<p>Observe that the preferences of <span class="math-container">$A$</span> and <span class="math-container">$B$</span> can be represented by the following utility functions:</p>
<p><span class="math-container">$u_A(x_A,y_A)=(x_A+3)(y_A+12)$</span></p>
<p><span class="math-container">$u_B(x_B,y_B)=(x_B+9)(y_B+8)$</span></p>
<p>For interior efficient allocations, the following conditions hold:</p>
<ul>
<li><span class="math-container">$\dfrac{y_A+12}{x_A+3}=\dfrac{y_B+8}{x_B+9}$</span></li>
<li><span class="math-container">$x_A+x_B=18$</span>, <span class="math-container">$y_A+y_B=40$</span></li>
</ul>
<p>This yields the following as set of interior efficient allocations as the feasible allocations satisfying</p>
<ul>
<li><span class="math-container">$2x_A-y_A=6, 3&lt; x_A &lt; 18$</span></li>
</ul>
<p>There are also efficient allocations at the boundaries. These are feasible allocations satisfying</p>
<ul>
<li><span class="math-container">$y_A=0, 0\leq x_A\leq 3$</span> or</li>
<li><span class="math-container">$x_A=18, 30\leq y_A\leq 40$</span></li>
</ul>
<p>These are the allocations on the above boundaries and here the following condition is true: <span class="math-container">$\text{MRS}_{X,Y}^A\geq \text{MRS}_{X,Y}^B$</span>.</p>
<p><a href="https://i.sstatic.net/Qs2Cmjvn.png" rel="nofollow noreferrer"><img src="https://i.sstatic.net/Qs2Cmjvn.png" alt="enter image description here" /></a></p>

CC BY-SA 4.0
2024-05-28T06:02:02.570Z
2024-05-28T06:02:02.570Z
Amit
<p>Suppose the agents A and B have the following utility functions <span class="math-container">$x_A y_A+12x_A+3y_A $</span> and <span class="math-container">$x_By_B +8x_B+9y_B$</span> respectively with endowments (8,30) and (10,10). </p>

<p>The contract curve's equation turns out to be <span class="math-container">$y_A=2x_A-6$</span> for interior Pareto efficient allocations. </p>

<p>My question is whether some portion of the edges and the origins are also Pareto efficient, and if yes, how do we arrive at them without graphing the box. Once I draw the box, it seems to me that these portions should be included. </p>

CC BY-SA 4.0
2020-02-02T21:27:04.617Z
Its clear that the origin cannot be included. Both agents start with positive endowments (and thus positive utility), so the origin, which gives 0 utility, cannot be a Pareto improving allocation.
Walrasian Auctioneer
CC BY-SA 4.0
2020-02-03T08:32:40.720Z
Are you asking about Pareto set or the contract curve?
Art
CC BY-SA 4.0
2020-02-03T10:51:00.807Z
You might want to edit your question accordingly.
Art
CC BY-SA 4.0
2020-02-03T10:51:52.390Z
@WalrasianAuctioneer I meant to ask if the origin will also be Pareto efficient. Have edited my question.
PGupta
CC BY-SA 4.0
2020-02-03T11:19:44.947Z
@Art Actually, to be precise I want to ask if the points will also be Pareto efficient. My edited post incorporates this change.
PGupta
<microeconomics><general-equilibrium><pareto-efficiency><edgeworth-box>
Pareto Set with strictly convex preferences
PGupta
# Pareto Set with strictly convex preferences

(Asked by: username_0 on 2020-02-02)

Suppose the agents A and B have the following utility functions $x\_A y\_A+12x\_A+3y\_A $ and $x\_By\_B +8x\_B+9y\_B$ respectively with endowments (8,30) and (10,10).

The contract curve's equation turns out to be $y\_A=2x\_A-6$ for interior Pareto efficient allocations.

My question is whether some portion of the edges and the origins are also Pareto efficient, and if yes, how do we arrive at them without graphing the box. Once I draw the box, it seems to me that these portions should be included.

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

> **Comments:**
> **username_1**: Its clear that the origin cannot be included. Both agents start with positive endowments (and thus positive utility), so the origin, which gives 0 utility, cannot be a Pareto improving allocation.  
> **username_2**: Are you asking about Pareto set or the contract curve?  
> **username_2**: You might want to edit your question accordingly.  
> **username_0**: @username_1 I meant to ask if the origin will also be Pareto efficient. Have edited my question.  
> **username_0**: @username_2 Actually, to be precise I want to ask if the points will also be Pareto efficient. My edited post incorporates this change.  

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## 1 answers

### Answer by username_3 on 2024-05-28. Score: 1

Observe that the preferences of $A$ and $B$ can be represented by the following utility functions:

$u\_A(x\_A,y\_A)=(x\_A+3)(y\_A+12)$

$u\_B(x\_B,y\_B)=(x\_B+9)(y\_B+8)$

For interior efficient allocations, the following conditions hold:

* $\dfrac{y\_A+12}{x\_A+3}=\dfrac{y\_B+8}{x\_B+9}$
* $x\_A+x\_B=18$, $y\_A+y\_B=40$

This yields the following as set of interior efficient allocations as the feasible allocations satisfying

* $2x\_A-y\_A=6, 3< x\_A < 18$

There are also efficient allocations at the boundaries. These are feasible allocations satisfying

* $y\_A=0, 0\leq x\_A\leq 3$ or
* $x\_A=18, 30\leq y\_A\leq 40$

These are the allocations on the above boundaries and here the following condition is true: $\text{MRS}\_{X,Y}^A\geq \text{MRS}\_{X,Y}^B$.

[![enter image description here](https://i.sstatic.net/Qs2Cmjvn.png)](https://i.sstatic.net/Qs2Cmjvn.png)


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