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I have some troubles in understanding how to find Pareto efficient allocation that are on the frontier of the Edgeworth box. I mean, the interior ones, can be found using the equality $MRS_A = MRS_B$, where $A$ and $B$ are the two agents of our pure exchange economy, whereas I cannot find anything about frontier allocations.

Is there any mathematical procedure I can use or it is only something I have to deal with using intuition (using the definition of Pareto efficiency I mean).

To be clearer, I cite an exercise from my workbook where there are two agents with preferences $u^A(x_1,x_2) =\sqrt{x_1x_2}$ and $u^B(x_1,x_2)=x_1+3x_2$, and endowments $\omega^A=(1,0)$ and $\omega^B=(0,1)$. The question is to identify the Pareto set.

This is the solution, where the blue lines are the Pareto set. Solution

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Let us take your example:

  • First, we note that both utility functions are differentiable and quasi-concave.
  • Noting this, we also know that the necessary and sufficient condition for internal Pareto optimality is that $MRS_{x_1,y_1}$=$MRS_{x_2,y_2}$ (as you have already correctly stated).

This condition will clearly coincide with the portion of the solution identifying the locus on internal P.O. allocations.

Now, for the P.O. points along the right edge:

  • We can find the bound of internal solutions by identifying the range over which the MRS condition noted above fails.
  • Because the equality fails, we know a strict inequality must prevail.
  • The directionality of the prevailing strict inequality identifies the edge along which we find our P.O. allocations.

So, there are two ways to answer your question, I think.

1.) For this type of graph, where one agent has linear preferences and the other has curvilinear and convex preferences, it is easy to see that the locus of P.O. allocations shifts toward the right edge of the Edgeworth box. Thus, corner solutions run along that edge across the range where points of tangency are no longer interior.

2.) If you encounter a situation where, for example, both agents have linear preferences, you can use the directionality of the strict inequality to identify along which edges of the Edgeworth box you have P.O. allocations.

An example for 2.) :

$U_1(x_1,y_1)=X_1+2Y_1$ and $U_2 = 2X_1+Y_1$

Now, you should do the following:

  • ensure you understand why $MRS_1 \neq MRS_2$
  • determine the direction of the strict inequality between the two
  • use this to identify along which edge of the box you will have P.O. allocations.

Hints:

  • $MRS_A<MRS_B$

  • the P.O. allocations are along the left and upper edges of the Edgeworth box

Edit:

I also think this is a good reference:

http://nicolalimodio.com/wp-content/uploads/2015/11/ec202d.pdf

Hope something in there helps. This was an issue for me as well when I began learning such things and it took me a bit of reading and practice to finally become comfortable with it all.

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