Can you please please help me with this question?

There are two people, Alice and Bob. Alice consumes quantityxAof goodx, with utility function

lnxAeA

(witheAto be explained; notice that the ln applies only toxA). Bob consumes

quantityxBof goodx, with utility function lnxBeB

(witheBto be explained, and with the ln applying only toxB). Each has incomeIand the price ofxis 1. If Alice consumers quantityxAof goodx, she generates an equal quantity of noxious fumes, which we denoteeB(and hencezB=xA). Alice does not care about the fumes, but they impose disutility on Bob. However, at costpper unit, Alice can reduce the amount of fumes she produces. Hence, we can think of Alice's budget constraint as

xA+p(xAeB)=I

Alice can spend all of her income onxif she does nothing about the fumes, and otherwise has her effective income reduced byptimes the quantity by which she reduces her fumes. (We assume that Alice cannot generate extra fumes, and cannot generate negative amounts.) Similarly, if Bob consumesxBof quantityx, he generates quantityeA=xBof infernal racket. Bob is not bothered by the noise, but the noise adversely affects Alice. Bob can reduce his racket at costp, so that his budget constraint is

xB+p(xBeA)=I

3.1 Find the utility-maximizing consumption bundles for Alice and Bob.

3.2 Is the allocation you found in 3.1 efficient? Explain precisely why not. Find an efficient allocation for Alice and Bob.

3.3 Suppose that Alice consumes her part of an effcient allocation. What would it be best for Bob to do? In light of this, is this situation like a prisoners' dilemma?

3.4 Suppose you are a regulator charged with overseeing fumes and rackets. Pro- pose a policy that would lead to an efficient allocation, and find the resulting allocation.