[If all individuals of a given group have the same consumption behavior described by a common demand function f(p,w), which is assumed to satisfy the weak axion of revealed preference, and if the distribution of individual total expenditure w is given by a decreasing density ρ, with ρ(O) extgreater 0, then we show that the market demand function F(p) = ∫f(p, w)ρ(w)dw is monotone, i.e., for any two price vectors p and q one has (q - p). (F(q) - F(p)) ≤ 0. Thus, all partial market demand curves are decreasing. Furthermore, if the expansion paths of f for two different price vectors are different then we show that F is strictly monotone, which implies that the market demand function F satisfies the weak axiom of revealed preference. The result is applied to prove uniqueness and global stability in distribution economies and special exchange economies.]

IT IS WELL KNOWN that for a finite exchange economy, where preferences are not assumed to be convex, there may be no price or even the core may be empty. For this reason it was proposed to enlarge the set of price and the core by introducing the concepts of equilibrium and core. The existence of approximate for exchange economies, where preferences are not assumed to be convex, has been investigated by R. Starr [6]. He showed that there exists a quasi-equilibrium, provided the number of participants is large enough and there is a bound on the degree of non-convexity [6, p. 30, Assumption D]. In this note we shall show the existence of equilibria (a stronger concept than the one considered by Starr [6, p. 31]) for large economies where the preferences are neither assumed to be convex nor complete. To obtain our result we shall assume that the preferences and the endowments of all participating agents belong to a compact set. In [5] Shapley and Shubik proved that, for a large replica of a given economy with transferable utility, the e-core is nonempty. We shall generalize this result to large economies without transferable utility by using the concept of ?-core as introduced by Kannai [2]. The nonemptiness of the e-core follows easily from the existence of approximate and a relationship between the set of approximate and e-core. The existence of c-core for large economies (with a fixed number of types) can also be deduced from Kannai's Theorem C' [2] in its stronger form (Theorem C in [3]).

Journal Article Size Removes Inequity Get access Werner Hildenbrand, Werner Hildenbrand University of Bonn and CORE Search for other works by this author on: Oxford Academic Google Scholar Alan P. Kirman Alan P. Kirman CORE Search for other works by this author on: Oxford Academic Google Scholar The Review of Economic Studies, Volume 40, Issue 3, July 1973, Pages 305–319, https://doi.org/10.2307/2296452 Published: 01 July 1973

A sufficient condition for market demand to satisfy the Law of Demand is that the mean of all households' income effect matrices be positive definite. We show how this mean income effect matrix can be estimated from cross section data under metonymy, an assumption about the distribution of households' characteristics. The estimation procedure uses the nonparametric method of average derivatives. Income effect matrices estimated this way from U.K. family expenditure data are in fact positive definite. This result can be explained by a special form of heteroskedasticity in the data: households' demands are more dispersed at higher income levels.