<p>We'll consider three cases:</p>
<ul>
<li>For <span class="math-container">$\alpha \in [\frac{1}{2}, 2]$</span>, there are only two Pareto efficient allocations: <span class="math-container">$\big\{\big((2,0),(0,2\alpha)\big), \big((0,2\alpha),(2,0)\big) \big\}$</span></li>
<li>For <span class="math-container">$\alpha &gt; 2$</span>, set of Pareto efficient allocations is <span class="math-container">$\big\{\big((2,0),(0,2\alpha)\big), \big((0,2\alpha),(2,0)\big) \big\} \bigcup \big\{\big((x_A,y_A),(x_B,y_B)\big)\in F: y_A &gt; 2, y_B &gt; 2 \big\} $</span>, where <span class="math-container">$F$</span> denotes the set of all feasible allocations.</li>
<li>For <span class="math-container">$\alpha &lt; \frac{1}{2}$</span>, set of Pareto efficient allocations is <span class="math-container">$\big\{\big((2,0),(0,2\alpha)\big), \big((0,2\alpha),(2,0)\big) \big\} \bigcup \big\{\big((x_A,y_A),(x_B,y_B)\big)\in F: x_A &gt; 2\alpha, x_B &gt; 2\alpha \big\} $</span>, where <span class="math-container">$F$</span> denotes the set of all feasible allocations.</li>
</ul>

CC BY-SA 4.0
2022-09-06T00:06:28.633Z
2022-09-06T00:06:28.633Z
Amit
<p>There is an exchange economy with two people and two goods.</p>
<p>Utility functions are</p>
<p><span class="math-container">$u_A(x_A, y_A)=\max\{x_A, y_A\}$</span></p>
<p><span class="math-container">$u_B(x_B, y_B)=\max\{x_B, y_B\}$</span></p>
<p>Endowments are <span class="math-container">$w_A(1,\alpha)$</span> and <span class="math-container">$w_B(1,\alpha)$</span> for <span class="math-container">$\alpha &gt;0$</span></p>
<p>Find the set of Pareto efficient allocations and show them in the Edgeworth box.</p>
<p>—-</p>
<p>For that I consider three cases in terms of <span class="math-container">$\alpha$</span></p>
<p>Case 1:  <span class="math-container">$\alpha=1$</span></p>
<p>Then we draw square-shaped edgeworth box. And the Pareto efficient allocations are only {<span class="math-container">$(0,2),(2,0)$</span>} and  {<span class="math-container">$(2,0),(0,2)$</span>}.</p>
<p>Case 2: <span class="math-container">$\alpha &gt;1$</span>. Let’s assume that <span class="math-container">$\alpha=2$</span></p>
<p>Then we draw rectangle Edgeworth box. And Pareto efficient allocations are</p>
<p>{<span class="math-container">$(0,2),(2,2)$</span>} and  {<span class="math-container">$(2,2),(0,2)$</span>} and the all point along the line between there two allocations. (Green line in the picture).</p>
<p>Case 3: <span class="math-container">$\alpha &lt;1$</span>. Let’s assume that <span class="math-container">$\alpha=1/2$</span></p>
<p>Then we draw rectangle Edgeworth box. And Pareto efficient allocations are</p>
<p>{<span class="math-container">$(1,0),(1,1)$</span>} and  {<span class="math-container">$(1,1),(1,0)$</span>} and the all point along the line between there two allocations. (Green line in the picture).</p>
<p>Sorry for hand-writing picture but could not draw this in latex format. The case 1 is the simple version. However, I am not sure about the case 2 and case 3. (<span class="math-container">$\alpha&gt;1 $</span> and <span class="math-container">$\alpha&lt;1$</span>). I think the Pareto efficient allocations which I found are wrong for cases 2 and 3. These seems not logical to me. Please discuss with me about the correct Pareto optimal allocations. Many thanks.</p>
<p><a href="https://i.sstatic.net/XNp4j.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/XNp4j.jpg" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2022-09-04T14:30:34.063Z
Hi! I made [this Desmos graph](https://www.desmos.com/calculator/aqhgw7tqmr) for you, so you can experiment with it! A note: your utility functions are somewhat unusual, max instead of min, so the graph displays for both consumers all points with lower utility than the current allocations rather than just the level curve.
Giskard
CC BY-SA 4.0
2022-09-05T00:18:27.370Z
For $\alpha = 2$, points on the green line are not Pareto efficient. Likewise for $\alpha=\frac{1}{2}$ also, points on the green line are not efficient.
Amit
CC BY-SA 4.0
2022-09-05T02:36:56.503Z
I see the reasons why these green lines are not Pareto efficient. Then, likewise the case of (2,2), for the case $\alpha =2$, {(0,4), (2,0)} and {(2,0), (0,4)} are Pareto efficient. And for the case of $\alpha =1/2$, { (0,1),(2,0)} and {(2,0),(0,1))} are Pareto efficient. 

in general, {(2,0),(0,2$\alpha$)} and {(0,2$\alpha$),(2,0)} are Pareto efficient allocations. 

