<p><strong>tldr:</strong> The core is a very general concept that can be used in a vast amount of models. Applying it to the setting of a general equilibrium, you can show that every competitive equilibrium is in the core.</p>
<h2>The core</h2>
<p>The core is a concept that can be defined for very abstract environments.</p>
<p>Consider a population of agents <span class="math-container">$N$</span> and an space <span class="math-container">$\Omega$</span> of outcomes. Assume that every agent has nice (transitive and complete) preference relation <span class="math-container">$\succeq_i$</span> over the elements in <span class="math-container">$\Omega$</span>.</p>
<p>In addition, agents can form coalitions <span class="math-container">$A \subseteq N$</span> and for every state <span class="math-container">$x \in \Omega$</span> and coalition <span class="math-container">$A$</span> there is a correspondence <span class="math-container">$\Gamma_A: \Omega \rightrightarrows \Omega$</span> that determines to which states the coalition <span class="math-container">$A$</span> can deviate to when starting at <span class="math-container">$x$</span>.</p>
<p>For example if <span class="math-container">$y \in \Gamma_A(x)$</span> this means that at state <span class="math-container">$x$</span> the coalition <span class="math-container">$A$</span> has the power to move to state <span class="math-container">$y$</span>.</p>
<p>We say that a state <span class="math-container">$x$</span> <strong>dominates</strong> the state <span class="math-container">$y$</span> by coalition <span class="math-container">$A$</span> if:</p>
<ol>
<li><span class="math-container">$y \in \Gamma_A(x)$</span></li>
<li>for all <span class="math-container">$i \in A$</span>: <span class="math-container">$y \succeq_i x$</span> and there is at least one <span class="math-container">$j \in A$</span> such that <span class="math-container">$y \succ_j x$</span>.</li>
</ol>
<p>Condition 1 states that <span class="math-container">$A$</span> has the power to move from <span class="math-container">$x$</span> to <span class="math-container">$y$</span>. Condition 2 requires that every member of <span class="math-container">$A$</span> (weakly) prefers <span class="math-container">$y$</span> over <span class="math-container">$x$</span> and at least one member strictly prefers <span class="math-container">$y$</span> over <span class="math-container">$x$</span>. If both conditions hold, then you can say that <span class="math-container">$x$</span> is not a stable situation as <span class="math-container">$A$</span> can find a better state <span class="math-container">$y$</span>.</p>
<p>Now given such structure <span class="math-container">$(N, \Omega, (\succeq_i|i \in N), (\gamma_A|A \subseteq N))$</span> we can define a the <strong>core</strong> as the set of all states <span class="math-container">$x \in \Omega$</span> that are not dominated by some other state. In other words, no coalition can find a profitable deviation from <span class="math-container">$x$</span>.</p>
<h2>Exchange economies and perfect equilibria</h2>
<p>Now give this, let us show that for an exchange economy, the competitive equilibrium is in the core. An exchange economy consists of a set of agents, <span class="math-container">$N$</span>, a set of goods <span class="math-container">$J$</span>. Each agent has a locally non-satiated utility function <span class="math-container">$u_i: \mathbb{R}^J_+ \to \mathbb{R}$</span> and an endowment <span class="math-container">$\omega_i \in \mathbb{R}^J_+$</span>. A price vector <span class="math-container">$p \in \mathbb{R}^J_+$</span> together with an allocation <span class="math-container">$(x_1, \ldots, x_N)$</span> for the agents in the economy is a <strong>competitive equilibrium</strong> if for all agents <span class="math-container">$i \in N$</span>:
<span class="math-container">$$
x_i \in \arg \max_{q} u(q) \text{ subject to } p' q = p' \omega_i
