<p>The diagram of the edge worth box is correct and yes indeed the points on two red lines are Pareto efficient but in addition to that the points contained in the area between these two red lines (two triangles formed only by the red lines) are also Pareto efficient allocations.</p>

CC BY-SA 4.0
2023-01-07T18:57:29.770Z
2023-01-07T18:57:54.043Z
2023-01-07T18:57:54.043Z
Amit Ranjan
<p>Consider a very simple economy with two agents (A, B) and two goods (<span class="math-container">$x$</span>, <span class="math-container">$y$</span>). Agent A has utility <span class="math-container">$u_a = \min \{x_a, 2y_a\}$</span>; agent B symmetrically has utility <span class="math-container">$u_b = \min \{2x_b, y_b\}$</span>. Suppose that there are 3 units of <span class="math-container">$x$</span> and 3 units of <span class="math-container">$y$</span> in total.</p>
<p>It seems that, if <span class="math-container">$x_a = 2y_a$</span> and endowments are exhausted, then the allocation is efficient. In addition, if <span class="math-container">$2x_b = y_b$</span> and endowments are exhausted, i.e. <span class="math-container">$2(3 - x_a) = 3 - y_a$</span>, then the allocation is also efficient. Finally, there seem to be some efficient allocations on the boundary of the feasible set. Plotting these yields</p>
<p><a href="https://i.sstatic.net/WYQ6r.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/WYQ6r.png" alt="enter image description here" /></a></p>
<p>Can I confirm that the red lines do indeed depict the set of efficient allocations? I'm not looking for a formal proof, but just a confirmation that I haven't overlooked anything in this problem.</p>

CC BY-SA 4.0
2023-01-07T16:07:16.297Z
You can follow the approach posted here. I think it'll help: https://youtu.be/HzJyV5zBaPg
Amit
<general-equilibrium>
Efficient allocations with perfect complements
afreelunch
# Efficient allocations with perfect complements

(Asked by: username_0 on 2023-01-07)

Consider a very simple economy with two agents (A, B) and two goods ($x$, $y$). Agent A has utility $u\_a = \min \{x\_a, 2y\_a\}$; agent B symmetrically has utility $u\_b = \min \{2x\_b, y\_b\}$. Suppose that there are 3 units of $x$ and 3 units of $y$ in total.

It seems that, if $x\_a = 2y\_a$ and endowments are exhausted, then the allocation is efficient. In addition, if $2x\_b = y\_b$ and endowments are exhausted, i.e. $2(3 - x\_a) = 3 - y\_a$, then the allocation is also efficient. Finally, there seem to be some efficient allocations on the boundary of the feasible set. Plotting these yields

[![enter image description here](https://i.sstatic.net/WYQ6r.png)](https://i.sstatic.net/WYQ6r.png)

Can I confirm that the red lines do indeed depict the set of efficient allocations? I'm not looking for a formal proof, but just a confirmation that I haven't overlooked anything in this problem.

**Tags:** general-equilibrium

> **Comments:**
> **username_1**: You can follow the approach posted here. I think it'll help: https://youtu.be/HzJyV5zBaPg  

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## 1 answers

### Answer by username_2 on 2023-01-07. Score: 2 | Accepted answer

The diagram of the edge worth box is correct and yes indeed the points on two red lines are Pareto efficient but in addition to that the points contained in the area between these two red lines (two triangles formed only by the red lines) are also Pareto efficient allocations.


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