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Suppose there are 2 individuals in a simple exchange economy with utilities $U_{1}= ax_{1} + x_{2}$ and $U_{2}= y_{1}y_{2}$.

Endowments are $(x_{1},x_{2})=(4,0)$ and $(y_{1},y_{2})=(1,5)$.

We are asked the values of $a$ such that the above allocation is pareto optimal.

The answer as given is $MRS_{A} \ge MRS_{B}$ implies pareto optimal.

How did we arrive here?

URL for reference: http://www.econschool.in/stuff-of-interest/anotherpost/dse-2013-q34

URL for actual problem: http://www.econschool.in/stuff-of-interest/anotherpost/dse-2013-q34

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  • $\begingroup$ Have you tried to understand how to derive the answer? $\endgroup$ – Herr K. Apr 13 '17 at 18:07
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    $\begingroup$ @denesp I wonder why this post is voted to close as off-topic when the one who posted the question has mentioned the source of the problem which contains both the answer and the question, and the intention of the poster is to clarify his doubts - clearly indicating it is not a homework question and the intention of the poster is not to cheat but to understand. $\endgroup$ – Amit May 12 '17 at 3:50
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    $\begingroup$ @Amit I did not get a notification of this comment, I guess because I did not post anything under this question. As it has been explained by others on meta "how" is not a very clear question. As a side note: Disclosing your conflicts of interest would be nice when launching such a petition. $\endgroup$ – Giskard May 14 '17 at 20:48
  • $\begingroup$ I only posted this because I wanted to understand how we arrived at the solution. I already had the solution but had no idea of the underlying concept. The solution does not help me in any way in the sense that I'm not an economics student and hence wasn't giving any test etc. At that point of time. @Amit thank you for the elegant solution. Drawing ICs in the edgeworth box is a beautiful way to solve. And I cannot get more detailed than "how", because solution is just one line. There's no part of solution I can specifically point out that I didn't understand. $\endgroup$ – Arshdeep Singh Duggal May 16 '17 at 6:04
  • $\begingroup$ With regards to reopening the question, I think that this one is a close one. However, I think that the OP should explain which part of the answer they don't understand. Otherwise, the answer posted here might just restate the provided answer. $\endgroup$ – jmbejara Jun 6 '17 at 3:28
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The following plot has the answer to your question, observe it carefully: enter image description here

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A parateo optimal solution can be found setting

$$MRS_1=MRS_2$$

and then solving for $a$

We know, that $MRS_1=$ $MU_{x_1}\over{MU_{x_2}}$ $=$ $\partial U_1 \over{\partial x_1}$$\div$ $\partial U_1\over\partial x_2$ $=$ $a\over1$ $=a$

and similarly $MRS_2=$ $MU_{y_1} \over{MU_{y_2}}$ $=$ $y_2\over{y_1}$

now plug in endowment values respectively which gives

$$a= {5\over{1}} \Longrightarrow a=5$$

$\therefore$ the above allocation is pareto optimal when $a=5$

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  • $\begingroup$ $$MRS_1=MRS_2$$ is only required in interior points. The allocation given in the question is not an interior point. $\endgroup$ – Giskard Apr 13 '17 at 21:46
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    $\begingroup$ @denesp the question cited is talking about envy free allocation and I'm giving the answer for Pareto optimal $\endgroup$ – FreakconFrank Apr 13 '17 at 22:26
  • $\begingroup$ So am I. Consider the extreme example of $U_A(x_1,x_2) = 2x_1 + x_2$ and $U_B(x_1,x_2) = x_1 + 2x_2$. The allocation where $A$ gets (1,0) and $B$ get (0,1) is clearly PO but $MRS_A = 2 \neq 1/2 = MRS_B$. The same will be true in some cases in this question. $\endgroup$ – Giskard Apr 14 '17 at 7:51
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    $\begingroup$ At the corner, condition for Pareto optimality is MRS $ = a\geq 5$. Refer to my answer for the picture proof: economics.stackexchange.com/a/16270/11824 $\endgroup$ – Amit Apr 14 '17 at 7:52

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