Smith and Jones are trying to decide whether to share an apartment. To live separately, each would have to pay $300 per month in rent. An apartment large enough to share can be rented for $450 per month. Costs aside, they are indifferent between living together and living separately except for these two problems: Smith likes to listen to music from his stereo speakers late at night, which disturbs Jones's sleep, and Jones likes to sing in the shower at 6am, which awakens Smith. Jones would sacrifice up to $80 per month rather than stop singing in the shower, and Smith would sacrifice up to $155 per month rather than stop listening with his speakers late at night. Smith can tolerate Jones's singing and Jones would tolerate Smith's playing stereo in return for compensation payments of not less than $75 per month and $80 per month, respectively.

a) Should they live together? If so, indicate how they can split the rent so that each does better than by living alone. If not, explain why no such arrangement is feasible.

b)Suppose now Smith wins a free pair of headphones. If he wears them late at night, Jones's sleep is not disturbed. Smith likes the headphones well enough, but would be willing to pay $40 per month to keep on listing to late-night music without them. b) How does the existence of the new option affect your answer to a)? Explain carefully