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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



A characterization of Pareto surfaces

Authors: Louis J. Billera and Robert E. Bixby
Journal: Proc. Amer. Math. Soc. 41 (1973), 261-267
MSC: Primary 90D12; Secondary 90D15
MathSciNet review: 0325163
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Abstract: Given $ n$ concave continuous functions $ {u_i}$ defined over the unit $ m$-cube $ {I^m}$, the corresponding attainable set $ V$ and Pareto surface $ P$ are defined. In the economic interpretation, $ V$ corresponds to the set of attainable utility outcomes realized through trading, and $ P$ the set of such outcomes for which no trader can attain more without another getting less. Sets of the form of $ V$ and $ P$ are characterized among all subsets of $ {R^n}$. The notion of complexity (the smallest $ m$ for which a given $ V$ can be realized) is briefly discussed, as is the idea of a ``market game".

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Keywords: Pareto surface, concave function, utility function, economic market, trading economy, game without side-payments, convex set
Article copyright: © Copyright 1973 American Mathematical Society