Market Demand: Theory and Empirical Evidence.

• Author: Neuberg, Leland G.

With [p, D(p)] = [prices, market demand function], the book studies the "law of demand" (LD): ([p.sub.1] - [p.sub.2]). [D([p.sub.1]) - D([p.sub.2])] [less than] 0 for all [p.sub.2] [not equal to] [p.sub.1]. (Price and market demand vectors always point in opposite directions or the market demand function is strictly monotone.) In 1874, Walras used additive utility functions with decreasing marginal utilities to show that partial demand curves are decreasing. Kannai [2] weakened the cardinal utilities assumption, giving necessary and sufficient conditions on ordinal preferences for strictly monotone individual demand functions (where they are differentiable). Until 1983, this research stream based strictly monotone market demand on strictly monotone individual demands. In an earlier paper [1], Hildenbrand used an implausible assumption of consumers with a decreasing income distribution density and identical preferences to show that LD follows without strictly monotone individual demands.

Seeking to improve on the earlier work [1], Market Demand focuses on some assumptions that imply LD and on LD's role in general equilibrium theory (GET). Chapter One is an overview. Chapter Two gives assumptions (A) under which LD holds for elementary commodities if it holds for composite commodities (e.g., "durables," "food") of household consumption surveys. Chapter Three argues that A and two further assumptions imply an axiom (weaker than LD) that Wald used in his 1936 GET version, and gives evidence for one of the new assumptions. Chapter Four strengthens that assumption, gives evidence for it, and argues that A and the strengthened and other new assumption imply LD. Six mathematical appendices prove propositions stated earlier and further relate LD to GET.

A consists of:

(A1) a probability measure characterizes household (demand, income) = (q, Y);

(A2) a probability density with a finite mean characterizes the marginal distribution of Y;

(A3) with i indexing consumers, [q.sub.i](p, [Y.sub.i]) is a continuously differentiable individual demand function in (p, [Y.sub.i]) with interchangeable integration and differentiation with respect to p;

(A4) the average Slutsky substitution effect is nonpositive.

To A, Hildenbrand adds that for a prevailing ([p.sub.1], Y) and any [p.sub.2] [not equal to] [p.sub.1]:

(A5) three observable well approximate three unobservable covariance matrices involving ([p.sub.1], [p.sub.2], Y).

He also adds, and presents data to...