# How to find the Utility Possibility Frontier when there are Perfect Substitutes?

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.

• 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 9 '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 9 '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 9 '20 at 12:09
• You might need to edit it again, look at your $u_B$... – VARulle May 9 '20 at 12:50
• Thanks. It should be in order now. Sorry for the sloppiness. – JKL May 9 '20 at 12:55

## 1 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.

• 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 7:12
• 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 9:44
• 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