hildenbrand aggregatio theory

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Aggregation: Theory

Werner Hildenbrand

University of Bonn

Abstract:The aim of the aggregation theory is to link the micro and macroeconomic

notions of aggregate demand. One would like such a link to exist for anyheterogeneous population, for a large set of all conceivable income assign-ments, and for a small number of statistics of the income distribution. Thiscannot be achieved. What can be achieved is critically discussed in Section2. In Section 3, another important topic of aggregation theory is consid-ered: how does mean demand react to price changes? As an example, thelaw of demand is discussed.

1. Introduction

Aggregation theory of demand aims at identifying observable explanatory

variables for aggregate demand starting from a microeconomic descrip-tion of the underlying population of households. In the simple case, wherethe demand decision of a household is the choice of a commodity vectorin a budget set, which is determined by the price vector p and income x(total expenditure), the demand behaviour of a household h is modeled

by a demand function fh(p,x) Rl+ (commodity space), which is definedfor every strictly positive price vector p P and every income level x 0.The demand function fh might, but need not be derived from preferencemaximization under the budget constraint.

Aggregate demand is defined as mean demand across the population H,that is to say, 1#H

hHf

h(p,xh). The population H is viewed as hetero-

Entry for The New Palgrave Dictionary of Economics, 2nd Edition.Forschungsgruppe Hildenbrand, University of Bonn, Lennestr. 37, D-53113 Bonn,

Germany. E-mail: [email protected].

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geneous in income and demand behaviour. Thus, mean demand is deter-

mined by the price vector p and the joint distribution of income xh

anddemand function fh across the population H.

This general microeconomic definition of mean demand is sufficientlyspecific for certain problems in pure theory, e.g., for the existence problemin general equilibrium theory.

In macroeconomics or in applied demand analysis the notion of aggre-gate demand is quite different. There the explanatory variables for aggre-gate demand are the price vector and certain statistics S(Gx) of the incomedistribution function Gx such as mean income, a measure of income in-

equality (e.g., the variance of log income) or higher moments of the incomedistribution. In any case, no household specific variable is used in the ag-gregate demand function. The aim of the aggregation theory is to link themicro and macroeconomic notions of aggregate demand. More specifi-cally, given an assignment (fh)hH of demand functions and a set X RH+of income assignments (xh)hH, one seeks for a representation of mean de-mand of the following form: there exists a function F from P RN intoRl+ and N statistics S1(Gx), . . . , S N(Gx) of the income distribution function

Gx, such that

1

#HhH

fh(p,xh) = F(p,S1(G

x), . . . , S

N(G

x)) (1)

for all income assignments (xh)hH in X and all price vectors p in P.

One would like such a representation to exist for any heterogeneouspopulation H, for a large set X, ideally for all conceivable income assign-ments, i.e., X = RH+ and for a small number N of statistics. This, of course,cannot be achieved.

The theory of income aggregation is surveyed in section I, where alsobasic references are given. The main results are:

a representation of the form (1), which must hold in the case X = RH+is an unreasonable strong requirement. Indeed, if a representationexists, then the population H must be homogeneous in demand be-haviour, i.e., fh = f for all h H, and furthermore

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if N is less than the number of households in H and the common

demand function f has the basic properties of demand theory (bud-get identity and homogeneity), then either f is linear in income orat least for one commodity i, the income share function wi(p,x) :=

pifi(p,x)/x is oscillating (i.e., the derivative xwi(p, ) changes its signinfinitely often).

Thus, households behaviour which is modeled by the common demandfunction is either unreasonably simple or incredibly sophisticated. Theseresults clearly show that the requirement X = RH+ leads to an ill-posedproblem.

For a heterogeneous population H there exists (see example 3) a finitepartition {Xk}kK of the set RH+ of all conceivable income assignments andfor every k K there is a function Fk(p,G), where G denotes an incomedistribution function, such that

1

#H

hH

fh(p,xh) = Fk(p,Gx) (2)

for every income assignment (xh)hH in the set Xk and for every p P.

Thus, for a heterogeneous population H, there is no closed-form def-

inition of an aggregate demand function; there is only a piecewise one,since the aggregate demand functions Fk and Fj are different for k = j.The less heterogeneous the population the coarser the partition, i.e., thesmaller is #K. The sets Xk of the partition are large (see Example 3), inparticular, if (xh0) X

k, then for every strictly increasing function theincome assignment xh = (xh0), h H, also belongs to X

k (see Figures 3and 4).

