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.