Abstract: I study a market where agents with jointly owned heterogeneous goods trade subject to the constraint that each agent ends up with one good. In this market the existence of an efficient, incentive compatible, individually rational and budget balanced mechanism depends on the shares of the agents. I characterize the set of shares for which having such a mechanism is possible. This set includes the symmetric allocation and excludes the allocation in which every agent owns a separate good. An implication of the results is that partnerships should be formed on equal shares ex-ante to make it possible for future dissolution.

Abstract: We study many-to-one matchings,  such as the assignment of students to colleges, where the students have preferences over the other students who would attend the same college. It is well known that the core of this model may be empty, without strong assumptions on agents' preferences. We introduce a method that finds all core matchings, if any exist. The method requires no assumptions on preferences. Our method also finds certain partial solutions that may be useful when the core is empty.