$$</span>
(equality of the budget constraint holds due to locally non-satiation) and:
<span class="math-container">$$
\sum_{i \in N} x_i = \sum_{i \in N} \omega_i
$$</span>
This second condition requires demand to be equal to supply.</p>
<p>To show that every allocation of a competitive equilibrium is in the core, we first need to get a mapping from our exchange economy to the structure given above:</p>
<ol>
<li><p>The set of agents is simply the set <span class="math-container">$N$</span></p>
</li>
<li><p>The set of states <span class="math-container">$\Omega$</span> can be any allocation <span class="math-container">$(x_1, \ldots, x_N)$</span> such that <span class="math-container">$\sum_{i \in N} x_i = \sum_{i \in N} \omega_i$</span>.</p>
</li>
<li><p>The preferences <span class="math-container">$\succeq_i$</span> are the ones that correspond to the utility functions.
<span class="math-container">$$
   (x_1, \ldots, x_n) \succeq_i (y_1, \ldots, y_N) \iff u(x_i) \ge u(y_i).
   $$</span></p>
</li>
<li><p>Now for the possible deviations of a coalition <span class="math-container">$A$</span>, we assume that each coalition can move to an allocation by simply reallocating their own endowments. This means that they can move to a new allocation for their members as long as the total consumption within the coalition does not exceed the total endowment of the members in the coalition. (There is an issue here as I am not specifying what happens to the consumption of the members outside the coalition once a coalition deviates, but we will see that this does not really matter for the argument).</p>
</li>
</ol>
<p>Given this, let us show that any competitive equilibrium is in the core.  Towards a contradiction, assume that <span class="math-container">$(x_1, \ldots, x_N)$</span> is a competitive equilibrium with price vector <span class="math-container">$p$</span>, and assume that coalition <span class="math-container">$A$</span> has found a profitable deviation <span class="math-container">$(y_i| i \in A)$</span>. This means that:</p>
<ol>
<li><p>for all members <span class="math-container">$i \in A$</span>:
<span class="math-container">$$
   u_i(y_i) \ge u_i(x_i),
   $$</span></p>
</li>
<li><p>for at least one member <span class="math-container">$j \in A$</span>
<span class="math-container">$$
   u_j(y_j) &gt; u_j(x_j).
   $$</span></p>
</li>
<li><p>The total consumption in <span class="math-container">$A$</span> must not exceed the total endowment of members in <span class="math-container">$A$</span>
<span class="math-container">$$
   \sum_{i \in A} y_i \le \sum_{i \in A} \omega_i
   $$</span></p>
</li>
</ol>
<p>Let us show that these three conditions lead to a contradiction. First, using a simple revealed preference argument it follows from  <span class="math-container">$u_i(y_i) \ge u_i(x_i)$</span>  that:
<span class="math-container">$$
p' y_i \ge p' x_i.
$$</span>
Indeed, otherwise <span class="math-container">$p' y_i &lt; p' x_i \le p' \omega_i$</span> which means that <span class="math-container">$y_i$</span> was cheaper than <span class="math-container">$x_i$</span>. This however contradicts the assumption that <span class="math-container">$x_i$</span> was utility maximising for <span class="math-container">$i$</span> at prices <span class="math-container">$p$</span>. Also using a similar revealed preference argument, we can show that <span class="math-container">$u_j(y_j) &gt; u_j(x_j)$</span> implies:
<span class="math-container">$$
p' y_j &gt; p' x_j.
$$</span>
So adding these inequalities across <span class="math-container">$i \in A$</span>, we get:
<span class="math-container">$$
\sum_{i \in A} p' y_i &gt; \sum_{i \in A} p' x_i.
$$</span>
On the other hand from the budget constraint conditions and 3. we also have that:
<span class="math-container">$$
\sum_{i \in A} p' x_i = \sum_{i \in A} p' \omega_i = p' \sum_{i \in A} \omega_i \ge p' \sum_{i \in A} y_i = \sum_{i \in A} p' y_i
$$</span>
a contradiction.</p>

