When utilities are additive, we uncovered in our previous paper (Bogomolnaia et al. "Dividing Goods or Bads under Additive Utilities") many similarities but also surprising differences in the behavior of the familiar Competitive rule (with equal incomes), when we divide (private) goods or bads....

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...

The Competitive Equilibrium with Equal Incomes is an especially appealing efficient and envy-free division of private goods when utilities are additive: it maximizes the Nash product of utilities and is single-valued and continuous in the marginal rates of substitution. The CEEI to divide bads...