There are two people, Alice and Bob. Alice consumes quantity xA of good x, with utility function

lnxA eA

(with eA to be explained; notice that the ln applies only to xA). Bob consumes

quantity xB of good x, with utility function lnxB eB

(with eB to be explained, and with the ln applying only to xB). Each has income I and the price of x is 1. If Alice consumers quantity xA of good x, she generates an equal quantity of noxious fumes, which we denote eB (and hence zB = xA). Alice does not care about the fumes, but they impose disutility on Bob. However, at cost p per unit, Alice can reduce the amount of fumes she produces. Hence, we can think of Alice's budget constraint as

xA +p(xA eB)=I

Alice can spend all of her income on x if she does nothing about the fumes, and otherwise has her effective income reduced by p times the quantity by which she reduces her fumes. (We assume that Alice cannot generate extra fumes, and cannot generate negative amounts.) Similarly, if Bob consumes xB of quantity x, he generates quantity eA = xB of infernal racket. Bob is not bothered by the noise, but the noise adversely affects Alice. Bob can reduce his racket at cost p, so that his budget constraint is

xB +p(xB eA)=I

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