Equal Distribution of Consumers astes and Perfect Behavioral Heterogeneity by Reinhard John January 2001 irtschaftstheoretische Abteilung II Rheinische Friedrich-ilhelms-Universität Bonn Adenauerallee 24-42 D-53113 Bonn Germany E-mail: rjohn@wiwi.uni-bonn.de his note is a slightly revised version of an unpublished manuscript that was written in 1984 under the title: Equal Distribution of Consumers astes Leads to ymmetric Cobb-Douglas Aggregate Demand. I would like to thank Hans Haller and alter rockel for very helpful discussions.
1 Introduction In this note we will prove the following result: If individual consumption behavior is equally distributed (in a sense that will be defined below), then the market demand is of the symmetric Cobb-Douglas type 1. Of course, making precise the idea of equal distribution of consumers tastes is known to be difficult (see e.g.. rockel (1984)). Here we solve this problem in a surprisingly easy way: By identifying a consumer s taste with his consumption behavior (described by his budget share function) one observes that the set of tastes can be considered as an infinite product of a finite-dimensional simplex. Consequently, the infinite product of the corresponding Lebesgue probabilities is a natural candidate to represent equal distribution of tastes. However, it is important to point out that there is no restriction on individual demand except that it satisfies the budget identity. In other words, the consumption behavior can be completely irregular in the sense that it is extremely discontinuous and irrational from a choicetheoretic viewpoint. Moreover, such a behavior is far from beeing a rare event. On the contrary, it occurs typically. Of course, we do not claim that this is very realistic. he crucial point is to provide an example which shows that extreme diversification of possibly chaotic individual behavior can lead to an extremely regular aggregate behavior. Moreover, this note demonstrates the possibility to define a notion of absolute heterogeneity (I. Maret (2000)). 2 Individual demand he consumption behavior of an individual is completely described by a budget share function, i.e. a function t : P, where = IR ++, P = IR n ++, and = {x IR n + n i=1 x i = 1} is the (n 1)-dimensional standard simplex. t assigns to any price-wealth pair (p, w) P the budget shares t(p, w) = (s 1,..., s n ) that are spent on n commodities, i.e. the demand for commodity i is given by f i (p, w) = s i w p i. 1 he essential idea is due to G. Becker (1962) although his interpretation ( random behavior ) is quite different. 1
he set P of all budget share functions is denoted by. Clearly, the individual demand of a consumer with characteristics t and w at the price system p is given by f(p, w, t) := D(p, w) t (p, w), where D(p, w) is the diagonal matrix of order n with w/p i as the i- th diagonal element. By definition, the demand function satisfies the budget identity p f(p, w, t) = w. 3 Aggregate demand e consider the measurable space (, ), where is endowed with the σ algebra := B() P, i.e. the infinite product of the Borel sets B() in. Definition 1: A taste distribution is a probability measure τ on. he mean demand of the consumption sector (w, τ), consisting of consumers with identical income w and tastes distributed according to τ, at the price system p is then given by F τ (p) := f(p, w, t)τ(dt) = D(p, w) t (p, w)τ(dt) = D(p, w) proj (p,w) dτ. here is of course a distinguished probability measure on. his is the product measure λ := λ P, where λ denotes the probability defined as the normalized Lebesgue measure on B(). ince λ is the uniform distribution on, the measure λ naturally represents equal distribution on. Now we obtain the following Proposition: he mean demand of the consumption sector (w, λ) consisting of consumers with identical income w and equally distributed tastes is the symmetric Cobb-Douglas demand, i.e. ( w ). F λ (p) = 1 n,..., w p n Proof : F λ (p) = f(p, w, t)λ(dt) = D(p, w) proj (p,w) dλ = D(p, w) id d(proj (p,w) (λ)) = D(p, w) 2 xλ (dx)
( 1 = D(p, w) n,..., 1 ) = 1 ( w,..., w ). n n p n his result can be easily extended to a special class of consumption sectors with not necessarily identical incomes. In general, a consumption sector is described by a probability measure µ on (, B( ) ) with proj dµ <. For example, the consumption sector in the Proposition is δ w λ, where δ w denotes the Dirac measure supported by w. Now, for any income distribution ρ with finite mean there is a consumption sector with equal distribution of tastes for each income class. his is given by the measure ρ λ. For such consumption sectors we obtain following Corollary: Let ρ be an income distribution with finite mean w = wρ(dw) and let λ be the equal distribution on. hen the mean demand of the consumption sector µ = ρ λ at the price system p is F µ (p) = 1 ( ) w w,...,. n Proof: F µ (p) = = = 1 n ( ( 1 f(p,, )dµ = p n f(p,, )d(ρ λ) ) f(p, w, t)λ(dt) ρ(dw) =,..., 1 p n ) wρ(dw) = 1 n ( 1 w n ( w,...,,..., w ) ρ(dw) p n ) w. he statements of the Proposition and the Corollary remain true for all measures τ on with marginal distributions proj (p,w) (τ) = λ. It is suggested that any such measure should be considered to represent perfect behavioral heterogeneity, i.e. we propose the following Definition 2: A taste distribution τ is called perfectly heterogeneous if proj (p,w) (τ) = λ for all (p, w) P. Apart from λ, an obvious further example is provided by the image measure of λ under the product mapping id P : which is defined by proj (p,w) id P = id for all (p, w) P. Clearly, this measure describes the uniform distribution on the set of all Cobb- Douglas-type demands which are parametrized by their constant budget shares as elements in. p n 3
References Becker, G. (1962). Irrational Behavior and Economic heory, Journal of Political Economy 70, 1-13 Maret, I. (2000). Modelling Behavioral Heterogeneity in Demand heory under Absence of Money Illusion, Disc. Paper B.E..A., trasbourg rockel,. (1984). Market Demand. Lecture Notes in Economics and Mathematical ystems No. 223. pringer Verlag. 4