<p>Consider the following example of a two-consumer, two-goods pure exchange economy:</p>
<ul>
<li>Utility functions: <span class="math-container">$u_1(x_1,y_1)=x_1+y_1$</span>, <span class="math-container">$u_2(x_2,y_2)=x_2+y_2$</span></li>
<li>Endowments: <span class="math-container">$\omega_1=(2,0)$</span>, <span class="math-container">$\omega_2=(0,2)$</span></li>
</ul>
<p>Set of feasible allocations is</p>
<p><span class="math-container">$\mathcal{F}=\{((x_1,y_1),(x_2,y_2))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_1+x_2=2 \ \wedge \ y_1+y_2=2\}$</span></p>
<p>Observe that the set of Pareto efficient allocations equals the set of feasible allocations:</p>
<p><span class="math-container">$\mathcal{PE}=\mathcal{F}$</span></p>
<p>So, endowment is Pareto efficient. But core consists of all allocations on the diagonal of the Edgeworth box that satisfy <span class="math-container">$x_1+y_1=2$</span>. So, Core equals</p>
<p><span class="math-container">$\mathcal{C}=\{((x_1,y_1),(x_2,y_2))\in\mathcal{F}|x_1+y_1=2\}$</span></p>
<p>and it is not singleton.</p>
<p>So, you cannot prove that if the endowment is Pareto optimal then the core is singleton.</p>
<p>I suggest that you try to prove the following proposition:</p>
<p><strong>In a 2 consumer pure exchange economy, if the endowment allocation is Pareto optimal and all utility functions are strictly quasi-concave (or preferences are strictly convex) then the core is singleton and consists of the endowment allocation</strong></p>

