<p>Given the endowments
Individual 1 : <span class="math-container">$(4,5); U(4,5) = 20$</span> and Individual 2 : <span class="math-container">$(6,5); U(6,5)=30$</span></p>

<p>Given that there is no more than and 10 units of Good X and 10 units of Good Y in the economy. If I allocate Individual 1 one with <span class="math-container">$X$</span> units of Good X and <span class="math-container">$Y$</span> units of Good Y then Individual 2 has <span class="math-container">$(10-X)$</span> units of Good X (<span class="math-container">$10-Y$</span>) units of Good Y.</p>

<p>Now the set of Pareto superior allocations to the given endowments, if they exist will satisfy these two simultaneous non linear inequalities.</p>

<p><span class="math-container">$ XY \geq 20$</span> Red region</p>

<p><span class="math-container">$ (10-X)(10-Y) \geq 30$</span> Blue region
<a href="https://i.sstatic.net/bAkAW.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/bAkAW.jpg" alt="enter image description here"></a>
Graphically plotting these inequalities we observe: <strong>The small common region which you see between red and blue areas is actually the infinite set of all Pareto superior allocations to the given endowment.</strong></p>

<p>Now our job is to find out that allocation which is Pareto optimal, that is that convex lens shaped common region should not be present.</p>

<p>This point could be achieved where both curves are tangent to each other.</p>

<p><a href="https://i.sstatic.net/UaFTk.jpg" rel="nofollow noreferrer"><img src="https://i.sstatic.net/UaFTk.jpg" alt="enter image description here"></a></p>

<p>Which are required Pareto <del>optimal</del> efficient/superior allocations.</p>

CC BY-SA 4.0
2018-11-16T17:27:43.980Z
This helps a lot. Thank you for the elaborate explaination.
S.Rana
CC BY-SA 4.0
2019-01-20T07:09:41.333Z
I am sorry, I made a blunder. The allocations are not Pareto optimal rather they are pareto superior and pareto efficient.
DrStrangeLove
CC BY-SA 4.0
2019-01-22T16:45:50.607Z
@DrStrangeLove Could you tell me how did you get the graph? Did you use Desmos? If so, how did you start the curves from the "other origin"?
PGupta
CC BY-SA 4.0
2019-01-23T06:23:13.277Z
@PGupta Graph (10-x) (10-y) > 30 in desmos
DrStrangeLove
CC BY-SA 4.0
2018-11-16T15:16:40.657Z
2019-01-20T07:11:21.267Z
2019-01-20T07:11:21.267Z
DrStrangeLove
<p>There exists a Pareto Superior allocation, since you already know that <span class="math-container">$MRS_1=MRS_2$</span> which gives the equation <span class="math-container">$\frac{y_1}{x_1}=\frac{y_2}{x_2}$</span>.</p>

<p>Consider an allocation for individual-1, <span class="math-container">$(\sqrt{20},\sqrt{20})$</span> which gives the same utility to individual-1 as his endowment of <span class="math-container">$U_1=20$</span>, while individual-2's bundle will be <span class="math-container">$(10-\sqrt{20},10-\sqrt{20})$</span>. Utility of individual-2 at this bundle will be <span class="math-container">$U_2=$$(10-\sqrt{20})^2&gt;30$</span>. This bundle is pareto superior to endowment bundle. </p>

<p>Thus, endowment is not Pareto efficient, as correctly pointed out, it doesn't even satisfy the <span class="math-container">$MRS$</span> condition in case of standard nice and smooth <span class="math-container">$ICs$</span>. </p>

CC BY-SA 4.0
2018-11-14T16:09:51.697Z
So it was, in fact, a matter of finding a Pareto Superior allocation against the one specified. I should have been more persistent with my calculations. Thanks!
S.Rana
CC BY-SA 4.0
2018-11-14T16:10:23.983Z
Also, does this mean that equalization of MRS is, in fact, a necessary condition for Pareto Optimality?
S.Rana
CC BY-SA 4.0
2018-11-14T16:14:09.173Z
Yeah, it can be a necessary condition but there are several conditions which has to be imposed on the $ICs$. So, it's a better way to actually find out the Pareto efficient allocations by drawing the Edgeworth Box.
Henam
CC BY-SA 4.0
2018-11-14T15:41:10.523Z
2018-11-14T15:41:10.523Z
Henam
<p>Consider a <span class="math-container">$2*2$</span> exchange economy where individual <span class="math-container">$1$</span> has an endowment <span class="math-container">$(4,5)$</span> and individual 2 has an endowment <span class="math-container">$(6,5)$</span>. 
The utility functions of individuals are represented by <span class="math-container">$U({x}_i{y}_i)$</span>=<span class="math-container">${x}_i{y}_i$</span> where <span class="math-container">$i$</span> refers to individual <span class="math-container">$1$</span> or <span class="math-container">$2$</span>.
I know that the allocations which equate <span class="math-container">$MRS$</span> for both individuals would be the Pareto Optimal allocations. However, I seem to be very confused regarding the endowments. I cannot find an allocation that is Pareto Superior to the endowments. Also, the endowment (if it is, in fact, Pareto Optimal) does not satisfy the equality of <span class="math-container">$MRS$</span> condition. 
Is it the case that equality of <span class="math-container">$MRS$</span> is not a necessary condition for pareto optimality? I can think of examples, such as Lexicographic Preferences which yield Pareto Efficient points without the condition of <span class="math-container">${MRS}_i=$${MRS}_j$</span>.
I think I'm missing out on something. 
My concepts seem quite shaky so please bear with me. </p>

