<p>Given a pure-exchange economy:</p>
<ul>
<li><span class="math-container">$u_1(x_1,y_1)=\max(x_1,y_1)$</span>, <span class="math-container">$\omega_1=(4,0)$</span></li>
<li><span class="math-container">$u_2(x_2,y_2)=\min(2x_2+y_2,x_2+2y_2)$</span>, <span class="math-container">$\omega_2=(0,3)$</span></li>
</ul>
<p>Set of feasible allocations is given by</p>
<p><span class="math-container">$\mathcal{F}=\{((x_1,y_1),(x_2,y_2))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_1+x_2=4 \ \wedge \ y_1+y_2=3\}$</span></p>
<p><a href="https://i.sstatic.net/Ms6IwipB.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Ms6IwipB.png" alt="enter image description here" /></a></p>
<p>As can be seen in the picture above, set of Pareto efficient allocations is given by</p>
<p><span class="math-container">$\mathcal{P}=\{((x_1,y_1),(x_2,y_2))\in\mathcal{F}|y_1=0\}$</span></p>
<p>Utility Possibility Frontier is therefore,
<span class="math-container">\begin{eqnarray*} u_2=\begin{cases} 10-u_1 &amp; \text{if } 0\leq u_1\leq 1 \\ 11-2u_1 &amp; \text{if } 1\leq u_1\leq 4 \end{cases}\end{eqnarray*}</span></p>
<p>which can also be expressed as <span class="math-container">$u_2=\min(10-u_1,11-2u_1)$</span> when <span class="math-container">$u_1\in [0,4]$</span>. Alternatively, we can also express it as <span class="math-container">$\max(u_1+u_2-10,2u_1+u_2-11)=0$</span> where <span class="math-container">$u_1\in [0,4]$</span>.</p>
<p>Here is the graph of it:</p>
<p><a href="https://i.sstatic.net/yrWXnUh0.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/yrWXnUh0.png" alt="enter image description here" /></a></p>
<p>Here is how we can write the utility possibility set:</p>
<p><span class="math-container">$\mathcal{U}=\{(u_1,u_2)\in\mathbb{R}^2|u_1\leq 4 \wedge u_2\leq 10 \wedge \max(u_1+u_2-10,2u_1+u_2-11)\leq 0\}$</span></p>
<p>Here is the picture:</p>
<p><a href="https://i.sstatic.net/8MlnszcT.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/8MlnszcT.png" alt="enter image description here" /></a></p>
<p>Regarding the competitive equilibrium, you are right that it does not exist.</p>

CC BY-SA 4.0
2024-08-15T18:42:13.513Z
2024-08-15T18:49:56.577Z
2024-08-15T18:49:56.577Z
Amit
<blockquote>
<p>Consider an Exchange economy with two individuals <span class="math-container">$a$</span> and <span class="math-container">$b$</span> and two goods <span class="math-container">$1$</span> and <span class="math-container">$2$</span>.<br />
Denote consumption of good <span class="math-container">$j$</span> by individual <span class="math-container">$i$</span> as <span class="math-container">$x_{ij}$</span> for <span class="math-container">$i\in\{a,b\}$</span> and <span class="math-container">$j\in\{1,2\}$</span></p>
<p>The utility function of agent <span class="math-container">$a$</span> is <span class="math-container">$u_a=\max(x_{a1},x_{a2})$</span>  and his endowments are <span class="math-container">$\omega_a=(4,0)$</span><br />
The utility function of agent <span class="math-container">$b$</span> is <span class="math-container">$u_b=\min(2x_{b1}+x_{b2},x_{b1}+2x_{b2})$</span> with endowments <span class="math-container">$\omega_b=(0,3)$</span></p>
<p>Find the set of Pareto Efficient Allocations and then find Utility Possibility Set as well as the Utility Possibility Frontier(UPF). Also find the set of competitive equilibrium</p>
</blockquote>
<p>In the above problem using the Edgeworth box and graphical inspection, I was able to find out the set of Pareto efficient allocation as: <span class="math-container">$\mathcal{P}=\{((x_{a1},x_{a2}),(x_{b1},x_{b2}))\in \mathcal{F} \mid x_{a2}=0\}$</span> where <span class="math-container">$\mathcal{F}=\{(x_{a1},x_{a2}),(x_{b1},x_{b2}))\in \mathbb{R}_+^4 \mid x_{a1}+x_{b1}=4 \land x_{a2}+x_{b2}=3\}$</span> is the set of feasible allocations</p>
<p>I know that in order to get the UPF we need to solve the following to get a relationship between utilities and endowments:<br />
<span class="math-container">$\bullet$</span> Feasibility: <span class="math-container">$x_{a1}+x_{b1}=4 $</span> and <span class="math-container">$x_{a2}+x_{b2}=3$</span><br />
<span class="math-container">$\bullet$</span> Utilities: <span class="math-container">$u_a=\max(x_{a1},x_{a2})$</span> and <span class="math-container">$u_b=\min(2x_{b1}+x_{b2},x_{b1}+2x_{b2})$</span><br />
<span class="math-container">$\bullet$</span> Efficiency: <span class="math-container">$x_{a2}=0$</span></p>
<p>Solving the above gives the UPF:  <span class="math-container">$u_b=\min(11-2u_a,10-u_a)$</span>
However, I am unsure of how we express the utility possibility set.</p>
<p>My guess is that analytically it can be expressed as <span class="math-container">$$\mathcal{U}=\{(u_a,u_b)\mid u_a=\max(x_{a1},x_{a2}),\; \min(2x_{b1}+x_{b2}, \; x_{b1}+2x_{b2}),\; x_{b1}+2x_{b2},\; x_{a1}+x_{b1}=4,\; x_{a2}+x_{b2}=3\}$$</span> and graphically it is the set of the area under the UPF as the UPF is the upper frontier of the utility possibility set.</p>
<p>For the competitive equilibrium part, I am not able to see any price such that both markets clear therefore I don't think there exists a CE but I am not sure.</p>
<p>Please verify whether my claims are correct or not and if not please help me out</p>

