<p>Assume there are three individuals <span class="math-container">$A, B, C$</span> and three goods <span class="math-container">$x, y, z$</span>. Endowment for <span class="math-container">$A$</span> is <span class="math-container">$(1,0,0)$</span> for <span class="math-container">$B$</span> is <span class="math-container">$(0,1,0)$</span> and for <span class="math-container">$C$</span> is <span class="math-container">$(0,0,1)$</span>.</p>
<p>Assume the utility function for the individuals are:
<span class="math-container">$$
u^A(x,y,z) = xy + z, u^B(x,y,z) = x + yz, U^C(x,y,z) = xz + y
$$</span>
The utilities at the endowments are
<span class="math-container">$$
u^A(1,0,0) = 0, u^B(0,1,0) = 0, u^C(0,0,1) = 0.
$$</span>
Now consider the allocation where <span class="math-container">$A$</span> gets <span class="math-container">$(0,0,0)$</span> for <span class="math-container">$B$</span> receives <span class="math-container">$(1,1,1)$</span> and for <span class="math-container">$C$</span> receives <span class="math-container">$(0,0,0)$</span>.
This gives utilities:
<span class="math-container">$$
u^A(0,0,0) = 0, u^B(1,1,1) = 2, u^C(0,0,0) = 0,
$$</span>
So these are individual rational. It is also clear that this is Pareto efficient as any reallocation would decrease the utility of <span class="math-container">$B$</span>.</p>
<p>Note however that the coalition <span class="math-container">$\{A, C\}$</span> has a profitable deviation. For example, they could switch their endowments, so <span class="math-container">$A$</span> consumes <span class="math-container">$(0, 0, 1)$</span> and <span class="math-container">$C$</span> consumes <span class="math-container">$(1, 0,0)$</span>, giving both a utility equal to 1. So the above Pareto efficient and individual rational allocation is not in the core.</p>

CC BY-SA 4.0
2024-06-18T10:05:21.047Z
Hi! the utility function $u^A(x,y,z)=xy + z$ is not quasi-concave. The reason is $u^A(\frac{1}{2}(1,1,0)+\frac{1}{2}(0,0,1))=u^A(\frac{1}{2},\frac{1}{2},\frac{1}{2})=\frac{3}{4}<1=\min(u^A(0,0,1),u^A(1,1,0))$. To get quasi-concavity, you can modify the utility function to $u^A(x,y,z)=\sqrt{xy} + z$. Likewise, for B and C.
Amit
CC BY-SA 4.0
2024-02-09T15:08:52.143Z
2024-02-09T15:08:52.143Z
tdm
<p>Here is an example of a pure-exchange economy with two goods (X and Y) and three consumers (1,2 and 3):</p>
<ul>
<li>Utilities: <span class="math-container">$u_1(x_1,y_1)=\min(x_1,y_1)$</span>, <span class="math-container">$u_2(x_2,y_2)=\min(x_2,y_2)$</span>, <span class="math-container">$u_3(x_3,y_3)=\min(x_3,y_3)$</span></li>
<li>Endowments: <span class="math-container">$\omega_1=(1,0)$</span>, <span class="math-container">$\omega_2=(0,1)$</span>, <span class="math-container">$\omega_3=(0,0)$</span></li>
</ul>
<p>Note that <span class="math-container">$u_1(1,0)=u_2(0,1)=u_3(0,0)=0$</span>. Consider an allocation <span class="math-container">$((x_1,y_1),(x_2,y_2),(x_3,y_3))=((0,0),(0,0),(1,1))$</span>. At this allocation, utility of <span class="math-container">$1$</span> and <span class="math-container">$2$</span> is <span class="math-container">$0$</span> each and <span class="math-container">$3$</span> gets the utility <span class="math-container">$1$</span>. Observe that it is Pareto efficient and also individually rational, but it is not in the core because it can be blocked by the coalition of <span class="math-container">$1$</span> and <span class="math-container">$2$</span> in a way such that <span class="math-container">$1$</span> and <span class="math-container">$2$</span> can consume <span class="math-container">$\left(\frac{1}{2},\frac{1}{2}\right)$</span> each and get utility <span class="math-container">$\frac{1}{2}$</span> each.</p>
<p>Note that the same argument also works for another economy</p>
<ul>
<li>Utilities:
<span class="math-container">$u_1(x_1,y_1)=\sqrt{x_1y_1}$</span>, <span class="math-container">$u_2(x_2,y_2)=\sqrt{x_2y_2}$</span>, <span class="math-container">$u_3(x_3,y_3)=\sqrt{x_3y_3}$</span></li>
<li>Endowments: <span class="math-container">$\omega_1=(1,0)$</span>, <span class="math-container">$\omega_2=(0,1)$</span>, <span class="math-container">$\omega_3=(0,0)$</span></li>
</ul>

