Homothetic Preferences, Homothetic Transformations, and the Law of Demand in Exchange Economies

There are four parts to this working paper. In the first part, we show that in an exchange economy where all agents have homothetic preferences, and where preferences and endowments are independently distributed, the economy's demand may be represented by a single agent. The proof involves extending an earlier theorem by Eisenberg. In the second part, we show that with additional smoothness assumptions, the economy will satisfy the Restricted Monotonicity Property (RMP), and so have, amongst other virtues, only one equilibrium price. Note that RMP could be thought of as a restricted form of the Law of Demand, hence the title (see Hildenbrand and Kirman, 1988). RMP also guarantees that the equilibrium price will be stable under some plausible tatonnement processes, these will be discussed in the third part. Finally, we will be considering a sequence of exchange economies, î , without homothetic preferences, but with increasing heterogeneity, this heterogeneity captured through the use of homothetic transformations, somewhat similar to the use of affine transformations in Grandmont (1992). We show that increasing heterogeneity will guarantee that the sequence î eventually satisfies RMP.