Pareto Optimality of Endowments

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.

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 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.

Which are required Pareto optimal efficient/superior allocations.

• This helps a lot. Thank you for the elaborate explaination. Commented Nov 16, 2018 at 17:27
• I am sorry, I made a blunder. The allocations are not Pareto optimal rather they are pareto superior and pareto efficient. Commented Jan 20, 2019 at 7:09
• @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"? Commented Jan 22, 2019 at 16:45
• @PGupta Graph (10-x) (10-y) > 30 in desmos Commented Jan 23, 2019 at 6:23

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 $$ICs$$.

• 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! Commented Nov 14, 2018 at 16:09
• Also, does this mean that equalization of MRS is, in fact, a necessary condition for Pareto Optimality? Commented Nov 14, 2018 at 16:10
• 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. Commented Nov 14, 2018 at 16:14