CC BY-SA 4.0
2024-10-12T17:39:22.330Z
2024-10-12T17:39:22.330Z
Amit
<p><strong>Claim</strong>: If the endowment is Pareto optimal, then the core is a singleton set.</p>
<hr />
<p><strong>Notations:</strong></p>
<ol>
<li><span class="math-container">$\mathbb{N} = \{1, 2\}$</span> denotes the set of all individuals.</li>
<li><span class="math-container">$\mathbb{M} = \{1, 2\}$</span> denotes the set of all goods.</li>
<li><span class="math-container">$\mathbb{C}$</span> denotes the set of all allocations that satisfy the core property.</li>
<li><span class="math-container">$u^i(\mathbf{x})$</span>, where <span class="math-container">$\mathbf{x} \in \mathbb{R}^{2}_{+}$</span>, is the utility function of the <span class="math-container">$i^{\text{th}}$</span> agent.
<ul>
<li>The utility function of every agent is assumed to be strictly quasi-concave (implying strictly convex indifference curves), continuous, and strongly increasing.</li>
</ul>
</li>
<li><span class="math-container">$\mathbf{e} \in \mathbb{R}^{2 \times 2}_{+}$</span> is the endowment vector, where <span class="math-container">$\mathbf{e}^i_j \in \mathbb{R}_{+}$</span> and <span class="math-container">$\mathbf{e}^i \in \mathbb{R}^2_{+}$</span> denote the endowment of the <span class="math-container">$j^{\text{th}}$</span> good available to the <span class="math-container">$i^{\text{th}}$</span> agent and the endowment vector of the <span class="math-container">$i^{\text{th}}$</span> agent, respectively.</li>
</ol>
<hr />
<p><strong>Proof:</strong></p>
<p>The endowment <span class="math-container">$\mathbf{e}$</span> is <strong>Pareto optimal</strong>. Since <span class="math-container">$\mathbf{e}$</span> is at least as good as itself and feasible, it satisfies the core property. Hence, <span class="math-container">$\mathbf{e} \in \mathbb{C}$</span>.</p>
<p>Now, let <span class="math-container">$\mathbf{x} \in \mathbb{R}^{2 \times 2}_{+}$</span> be another feasible allocation such that <span class="math-container">$\mathbf{x} \in \mathbb{C}$</span> and <span class="math-container">$\mathbf{x} \neq \mathbf{e}$</span>.<br />
Then <span class="math-container">$\mathbf{x}$</span> must also be Pareto optimal and at least as good as the endowment. That is, the following must hold:</p>
<p>1.<br />
<span class="math-container">$$
   \neg \left(\exists \; \mathbf{y} \in \mathbb{R}^{2 \times 2}_{+} \right) \left[ (\forall i \in \mathbb{N}) \big(u(\mathbf{y}^i) \geq u(\mathbf{x}^i) \land (\exists j \in \mathbb{N}) (u(\mathbf{y}^j) &gt; u(\mathbf{x}^j))\big) \right]
   $$</span></p>
<p>2.<br />
<span class="math-container">$$u(\mathbf{x}^i) \geq u(\mathbf{e}^i) \quad \forall i \in \mathbb{N}$$</span></p>
<p>From (2), consider the following cases:</p>
<hr />
<p><strong>Case 1:</strong><br />
<span class="math-container">$\exists \; k \in \mathbb{N}$</span> such that <span class="math-container">$u(\mathbf{x}^k) &gt; u(\mathbf{e}^k)$</span>.</p>
<p>This implies:<br />
<span class="math-container">$\mathbf{x}$</span> is Pareto superior to <span class="math-container">$\mathbf{e} \implies \mathbf{e}$</span> is not Pareto optimal.<br />
This is a contradiction.</p>
<hr />
<p><strong>Case 2:</strong><br />
<span class="math-container">$u(\mathbf{x}^i) = u(\mathbf{e}^i) \quad \forall i \in \mathbb{N}$</span>.</p>
<p>This implies:<br />
<span class="math-container">$u(\mathbf{x})$</span> and <span class="math-container">$u(\mathbf{e})$</span> lie on the same indifference curve.</p>
<p>Moreover, since both <span class="math-container">$\mathbf{e}$</span> and <span class="math-container">$\mathbf{x}$</span> are Pareto optimal:<br />
The indifference curves of all individuals are tangent at both <span class="math-container">$\mathbf{x}$</span> and <span class="math-container">$\mathbf{e}$</span>.</p>
<p>However, since indifference curves are strictly convex:<br />
This implies either <span class="math-container">$\mathbf{x} = \mathbf{e}$</span>, or there are multiple points of tangency.<br />
The latter cannot occur because of strict convexity, and the former contradicts the assumption that <span class="math-container">$\mathbf{x} \neq \mathbf{e}$</span>.</p>
<p>In either case, we arrive at a contradiction.</p>
<hr />
<p>Therefore, the core consists of a single allocation, which is the endowment itself.</p>
<p>As a corollary, we conclude that <strong>if the endowment satisfies the core property, the Walrasian equilibrium is the endowment</strong>.</p>

CC BY-SA 4.0
2024-12-07T21:43:47.680Z
2024-12-07T21:43:47.680Z
Yash Burman
<p>I have been asked to state and prove the conditions on endowments to ensure core to be a singleton set.</p>
<p>I claim that if the endowment itself is Pareto Optimal then the core would be a singleton set consisting of the endowment itself, however I am having a hard time proving my claim.(This idea might be completely wrong but this is what I've got so far)</p>
<p><span class="math-container">$
\text{Claim: if the endowment } \textit{e} \text{ is PO then } \mathbb{C} = \{e\}\\
\text{Proof:} \\ \text{Let the endowment point $e \in \mathbb{C} \Rightarrow e$ is Pareto Optimal and} \color{red}{ \text{ e is Pareto Superior to itself. }} \\
\text{Let $x$ be an allocation such that $x \neq e$ and $x \in \mathbb{C}$}\\
\text{$\Rightarrow x$ is Pareto Optimal and $x$ is Pareto Superior to the endowment $e$}\\
\text{But since $e$ is assumed to be Pareto Optimal there cant exist any Allocation that is Pareto Superior to e.}\\
\text{And hence we arrive at a Contradiction}
$</span></p>
<p>My proof doesn't work because how can <span class="math-container">$e$</span> be Pareto Superior to itself.
Moreover my proof starts from the fact that <span class="math-container">$e$</span> is in the Core, but I need to show that if <span class="math-container">$e$</span> is PO then <span class="math-container">$e$</span> will be in the core.
How do I proceed further?
Thank you.</p>