The aggregate demand functions Fk(p,G) in (2) require the knowledgeof the entire income distribution. In many applications one might assumethat the distribution of relevant income assignments in the set Xk can be

modeled by some few parameters (structural stability of income distribu-tions). For example, if the population is very large one might restrictattention to those (xh) in Xk whose distributions are (approximately) lognormal. Then, on this subset ofXk, mean demand has a representation ofthe form Fk(p, x, ), where x denotes mean income across the population

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and 2 is the variance of log income, which can be interpreted as a mea-

sure of income inequality.

Another important topic of aggregation theory is to analyse how meandemand of a heterogeneous population reacts to price changes under theceteris paribus clause that households income and demand functions re-main fixed. In this case mean demand is denoted by F(p). Among thevarious desirable dependence structures is certainly the law of demand,which asserts that the vector p Rl of price changes and the resultingvector F Rl of mean demand changes point in opposite directions, i.e.,

the scalar product p F :=l

i=1 piFi is negative.

Certainly, the law is not meant to be an empirical law, but a mono-tonicity property of the mean demand function F(p) which is defined un-der a ceteris paribus clause in a mathematical model of a population ofhouseholds. Thus, the law asserts that the mean demand function F isstrictly monotone, i.e.,

(p q) (F(p) F(q)) < 0 for all p = q in P.

In particular, every partial mean demand curve is strictly decreasing. Thispartial monotonicity property, however, is not sufficient for proving theuniqueness and stability of the equilibria for a multi-commodity demand-

supply system; one needs strict monotonicity in the multi-commodity ver-sion.

Which behavioural assumption on the household level and/or whichform of heterogeneity of the population lead to monotone mean demand?To answer this question one assumes that demand functions fh satisfy theweak axiom of revealed preferences or, more specifically, that they are de-rived from preference maximization. Then, partial monotonicity is eas-ily obtained, for example, by excluding inferior goods. However, multi-commodity monotonicity is more difficult to obtain. Trivially, mean de-

mand is monotone if all demand functions fh

(p,xh

) were monotone in p.This, however, requires that either fh(p, ) is linear in income or that theSlutzky substitution effect is sufficiently strong. (For a precise formula-tion, see the Theorem of Mitjuschin and Polterovich, 1978; LAW OF DE-MAND.) Since the Slutzky substitution effect might be arbitrarily small,

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one is interested in finding alternative assumptions, which do not rely

on a strong Slutzky substitution effect. These assumptions should not re-quire that households demand functions are monotone. Obviously, toobtain the desirable aggregation effect, the population must be hetero-geneous. Thus, in contrast to the problem of income aggregation, het-erogeneity does not complicate the analysis, yet it is necessary to obtainmonotonicity of mean demand by aggregation. More details are given insection II. For example, let H be a population which is homogeneous indemand behaviour, i.e., fh = f, h H and the common demand functionis not monotone. However, the population is heterogeneous in income.Then, for a given income assignment (xh)hH, mean demand F

H(p) is notmonotone in p. If one increases now the population size, i.e., the number

#H of households tends to infinity and if for increasing #H the incomedistribution functions GH of households in Hconverge to a concave distri-

bution function G, then, for #H sufficiently large, mean demand FH(p) isapproximately monotone, that is to say, FH(p) converges to a monotonefunction. Consequently, in the limit, i.e., for an indefinitely large popula-tion which admits a concave income distribution function, mean demandis monotone. The mathematical model for such a limit population cannot

be a finite or countably infinite set; it must be an atomless measure spaceof households, e.g., the unit interval [0, 1] with Lebesgue measure (contin-uum of households).

If these large populations are heterogeneous in income and demandbehaviour, then one can meaningfully pose the problem of smoothing byaggregation: is mean demand continuous or differentiable without as-suming these properties on the household level? The basic reference isTrockel (1984).

Finally, one should mention the literature on behavioural heterogene-ity initiated by Grandmont (1992). Here the goal is to obtain a strongerproperty than strict monotonicity of mean demand: diagonal dominanceof the Jacobian pF(p) of mean demand in the sense that

pipiFi(p) >j=i

pj |pjFi(p)|

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and

pipiFi(p) >j=i

pj|piFj(p)|.

This diagonal dominance models a strong restriction on the interdepen-dence among the various commodity markets and is the basis for partialequilibrium analysis. For a general discussion of behavioural heterogene-ity see Hildenbrand and Kneip (2005).

2. Income Aggregation

The demand behaviour of every household h in a population H is mod-eled by a demand function fh F. In this section it is not required thatdemand functions are derived by preference maximization under budgetconstraints. One only needs that demand functions f F are continuousfunctions from P R+ into R

l+ with f(p, 0) = 0, where P denotes the set

of all strictly positive price vectors in Rl.

