Demand functions and reflexivity. (English) Zbl 1222.91034

Polyrakis, Ioannis A.

J. Math. Anal. Appl. 338, No. 1, 695-704 (2008).

Summary: In the theory of ordered spaces and in microeconomic theory two important notions, the notion of the base for a cone which is defined by a continuous linear functional and the notion of the budget set are equivalent. In economic theory the maximization of the preference relation of a consumer on any budget set defines the demand correspondence which at any price vector indicates the preferred vectors of goods and this is one of the fundamental notions of this theory. Contrary to the finite-dimensional economies, in the infinite-dimensional ones, the existence of the demand correspondence is not ensured. In this article we show that in reflexive spaces (and in some other classes of Banach spaces), there are only two classes of closed cones, i.e. cones whose any budget set is bounded and cones whose any budget set is unbounded. Based on this dichotomy result, we prove that in the first category of these cones the demand correspondence exists and that it is upper hemicontinuous. We prove also a characterization of reflexive spaces based on the existence of the demand correspondences.

MSC:

91B42	Consumer behavior, demand theory
46B10	Duality and reflexivity in normed linear and Banach spaces
46N10	Applications of functional analysis in optimization, convex analysis, mathematical programming, economics
91B08	Individual preferences

Keywords:

demand functions; budget sets; cones; bases for cones; reflexive Banach spaces

References:

[1]	Aliprantis, C.D.; Border, K.C., Infinite dimensional analysis, (1999), Springer-Verlag · Zbl 0938.46001
[2]	Aliprantis, C.D.; Brown, D.; Burkinshaw, O., Existence and optimality in competitive equilibria, (1990), Springer-Verlag Heidelberg
[3]	Araujo, A., The non-existence of smooth demand in general Banach spaces, J. math. econom., 17, 309-319, (1988) · Zbl 0666.90004
[4]	Jameson, G.J.O., Ordered linear spaces, Lecture notes in math., vol. 142, (1970), Springer-Verlag Heidelberg · Zbl 0234.46007
[5]	Mas-Colell, A., The price equilibrium existence problem in topological vector lattices, Econometrica, 54, 1039-1053, (1986) · Zbl 0602.90028
[6]	Megginson, R.E., An introduction to Banach space theory, (1998), Springer-Verlag Heidelberg · Zbl 0910.46008
[7]	Mil’man, D.P.; Mil’man, V.D., Some properties of non-reflexive Banach spaces, Mat. sb., 65, 486-497, (1964), MR 30, 1383 · Zbl 0121.09303
[8]	Mil’man, V.D., Geometric theory of Banach spaces, I, Russian math. surveys, 05, 111-170, (1970) · Zbl 0221.46015
[9]	Singer, I., Bases in Banach spaces, I, (1970), Springer-Verlag Heidelberg · Zbl 0198.16601