I have the following question on my problem set:
It's clear to me, since consumer 2 does not care about good 2, that we should give all the economy's endowment of good 2 to consumer 1. In the other hand, both consumers care about good 1.
For item a), I think that in any optimal allocation we should have $x_{2,1} = 1$, since consumer 2 does not care about good 2.
Also, with this in mind, any allocation that has $x_{1,1} + x_{1,2} = 1$ is Pareto optimal, because we can only make one better hurting the other (assuming we have already exhausted good 2, giving all the economy's endowment to consumer 1). Question: is this reasoning right?
Another question: assuming I got it right, I have no idea how to find the vector price for each case.
For item b), the 1st Welfare Theorem needs to hold because there's local non-satiation for both consumers. On the other hand, lexicographic preferences are not convex. So, there's no reason for the 2nd theorem to hold. Is that sound?
Thanks a lot in advance!!