For every income assignment (xh)hH, xh 0, we consider mean de-

mand 1#H

hHf

h(p,xh). The problem of income aggregation has beendefined in the literature by the question: does there exist a function Ffrom P R+ into Rl+ such that

1

#HhH

fh(p,xh) = F(p, x), where x =1

#HH

xh, (3)

for all income assignments in a given set X RH+ and all p P?

If one asks this question for all conceivable income assignments, i.e.,X = RH+ , then this is an ill-posed problem since it allows only a trivial so-lution.

Theorem (Antonelli, 1886): There exists a function F(p, x) such that (1) holdson RH+ P if and only if the population H is homogeneous in demand behaviour,

i.e., fh

= f, and furthermore f(p,x) is linear in x, i.e., f(p,x) = (p)x,(p) Rl+. Thus, F(p, x) = (p)x.

One might ask whether a less restrictive condition than (3) allows for anontrivial solution. That is to say, one might consider mean demand func-

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tions that depend on a wider set of aggregate income variables than just

mean income, for example, the variance or higher moments of the distri-bution of income. The answer is definitely negative.

For every income assignment (xh)hH, let Gx denote its distributionfunction, i.e.,

Gx() :=1

#H#{h H | xh }, R.

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Proposition 1. There exists a function F(p,G) such that

1

#H

hH

fh(p,xh) = F(p,Gx) (4)

for all conceivable income assignments, i.e., X = RH+ and all p P, if and only ifthe population H is homogeneous in demand behaviour, i.e., all households in Hhave the same demand function. Then F(p,Gx) =

f(p,)dGx().

Proof: Consider any two households k and j in H, and an income assign-ment (xh)hH with x

k > 0 and xj = 0. Now one interchanges the incomeof households k and j. This does not change the distribution function of

income. Hence property (4) and the fact that fk(p, 0) = fj(p, 0) = 0 im-plies that fk(p,xk) = fj(p,xk). Since this holds for all xk > 0 and p Pone obtains fk = fj . On the other hand, if fh = f for all h H, then1

#H

hHf

h(p,xh) =

f(p,x)dGx =: F(p,Gx).

The justification for considering the generalized problem of income ag-gregation as defined by (4) is based on the view that for large populations,which this survey emphasizes, income distribution functions can often bemodeled by some few parameters, e.g. log-normal distributions.

By Proposition 1 it is clear that one is forced to restrict the set X ofadmissible income assignments if one wants to escape the case of trivialsolutions, fh = f, to the aggregation problem as defined by (4). Motivated

by the special role which zero income and the assumption f(p, 0) = 0 playin the proofs of Antonellis Theorem or Proposition 1 one has consideredin the literature (e.g. Nataf, 1948 or Gorman, 1953) a restriction on thedomain of individual income:

X(a, b) := {(xh) RH+ | 0 < a xh b }, a < b.

Proposition 2 shows that this restriction allows merely for some verylimited and quite special heterogeneity in demand behaviour of the pop-ulation H.

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Proposition 2.

(i) There exists a function F(p,G) such that (2) holds on X(a, b) P if andonly if for every commodity i and p P the income expansion pathsfhi (p, ), h H, are parallel (vertically) on the interval (a,b); (with dif-

ferentiability) xfhi (p,x) does not depend on h H (Figure 1).

(ii) There exists a function F(p, x) such that (1) holds on X(a, b) P if andonly if for every commodity i and p P the income expansion pathsfhi (p, ), h H, are affine and parallel on the interval (a,b); (with differ-entiability) xf

hi (p,x) does not depend on h H and x (a, b) (Figure

2).

(iii) If all individual demand functions fh belong to F and are homogeneous in(p,x), then the necessary condition in (i) implies that fh f.

(Insert Figure 1 about here)


Proof:

(i) Consider any two households k and j in H and an income assignment

in X(a, b) with xk = xj . Now one interchanges the income of house-holds k and j. This does not change the income distribution function.Hence, property (2) implies fk(p,xk)+fj(p,xj) = fk(p,xj)+fj(p,xk). Thusfk(p,xk)fk(p,xj) = fj(p,xk)fj(p,xj). Since it holds for all xk, xj (a, b)and all p P one obtains the claimed property in (i). The converse is triv-ial.

(ii) Instead of interchanging the income of households k and j one choosesxk+ and xj (a, b) for sufficiently small . Property (1) then implies

fk(p,xk+)fk(p,xk) = fj(p,xj)fj(p,xj ) = fk(p,xj)fk(p,xj )

by (i), which implies the claimed property in (ii). The converse is trivial.

(iii) If the expansion paths fki (p, ), h H, are parallel on (a, b) for ev-ery p P, then homogeneity implies that they are also parallel on (a,b)

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for all > 0 and p P. Hence they are parallel on (0, ) for all p P.

Continuity and fh

(p, 0) = 0 then implies the claim.

