Resumé

SUMMARY: In this paper it is shown that nice properties of the aggregated demand
function can be obtained from assumptions on the distribution of consumers' characteristics.

Introduction

In traditional micro economic theory the starting point is a description of the economy a set of agents: consumers, producers, the state etc., and individual characteristics these agents, e.g. preferences and production technologies. In order to make the economic analysis one makes assumptions on the individual characteristics such as to obtain that each of the agents is well behaved.

However, when looking at an economy it is not only important how well the individual behave, but also how the individual characteristics are distributed over the space of agents' characteristics. Even if all the individual agents did not behave well, the aggregated system i.e. the total economy might be well behaved, due to the way the individual characteristics were distributed.

To involve distrubutional assumptions is still more important since even when the individual agents behave well, it is not necessarily the case that the aggregated system is well behaved. A classical example is the weak Axiom of Revealed Preference. Of course, a consumer characterized by a consumption set and a total, strictly convex preordering the consumption set will give rise to a demand functon, which satisfies the weak Axiom. However, the aggregated demand function will in general not satisfy the weak Axiom. In Grodal and Hildenbrand (1989) and Hildenbrand (1989) it is even proved that generically the weak Axiom will not be satisfied for the aggregated demand function, if the economy has a special structure.

The need for distributional assumptions was highlighted in the important Sonnenschein Debreu - Mantel Theorem (see e.g. Mas - Colell (1985)) on the arbitrariness of the excess demand function of an exchange economy. It was shown that any continuousfunction satisfies Walras law and which is homogeneous of degree zero can be obtained (except for the boundary) as excess demand function for a pure exchange economy with £ consumers, who satisfy even the most restrictive of the individual

assumptions of general equilibrium theory (smoothness, strict convexity, monotonicity, etc.). This especially implies that in general the Jacobian matrix of the total demand of an exchange economy will not have any other properties than the ones, which can be obtained for any continuous differentiable function F, which is homogeneous of degree zero and for which the mapping pF(p) is linear.

The reason for this negative result is that the Jacobian matrix of the total demand function only gets nice properties on the intersection of the subspaces orthogonal to the individual demands. For each consumer it is on the subspace orthorgonal to his demand the income effects disappear. By constructing the individual consumers nicely, but with different demand behavior, one obtains that the intersection of the subspaces {o}, and all the nice properties of the individual Jacobian matrices, the part coming from the Slutsky substitution matrices, have disappeared.

It is the purpose of the present paper to point at some examples within demand theory, results on the aggregated behavior have been obtained, not due to strong assumptions the individual behavoir, but due to distributional assumptions on the consumption of the economy.

A Consumption Sector

We shall define a consumption sector as a measure on the space of consumers' characteristics. C be a separable metric space of consumers' characteristics. We could e.g. have C=p*W, where IR' is the commodity space and p the set of continuous irreflexive relations with the topology of closed convergence, or C=v7_xIR+, where fi is a set of individual demand functions with some separable metric. In the first case a consumer characteristic would be a pair (>, oj), where w is a vector of initial and > a preference relation. In the second case a consumer characteristic be if, x), where/is a demand function and x an expenditure level or income. order not to complicate notation we shall always assume that the consumption set is the positive orthant of IR' i.e. IR(. We shall also assume that individual demand functions feft, f IRl+*IR +—»IRI, are continuous differentiable and that they satisfy pf(p,w)=w. We do not assume that the individual demand functions are obtained by maximizing a preference relation.

The actual consumers in an economy % can the be dscribed by a measureable mapping (I, $,y)^>C, where (I, $,y) is a measure space of consumers and where for all i el, %(i)eC is the individual characteristics of consumer /. In the case, where / is a finite set, we can take 3 to be all subsets of/ and y to be counting measure

Given the mapping % we can define the corresponding distribution /a^ on C by
= y{iel I for all measurable sets FCC. ix%(F) is the relative number of
consumers, who have characteristics in the set F.

We define a consumption sector as a measure \x on the space of consumers' characteristics.

