Topological complexity of manifolds of preferences

Chichilnisky, Graciela (1986): Topological complexity of manifolds of preferences. Published in: Contributions to Mathematical Economics- In honor of Gerald Debreu (1986): pp. 131-142.

Abstract

The problem of endowing preferences with manifold structures emerged from discussions with Gerard Debreu in 1975 . Time has shown that such structures can be useful in understanding the behavior of economic systems .

In Chichilnisky (1976) spaces of smooth preferences were endowed with a Hilbert manifold structure, and this was used to study the existence and structural stability of competitive equilibria in economies where preferences might be non-monotonic and non-convex . This paper constructs manifolds of preferences and applies this construction to the aggregation of preferences . We examine the topological complexity of manifolds of smooth preferences and use this to determine when appropriate aggregation rules exist and when they do not .

• Item Type: MPRA Paper
• Language: English
• Keywords: preferences; manifold structures; debreu; economic systems; Hilbert spaces; manifold; topological; smooth preferences; aggregation; convex; monotonic; continuous; modelling; modeling; mathematical models; topology
• Subjects: C - Mathematical and Quantitative Methods > C0 - General > C02 - Mathematical Methods

References

R. Abraham and J. Robbin, "Transversal Mappings and Flows," Benjamin, New York, 1967.

G. CHICHILNISKY, "Manifolds of preferences and equilibria," Technical Report No. 29, Project on Efficiency of Decision Making in Economic Systems, Harvard Univ, 1976.

CHICHILNISKY, G., (1977), "Spaces of Economic Agents", Journal of Economic Theory, 16 (1), 160-173.

G. CHICHILNISKY, Nonlinear analysis and optimal economic growth, J. Math. Anal. and App., (1977).

Chichilnisky, G . (Jan. 1980) "Social Choice and the Topology of Spaces of Preferences", Advances in Mathematics. 27, 19-32.

Chichilnisky, G . (1981) "Continuous Representation of Preferences", Review of Economic Studies

Chichilnisky, G . and G . Heal (1983) "Necessary and Sufficient Conditions for a Resolution of the Social Choice Paradox", Journal of Economic Theory.

G. DEBREU, Smooth preferences, Econometrica (1972).

Debreu, G., 1964, Representation of a preference ordering by a numerical function, International Economic Review.

M. W. HIRSCH, "Differential Topology," Springer-verlag, New York/Heidelberg/Berlin, 1976.

Hirsch, M. and B. Mazur, 1974, Smoothings of Piecewise Linear Manifolds, Annals of Mathematical Studies (Princeton University Press, Princeton, NJ) .

Skorohod, A.V ., 1974, Integration in Hilbert Space (Springer-Verlag, New York) .

Milnor, J ., 1965, Topology from the Differentiable Viewpoint (The University Press of Virginia, Charlottesville, VA) .

S. L. SOBOLEV, "Applications of Functional Analysis in Mathematical Physics," Translations of Mathematical Monographs, America Mathematics Society, Providence, R. I., 1963.

Spanier, E . (1966) Algebraic Topology (McGraw-Hill, New York) .