CC BY-SA 4.0
2024-06-18T11:34:31.600Z
2024-06-18T11:34:31.600Z
Amit
<p>Suppose that an economy has <span class="math-container">$n$</span> agents, each with endowment <span class="math-container">$\omega_i$</span>.
Their preferences are represented by the quasiconcave and increasing function <span class="math-container">$u_i$</span>.
Let the set of Pareto efficient allocations be <span class="math-container">$E$</span> and let <span class="math-container">$C_i$</span> be the upper-contour set for <span class="math-container">$u_i(\omega_i)$</span> for all <span class="math-container">$i$</span>.</p>
<p>My question is: can we define the core as the intersection of <span class="math-container">$E$</span> and all <span class="math-container">$C_i$</span>, so <span class="math-container">$core = E \cap \left(\bigcap_{i = 1}^n C_i\right)$</span>?
I think this definition works when <span class="math-container">$n = 2$</span>, but what about when <span class="math-container">$n &gt; 2$</span>?</p>

<general-equilibrium><pareto-efficiency>
Can we define the core as the intersection of Pareto efficient allocations and upper-contour sets?
RobinsonWM
# Can we define the core as the intersection of Pareto efficient allocations and upper-contour sets?

(Asked by: username_0 on 2024-02-09)

Suppose that an economy has $n$ agents, each with endowment $\omega\_i$.
Their preferences are represented by the quasiconcave and increasing function $u\_i$.
Let the set of Pareto efficient allocations be $E$ and let $C\_i$ be the upper-contour set for $u\_i(\omega\_i)$ for all $i$.

My question is: can we define the core as the intersection of $E$ and all $C\_i$, so $core = E \cap \left(\bigcap\_{i = 1}^n C\_i\right)$?
I think this definition works when $n = 2$, but what about when $n > 2$?

**Tags:** general-equilibrium, pareto-efficiency

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## 2 answers

### Answer by username_1 on 2024-02-09. Score: 2 | Accepted answer

Assume there are three individuals $A, B, C$ and three goods $x, y, z$. Endowment for $A$ is $(1,0,0)$ for $B$ is $(0,1,0)$ and for $C$ is $(0,0,1)$.

Assume the utility function for the individuals are:
$$
u^A(x,y,z) = xy + z, u^B(x,y,z) = x + yz, U^C(x,y,z) = xz + y
$$
The utilities at the endowments are
$$
u^A(1,0,0) = 0, u^B(0,1,0) = 0, u^C(0,0,1) = 0.
$$
Now consider the allocation where $A$ gets $(0,0,0)$ for $B$ receives $(1,1,1)$ and for $C$ receives $(0,0,0)$.
This gives utilities:
$$
u^A(0,0,0) = 0, u^B(1,1,1) = 2, u^C(0,0,0) = 0,
$$
So these are individual rational. It is also clear that this is Pareto efficient as any reallocation would decrease the utility of $B$.

Note however that the coalition $\{A, C\}$ has a profitable deviation. For example, they could switch their endowments, so $A$ consumes $(0, 0, 1)$ and $C$ consumes $(1, 0,0)$, giving both a utility equal to 1. So the above Pareto efficient and individual rational allocation is not in the core.

> **Comments:**
> **username_2**: Hi! the utility function $u^A(x,y,z)=xy + z$ is not quasi-concave. The reason is $u^A(\frac{1}{2}(1,1,0)+\frac{1}{2}(0,0,1))=u^A(\frac{1}{2},\frac{1}{2},\frac{1}{2})=\frac{3}{4}<1=\min(u^A(0,0,1),u^A(1,1,0))$. To get quasi-concavity, you can modify the utility function to $u^A(x,y,z)=\sqrt{xy} + z$. Likewise, for B and C.  

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### Answer by username_2 on 2024-06-18. Score: 1

Here is an example of a pure-exchange economy with two goods (X and Y) and three consumers (1,2 and 3):

* Utilities: $u\_1(x\_1,y\_1)=\min(x\_1,y\_1)$, $u\_2(x\_2,y\_2)=\min(x\_2,y\_2)$, $u\_3(x\_3,y\_3)=\min(x\_3,y\_3)$
* Endowments: $\omega\_1=(1,0)$, $\omega\_2=(0,1)$, $\omega\_3=(0,0)$

Note that $u\_1(1,0)=u\_2(0,1)=u\_3(0,0)=0$. Consider an allocation $((x\_1,y\_1),(x\_2,y\_2),(x\_3,y\_3))=((0,0),(0,0),(1,1))$. At this allocation, utility of $1$ and $2$ is $0$ each and $3$ gets the utility $1$. Observe that it is Pareto efficient and also individually rational, but it is not in the core because it can be blocked by the coalition of $1$ and $2$ in a way such that $1$ and $2$ can consume $\left(\frac{1}{2},\frac{1}{2}\right)$ each and get utility $\frac{1}{2}$ each.

Note that the same argument also works for another economy

* Utilities:
  $u\_1(x\_1,y\_1)=\sqrt{x\_1y\_1}$, $u\_2(x\_2,y\_2)=\sqrt{x\_2y\_2}$, $u\_3(x\_3,y\_3)=\sqrt{x\_3y\_3}$
* Endowments: $\omega\_1=(1,0)$, $\omega\_2=(0,1)$, $\omega\_3=(0,0)$


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