<p>Allocation <span class="math-container">$((x_A, y_A), (x_B, y_B)) = \left(\left(\frac{1}{4}, 0\right), \left(\frac{3}{4}, 1\right)\right)$</span> is Pareto superior to <span class="math-container">$\left(\left(\frac{1}{4}, \frac{1}{2}\right), \left(\frac{3}{4}, \frac{1}{2}\right)\right)$</span>, so option A is ruled out.</p>
<p>Allocation <span class="math-container">$\left(\left(\frac{1}{2}, 0\right), \left(\frac{1}{2}, 1\right)\right)$</span> is Pareto superior to <span class="math-container">$\left(\left(\frac{1}{2}, \frac{1}{4}\right), \left(\frac{1}{2}, \frac{3}{4}\right)\right)$</span>, so option B is ruled out.</p>
<p>Allocation <span class="math-container">$\left(\left(\frac{1}{2}, 0\right), \left(\frac{1}{2}, 1\right)\right)$</span> is Pareto superior to <span class="math-container">$\left(\left(\frac{1}{4}, 1\right), \left(\frac{3}{4}, 0\right)\right)$</span>, so option C is ruled out.</p>
<p>So, allocations in options A, B and C are not Pareto efficient.</p>
<p>Here is the set of all efficient allocations:</p>
<p><a href="https://i.sstatic.net/be4I5.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/be4I5.png" alt="enter image description here" /></a></p>
<p>Set of feasible allocations is
<span class="math-container">$\mathcal{F} = \left\{((x_A,y_A),(x_B,y_B))\in\mathbb{R}^2_+\times\mathbb{R}^2_+|x_A+x_B=1\wedge y_A+y_B=1\right\}$</span></p>
<p>Set of efficient allocations is
<span class="math-container">$\mathcal{E} = \left\{((x_A,y_A),(x_B,y_B))\in\mathcal{F}|\left(y_A=0\wedge 0\leq x_A\leq \frac{2}{3}\right)\vee\left(y_A=1\wedge \frac{1}{3}\leq x_A\leq 1\right)\right\}$</span></p>

CC BY-SA 4.0
2022-05-31T20:45:10.600Z
i was trying to solve this through edgeworth box, could you please see to it, i have just added that.
petearGriffin
CC BY-SA 4.0
2022-05-31T23:35:50.257Z
Shape of the IC looks correct to me. If you want to answer this MCQ using Edgeworth box, then draw ICs of the two individuals passing through the allocation in option D and check that it is efficient in the usual way.
Amit
CC BY-SA 4.0
2022-05-31T23:43:35.937Z
Also note that not all boundaries of the Edgeworth box are efficient. First argue that if a feasible allocation is efficient then it must satisfy $y_A=0$ or $y_A=1$. Note that the converse is not true (see option C).
Amit
CC BY-SA 4.0
2022-05-31T19:12:36.390Z
2023-07-29T16:45:18.810Z
2023-07-29T16:45:18.810Z
Amit
<p>A two-person two commodity economy has social endowment of x = 1 unit of food and y = 1 unit of wine. Agents preferences are increasing in own consumption but decreasing in wine consumption of the other person. Preferences of agents A and B are as follows,
<span class="math-container">$$u_A(x_A, y_A, y_B) = x_A[1 + max(y_A − y_B, 0)],$$</span>
<span class="math-container">$$u_B(x_B, y_B, y_A) = x_B[1 + max(y_B − y_A, 0)],$$</span>
where A consumes <span class="math-container">$x_A$</span> and <span class="math-container">$y_A$</span> units of x and y respectively, similarly B’s consumption
is <span class="math-container">$x_B$</span> and <span class="math-container">$y_B$</span>. Which of the following is a Pareto optimum allocation.</p>
<p><span class="math-container">$A. (1/4,1/2) \ (3/4,1/2)$</span></p>
<p><span class="math-container">$B. (1/2,1/4) \ (1/2,3/4)$</span></p>
<p><span class="math-container">$C. (1/4,1) \ (3/4,0)$</span></p>
<p><span class="math-container">$D. (1/4,0) \ (3/4,1)$</span></p>
<p><strong>Answer is supposed to be the option D</strong></p>
<p>My approach: if we take out the utilities associated with the given allocations then we'll get that Option B and C are providing the same utilities and for Agent B, option D is providing the best satisfaction.
<a href="https://i.sstatic.net/Lavii.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Lavii.jpg" alt="enter image description here" /></a></p>
<p>the answer must be a boundary solution according to me</p>
<p>i absolutely do not understand how to proceed after that. can someone please guide me?
thanks a lot.</p>

