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I am trying to derive the Utility Possibility Frontier (UPF) when both utility functions display perfect substitutes (in an Edgeworth economy with to consumers and two goods). The specific problem:

$u_A = 2x_{1A}+x_{2A}$

$u_B = x_{1B}+2x_{2B}$

The total endowments being five of each good. Thus:

$x_{1A}+x_{1B}=5$

$x_{2A}+x_{2B}=5$

We are given the hint that "the Utility Possibility Set can be described as the set of $(u_A, u_B)$ in the non-negative orthant of $R^2$ satisfying: $u_A ≥ 0, u_B ≥ 0$".

I keep trying different approaches, but always end up with imposible conditions. When following the suggestions from this another post (How to derive utility possibility frontier?), I keep getting weird first order conditions such as "1=0".

Once the UPF is found, we are asked to maximize social welfare through the following Social Welfare Function (SWF):

$W(u_A,u_B) = 3u_A + u_b$

Thank you in advance.

JKL
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  • Forget about the FOC, the indifference curves are linear. Draw them in the Edgeworth box. Add $u_A$ and $u_B$. Do you notice something special here? – VARulle May 09 '20 at 11:30
  • No. This is what you'd get if you had $u_B=x_{1B}+2x_{2B}$. But you have $u_B=2x_{1B}+x_{2B}$. It actually gets easier that way! – VARulle May 09 '20 at 12:02
  • I accidently deleted my former comment... But I realize that I wrote the problem incorrectly! I have edited it now. My bad.

    I solved for efficient states and then added up utilities. I figured out, that the contract curve must fulfil ${{(x_{1A},x_{2A})|x_{1A}=5∨x_{2A}=0}}$. Thus only corner solutions, as the indifference curves can never be tangent.

    Summing up the utilities along the contract curve, it seems that

    $u_B=15-0,5u_A$

    If I keep $x_{2A}=0$

    OR

    $u_A=15-0,5u_B$

    If I keep If I keep $x_{1A}=5$.

     – JKL May 09 '20 at 12:09
  • You might need to edit it again, look at your $u_B$... – VARulle May 09 '20 at 12:50
  • Thanks. It should be in order now. Sorry for the sloppiness. – JKL May 09 '20 at 12:55
  • O.k., then for the corrected question there's no need to sum up the utilities. Just look at your utility possibility frontier and check where welfare is maximized. This remaining problem should be easy. – VARulle May 09 '20 at 14:13
  • Maybe I am missing an obvious point.. But I can't figure how to find said utility possibility frontier. – JKL May 09 '20 at 15:13
  • You already figured out the contract curve. Now just use your utility functions to calculate the pair of utilities along the contract curve. This is your UPF. – VARulle May 10 '20 at 18:15
  • I did exactly this, but since the contract curve consists of two seperate straight lines (${{(x_{1A},x_{2A})|x_{1A}=5∨x_{2A}=0}}$), I came to the conclusion, that the UPF must be either $u_B=15-0.5u_A$ (if I am using the part of the contract curve in which $x_{2A}=0$) OR $u_A=15-0.5u_B$ (if I am using the part of the contract curve in which $x_{1A}=5$). Is this correct? If so: should I maximize my SWF with respect to each UPF, and then compare maximum potential utility? – JKL May 10 '20 at 19:51
  • It's better to write both utilities as functions of $x_{1A}$ (for the horizontal part of the contract curve) and $x_{2A}$ (for the vertical part of the contract curve). Then draw the $(u_A,u_B)$-curves for these two parts by letting $x_{1A}$ and $x_{2A}$, respectively, go from 0 to 5. You'll see that both these curves are line segments: A segment from (0,15) to (10,10) and a segment from (10,10) to (15,0). Then you can either draw the indifference lines given by the welfare function or just subtitute and compare the two endpoints and the kink at (10,10) to find... – VARulle May 11 '20 at 10:36
  • ... that social welfare is maximized where $(u_A,u_B)=(15,0)$, corresponding to B's origin in the Edgeworth box, $x_{1A}=x_{2A}=5$. – VARulle May 11 '20 at 10:36
  • I added a faster proof circumventing the construction of the UPF as an answer. – VARulle May 11 '20 at 10:48
  • I ended up using $u_B=30-2u_A$ and $u_B=15-0.5u_A$. From here it is obvious, that that SWF maximizes, when alle utility is given to consumer A. It is equivalent to solving for the actual allocation of each resource. Since we only get corner solution, the usual math breaks down. Using a graph and/or intution, however, makes it farily easy to solve. – JKL May 13 '20 at 07:15

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In addition to the solution worked out in the comments, there's also a faster way to find the welfare maximizing distribution:

Note that social welfare can also be written in terms of $x_{1A}$ and $x_{2A}$ as $W(x_{1A},x_{2A})=15+5x_{1A}+x_{2A}$, which is maximized in the Edgeworth box at $(x_{1A},x_{2A})=(5,5)$. As long as utilities are increasing in the quantities of both goods, this point is always on the contract curve and is therefore the maximizer of social welfare independently of the specific forms of the individuals' preferences.

VARulle
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  • But this gets you to the optimal point (5,5). However, to optimal point is actually (0,15). Thus this answer is a contradiction to your other answer? – JKL May 13 '20 at 07:12
  • No, it's (5,5) in A's quantities of good 1 and good 2, corresponding to (15,0) in utilities for A and B. – VARulle May 13 '20 at 09:44
  • I see. I guess this provide two kinds of solutions. One that helps you find the utility distribution, and one that helps you find the distribution of the goods. Essentially different paths to the same goal. – JKL May 13 '20 at 10:20