) firms
\(l = 1, ..., L\) goods
\(x_{i} = (x_{1, i}, ... x_{L, i}) \in X_{i}\): consumer \(i\)’s consumption bundles
\(u_{i}(\cdot)\): consumer \(i\)’s utility function that represents consumer \(i\)’s preferences over consumption bundles
\(\omega_{l} \geq 0\) for \(l = 1, ..., L\): endowment of good \(l\)
\(Y_{j} \subset \mathbb{R}^{L}\): firm \(j\)’s production set
\(y_{j} = (y_{1j}, ..., y_{Lj}) \in Y_{j}\): firm \(j\)’s production vector
The total net amount of goo \(l\) available to the economy is \(\omega_{l} + \sum_{j}y_{lj}\)

Definition 10.B.1 An economic allocation \((x_{1}, ..., x_{I}, y_{1}, ..., y_{J})\) is a specification of a consumption of vector \(x_{i} \in X_{i}\) for each consumer \(i = 1, ..., I\) and a production vector \(y_{j} \in Y_{j}\) for each firm \(j = 1, ..., J\). The allocation \((x_{1}, ..., x_{I}, y_{1}, ..., y_{J})\) is feasible if

\[ \sum_{i = 1}^{I}x_{li} \leq \omega_{l} + \sum_{j}y_{lj} \]

2.1 Pareto Optimality

Definition 10.B.2 A feasible allocation \((x_{1}, ..., x_{I}, y_{1}, ..., y_{J})\) is Pareto optimal if there is no other feasible allocation \((x'_{1}, ..., x'_{I}, y'_{1}, ..., y'_{J})\) such that \(u_{i}(x'_{i}) \geq u_{i}(x_{i})\) for all \(i = 1, ..., I\) and \(u_{i}(x'_{i}) > u_{i}(x_{i})\) for some \(i\)

2.2 Competitive Equilibria

Suppose that consumer \(i\) initially owns \(\omega_{li}\) of good \(l\), where \(\sum_{i}\omega_{li} = \omega_{l}\)
\(\omega_{i} = (\omega_{1i}, ..., \omega_{Li})\): consumer \(i\)’s vector of endowments
Suppose that consumer \(i\) owns a share \(\theta_{if}\) of firm \(j\) (where \(\sum_{i}\theta_{ij} = 1\)), giving him a claim to fraction \(\theta_{ij}\) of firm \(j\)’s profits

Definition 10.B.3 The allocation \((x^{*}_{1}, ..., x^{*}_{I}, y^{*}_{1}, ..., y^{*}_{J})\) and price vector \(p^{*} \in \mathbb{R}^{L}\) constitute a competitive equilibrium if the following conditions are satisfied:

Profit maximization: For each firm \(j\), \(y^{*}_{j}\) solves

\[ \max_{y_{j} \in Y_{j}} p^{*} \cdot y_{j} \]

Utility maximization: For each consumer \(i\), \(x^{*}_{i}\) solves

\[ \begin{align} \max_{x_{i} \in X_{i}} u_{i}(x_{i}) \\ \text{s.t.} \quad p^{*} \cdot x_{i} \leq p^{*} \cdot \omega_{i} + \sum_{j = 1}^{J}\theta_{ij}(p^{*} \cdot y^{*}_{j}) \end{align} \]

Market clearing: For each good \(l = 1, ..., L\),

\[ \sum_{i = 1}^{I}x^{*}_{li} = \omega_{l} + \sum_{j}y^{*}_{lj} \]

Lemma 10.B.1 If the allocation \((x_{1}, ..., x_{I}, y_{1}, ..., y_{J})\) and price vector \(p >> 0\) satisfy the market clearing condition for all goods \(l \neq k\), and if every consumer’s budget constraint is satisfied with equality, so that \(p \cdot x_{i} = p \cdot \omega_{i} + \sum_{j = 1}^{J}\theta_{ij}(p \cdot y_{j})\) for all \(i\), then the marked for good \(k\) also clears