CC BY-SA 4.0
2024-10-12T15:27:20.820Z
Why do you need $e$ to be Pareto Superior to itself?
Brennan
CC BY-SA 4.0
2024-10-12T16:15:58.807Z
According to what I have studied, an allocation belongs to the core if two conditions hold, *first* the allocation needs to be Pareto Optimum and *Second* it needs to be pareto superior to the endowment. Now if i claim that e belongs to the core shouldn't I require that it must be Pareto Sup to itself? I might be wrong here, thank you for your help anyway.
Yash Burman
CC BY-SA 4.0
2024-10-12T17:09:08.987Z
@YashBurman Are you assuming its a two consumer pure exchange economy?
Amit
CC BY-SA 4.0
2024-10-12T22:10:46.647Z
@YashBurman I don’t think this is how you define Core. Given the way you have defined Core in the comments - that it consists of Pareto optimal allocations that are Pareto Superior to the endowment allocation, the core will be empty if endowment allocation is Pareto efficient.
Amit
<microeconomics><general-equilibrium><pareto-efficiency>
Conditions on the Endowment to make the core a singleton set
Yash Burman
# Conditions on the Endowment to make the core a singleton set

(Asked by: username_0 on 2024-10-12)

I have been asked to state and prove the conditions on endowments to ensure core to be a singleton set.

I claim that if the endowment itself is Pareto Optimal then the core would be a singleton set consisting of the endowment itself, however I am having a hard time proving my claim.(This idea might be completely wrong but this is what I've got so far)

$
\text{Claim: if the endowment } \textit{e} \text{ is PO then } \mathbb{C} = \{e\}\\
\text{Proof:} \\ \text{Let the endowment point $e \in \mathbb{C} \Rightarrow e$ is Pareto Optimal and} \color{red}{ \text{ e is Pareto Superior to itself. }} \\
\text{Let $x$ be an allocation such that $x \neq e$ and $x \in \mathbb{C}$}\\
\text{$\Rightarrow x$ is Pareto Optimal and $x$ is Pareto Superior to the endowment $e$}\\
\text{But since $e$ is assumed to be Pareto Optimal there cant exist any Allocation that is Pareto Superior to e.}\\
\text{And hence we arrive at a Contradiction}
$

My proof doesn't work because how can $e$ be Pareto Superior to itself.
Moreover my proof starts from the fact that $e$ is in the Core, but I need to show that if $e$ is PO then $e$ will be in the core.
How do I proceed further?
Thank you.

**Tags:** microeconomics, general-equilibrium, pareto-efficiency
> **Comments:**
> **username_1**: Why do you need $e$ to be Pareto Superior to itself?  
> **username_0**: According to what I have studied, an allocation belongs to the core if two conditions hold, *first* the allocation needs to be Pareto Optimum and *Second* it needs to be pareto superior to the endowment. Now if i claim that e belongs to the core shouldn't I require that it must be Pareto Sup to itself? I might be wrong here, thank you for your help anyway.  
> **username_2**: @YashBurman Are you assuming its a two consumer pure exchange economy?  
> **username_2**: @YashBurman I don’t think this is how you define Core. Given the way you have defined Core in the comments - that it consists of Pareto optimal allocations that are Pareto Superior to the endowment allocation, the core will be empty if endowment allocation is Pareto efficient.  

---

## 2 answers

### Answer by username_2 on 2024-10-12. Score: 2

Consider the following example of a two-consumer, two-goods pure exchange economy:

* Utility functions: $u\_1(x\_1,y\_1)=x\_1+y\_1$, $u\_2(x\_2,y\_2)=x\_2+y\_2$
* Endowments: $\omega\_1=(2,0)$, $\omega\_2=(0,2)$

Set of feasible allocations is

$\mathcal{F}=\{((x\_1,y\_1),(x\_2,y\_2))\in\mathbb{R}^2\_+\times\mathbb{R}^2\_+|x\_1+x\_2=2 \ \wedge \ y\_1+y\_2=2\}$

Observe that the set of Pareto efficient allocations equals the set of feasible allocations:

$\mathcal{PE}=\mathcal{F}$

So, endowment is Pareto efficient. But core consists of all allocations on the diagonal of the Edgeworth box that satisfy $x\_1+y\_1=2$. So, Core equals

$\mathcal{C}=\{((x\_1,y\_1),(x\_2,y\_2))\in\mathcal{F}|x\_1+y\_1=2\}$

and it is not singleton.

