A Utility Representation for a Preference Relation on a @s-Algebra

Abstract

MODELS OF ECONOMIES with land used by urban economists often employ assumptions that are inappropriate for the commodity known as land (see Berliant [3]). It is therefore natural to attempt to formulate a model that does not suffer from this drawback. In particular, a framework with a finite number of consumers trading in measurable subsets of land (rather than densities or points) is constructed below. The purpose of the present study is to examine the preference relations and utilities used in this type of framework. Problems arise in generating utility representations of preference relations because the separability property of a oX-algebra (on a set of possibly infinite measure) endowed with the L' topology on indicator functions of elements of the a-algebra is not obvious. Let (L, A, m) be a nonatomic measure space with L a separable metric space, A its Borel v-algebra, and m regular and positive. Generally, script letters will represent subsets of $ while capital letters will represent elements of $. If A Be , define A/B={xc L xc A, xiB}. For example, L could be a measurable subset of R2, $ the c-algebra of all measurable subsets of L, and m Lebesgue measure restricted to L. The set L is then a representation of land in that anything immobile can be embedded in L, and consumers have complete preorderings over these immobile objects along with land through preference relations over subsets of L. There is no linear structure imposed on A. There are two reasons to consider all sets in A rather than only those with finite measure. First, it is natural to include L the totality of land, in the choice set of each agent of the example. Second, it is convenient to embed indicator functions of elements of $ in L' (see Bewley [6]), and completeness of the space requires that sets of infinite measure be included. Berliant [4] uses utility functions on L' to prove existence of an equilibrium in an economy with land. Next, a topology is imposed on A. Let the basis for the topology be given by T={$2cAi j$9={Bc$X 3jAcBccC m(B\A)>O wm(C\B)>O} for A, Cc X, Ac C, m(C\A)>O, m(A) O} for Cc$A open}u{$c9X |2 ={Bc$91| A'B. m(B\A)> O} for A c $ compact, m(A) <oo}. Thus, basis elements consist of subsets of $ that can be trapped (in a set containment sense) between various