<p>In the economy you provided, set of feasible allocations is</p>
<p><span class="math-container">$\mathcal{F}=\{((x_1,y_1),(x_2,y_2))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_1+x_2=2 \ \wedge \ y_1+y_2=1\}$</span></p>
<p>and is represented by points in the Edgeworth Box.</p>
<p>Set of Pareto efficient Allocations is</p>
<p><span class="math-container">$\mathcal{PE}=\{((x_1,y_1),(x_2,y_2))\in\mathcal{F}|x_1=0 \ \vee \ y_1=1\}$</span></p>
<p>and is represented by the left and the top boundaries of the box.</p>
<p>Here is the picture:</p>
<p><a href="https://i.sstatic.net/lHut4.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/lHut4.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2023-10-31T00:02:30.293Z
For additional details on the methodology, please watch https://youtu.be/oWXHbX9j0mc?feature=shared
Amit
CC BY-SA 4.0
2023-10-31T00:12:38.517Z
Thank you dear Amit! But how do I find the highest and lowest value of $p_y$?
Ludwig Gershwin
CC BY-SA 4.0
2023-10-31T00:27:59.150Z
In this case, Walrasian equilibrium price ratio $\frac{p_X}{p_Y}$ will always lie in the interval $[\frac{1}{2},1]$. Given that $p_X=1$, this means that $p_Y\in [1,2]$. Exact price ratio will, however, depend on the initial endowment allocation. For example: See this https://youtu.be/wWXLkgwWmgc?feature=shared
Amit
CC BY-SA 4.0
2023-10-31T00:28:09.563Z
very nice graph, did you made it using tikz? What settings did you use for font, and arrow axes?
1muflon1
CC BY-SA 4.0
2023-10-31T00:30:45.100Z
@1muflon1 Thanks. I used Mathcha to plot the graph in the answer : https://www.mathcha.io Please try it. I find it very fast and convenient to plot graphs. Let me know if you need any help.
Amit
CC BY-SA 4.0
2023-10-30T23:58:47.663Z
2023-10-30T23:58:47.663Z
Amit
<p>Suppose there are 2 individuals and 2 goods. <span class="math-container">$u_1(x_1,y_1)=x_1+2y_1$</span>, and <span class="math-container">$u_2(x_2,y_2)=x_2+y_2$</span>. There are 2 units of good <span class="math-container">$x$</span> and 1 unit of good <span class="math-container">$y$</span> in total.</p>
<p>How do I find the set of Pareto optimal allocations for this economy? Here's what I've got:
<span class="math-container">\begin{align*}
\max x_2+y_2\\
\text{s.t. }(2-x_2)+(1-y_2)=\bar{u}\implies x_2+2y_2=4-\bar{u}\\
\implies \max_x x+\frac{4-\bar{u}-x}{2}\\
\text{s.t. }0\leq x\leq 2
\end{align*}</span>
Thus, <span class="math-container">$x_2=2$</span> is the optimal. Is this correct? Also, suppose at the Walrasian equilibrium <span class="math-container">$p_x^*=1$</span>. What is the lowest and highest of <span class="math-container">$p_y^*$</span> could be?</p>

<pure-exchange-economy>
How do I find the set of pareto optimal allocations?
Ludwig Gershwin
# How do I find the set of pareto optimal allocations?

(Asked by: username_0 on 2023-10-30)

Suppose there are 2 individuals and 2 goods. $u\_1(x\_1,y\_1)=x\_1+2y\_1$, and $u\_2(x\_2,y\_2)=x\_2+y\_2$. There are 2 units of good $x$ and 1 unit of good $y$ in total.

How do I find the set of Pareto optimal allocations for this economy? Here's what I've got:
\begin{align\*}
\max x\_2+y\_2\\
\text{s.t. }(2-x\_2)+(1-y\_2)=\bar{u}\implies x\_2+2y\_2=4-\bar{u}\\
\implies \max\_x x+\frac{4-\bar{u}-x}{2}\\
\text{s.t. }0\leq x\leq 2
\end{align\*}
Thus, $x\_2=2$ is the optimal. Is this correct? Also, suppose at the Walrasian equilibrium $p\_x^\*=1$. What is the lowest and highest of $p\_y^\*$ could be?

**Tags:** pure-exchange-economy


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## 1 answers

### Answer by username_1 on 2023-10-30. Score: 4 | Accepted answer

In the economy you provided, set of feasible allocations is

$\mathcal{F}=\{((x\_1,y\_1),(x\_2,y\_2))\in\mathbb{R}^2\_+\times\mathbb{R}^2\_+|x\_1+x\_2=2 \ \wedge \ y\_1+y\_2=1\}$

and is represented by points in the Edgeworth Box.

Set of Pareto efficient Allocations is

$\mathcal{PE}=\{((x\_1,y\_1),(x\_2,y\_2))\in\mathcal{F}|x\_1=0 \ \vee \ y\_1=1\}$

and is represented by the left and the top boundaries of the box.

Here is the picture:

[![enter image description here](https://i.sstatic.net/lHut4.png)](https://i.sstatic.net/lHut4.png)

> **Comments:**
> **username_1**: For additional details on the methodology, please watch https://youtu.be/oWXHbX9j0mc?feature=shared  
> **username_0**: Thank you dear username_1! But how do I find the highest and lowest value of $p_y$?  
> **username_1**: In this case, Walrasian equilibrium price ratio $\frac{p_X}{p_Y}$ will always lie in the interval $[\frac{1}{2},1]$. Given that $p_X=1$, this means that $p_Y\in [1,2]$. Exact price ratio will, however, depend on the initial endowment allocation. For example: See this https://youtu.be/wWXLkgwWmgc?feature=shared  
> **username_2**: very nice graph, did you made it using tikz? What settings did you use for font, and arrow axes?  
> **username_1**: @username_2 Thanks. I used Mathcha to plot the graph in the answer : https://www.mathcha.io Please try it. I find it very fast and convenient to plot graphs. Let me know if you need any help.  

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