John and Karl can live together in a two-bedroom apartment for $500 per month, or each can rent a single-bedroom apartment for $350 per month. Aside from the rent, the two would be indifferent between living together and living separately, except for one problem: John leaves dirty dishes in the sink every night. Karl would be willing to pay up to $175 per month to avoid John's dirty dishes. John, for his part, would be willing to pay up to $225 to be able to continue his sloppiness.
a. Should John and Karl live together? If they do, will there be dirty dishes in the sink? Explain.
b. What if John would be willing to pay up to $30 per month to avoid giving up his privacy by sharing quarters with Karl? Should John and Karl live together?