---
tags: micro, lecture_note, book
---

# Core

In the following models, we assume there are $I$ consumers, $l$ commodities, and a publicly available constant returns convex technology $Y \subset \mathbb{R}^L$ in the economy. The consumption set of each consumer is $\mathbb{R}^L$, and each consumer has an endowment $\omega_i \in \mathbb{R}^L, \omega_i \ge 0$ and a continuous, strictly convex, strongly monotone preference relation $\succeq_i$.

## Allocation
**Definition** We use a set to represent an allocation, $\{x = (x_1, ...,x_I )\in \mathbb{R}^{LI}_+ ,y = (y_1, ...,y_I )\in \mathbb{R}^{LI}\}$, Where $y_i \in Y$ for each $i$. Let $y = \sum y_i$, the allocaiton can also represent as $\{x = (x_1, ...,x_I )\in \mathbb{R}^{LI}_+ ,y \in Y\}$. In this model, a Walrasian equilibrium can be represented as $\{x^* = (x_1^*, ...,x_I^* )\in \mathbb{R}^{LI}_+ ,y^* = (y_1^*, ...,y_I^* )\in \mathbb{R}^{LI}_+, p\}$.

## Coalition
**Definition** We also use $I$ to represent the set of consumers. (It is an abuse of notation. Formally, the set of consumers should be $C_I =\{1, 2, ..., I\}$.) A coalition $S$ is a subset of all consumers, that is, $S \subset I$.

## Improves Upon, Blocks
**Definition** A coalition $S \subset I$ improves upon, or blocks, the feasible allocation $x^* = (x_1^*, ...,x_I^* )\in \mathbb{R}^{LI}_+$ if for every $i \in S$ we can find a consumption $x_i \ge 0$ with the properties:

1. $x_i \succ_i x_1^*$ for every $i \in S.$
2. $\sum_{i \in S} x_i \in Y + \sum_{i \in S} \omega_i.$

## Core Property
**Definition** We say that the feasible allocation $x^* = (x_1^*, ...,x_I^* )\in \mathbb{R}^{LI}_+$ has the core property if there is no coalition of consumers $S \subset I$ that can improve upon $x^*$. The core is the set of allocations that have the core property.

## Walrasian Equilibrium is in Core
**Proposition** Any Walrasian equilibrium allocation has the core property in this model.

**Claim** Under the competitive equilibrium price, any production plan could not induce positive profit, and any implemented production plan should induce zero profit. That is, $p^* \cdot y^*_i = 0$, for all $i \in I$.

**Proof** Since $Y$ is a CRS technology, $y \in Y$ implies $ay \in Y$ for any $a \in \mathbb{R}_+$. If $y' \in Y$ can induce positive profit, we can use $ay'$ to induce arbitrary high profit. It could not constitute a competitive equilibrium. Under utility maximization, a consumer should not choose a production plan to induce negative profit.

**Proof** Let $\{x^* = (x_1^*, ...,x_I^* )\in \mathbb{R}^{LI}_+ ,y^* = (y_1^*, ...,y_I^* )\in \mathbb{R}^{LI}_+, p^*\}$ be a Walrasian equilibrium in this model. Suppose there is a coalition $S \subset I$ block this alllocation, that is, there exist another feasible allocaiton $x = (x_1, ...,x_I )$ such that $x_i \succ_i x_1^*$ for every $i \in S.$

The utility maximization condition of Walrasian equilibrium implies the $p^* \cdot x_i >p^* \cdot x_i^*$ for for every $i \in S.$ Hence,

$$\sum_{i \in S}p^* \cdot x_i > \sum_{i \in S} p^* \cdot x_i^* = \sum_{i \in S} p^* \cdot (\omega_i^* +y_i^*)$$

As the above claim, $p^* \cdot y^*_i = 0$, for all $i \in I$ and $p^* \cdot y \le 0$ for every $y \in Y.$ There does not exist a vector $y' \in Y$ such that $\sum_{i \in S} x_i = y' + \sum_{i \in S} \omega_i\in Y + \sum_{i \in S} \omega_i.$ Therefore, any Walrasian equilibrium allocation in this model has the core property.