Externality

There are two people, Alice and Bob. Alice consumes quantity IA of good r, with utility function In IA - eA (with ex to be explained; notice that the In applies only to IA). Bob consumes quantity Ig of good r, with utility function In1B - eB (with es to be explained, and with the In applying only to rg). Each has income / and the price of r is 1. If Alice consumers quantity FA of good r, she generates an equal quantity of noxious fumes, which we denote eg (and hence = = IA). 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 IA + P(IA - eB) = 1 Alice can spend all of her income on a 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 TB of quantity r, he generates quantity eA = Ig 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 IB + P(B - eA) = 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