An alternative approach to allow for heterogeneous populations con-sists of considering, in addition to income, further explanatory variablesfor household demand. For example, in applications it is standard practiceto stratify the whole population H by a certain profile a = (a1, a2, . . . ) ofobservable household attributes, such as household size, age of householdhead, etc. Let H(a) denote the subpopulation of all households in H withattribute profile a. Without loss of generality one can assume that a Rm.Let Gx,a denote the joint distribution of function ofx

h, ah across H. Analo-gously to Proposition 1 one shows

Proposition 1. There exists a function F(p,Gx,a) such that

1

#H

hH

fh(p,xh) = F(p,Gx,a)

for all conceivable income-attribute assignments and all p P if and only if allsubpopulations H(a) are homogeneous in demand behaviour, i.e., fh = fa for allh H(a).

Thus, the whole population need not be homogeneous, yet the joint

distribution of xh

and ah

across H has typically a complex dependencestructure, and hence, it cannot be modeled by some few parameters, as inthe case of income.

Exact income aggregation

In the literature on exact income aggregation, as initiated by Gorman(1953), Lau (1982), and Jorgenson et al. (1982), one seeks for a representa-tion of mean demand which is less restrictive than (3), yet more demand-ing than (4), that is to say, 1#HhH

fh(p,xh) = F(p,S1(Gx), . . . , S N(Gx))

on RH+ P for some continuous function F from P RN into Rl (the com-modity space) and some vector of distributional statistics S1(Gx), . . . , S N(Gx)with N < #H. This representation is more demanding than (4); it does notrequire the knowledge of the entire income distribution since N < #H.

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If such a representation exists, then by Proposition 1, fh = f, h H,

and f is called exactly aggregable. Thus, the question is whether thereare exactly aggregable demand functions which are not linear in incomeand satisfy the basic restrictions of demand theory?

To simplify the presentation one assumes that all distributional statis-tics are generalized moments, i.e., Sn(Gx) =

sn()dGx(), with con-

tinuous functions sn(). Without loss of generality one can require thatsn(0) = 0.

Proposition 3. There exists a representation of mean demand of the form

f(p,)dGx() = F

p,

s1()dGx(), . . . ,

sN()dGx()

, (5)

which holds for every income distribution function Gx of every finite populationH and every price vector in P if and only if the function f is of the form

f(p,) = 1(p)s1() + N(p)sN(), p P and R+, (6)

where n(p) Rl.

Proof: Trivially, (6) implies (5). Assume that (5) holds. Let G denote the

set of all income distribution functions for every finite population. Notethat for every G1, G2 G and any rational with 0 1 it follows thatG = G1 + (1 )G2 G.

The representation (5) implies for every commodity i

fi(p,) = Fip,s1(), . . . , sN()

, p P and R+. (7)

Now one shows that the function Fi(p, ) has a linear structure on its rel-evant domain D := {y RN| yn =

sn()dG(), G G, n = 1, . . . , N },

i.e.,

Fi(p, y1

+ (1 )y2

) = Fi(p,y1

) + (1 )Fi(p,y2

) (8)for every y1, y2 D and any rational with 0 1. Indeed, ykn =

sn()dGk(), k = 1, 2 for some G1, G2 G. Let G = G1 + (1 )G2.

Then

sn()dG() =

sn()dG

1() +( 1 )

sn()dG2(). Hence y1 +

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(1 )y2 D since G G for rational . Consequently, the closure D of

D is convex. Since G

G, one obtains from (5)fi(p,)dG

() = Fip,

s1()dG

(), . . . ,

sN()dG

()

= Fi(p, y1+(1)y2).

The left hand side is equal to

fi(p,)dG1() + (1 )

fi(p,)dG

2() =Fi(p,y

1) + (1 )Fi(p,y2) by (5), which proves (8). Since Fi is continu-

ous, the linear structure (8) also holds for any y1, y2 in the closure D ofD and any with 0 1. Since sn(0) = 0 and f(p, 0) = 0 it followsfrom (7) that Fi(p, 0) = 0. Consequently, by (8), the restriction of the func-tion Fi(p, ) on the convex domain D can be extended to a function Fi(p, ),which is linear in y, i.e., Fi(p,y) =

i1(p)y1 +

iN(p)yN. Thus (7) implies

(6). The extension is unique if the dimension of the convex domain D isequal to N.

Remark: The proof of Proposition 3 is quite simple since it was assumedthat the representation (5) must hold for all income distribution functionsfor all finite populations. This case is also treated in Heineke and Shefrin(1988), their proof, however, requires differentiability. If one only requires(5) to hold for all income distribution functions of a given population Hwith N < #H, then it is much more difficult to obtain (6). See Lau (1982)and Heineke and Shefrin (1988).