Any description % of a set of consumers will as shown above give rise to a uniquely defined consumption sector. However, a given consumption sector can be generated by means of different sets of consumers; any consumption sector /i can e.g. be obtained as the distribution of an atomless measure space of consumers. Indeed let the atomless measure space of consumers be given by /=Cx[o,l], with the product measure y=fx<S)\, where A denote the Lebesque measure on [o,l]. Now we let %: I^>C be defined %(c,s)=c. Then we obtain that the atomless measure space of consumers (1,3, y) with the discription % has exactly the distribution fx. Looking at a consumption sector /x we often have his standard representation with many consumers in mind.

We shall in the following consider consumption sectors (x defined on C=px\R+ or $*IR+. It should be noticed that in this setup the distribution of total expenditure (income), the marginal distribution fx₂ on IR+ of fx, is by definition independent of the price system/?.

We shall also assume that mean expenditure x =J r+jc djx2 is finite.

Continuous and Differentiable Demand

We now consider a consumption sector jx defined on #x|R+. Let /a, be the marginal
distribution of fx on p.

If we assume that for all preference relations >in the support of /x_b > is either transitive or convex, then we can for each characteristic (>, x) define the corresponding correspondence <p(>, x,-)-' IR++=^ iR^ and we can define that the mean market demand for the consumption sector by F(p,\x) =J IR+ (f>(>, x, p) d\x.

Under standard assumptions we shall have that the demand correspondence F(-,iJi):\Ri₊ => IR' is upper hemi-continuous, but a natural question to ask is, if there exist conditions on jx such that F(-,fi) is in fact a continuous function. Of course, this will be the case if we assume that for every consumer characteristic (>, x) in the support fx, the preference relation > corresponds to a strictly convex and total preordering. indeed <p(>, x, •) will be single valued and consequently F(-,fi), will be single valued, and therefore a continuous function. However, the main question is, if there exist conditions on fx such that F(-,\x) is single valued even when the individual demand correspondences do not have this property.

In the book W Trockel (1984), which partly is based on papers by E. and H. Dierker
and W. Trockel, is given a very profound discussion of the different ways to obtain

into a situation where the consumption sector fx is a product measure and where preferencesare (but not strictly convex). The main idea is that if we look at a fixed preference relation, then (for most preference relations) the set of price systems for which there is not a unique maximal element in the budget set is a null set. This null set of prices will vary systematically when income of the consumers vary, but the preferencerelation fixed. By making the assumption that income is dispersed they obtain that for all preferences, except a residual subset in the space of C^x utility functions,the correspondence after having just integrated over the wealth distribution,will a continuous function.

In order to obtain not only continuity of F(-,fji) but also that F(-,}x) is a continuous
function, one needs to introduce assumptions also on the measure
ixx on preferences.

The Weak Axiom and The Law of Demand

We shall go one step further. In order to make comparative statics and also to use the economic model for policy issues, one wants to have more information on the market demand. Indeed one should like to know that F(-,fx) satisfies the Law of Demand or at least the weak Axiom of Revealed Preference. E.g. it is well known that the weak Axiom for the market demand function is closely related to uniqueness of equilibria.

The Law of Demand, i.e. the monotonicity of the market demand function F, for the consumption sector, means that the price change and the demand change are in "opposite and therefore especially that the market demand for a commodity is a non-increasing function of its own price. This is a useful property when e.g. studying stability of the equilibrium or making comparative statics.

Formally the market demand function F(-,ix) satisfies the weak Axiom if for all price
systemsp,qe\R{+:qF(p,fji)^ x impliespF(q,^x)>x.

Correspondingly, the market demand function F(-,(i), satisfies the Law of Demand
if for all price systems p,qe\R{+:(p-q)(F(p,iJi) - F(q,\x)) 0.

We are interested in finding conditions on a consumption sector fx such that F( •, (jl)
satisfies respectively the weak Axiom and the Law of Demand.

Let us assume that each consumer is described by a demand function /and a level of total expenditure Jcelß+, and let us assume that the admissible demand functions are parametrized a parameter ae^. This is no restriction since the parameter space s4 can be any seperable metric space, for example, the space of all admissible demand functions.