So, you cannot prove that if the endowment is Pareto optimal then the core is singleton.

I suggest that you try to prove the following proposition:

**In a 2 consumer pure exchange economy, if the endowment allocation is Pareto optimal and all utility functions are strictly quasi-concave (or preferences are strictly convex) then the core is singleton and consists of the endowment allocation**


---

### Answer by username_0 on 2024-12-07. Score: 0

**Claim**: If the endowment is Pareto optimal, then the core is a singleton set.

---

**Notations:**

1. $\mathbb{N} = \{1, 2\}$ denotes the set of all individuals.
2. $\mathbb{M} = \{1, 2\}$ denotes the set of all goods.
3. $\mathbb{C}$ denotes the set of all allocations that satisfy the core property.
4. $u^i(\mathbf{x})$, where $\mathbf{x} \in \mathbb{R}^{2}\_{+}$, is the utility function of the $i^{\text{th}}$ agent.
   * The utility function of every agent is assumed to be strictly quasi-concave (implying strictly convex indifference curves), continuous, and strongly increasing.
5. $\mathbf{e} \in \mathbb{R}^{2 \times 2}\_{+}$ is the endowment vector, where $\mathbf{e}^i\_j \in \mathbb{R}\_{+}$ and $\mathbf{e}^i \in \mathbb{R}^2\_{+}$ denote the endowment of the $j^{\text{th}}$ good available to the $i^{\text{th}}$ agent and the endowment vector of the $i^{\text{th}}$ agent, respectively.

---

**Proof:**

The endowment $\mathbf{e}$ is **Pareto optimal**. Since $\mathbf{e}$ is at least as good as itself and feasible, it satisfies the core property. Hence, $\mathbf{e} \in \mathbb{C}$.

Now, let $\mathbf{x} \in \mathbb{R}^{2 \times 2}\_{+}$ be another feasible allocation such that $\mathbf{x} \in \mathbb{C}$ and $\mathbf{x} \neq \mathbf{e}$.  
Then $\mathbf{x}$ must also be Pareto optimal and at least as good as the endowment. That is, the following must hold:

1.  
$$
\neg \left(\exists \; \mathbf{y} \in \mathbb{R}^{2 \times 2}\_{+} \right) \left[ (\forall i \in \mathbb{N}) \big(u(\mathbf{y}^i) \geq u(\mathbf{x}^i) \land (\exists j \in \mathbb{N}) (u(\mathbf{y}^j) > u(\mathbf{x}^j))\big) \right]
$$

2.  
$$u(\mathbf{x}^i) \geq u(\mathbf{e}^i) \quad \forall i \in \mathbb{N}$$

From (2), consider the following cases:

---

**Case 1:**  
$\exists \; k \in \mathbb{N}$ such that $u(\mathbf{x}^k) > u(\mathbf{e}^k)$.

This implies:  
$\mathbf{x}$ is Pareto superior to $\mathbf{e} \implies \mathbf{e}$ is not Pareto optimal.  
This is a contradiction.

---

**Case 2:**  
$u(\mathbf{x}^i) = u(\mathbf{e}^i) \quad \forall i \in \mathbb{N}$.

This implies:  
$u(\mathbf{x})$ and $u(\mathbf{e})$ lie on the same indifference curve.

Moreover, since both $\mathbf{e}$ and $\mathbf{x}$ are Pareto optimal:  
The indifference curves of all individuals are tangent at both $\mathbf{x}$ and $\mathbf{e}$.

However, since indifference curves are strictly convex:  
This implies either $\mathbf{x} = \mathbf{e}$, or there are multiple points of tangency.  
The latter cannot occur because of strict convexity, and the former contradicts the assumption that $\mathbf{x} \neq \mathbf{e}$.

In either case, we arrive at a contradiction.

---

Therefore, the core consists of a single allocation, which is the endowment itself.

As a corollary, we conclude that **if the endowment satisfies the core property, the Walrasian equilibrium is the endowment**.


---