CC BY-SA 4.0
2022-05-31T19:14:53.643Z
Allocation in option D does not exhaust all of $y$. Is it  $\left(\left(\frac{1}{4}, 0\right), \left(\frac{3}{4}, 1\right)\right)$ ?
Amit
<general-equilibrium><pareto-efficiency>
what is the optimal allocation for this economy
petearGriffin
# what is the optimal allocation for this economy

(Asked by: username_0 on 2022-05-31)

A two-person two commodity economy has social endowment of x = 1 unit of food and y = 1 unit of wine. Agents preferences are increasing in own consumption but decreasing in wine consumption of the other person. Preferences of agents A and B are as follows,
$$u\_A(x\_A, y\_A, y\_B) = x\_A[1 + max(y\_A − y\_B, 0)],$$
$$u\_B(x\_B, y\_B, y\_A) = x\_B[1 + max(y\_B − y\_A, 0)],$$
where A consumes $x\_A$ and $y\_A$ units of x and y respectively, similarly B’s consumption
is $x\_B$ and $y\_B$. Which of the following is a Pareto optimum allocation.

$A. (1/4,1/2) \ (3/4,1/2)$

$B. (1/2,1/4) \ (1/2,3/4)$

$C. (1/4,1) \ (3/4,0)$

$D. (1/4,0) \ (3/4,1)$

**Answer is supposed to be the option D**

My approach: if we take out the utilities associated with the given allocations then we'll get that Option B and C are providing the same utilities and for Agent B, option D is providing the best satisfaction.
[![enter image description here](https://i.sstatic.net/Lavii.jpg)](https://i.sstatic.net/Lavii.jpg)

the answer must be a boundary solution according to me

i absolutely do not understand how to proceed after that. can someone please guide me?
thanks a lot.

**Tags:** general-equilibrium, pareto-efficiency

> **Comments:**
> **username_1**: Allocation in option D does not exhaust all of $y$. Is it  $\left(\left(\frac{1}{4}, 0\right), \left(\frac{3}{4}, 1\right)\right)$ ?  

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## 1 answers

### Answer by username_1 on 2022-05-31. Score: 4 | Accepted answer

Allocation $((x\_A, y\_A), (x\_B, y\_B)) = \left(\left(\frac{1}{4}, 0\right), \left(\frac{3}{4}, 1\right)\right)$ is Pareto superior to $\left(\left(\frac{1}{4}, \frac{1}{2}\right), \left(\frac{3}{4}, \frac{1}{2}\right)\right)$, so option A is ruled out.

Allocation $\left(\left(\frac{1}{2}, 0\right), \left(\frac{1}{2}, 1\right)\right)$ is Pareto superior to $\left(\left(\frac{1}{2}, \frac{1}{4}\right), \left(\frac{1}{2}, \frac{3}{4}\right)\right)$, so option B is ruled out.

Allocation $\left(\left(\frac{1}{2}, 0\right), \left(\frac{1}{2}, 1\right)\right)$ is Pareto superior to $\left(\left(\frac{1}{4}, 1\right), \left(\frac{3}{4}, 0\right)\right)$, so option C is ruled out.

So, allocations in options A, B and C are not Pareto efficient.

Here is the set of all efficient allocations:

[![enter image description here](https://i.sstatic.net/be4I5.png)](https://i.sstatic.net/be4I5.png)

Set of feasible allocations is
$\mathcal{F} = \left\{((x\_A,y\_A),(x\_B,y\_B))\in\mathbb{R}^2\_+\times\mathbb{R}^2\_+|x\_A+x\_B=1\wedge y\_A+y\_B=1\right\}$

Set of efficient allocations is
$\mathcal{E} = \left\{((x\_A,y\_A),(x\_B,y\_B))\in\mathcal{F}|\left(y\_A=0\wedge 0\leq x\_A\leq \frac{2}{3}\right)\vee\left(y\_A=1\wedge \frac{1}{3}\leq x\_A\leq 1\right)\right\}$

> **Comments:**
> **username_0**: i was trying to solve this through edgeworth box, could you please see to it, i have just added that.  
> **username_1**: Shape of the IC looks correct to me. If you want to answer this MCQ using Edgeworth box, then draw ICs of the two individuals passing through the allocation in option D and check that it is efficient in the usual way.  
> **username_1**: Also note that not all boundaries of the Edgeworth box are efficient. First argue that if a feasible allocation is efficient then it must satisfy $y_A=0$ or $y_A=1$. Note that the converse is not true (see option C).  

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