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Competitive division of a mixed manna

Author

Listed:
  • Anna Bogomolnaia

    (National Research University Higher School of Economics)

  • Herve Moulin

    (National Research University Higher School of Economics)

  • Fedor Sandomirskiy

    (National Research University Higher School of Economics)

  • Elena Yanovskaya

    (National Research University Higher School of Economics)

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 pro les), 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 pro le (before any manna) is Pareto dominated, the competitive pro le is unique and still maximizes the product of utilities. If the zero pro le is unfeasible, the competitive pro les are the critical points of the product of disutilities on the eciency 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 re ne 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 bene ts 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 Ecient divisions of bads can be disconnected so that it will admit no continuous selection.

Suggested Citation

  • Anna Bogomolnaia & Herve Moulin & Fedor Sandomirskiy & Elena Yanovskaya, 2017. "Competitive division of a mixed manna," HSE Working papers WP BRP 158/EC/2017, National Research University Higher School of Economics.
    • Handle: RePEc:hig:wpaper:158/ec/2017
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    JEL classification:

    • D61 - Microeconomics - - Welfare Economics - - - Allocative Efficiency; Cost-Benefit Analysis
    • D63 - Microeconomics - - Welfare Economics - - - Equity, Justice, Inequality, and Other Normative Criteria and Measurement
    • D82 - Microeconomics - - Information, Knowledge, and Uncertainty - - - Asymmetric and Private Information; Mechanism Design

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