Is this right Professor @Amit?
studentp
CC BY-SA 4.0
2022-09-06T00:06:49.013Z
posted the answer
Amit
CC BY-SA 4.0
2022-09-06T04:08:43.487Z
@Giskard That's a good graph.
Amit
<microeconomics><self-study><general-equilibrium><pareto-efficiency><edgeworth-box>
Find the set of Pareto efficient allocations
studentp
# Find the set of Pareto efficient allocations

(Asked by: username_0 on 2022-09-03)

There is an exchange economy with two people and two goods.

Utility functions are

$u\_A(x\_A, y\_A)=\max\{x\_A, y\_A\}$

$u\_B(x\_B, y\_B)=\max\{x\_B, y\_B\}$

Endowments are $w\_A(1,\alpha)$ and $w\_B(1,\alpha)$ for $\alpha >0$

Find the set of Pareto efficient allocations and show them in the Edgeworth box.

—-

For that I consider three cases in terms of $\alpha$

Case 1: $\alpha=1$

Then we draw square-shaped edgeworth box. And the Pareto efficient allocations are only {$(0,2),(2,0)$} and {$(2,0),(0,2)$}.

Case 2: $\alpha >1$. Let’s assume that $\alpha=2$

Then we draw rectangle Edgeworth box. And Pareto efficient allocations are

{$(0,2),(2,2)$} and {$(2,2),(0,2)$} and the all point along the line between there two allocations. (Green line in the picture).

Case 3: $\alpha <1$. Let’s assume that $\alpha=1/2$

Then we draw rectangle Edgeworth box. And Pareto efficient allocations are

{$(1,0),(1,1)$} and {$(1,1),(1,0)$} and the all point along the line between there two allocations. (Green line in the picture).

Sorry for hand-writing picture but could not draw this in latex format. The case 1 is the simple version. However, I am not sure about the case 2 and case 3. ($\alpha>1 $ and $\alpha<1$). I think the Pareto efficient allocations which I found are wrong for cases 2 and 3. These seems not logical to me. Please discuss with me about the correct Pareto optimal allocations. Many thanks.

[![enter image description here](https://i.sstatic.net/XNp4j.jpg)](https://i.sstatic.net/XNp4j.jpg)

**Tags:** microeconomics, self-study, general-equilibrium, pareto-efficiency, edgeworth-box

> **Comments:**
> **username_1**: Hi! I made [this Desmos graph](https://www.desmos.com/calculator/aqhgw7tqmr) for you, so you can experiment with it! A note: your utility functions are somewhat unusual, max instead of min, so the graph displays for both consumers all points with lower utility than the current allocations rather than just the level curve.  
> **username_2**: For $\alpha = 2$, points on the green line are not Pareto efficient. Likewise for $\alpha=\frac{1}{2}$ also, points on the green line are not efficient.  
> **username_0**: I see the reasons why these green lines are not Pareto efficient. Then, likewise the case of (2,2), for the case $\alpha =2$, {(0,4), (2,0)} and {(2,0), (0,4)} are Pareto efficient. And for the case of $\alpha =1/2$, { (0,1),(2,0)} and {(2,0),(0,1))} are Pareto efficient.   in general, {(2,0),(0,2$\alpha$)} and {(0,2$\alpha$),(2,0)} are Pareto efficient allocations.   Is this right Professor @username_2?  
> **username_2**: posted the answer  
> **username_2**: @username_1 That's a good graph.  

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## 1 answers

### Answer by username_2 on 2022-09-06. Score: 2 | Accepted answer

We'll consider three cases:

* For $\alpha \in [\frac{1}{2}, 2]$, there are only two Pareto efficient allocations: $\big\{\big((2,0),(0,2\alpha)\big), \big((0,2\alpha),(2,0)\big) \big\}$
* For $\alpha > 2$, set of Pareto efficient allocations is $\big\{\big((2,0),(0,2\alpha)\big), \big((0,2\alpha),(2,0)\big) \big\} \bigcup \big\{\big((x\_A,y\_A),(x\_B,y\_B)\big)\in F: y\_A > 2, y\_B > 2 \big\} $, where $F$ denotes the set of all feasible allocations.
* For $\alpha < \frac{1}{2}$, set of Pareto efficient allocations is $\big\{\big((2,0),(0,2\alpha)\big), \big((0,2\alpha),(2,0)\big) \big\} \bigcup \big\{\big((x\_A,y\_A),(x\_B,y\_B)\big)\in F: x\_A > 2\alpha, x\_B > 2\alpha \big\} $, where $F$ denotes the set of all feasible allocations.


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