<microeconomics><general-equilibrium><pareto-efficiency>
Pareto Optimality of Endowments
S.Rana
# Pareto Optimality of Endowments

(Asked by: username_0 on 2018-11-14)

Consider a $2\*2$ exchange economy where individual $1$ has an endowment $(4,5)$ and individual 2 has an endowment $(6,5)$.
The utility functions of individuals are represented by $U({x}\_i{y}\_i)$=${x}\_i{y}\_i$ where $i$ refers to individual $1$ or $2$.
I know that the allocations which equate $MRS$ for both individuals would be the Pareto Optimal allocations. However, I seem to be very confused regarding the endowments. I cannot find an allocation that is Pareto Superior to the endowments. Also, the endowment (if it is, in fact, Pareto Optimal) does not satisfy the equality of $MRS$ condition.
Is it the case that equality of $MRS$ is not a necessary condition for pareto optimality? I can think of examples, such as Lexicographic Preferences which yield Pareto Efficient points without the condition of ${MRS}\_i=$${MRS}\_j$.
I think I'm missing out on something.
My concepts seem quite shaky so please bear with me.

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


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## 2 answers

### Answer by username_1 on 2018-11-16. Score: 1 | Accepted answer

Given the endowments
Individual 1 : $(4,5); U(4,5) = 20$ and Individual 2 : $(6,5); U(6,5)=30$

Given that there is no more than and 10 units of Good X and 10 units of Good Y in the economy. If I allocate Individual 1 one with $X$ units of Good X and $Y$ units of Good Y then Individual 2 has $(10-X)$ units of Good X ($10-Y$) units of Good Y.

Now the set of Pareto superior allocations to the given endowments, if they exist will satisfy these two simultaneous non linear inequalities.

$ XY \geq 20$ Red region

$ (10-X)(10-Y) \geq 30$ Blue region
[![enter image description here](https://i.sstatic.net/bAkAW.jpg)](https://i.sstatic.net/bAkAW.jpg)
Graphically plotting these inequalities we observe: **The small common region which you see between red and blue areas is actually the infinite set of all Pareto superior allocations to the given endowment.**

Now our job is to find out that allocation which is Pareto optimal, that is that convex lens shaped common region should not be present.

This point could be achieved where both curves are tangent to each other.

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

Which are required Pareto ~~optimal~~ efficient/superior allocations.

> **Comments:**
> **username_0**: This helps a lot. Thank you for the elaborate explaination.  
> **username_1**: I am sorry, I made a blunder. The allocations are not Pareto optimal rather they are pareto superior and pareto efficient.  
> **username_2**: @username_1 Could you tell me how did you get the graph? Did you use Desmos? If so, how did you start the curves from the "other origin"?  
> **username_1**: @username_2 Graph (10-x) (10-y) > 30 in desmos  

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### Answer by username_3 on 2018-11-14. Score: 0

There exists a Pareto Superior allocation, since you already know that $MRS\_1=MRS\_2$ which gives the equation $\frac{y\_1}{x\_1}=\frac{y\_2}{x\_2}$.

Consider an allocation for individual-1, $(\sqrt{20},\sqrt{20})$ which gives the same utility to individual-1 as his endowment of $U\_1=20$, while individual-2's bundle will be $(10-\sqrt{20},10-\sqrt{20})$. Utility of individual-2 at this bundle will be $U\_2=$$(10-\sqrt{20})^2>30$. This bundle is pareto superior to endowment bundle.

Thus, endowment is not Pareto efficient, as correctly pointed out, it doesn't even satisfy the $MRS$ condition in case of standard nice and smooth $ICs$.

> **Comments:**
> **username_0**: So it was, in fact, a matter of finding a Pareto Superior allocation against the one specified. I should have been more persistent with my calculations. Thanks!  
> **username_0**: Also, does this mean that equalization of MRS is, in fact, a necessary condition for Pareto Optimality?  
> **username_3**: Yeah, it can be a necessary condition but there are several conditions which has to be imposed on the $ICs$. So, it's a better way to actually find out the Pareto efficient allocations by drawing the Edgeworth Box.  

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