Note that the global structural specification (6) is very restrictive if thedemand function f F has the basic properties of static demand theory.In fact, Heineke and Shefrin (1987) show the following result: iff F sat-isfies the budget-identity, is homogeneous in p and x and if no budget share func-tion wi(p,x) := pifi(p,x)/x is oscillating (i.e., the derivative xwi(p,x) changesinfinitely often its sign), then (6) implies f(p,x) = (p)x.

Indeed, if f F satisfies the budget identity, then 0 wi(p,x) 1.Let the budget share function wk(p, ) be non-constant and non-oscillating.Consider the function (), > 0, defined by (x) = wk(p,x), and

the linear function space which is generated by all functions (), > 0.Heineke and Shefrin (1987) argue that the dimension of this linear spaceis infinite. By homogeneity, (x) = wk(p/, x); thus, the linear space Lwhich is generated by all budget share functions wk(p, ), p P has infi-nite dimension. Consequently, the demand function f cannot satisfy (6),

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since (6) implies that dim L N. Thus, iff satisfies (6) and wk(p, ) is non-

oscillating, then it must be constant, i.e., fk(p, ) is linear.

As a consequence, for demand functions which have the basic proper-ties of atemporal demand theory including non-oscillating budget sharefunctions, one either has to be satisfied with a representation as in Propo-sition 1 or one is in the trivial case of Antonellis Theorem.

Heterogeneous populations

The representations (3), (4), and (5) of mean demand which have been con-

sidered up to now imply that the population of households must be ho-mogeneous in demand behaviour, i.e., fh f, h H. The reason for thisunsatisfactory fact is due to the very strong requirement that the represen-tations must hold for every conceivable income assignment. This is moredemanding than is needed in many applications, since there, changes inindividual income are not entirely arbitrary; they might be the result of anunderlying process. This point was emphasized by Malinvaud (1956) and(1993). To capture this idea, one starts from an initial income assignment(xh0) (status quo), and then one considers a sequence (x

hn), n = 1, 2, . . . or a

set X(x0) of income assignments which are viewed as the result of the un-derlying (unspecified) process. Which properties must the sequence (xhn)

or the set X(x0) have such that for any assignment of demand functionsfh the representations of mean demand hold along this sequence or on theset X(x0)?

We give three examples. The first one is well-known. The second andthird example generalize substantially the first one.

Example 1: Fixed income shares

Starting from an initial income assignment (xh0), one defines the set X() RH+

of income assignmentsX() :=

(xh) RH+ | x

h/x = xh0/x0 =: h

.

where x denotes mean income across H.

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Given any assignment of demand functions fh, h H, there exists a function

F from P R+ into

Rl

+ such that mean demand has the representation1

#H

hH

fh(p,xh) F(p, x) on X() P. (9)

The function F is defined by F(p, x) = 1#H

hHf

h(p,hx). If all fh are

linear in income then F(p, x) is linear in mean income x. Moreover, Eisen-berg (1961) and Chipman and Moore (1979) have shown: if all fh are gen-erated by a utility function homogeneous of degree one then F(p, x) is alsogenerated by a utility function homogeneous of degree one given by

u(z) = maxzhRl

+,

H zh=z

hH

(uh(zh))xh0/x0

Example 2: Rank preserving income changes.

Starting from an initial income assignment (xh0)hH one defines the setX(x0) RH+ of income assignments (x

h) which have the property thatevery household keeps his rank position of income, i.e., if for two house-holds j and k, xj0 = x

k0 then x

j = xk and ifxj0 < xk0 then x

j < xk. For any(xh1) and (x

h2) in X(x0) there is a strictly increasing function such that

(xh1) = xh2 , h H. Examples for () are given in Figure 3 (low income is

increased, high income is decreased) and Figure 4 (low and high incomesare decreased, middle ones increased) below.



Note that (xh) X(x0) implies X(x) = X(x0) and (xh) / X(x0) impliesX(x) X(x0) = . Thus, there is a finite partition {Xi} ofRH+ into sets Xiof rank preserving income assignments.

Note that for any rank preserving income assignments (xh) in X(x0)one can recover the income assignment from knowing only its distributionfunction Gx, since x

h = G1x G0(xh0) for every h H, where G

1 denotes thequantile function (quasi-inverse) of the distribution function G, which is

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defined by G1(q) := inf{x R+| G(x) q}. Consequently, one obtains:

Given any assignment of demand functions fh, h H, there exists a functionF(p,G) such that mean demand has the representation

1

#H

hH

fh(p,xh) F(p,Gx) on X(x0) P. (10)

The function F is defined by F(p,Gx) =1

#H

hHf

h(p,G1x G0(xh0)).

