<p>To answer your question, you can't use the Edgeworth box in this kind of setup because it is used for an economy where we have two goods and two agents. However, there exists an alternative graphical representation to solve such problems.</p>
<p><a href="https://i.sstatic.net/H4U1U.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/H4U1U.png" alt="enter image description here" /></a>
To see how we arrive at the above graph, first, plot the utility function for individual 1, i.e., <span class="math-container">$U_1(x_1)$</span>. Next, we plot the utility function for individual 2, but to get the picture above, we horizontally flip the graph of <span class="math-container">$U_2(x_2)$</span>. This sort of diagram helps us find the set of Pareto efficient allocations. In the picture above, notice that for <span class="math-container">$x_1\in[2,12]$</span> and <span class="math-container">$x_2\in[8,18]$</span>, an increase in the utility of individual 1 leads to a fall in the utility of individual 2 and vice versa.</p>
<p>Therefore, the green highlighted portion in the above diagram is the set of Pareto efficient locations, it can be written as <span class="math-container">$$\boxed{\mathcal{P}=\{(x_1,x_2)\in \mathbb{R}_+^2 \mid x_1\in [2,12] \land x_2=20-x_1\}}$$</span></p>
<p>Formally, the set of <em>Pareto Efficient</em> allocations is the set of all those allocations which maximize the utility of both individuals subject to the feasibility constraint.
In what follows, I use <span class="math-container">$\mathcal{F}$</span> to denote the set of all feasible allocations.</p>
<blockquote>
<p>A set <span class="math-container">$\mathcal{P} \subseteq \mathcal{F}=\{(x_1,x_2)\in \mathbb{R}_+^2 \mid x_1+x_2=20\}$</span> is called the set of Pareto Efficient allocations if for all <span class="math-container">$(x_1^*,x_2^*)\in\mathcal{P}$</span> the following holds:</p>
<ol>
<li><p>Given <span class="math-container">$x_2^*$</span>, <span class="math-container">$x_1^*$</span> is a solution to the problem <span class="math-container">$-$</span> <span class="math-container">$$\begin{align}\max_{x_1\geq 0} \quad &amp; U_1=-|x_1-2| \\ \textrm{s.t.} \quad &amp; x_1+x_2^*=20\end{align}$$</span></p>
</li>
<li><p>Given <span class="math-container">$x_1^*$</span>, <span class="math-container">$x_2^*$</span> is a solution to the problem <span class="math-container">$-$</span>
<span class="math-container">$$\begin{align} \max_{x_2\geq0} \quad &amp; U_2=-|x_2-8| \\ \textrm{s.t.} \quad &amp; x_1^*+x_2=20\end{align}$$</span></p>
</li>
</ol>
</blockquote>
<p>You can use the above definition to solve for Pareto Efficiency analytically.</p>
<p>For part b) we need to solve the following:
<span class="math-container">$$\begin{align}
\max_{x_1,x_2\geq 0} \quad &amp; \min(U_1(x_1),U_2(x_2)) \\
\textrm{s.t.} \quad &amp; x_1+x_2=20 \\ \\
\max_{x_1,x_2\geq 0} \quad &amp; \min(-|x_1-2|,-|x_2-8|) \\
\textrm{s.t.} \quad &amp; x_1+x_2=20 \\ \\
\max_{0 \leq x_1 \leq 20} \quad &amp; \min(-|x_1-2|, -|12-x_1|) &amp; [\text{using substitution}]
\end{align}$$</span></p>
<p>The above problem is solved at <span class="math-container">$x_1=7$</span>.
Thus, the professor chooses <span class="math-container">$\boxed{(x_1,x_2)=(7,13)}$</span> in order to maximise rawlsian social welfare function for the above setup.</p>

CC BY-SA 4.0
2023-06-23T06:54:30.643Z
According to the figure, isn't the Pareto efficient allocations for $x_1 \in [2,12]$
sachin
CC BY-SA 4.0
2023-06-23T19:09:40.017Z
@sachin Yes, you are right. I have edited the mistake, thanks for pointing it out!
mynameparv
CC BY-SA 4.0
2023-06-17T18:29:54.463Z
2023-06-23T19:08:39.687Z
2023-06-23T19:08:39.687Z
mynameparv
<p>A professor has 20 hours to allocate between two PhD students. Let x1 and x2 be the time allocated to the two students. The utility of each student is as follows:
<span class="math-container">$U_1 = −|x_1 − 2|$</span> and <span class="math-container">$U_2 = −|x_2 − 8|$</span></p>
<p>(a) Find the set of Pareto efficient allocations.</p>
<p>(b) What will be the optimal time allocation if the professor maximizes a social function given by : <span class="math-container">$min\{U_1, U_2\}$</span></p>
<p>I know about the Edgeworth box but don't know if it applies here as the utility of each is in one good only which is time, the max utility for <span class="math-container">$U_1$</span> is when <span class="math-container">$x_1=2$</span> and for <span class="math-container">$U_2$</span> is when <span class="math-container">$x_2=8$</span>, still 10 hours are left so I divide them equally and get <span class="math-container">$x_1=7$</span> and <span class="math-container">$x_2=13$</span> and <span class="math-container">$U_1=-5$</span> and <span class="math-container">$U_2=-5$</span>, any other combination will hurt at least one individual. But this is just one allocation, is there any else, also how to solve it properly?
No idea how to go about part (b).</p>