CC BY-SA 4.0
2021-05-19T20:11:14.107Z
What about the part where in an economy with $n$ players the $\lim_{n\to\infty}$ of the core is exactly the set of competitive equilibria? :)
Giskard
CC BY-SA 4.0
2021-05-19T18:41:20.953Z
2021-05-19T18:41:20.953Z
tdm
<p>Can someone please explain why the concept of a core is equivalent to a competitive equilibrium?</p>
<p>I am just writing a term paper for my matching theory class and I'm having a hard time following along with the literature.</p>
<p>In particular I'm referring to Gale and Shapley (1962), Shapley and Shubik (1971), and Crawford and Knoer (1981).</p>

CC BY-SA 4.0
2020-12-03T10:55:40.797Z
Gale and Shapley (1962) has nothing to do with competitive equilibrium, Crawford and Knoer (1981) very little.
Michael Greinecker
CC BY-SA 4.0
2020-12-03T11:15:38.553Z
@MichaelGreinecker On page 438 of Crawford and Knoer (1981), they write "Gale and
 Shapley take as their notion of equilibrium the natural one that no pair of agents
 on opposite sides of the market prefer each other to any of their current partners. As Shapley and Shubik [13, pp. 114-118] point out, this notion of equilibrium is
 equivalent to the core in such markets because the two-agent coalitions made up
 of agents on opposite sides of the market can accomplish anything for themselves
 that they could as members of larger coalitions, and more than they could acting
 separately."
Friendlyperson2020
CC BY-SA 4.0
2020-12-03T11:16:38.267Z
@MichaelGreinecker don't quite follow the passage above. Here is my confusion, my understanding is that in a competitive equilibrium, there are prices. But there are no prices in Gale and Shapley, nor matching theory in general
Friendlyperson2020
CC BY-SA 4.0
2020-12-03T11:17:45.267Z
That is my point. No prices, no equilibrium. However, in the model of Shapley and Shubik one can define a competitive equilibrium and relate it to the core.
Michael Greinecker
CC BY-SA 4.0
2020-12-03T13:55:44.067Z
In general, the two concepts are not equivalent. However, in specific settings this can be the case. You have to provide more context such that an answer to your question can be found.
Bayesian
CC BY-SA 4.0
2020-12-03T14:47:15.447Z
@Bayesian Are you familiar with the Kelso-Crawford labour model? In that one, the authors argue establish the existence of an equilibrium and argue its equivalent to the core...
Friendlyperson2020
CC BY-SA 4.0
2020-12-03T15:02:28.550Z
But they specifically limit this statement to their model, where utility is transferable through the wages. There are no monetary transfers in, e.g., Gale and Shapley 62.
Bayesian
CC BY-SA 4.0
2020-12-03T15:19:52.520Z
As it was already mentioned, both concepts are not equivalent. The core concept used by  Gale/Shubik Shapley is based on transferable utilities, whereas  a competitive equilibrium (Walras equilibrium) is obtained from an exchange economy where non-transferable utility is present. Of course, there exist also a core concept that is not equivalent to that used by the mentioned authors. In this respect, the following post may clarify the facts https://math.stackexchange.com/questions/2514861/connection-between-shapley-value-and-competitive-equlibirium-payoff/2516418#2516418
Holger I. Meinhardt
<microeconomics><game-theory><cooperative-game-theory><matching>
The core and competitive equilibrium
Friendlyperson2020
# The core and competitive equilibrium

(Asked by: username_0 on 2020-12-03)

Can someone please explain why the concept of a core is equivalent to a competitive equilibrium?

I am just writing a term paper for my matching theory class and I'm having a hard time following along with the literature.

In particular I'm referring to Gale and Shapley (1962), Shapley and Shubik (1971), and Crawford and Knoer (1981).

**Tags:** microeconomics, game-theory, cooperative-game-theory, matching

> **Comments:**
> **username_1**: Gale and Shapley (1962) has nothing to do with competitive equilibrium, Crawford and Knoer (1981) very little.  
> **username_0**: @username_1 On page 438 of Crawford and Knoer (1981), they write "Gale and  Shapley take as their notion of equilibrium the natural one that no pair of agents  on opposite sides of the market prefer each other to any of their current partners. As Shapley and Shubik [13, pp. 114-118] point out, this notion of equilibrium is  equivalent to the core in such markets because the two-agent coalitions made up  of agents on opposite sides of the market can accomplish anything for themselves  that they could as members of larger coalitions, and more than they could acting  separately."  
> **username_0**: @username_1 don't quite follow the passage above. Here is my confusion, my understanding is that in a competitive equilibrium, there are prices. But there are no prices in Gale and Shapley, nor matching theory in general  
> **username_1**: That is my point. No prices, no equilibrium. However, in the model of Shapley and Shubik one can define a competitive equilibrium and relate it to the core.  
> **username_2**: In general, the two concepts are not equivalent. However, in specific settings this can be the case. You have to provide more context such that an answer to your question can be found.  
> **username_0**: @username_2 Are you familiar with the Kelso-Crawford labour model? In that one, the authors argue establish the existence of an equilibrium and argue its equivalent to the core...  
> **username_2**: But they specifically limit this statement to their model, where utility is transferable through the wages. There are no monetary transfers in, e.g., Gale and Shapley 62.  
> **username_3**: As it was already mentioned, both concepts are not equivalent. The core concept used by  Gale/Shubik Shapley is based on transferable utilities, whereas  a competitive equilibrium (Walras equilibrium) is obtained from an exchange economy where non-transferable utility is present. Of course, there exist also a core concept that is not equivalent to that used by the mentioned authors. In this respect, the following post may clarify the facts https://math.stackexchange.com/questions/2514861/connection-between-shapley-value-and-competitive-equlibirium-payoff/2516418#2516418  

---

## 1 answers

### Answer by username_4 on 2021-05-19. Score: 5

**tldr:** The core is a very general concept that can be used in a vast amount of models. Applying it to the setting of a general equilibrium, you can show that every competitive equilibrium is in the core.

## The core

The core is a concept that can be defined for very abstract environments.

Consider a population of agents $N$ and an space $\Omega$ of outcomes. Assume that every agent has nice (transitive and complete) preference relation $\succeq\_i$ over the elements in $\Omega$.

In addition, agents can form coalitions $A \subseteq N$ and for every state $x \in \Omega$ and coalition $A$ there is a correspondence $\Gamma\_A: \Omega \rightrightarrows \Omega$ that determines to which states the coalition $A$ can deviate to when starting at $x$.