There might be larger sets than X(x0) for which the representation (10)holds. For example, if households k and j have the same demand function

then one can interchange their rank position. Thus, in defining a set X forwhich (10) holds, one should take into account the heterogeneity structureof(fh)hH. This is done in the next example

Example 3: Common copula

Let {f1, . . . , f N} be the set of distinct demand functions of the given as-signment (fh)hH. Thus, for h H there is an integer n(h) N such thatfh = fn(h). For every income assignment (x

h)hH consider the bivariatedistribution function Dx, which is defined by

Dx(, ) := 1#H

#{h H| xh and n(h) }, , R.

The distribution function Dx and the price vector p determines mean de-mand 1

#H

hHf

h(p,xh). The marginal distribution functions of Dx aredenoted by Gx and V.

By Sklars Theorem (see, e.g., Nelson, 1999), for every bivariate dis-tribution function D with marginals G and V, there exists a copula C (afunction from [0, 1]2 into [0, 1] with certain properties) such that D(, ) =C(G(), V()) for all , R. Conversely, ifC is a copula and G and V are

distribution functions, then C(G(), V()) is a bivariate distribution func-tion. Thus, a copula couples the marginals to the bivariate distribution.The copula models the dependence structure of the bivariate distributionfunction.

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Starting from an initial income assignment (xh0), one considers the set

X(x0, f) RH

+ of income assignments (xh

) such that the correspondingbivariate distribution functions Dx have a common copula. Thus, the de-pendence structure of(xh, fh) across H is the same for all (xh) in X(x0, f).It follows that income assignments in the set X(x0) of rank preserving in-come assignments is contained in the set X(x0, f). Furthermore,

given any assignment of demand functions (fh)hH, there exists a function F(p,G)such that mean demand has the representation

1

#H

hH

fh(p,xh) F(p,Gx) on X(x0, f) P.

There is a very simple, however, special case which is worthwhile tobe mentioned (and could have been discussed at the beginning). If theinitial income xh0 and the demand function f

h of household h are indepen-dently distributed across H, i.e., Dx0(, ) Gx0()V() (the copula ofDxois equal to C(u, v) = u v), then the set X(x0, f) =: Z(x0) is very large; itconsists of all income assignments (xh) RH+ with the property: x

k0 = x

j0

implies xk = xj . Then, one obtains

1

#H

hH

fh(p,xh) F(p,Gx) on Z(x0) P

with F(p,G) =

f(p,)dG() where

f(p,) =

1

#H

hHf

h

(p,).

3. Monotone mean demand

The law of demand for a population of households asserts that the vec-tor of price changes p Rl and the resulting vector of mean demandchanges F Rl point in opposite directions, provided the price changesdo not affect households incomes (total expenditure) and demand func-tions (preferences). Thus, the law asserts that the mean demand functionF(p) is strictly monotone, i.e.

(p q) (F(p) F(q)) < 0 for all p, q Rl++, p = q

Strict monotonicity of mean demand implies, in particular, that for ev-ery commodity i the partial mean demand function Fi is strictly decreas-ing in its own pricepi and that the mean demand function F() is invertible

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(existence of an inverse demand function).

The goal of aggregation theory is to derive strict monotonicity of meandemand without assuming that households demand functions fh(p,x) arestrictly monotone in p.

Demand functions fh F are assumed to be continuous in p and xand satisfy the budget-identity p f(p,x) = x. The function f F satisfiesthe Weak Axiom of revealed preferences if for every price-income pair (p,x)and (p, x), p f(p, x) x implies p f(p,x) x, and satisfies the Axiomof revealed preferences, if f(p,x) = f(p, x) and p f(p, x) x implies

p f(p,x) > x.

Every demand function which is derived from a continuous, strictlyconvex and non-saturated preference relation satisfies the Axiom, yet it isnot necessarily monotone.

Theorem (Hildenbrand, 1983)

1. The function F(p) :=0 f(p,x)(x)dx is monotone, i.e.,

(p q) (F(p) F(q)) 0 for all p, q in Rl++, iff F satisfiesthe Weak Axiom of revealed preferences and is a density which isnon-increasing on R+ with

0

(x)dx <

2. The functionF

is strictly monotone, if, in addition,f

satisfies theAxiom of revealed preferences and the expansion paths f(p,.) andf(q, .) have only 0 in common for any p, q that are not collinear.

Interpretation: The underlying micro-model is a population H of house-holds which is indefinitely large; mathematically, an atomless measurespace, e.g. the unit interval [0, 1] with Lebesgue measure. Every householdh [0, 1] is modeled by its income x(h) 0 and the common demand func-tion f. The income assignment x() is an integrable function whose distri-

bution admits a density . Thus, mean demand F(p) =10

f(p,x(h))dh =

0

f(p,x)(x)dx.

