Conditions on the Endowment to make the core a singleton set

Question

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.

Answer 1

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 2

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

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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.

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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:

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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.

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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.

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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.