For example if $y \in \Gamma\_A(x)$ this means that at state $x$ the coalition $A$ has the power to move to state $y$.

We say that a state $x$ **dominates** the state $y$ by coalition $A$ if:

1. $y \in \Gamma\_A(x)$
2. for all $i \in A$: $y \succeq\_i x$ and there is at least one $j \in A$ such that $y \succ\_j x$.

Condition 1 states that $A$ has the power to move from $x$ to $y$. Condition 2 requires that every member of $A$ (weakly) prefers $y$ over $x$ and at least one member strictly prefers $y$ over $x$. If both conditions hold, then you can say that $x$ is not a stable situation as $A$ can find a better state $y$.

Now given such structure $(N, \Omega, (\succeq\_i|i \in N), (\gamma\_A|A \subseteq N))$ we can define a the **core** as the set of all states $x \in \Omega$ that are not dominated by some other state. In other words, no coalition can find a profitable deviation from $x$.

## Exchange economies and perfect equilibria

Now give this, let us show that for an exchange economy, the competitive equilibrium is in the core. An exchange economy consists of a set of agents, $N$, a set of goods $J$. Each agent has a locally non-satiated utility function $u\_i: \mathbb{R}^J\_+ \to \mathbb{R}$ and an endowment $\omega\_i \in \mathbb{R}^J\_+$. A price vector $p \in \mathbb{R}^J\_+$ together with an allocation $(x\_1, \ldots, x\_N)$ for the agents in the economy is a **competitive equilibrium** if for all agents $i \in N$:
$$
x\_i \in \arg \max\_{q} u(q) \text{ subject to } p' q = p' \omega\_i
$$
(equality of the budget constraint holds due to locally non-satiation) and:
$$
\sum\_{i \in N} x\_i = \sum\_{i \in N} \omega\_i
$$
This second condition requires demand to be equal to supply.

To show that every allocation of a competitive equilibrium is in the core, we first need to get a mapping from our exchange economy to the structure given above:

1. The set of agents is simply the set $N$
2. The set of states $\Omega$ can be any allocation $(x\_1, \ldots, x\_N)$ such that $\sum\_{i \in N} x\_i = \sum\_{i \in N} \omega\_i$.
3. The preferences $\succeq\_i$ are the ones that correspond to the utility functions.
   $$
   (x\_1, \ldots, x\_n) \succeq\_i (y\_1, \ldots, y\_N) \iff u(x\_i) \ge u(y\_i).
   $$
4. Now for the possible deviations of a coalition $A$, we assume that each coalition can move to an allocation by simply reallocating their own endowments. This means that they can move to a new allocation for their members as long as the total consumption within the coalition does not exceed the total endowment of the members in the coalition. (There is an issue here as I am not specifying what happens to the consumption of the members outside the coalition once a coalition deviates, but we will see that this does not really matter for the argument).

Given this, let us show that any competitive equilibrium is in the core. Towards a contradiction, assume that $(x\_1, \ldots, x\_N)$ is a competitive equilibrium with price vector $p$, and assume that coalition $A$ has found a profitable deviation $(y\_i| i \in A)$. This means that:

1. for all members $i \in A$:
   $$
   u\_i(y\_i) \ge u\_i(x\_i),
   $$
2. for at least one member $j \in A$
   $$
   u\_j(y\_j) > u\_j(x\_j).
   $$
3. The total consumption in $A$ must not exceed the total endowment of members in $A$
   $$
   \sum\_{i \in A} y\_i \le \sum\_{i \in A} \omega\_i
   $$

Let us show that these three conditions lead to a contradiction. First, using a simple revealed preference argument it follows from $u\_i(y\_i) \ge u\_i(x\_i)$ that:
$$
p' y\_i \ge p' x\_i.
$$
Indeed, otherwise $p' y\_i < p' x\_i \le p' \omega\_i$ which means that $y\_i$ was cheaper than $x\_i$. This however contradicts the assumption that $x\_i$ was utility maximising for $i$ at prices $p$. Also using a similar revealed preference argument, we can show that $u\_j(y\_j) > u\_j(x\_j)$ implies:
$$
p' y\_j > p' x\_j.
$$
So adding these inequalities across $i \in A$, we get:
$$
\sum\_{i \in A} p' y\_i > \sum\_{i \in A} p' x\_i.
$$
On the other hand from the budget constraint conditions and 3. we also have that:
$$
\sum\_{i \in A} p' x\_i = \sum\_{i \in A} p' \omega\_i = p' \sum\_{i \in A} \omega\_i \ge p' \sum\_{i \in A} y\_i = \sum\_{i \in A} p' y\_i
$$
a contradiction.

> **Comments:**
> **username_5**: What about the part where in an economy with $n$ players the $\lim_{n\to\infty}$ of the core is exactly the set of competitive equilibria? :)  

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