Three questions are relevant:

1. Why a continuum of households? Does the result still hold approxi-mately for a large but finite population?

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2. Why a non-increasing income density? Does monotonicity ofF fail

if the density is first increasing and then decreasing?3. Why a common demand function? Does the result extend to hetero-

geneous populations in income and demand behaviour?

The discussion of these questions is simplified by assuming that f iscontinuously differentiable in p and x. Then monotonicity ofF is equiva-lent with negative semi-definiteness (n.s.d.) of the Jacobian matrix pF(p)

for all p, i.e.,l

i,j=1 vivjpiFj(p) 0 for all v Rl, and the Weak Axiom

for f is equivalent with n.s.d. of the Slutzky substitution matrix. Con-

sequently, monotonicity of F follows from the positive semi-definiteness(p.s.d.) of the mean income effect matrix I(f, ) =

I(f, x)(x)dx, where

I(f, x) = (fi(p,x)xfj(p,x))i,j=1,...,l.

Question 1: The mean income effect matrix for a finite population H,i.e., 1

#H

H(fi(p,x

h) xfj(p,xh))i,j = IH is p.s.d. if and only if for ev-

ery v Rl, v IHv =1

#H

Hg

(xh) 0 where g(x) := 12(v f(p,x))2.

Assume that income xh is measured in multiples of (euro). Let n :=1

#H#{h H | xh = n =: xn}, n = 0, 1, . . . Then

(1)

1

#H

Hg

(x

h

) =

n=0 ng

(xn) =

n=1

1

(n1 n)g(xn) + o()using the approximation

(2) g(xn) =1

(g(xn+1) g(xn)) + o().

Consequently, one needs n1 n, n = 1,..., to obtain a non-negativefirst term on the right hand side of (1); this is the finite analogue of anon-increasing density. Thus, for a finite population with a small (whichrequires by n1 n a large population) one obtains the desired resultup to the small term o(). For a population H = [0, 1] one does not needthe approximation (2) and hence o(), since (1) becomes g

(x)(x)dx =

g(x)

(x)dx (by partial integration), which is non-negative for a non-increasing differentiable density .

Question 2: The mean income effect matrix I(f, ) is p.s.d. in each of thetwo extreme cases: either, is non-increasing and no assumption on the

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shape of the income expansion path fi(p, ) or, no assumption on yet lin-

earity offi(p, ). There must be results in between. Indeed, if the curvatureof all income expansion paths fi(p, ) is limited and the unimodal density is sufficiently skewed, then I(f, ) is p.s.d.

Example: All income expansion paths restricted to the interval [0, x] arepolynomials of degree n (note that, no non-linear fi(p, ) can be a poly-nomial on R+) and is concentrated on [0, x]. Then, I(f, g) is p.s.d. ifand only if the matrix M(n, ) := ((i + j)mi+j1)i,j=1,...,n is p.s.d. wheremk :=

xk(x)dx (Hildenbrand, 1994, Appendix 6).

Let the densities m be as in Figure 5.

< Figure 5 here >

For every n there exists m(n) > 0 such that I(f, m) is p.s.d. if m m(n);e.g. n = 2, m(2) = 0.38x or n = 3, m(3) = 0.14x.

For a more general analysis see Chiappori (1985) and Hildenbrand(1994).

Question 3: A population of households that is heterogeneous in income

and demand functions is described by a joint distribution of income anddemand functions, i.e., is a distribution onR+F. (A reader not familiarwith distributions on function spaces might replace F by a finite set F0).As before, the marginal distribution of income admits a density . Theconditional distribution of demand functions given the income level x isdenoted by (x). Then mean demand

F(p) :=

R+F

f(p,x)d =

0

f(p,x)(x)dx

where f(p,x) :=

Ff(p,x)d(x). Consequently, the Theorem or the exten-

sions discussed under Question 2 imply that F(p) is monotone providedthe function f satisfies the Weak Axiom. This approach to derive mono-tonicity for a heterogeneous population is the most direct, yet not the mostgeneral way (see Hildenbrand, 1994).

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It is well-known (Hicks, 1956, p.53) that f does not necessarily satisfy

the Weak Axiom, even if individual demand functions are derived fromutility maximization. The following two assumptions (which, again, arenot the most general ones) imply that f satisfies the Weak Axiom

(a) independence: (x) does not depend on x

(b) increasing dispersion: the distribution D(x + ), > 0, is more dis-persed than the distribution D(x), where D() denotes the distribu-tion (in the commodity space Rl) of individual demand of all house-holds with income at the price p (i.e., D() is the image distributionof under the mapping f f(p,)).