<microeconomics><utility><pareto-efficiency><edgeworth-box>
Find the set of Pareto efficient allocations. $U_1 = -|x_1-2|$ and $U_2 = −|x_2 − 8|$
sachin
# Find the set of Pareto efficient allocations. $U_1 = -|x_1-2|$ and $U_2 = −|x_2 − 8|$

(Asked by: username_0 on 2023-06-17)

A professor has 20 hours to allocate between two PhD students. Let x1 and x2 be the time allocated to the two students. The utility of each student is as follows:
$U\_1 = −|x\_1 − 2|$ and $U\_2 = −|x\_2 − 8|$

(a) Find the set of Pareto efficient allocations.

(b) What will be the optimal time allocation if the professor maximizes a social function given by : $min\{U\_1, U\_2\}$

I know about the Edgeworth box but don't know if it applies here as the utility of each is in one good only which is time, the max utility for $U\_1$ is when $x\_1=2$ and for $U\_2$ is when $x\_2=8$, still 10 hours are left so I divide them equally and get $x\_1=7$ and $x\_2=13$ and $U\_1=-5$ and $U\_2=-5$, any other combination will hurt at least one individual. But this is just one allocation, is there any else, also how to solve it properly?
No idea how to go about part (b).

**Tags:** microeconomics, utility, pareto-efficiency, edgeworth-box

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## 1 answers

### Answer by username_1 on 2023-06-17. Score: 2 | Accepted answer

To answer your question, you can't use the Edgeworth box in this kind of setup because it is used for an economy where we have two goods and two agents. However, there exists an alternative graphical representation to solve such problems.

[![enter image description here](https://i.sstatic.net/H4U1U.png)](https://i.sstatic.net/H4U1U.png)
To see how we arrive at the above graph, first, plot the utility function for individual 1, i.e., $U\_1(x\_1)$. Next, we plot the utility function for individual 2, but to get the picture above, we horizontally flip the graph of $U\_2(x\_2)$. This sort of diagram helps us find the set of Pareto efficient allocations. In the picture above, notice that for $x\_1\in[2,12]$ and $x\_2\in[8,18]$, an increase in the utility of individual 1 leads to a fall in the utility of individual 2 and vice versa.

Therefore, the green highlighted portion in the above diagram is the set of Pareto efficient locations, it can be written as $$\boxed{\mathcal{P}=\{(x\_1,x\_2)\in \mathbb{R}\_+^2 \mid x\_1\in [2,12] \land x\_2=20-x\_1\}}$$

Formally, the set of *Pareto Efficient* allocations is the set of all those allocations which maximize the utility of both individuals subject to the feasibility constraint.
In what follows, I use $\mathcal{F}$ to denote the set of all feasible allocations.

> A set $\mathcal{P} \subseteq \mathcal{F}=\{(x\_1,x\_2)\in \mathbb{R}\_+^2 \mid x\_1+x\_2=20\}$ is called the set of Pareto Efficient allocations if for all $(x\_1^\*,x\_2^\*)\in\mathcal{P}$ the following holds:
>
> 1. Given $x\_2^\*$, $x\_1^\*$ is a solution to the problem $-$ $$\begin{align}\max\_{x\_1\geq 0} \quad & U\_1=-|x\_1-2| \\ \textrm{s.t.} \quad & x\_1+x\_2^\*=20\end{align}$$
> 2. Given $x\_1^\*$, $x\_2^\*$ is a solution to the problem $-$
>    $$\begin{align} \max\_{x\_2\geq0} \quad & U\_2=-|x\_2-8| \\ \textrm{s.t.} \quad & x\_1^\*+x\_2=20\end{align}$$

You can use the above definition to solve for Pareto Efficiency analytically.

For part b) we need to solve the following:
$$\begin{align}
\max\_{x\_1,x\_2\geq 0} \quad & \min(U\_1(x\_1),U\_2(x\_2)) \\
\textrm{s.t.} \quad & x\_1+x\_2=20 \\ \\
\max\_{x\_1,x\_2\geq 0} \quad & \min(-|x\_1-2|,-|x\_2-8|) \\
\textrm{s.t.} \quad & x\_1+x\_2=20 \\ \\
\max\_{0 \leq x\_1 \leq 20} \quad & \min(-|x\_1-2|, -|12-x\_1|) & [\text{using substitution}]
\end{align}$$

The above problem is solved at $x\_1=7$.
Thus, the professor chooses $\boxed{(x\_1,x\_2)=(7,13)}$ in order to maximise rawlsian social welfare function for the above setup.

> **Comments:**
> **username_0**: According to the figure, isn't the Pareto efficient allocations for $x_1 \in [2,12]$  
> **username_1**: @sachin Yes, you are right. I have edited the mistake, thanks for pointing it out!  

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