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

<p>Note that social welfare can also be written in terms of <span class="math-container">$x_{1A}$</span> and <span class="math-container">$x_{2A}$</span> as <span class="math-container">$W(x_{1A},x_{2A})=15+5x_{1A}+x_{2A}$</span>, which is maximized in the Edgeworth box at <span class="math-container">$(x_{1A},x_{2A})=(5,5)$</span>. 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.</p>

CC BY-SA 4.0
2020-05-13T07:12:19.323Z
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
CC BY-SA 4.0
2020-05-13T09:44:16.840Z
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
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2020-05-13T10:20:12.583Z
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
CC BY-SA 4.0
2020-05-11T10:47:08.140Z
2020-05-11T10:47:08.140Z
VARulle
<p>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:</p>

<p><span class="math-container">$u_A = 2x_{1A}+x_{2A}$</span></p>

<p><span class="math-container">$u_B = x_{1B}+2x_{2B}$</span></p>

<p>The total endowments being five of each good. Thus:</p>

<p><span class="math-container">$x_{1A}+x_{1B}=5$</span></p>

<p><span class="math-container">$x_{2A}+x_{2B}=5$</span></p>

<p>We are given the hint that "the Utility Possibility Set can be described as the set of
<span class="math-container">$(u_A, u_B)$</span> in the non-negative orthant of <span class="math-container">$R^2$</span> satisfying: <span class="math-container">$u_A ≥ 0, u_B ≥ 0$</span>".</p>

<p>I keep trying different approaches, but always end up with imposible conditions. When following the suggestions from this another post (<a href="https://economics.stackexchange.com/questions/6173/how-to-derive-utility-possibility-frontier/36519#36519">How to derive utility possibility frontier?</a>), I keep getting weird first order conditions such as "1=0". </p>

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

<p><span class="math-container">$W(u_A,u_B) = 3u_A + u_b$</span></p>

<p>Thank you in advance.</p>

CC BY-SA 4.0
2020-05-09T11:30:22.983Z
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
CC BY-SA 4.0
2020-05-09T12:02:01.470Z
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
CC BY-SA 4.0
2020-05-09T12:09:43.970Z
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
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2020-05-09T12:50:40.013Z
You might need to edit it again, look at your $u_B$...
VARulle
CC BY-SA 4.0
2020-05-09T12:55:04.767Z
Thanks. It should be in order now. Sorry for the sloppiness.
JKL
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2020-05-09T14:13:45.803Z
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
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2020-05-09T15:13:50.930Z
Maybe I am missing an obvious point.. But I can't figure how to find said utility possibility frontier.
JKL
CC BY-SA 4.0
2020-05-10T18:15:19.770Z
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
CC BY-SA 4.0
2020-05-10T19:51:17.340Z
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
CC BY-SA 4.0
2020-05-11T10:36:45.167Z
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
CC BY-SA 4.0
2020-05-11T10:36:55.780Z
...  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
CC BY-SA 4.0
2020-05-11T10:48:21.670Z
I added a faster proof circumventing the construction of the UPF as an answer.
VARulle
CC BY-SA 4.0
2020-05-13T07:15:30.233Z
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
<microeconomics><utility><optimization><edgeworth-box>
How to find the Utility Possibility Frontier when there are Perfect Substitutes?
JKL
# How to find the Utility Possibility Frontier when there are Perfect Substitutes?

(Asked by: username_0 on 2020-05-08)

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?](https://economics.stackexchange.com/questions/6173/how-to-derive-utility-possibility-frontier/36519#36519)), 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.

**Tags:** microeconomics, utility, optimization, edgeworth-box

> **Comments:**
> **username_1**: 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?  
> **username_1**: 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!  
> **username_0**: 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 $x_{1A}=5$.  
> **username_1**: You might need to edit it again, look at your $u_B$...  
> **username_0**: Thanks. It should be in order now. Sorry for the sloppiness.  
> **username_1**: 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.  
> **username_0**: Maybe I am missing an obvious point.. But I can't figure how to find said utility possibility frontier.  
> **username_1**: 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.  
> **username_0**: 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?  
> **username_1**: 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...  
> **username_1**: ...  that social welfare is maximized where $(u_A,u_B)=(15,0)$, corresponding to B's origin in the Edgeworth box, $x_{1A}=x_{2A}=5$.  
> **username_1**: I added a faster proof circumventing the construction of the UPF as an answer.  
> **username_0**: 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.  

---

## 1 answers

### Answer by username_1 on 2020-05-11. Score: 0 | Accepted answer

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.

> **Comments:**
> **username_0**: 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?  
> **username_1**: 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.  
> **username_0**: 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.  

---