Generalizing the one-dimensional case where the variance is a mea-sure of dispersion one chooses the positive definiteness of the covariancematrix as a measure of dispersion for distributions on Rl. Thus, increasingdispersion means that for > 0, covD(x + ) covD(x) is positive semi-definite.

Assumptions (a) and (b) are quite restrictive, in particular, the inde-pendence assumption. Therefore one partitions the whole population Hinto subpopulations H(a) by stratifying with respect to a certain vector aof household attributes (household size, age, ...) and then one requires as-

sumptions (a) and (b) for each subpopulation H(a). The role of stratifyingis to reduce the heterogeneity in demand behaviour. In the extreme case,where stratifying leads to a homogeneous subpopulation in demand be-haviour, assumptions (a) and (b) are trivially satisfied. If the income den-sity of each subpopulation H(a) is non-increasing onR+ or if the extensiondiscussed in Question 2 apply, the mean demand of each subpopulationis monotone and hence also the mean demand of the whole population,since monotonicity is additive.

A more general definition of increasing dispersion and a detailed dis-cussion is given in Hildenbrand (1994). For an empirical study of the Lawof Demand see Hardle et al. (1991).

A broader discussion of the law of demand and related properties in-cluding cases where income is price dependent is contained in the entry

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LAW OF DEMAND.

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Bibliography

Antonelli, G.B. 1886. Sulla teoria matematica della economia politica. Pisa:Nella Tipografia del Folchetto (translated by J.S. Chipman and A.P. Kir-man in Preferences, utility and demand, ed. J.S. Chipman et al. NewYork: Harcourt Brace Jovanovich, 33360.)

Chiappori, P. 1985. Distribution of income and the law of demand. Econo-metrica 53, 109127.

Chipman, J.S. and Moore, J. 1979. On social welfare functions and the ag-gregation of preferences. Journal of Economic Theory 21, 11139.

Eisenberg, B. 1961. Aggregation of utility functions. Management Science7, 33750.

Gorman, W.M. 1953. Community preference fields. Econometrica 21, 6380.

Grandmont, J.M. 1992. Transformations of the commodity space, behaviouralheterogeneity, and the aggregation problem. Journal of Economic Theory57(1), 1-35.

Hardle, W., Hildenbrand, W. and Jerison, M. 1991. Empirical evidence onthe law of demand. Econometrica 59, 15251549.

Heineke, J. and Shefrin, H. 1987. On some global properties of Gormanclass demand systems. Economics Letters 25, 155160.

Heineke, J. and Shefrin, H. 1988. Exact aggregation and the finite basisproperty. International Economic Review 29(3), 525538.

Hicks, J.R. 1956. A Revision of Demand Theory. London: Oxford Univer-

sity Press.

Hildenbrand, W. 1983. On the law of Demand. Econometrica 51, 9971019.

Hildenbrand, W. 1994. Market Demand. Princeton: Princeton University

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Press.

Hildenbrand, W. and Kneip, A. (2005). On behavioral heterogeneity. Eco-nomic Theory 25, 155169.

Jorgensen, D.W., Lau, L. and Stoker, T.M. 1982. The transcendental loga-rithmic model of aggregate consumer behavior. (in: Basmann and Rhodeseds. Advances in Econometrics. Greenwich: JAI Press), 97238.

Lau, L. 1982. A note on the fundamental theorem of exact aggregation.Economics Letters 9, 119126.

Malinvaud, E. 1956. Lagregation dans les modeles economiques. Cahiersdu Seminaire dEconometrie 4, 69146.

Malinvaud, E. 1993. A framework for aggregation theories. Ricerche Eco-nomiche 47(2), 107135.

Mitjuschin, L.G. and Polterovich, W.M. 1978. Criteria for monotonicity ofdemand functions. Ekonomika i Matematicheskie Metody 14, 122128. (InRussian.)

Nataf, A. 1948. Sur la possibilite de construction de certains macromode-les. Econometrica 16, 232244.

Nelson, R. 1999. An introduction to copulas. Lecture notes in statistics139. New York: Springer Verlag.

Trockel. W. 1984. Market demand, an analysis of large economies withnon-convex preferences. Lecture notes in economics and mathematicalsystems. Vol. 223. Heidelberg: Springer Verlag.

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-

6fh

i (p,)

incomea b

Figure 1:

-

6

fhi

(p, )

incomea b

Figure 2:

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-

6

income

xh

xh0

6

xj0

?strictly increasing function

xj

Figure 3:

-

6

xl0

?

xh0x

m0

6

?

xl

xm

xh

Figure 4:

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-

6

ZZZZZZZZZZZZZZZZ

xm

2

x

m

-

Figure 5:

26

hildenbrand aggregatio theory

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