Title: Competitive division of a mixed manna,
Abstract: A mixed manna contains goods (that everyone likes), bads (that everyone
dislikes), as well as items that are goods to some agents, but bads or satiated
to others.
If all items are goods and utility functions are homothetic, concave (and
monotone), the Competitive Equilibrium with Equal Incomes maximizes the Nash
product of utilities: hence it is welfarist (determined utility-wise by the
feasible set of profiles), single-valued and easy to compute.
We generalize the Gale-Eisenberg Theorem to a mixed manna. The Competitive
division is still welfarist and related to the product of utilities or
disutilities. If the zero utility profile (before any manna) is Pareto
dominated, the competitive profile is unique and still maximizes the product of
utilities. If the zero profile is unfeasible, the competitive profiles are the
critical points of the product of disutilities on the efficiency frontier, and
multiplicity is pervasive. In particular the task of dividing a mixed manna is
either good news for everyone, or bad news for everyone.
We refine our results in the practically important case of linear
preferences, where the axiomatic comparison between the division of goods and
that of bads is especially sharp. When we divide goods and the manna improves,
everyone weakly benefits under the competitive rule; but no reasonable rule to
divide bads can be similarly Resource Monotonic. Also, the much larger set of
Non Envious and Efficient divisions of bads can be disconnected so that it will
admit no continuous selection.