<microeconomics><mathematical-economics><self-study><pareto-efficiency><pure-exchange-economy>
Utility Possibility Set and Utility Possibility Frontier
mynameparv
# Utility Possibility Set and Utility Possibility Frontier

(Asked by: username_0 on 2023-06-02)

> Consider an Exchange economy with two individuals $a$ and $b$ and two goods $1$ and $2$.  
> Denote consumption of good $j$ by individual $i$ as $x\_{ij}$ for $i\in\{a,b\}$ and $j\in\{1,2\}$
>
> The utility function of agent $a$ is $u\_a=\max(x\_{a1},x\_{a2})$ and his endowments are $\omega\_a=(4,0)$  
> The utility function of agent $b$ is $u\_b=\min(2x\_{b1}+x\_{b2},x\_{b1}+2x\_{b2})$ with endowments $\omega\_b=(0,3)$
>
> Find the set of Pareto Efficient Allocations and then find Utility Possibility Set as well as the Utility Possibility Frontier(UPF). Also find the set of competitive equilibrium

In the above problem using the Edgeworth box and graphical inspection, I was able to find out the set of Pareto efficient allocation as: $\mathcal{P}=\{((x\_{a1},x\_{a2}),(x\_{b1},x\_{b2}))\in \mathcal{F} \mid x\_{a2}=0\}$ where $\mathcal{F}=\{(x\_{a1},x\_{a2}),(x\_{b1},x\_{b2}))\in \mathbb{R}\_+^4 \mid x\_{a1}+x\_{b1}=4 \land x\_{a2}+x\_{b2}=3\}$ is the set of feasible allocations

I know that in order to get the UPF we need to solve the following to get a relationship between utilities and endowments:  
$\bullet$ Feasibility: $x\_{a1}+x\_{b1}=4 $ and $x\_{a2}+x\_{b2}=3$  
$\bullet$ Utilities: $u\_a=\max(x\_{a1},x\_{a2})$ and $u\_b=\min(2x\_{b1}+x\_{b2},x\_{b1}+2x\_{b2})$  
$\bullet$ Efficiency: $x\_{a2}=0$

Solving the above gives the UPF: $u\_b=\min(11-2u\_a,10-u\_a)$
However, I am unsure of how we express the utility possibility set.

My guess is that analytically it can be expressed as $$\mathcal{U}=\{(u\_a,u\_b)\mid u\_a=\max(x\_{a1},x\_{a2}),\; \min(2x\_{b1}+x\_{b2}, \; x\_{b1}+2x\_{b2}),\; x\_{b1}+2x\_{b2},\; x\_{a1}+x\_{b1}=4,\; x\_{a2}+x\_{b2}=3\}$$ and graphically it is the set of the area under the UPF as the UPF is the upper frontier of the utility possibility set.

For the competitive equilibrium part, I am not able to see any price such that both markets clear therefore I don't think there exists a CE but I am not sure.

Please verify whether my claims are correct or not and if not please help me out

**Tags:** microeconomics, mathematical-economics, self-study, pareto-efficiency, pure-exchange-economy


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## 1 answers

### Answer by username_1 on 2024-08-15. Score: 1 | Accepted answer

Given a pure-exchange economy:

* $u\_1(x\_1,y\_1)=\max(x\_1,y\_1)$, $\omega\_1=(4,0)$
* $u\_2(x\_2,y\_2)=\min(2x\_2+y\_2,x\_2+2y\_2)$, $\omega\_2=(0,3)$

Set of feasible allocations is given by

$\mathcal{F}=\{((x\_1,y\_1),(x\_2,y\_2))\in\mathbb{R}^2\_+\times\mathbb{R}^2\_+|x\_1+x\_2=4 \ \wedge \ y\_1+y\_2=3\}$

[![enter image description here](https://i.sstatic.net/Ms6IwipB.png)](https://i.sstatic.net/Ms6IwipB.png)

As can be seen in the picture above, set of Pareto efficient allocations is given by

$\mathcal{P}=\{((x\_1,y\_1),(x\_2,y\_2))\in\mathcal{F}|y\_1=0\}$

Utility Possibility Frontier is therefore,
\begin{eqnarray\*} u\_2=\begin{cases} 10-u\_1 & \text{if } 0\leq u\_1\leq 1 \\ 11-2u\_1 & \text{if } 1\leq u\_1\leq 4 \end{cases}\end{eqnarray\*}

which can also be expressed as $u\_2=\min(10-u\_1,11-2u\_1)$ when $u\_1\in [0,4]$. Alternatively, we can also express it as $\max(u\_1+u\_2-10,2u\_1+u\_2-11)=0$ where $u\_1\in [0,4]$.

Here is the graph of it:

[![enter image description here](https://i.sstatic.net/yrWXnUh0.png)](https://i.sstatic.net/yrWXnUh0.png)

Here is how we can write the utility possibility set:

$\mathcal{U}=\{(u\_1,u\_2)\in\mathbb{R}^2|u\_1\leq 4 \wedge u\_2\leq 10 \wedge \max(u\_1+u\_2-10,2u\_1+u\_2-11)\leq 0\}$

Here is the picture:

[![enter image description here](https://i.sstatic.net/8MlnszcT.png)](https://i.sstatic.net/8MlnszcT.png)

Regarding the competitive equilibrium, you are right that it does not exist.


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