Question regarding General Equilibrium under non-convexities

Question

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!!

Answer 1

You are right. Set of Pareto efficient allocations consist of all feasible allocations $((x_{11}, x_{21}), (x_{12}, x_{22}))$ satisfying the property that individual 1 consumes all of good 2 i.e. $x_{21} = 1$ and $x_{22} = 0$.

Competitive (or Walrasian) equilibrium in such an economy does not exist. At all price vectors $(p_1, p_2)$ satisfying $p_1 > 0$ and $p_2 > 0$, both the consumers will only demand good 1. As a result we will always have excess demand for good 1 and excess supply of good 2. When we consider a price vector $(p_1, p_2)$ of the form $p_1 > 0$ and $p_2 = 0$, then consumer 1 will demand infinite amount of good 2 leading to excess demand for good 2. Therefore, there does not exist a price vector that clears both the markets.

First Welfare Theorem holds because preferences satisfy Local Non Satiation. Second Welfare Theorem does not hold because no matter how we distribute endowments, competitive equilibrium in such an economy does not exist, but efficient allocations exist.

Also, please note that the lexicographic preferences are convex.