The space of consumers' characteristics is then +, and the market demand
function F(\\x): R(+—>;Rl is

We shall assume that the marginal distribution fi2 of expenditure has a density p and
that the conditional distribution jjllx on sd exists for every expenditure jc in the support
of p. We can then define the cross-section demand function f: !R(+xlR+—>IRf

Using the cross-section demand function/ we obtain

Now it is well known that the Law of Demand as well as the weak Axiom for the
market demand function F(-,fx) can be expressed in terms of the €*■€ Jacobian matrix

Indeed

(1) F(-,ix) satisfies the weak Axiom, if and only if dp F(p,fi) is negative semi-definite

and

(2) F(-,ix) satisfies the Law of Demand (monotonicity), if and only if dpF(p,ix) is
negative

Therefore in order to find out whether F satisfies the weak Axiom and the Law of
Demand one has to investigate if the Jacobian matrix dpF(p,fx) has the above properties.

We first notice that the Slutsky decomposition of the cross-section demand function
/ leads to

where Sf(p,x) is the Slutsky substitution matrix of the cross-section demand function

Hence we obtain

Therefore, if S is negative semi-definite and A is positive semi-definite, we have
that dp F(p, n) is negative semi-definite.

The matrix S is studied in Hårdie (1991) and Hildenbrand (1991). They show that if the individual demand functions satisfy the weak Axiom, and the dispersion of the distribution of the demands of the consumers with total expenditure x would increase if they have had the expenditure x+A, then Sf(p,x), and therefore S, is negative semi-definite.

The other matrix A =\ +^{d x} f(p,x) f(p,x)^Tp(x) dx, which is the most delicate one, is analyzed in details in the paper by Grodal and Hildenbrand (1991). We want to find conditions such that A is positive semi-definite (on the hyperplan determined by mean demand).

First we remark that the matrix A is positive semi-definite (on the hyperplan determined
mean demand) if and only if the symmetrized matrix

is positive semi-definite (on the hyperplan determined by mean demand). Let us denote
demand with a (f,p) i.e. a(f,p) = J]R+ f(p,x) p(x) dx.

Looking at B(f,p) we see by partial integration that if the density p has the support
IR+ and is differentiable, then

Now we notice that the matrix (f(p,x) f(p,x)^T) is a positive semi-definite for all ;telR₊ (it has rank 1 and the only non-zero eigenvalue is jf(p, x)/²). Therefore if the density non-increasing, then B(f,p) is indeed positive semi-definite. This gives essentially example in Hildenbrand (1983).

In the general case we have B(f,p) =£,- 82,B2, where

B_x and B2B₂ are positive semi-definite matrixes, and we get that B(f,n) will be positive if every eigenvalue of B_x is larger or equal to every eigenvalue of 82.B₂. However, if B_{ has eigenvalue 0, e.g. since {f(p,x)\x: p'(x)^o} is contained in a lower dimensional subspace of IR^, then even small eigenvalue of B2B₂ can prevent the positive semi-definiteness of B.

When we in the general case want to investigate the matrix B(f,p), we have two different namely the cross-section demand function for a fixed p i.e. the cross- section Engelfunction/(/?, •), and the distribution p of consumers' total expenditure. is the shape of the various cross-section Engelfunctions as well as the shape of the expenditure distribution, which play an important role in deriving the positive semidefiniteness the hyperplan determined by mean demand) of the matrix B(f,p). In Grodal and Hildenbrand (1991) it is shown that B(f,p), and therefore the mean income effect matrix, tends to be positive semi-definite if the cross-section Engelfunctions and the expenditure distribution have the following characteristics:

(1) The cross-section Engelfunctions "bend slowly", in the sense that higher derivatives
be neglected.

(2) There is sufficient "dispersion" in the distribution of total expenditure.

Grodal and Hildenbrand consider cross-section Engelfunctions, which can be generated
given family of base function bx(-, •), >bs(-, •) from IR++x|R+ into IR. I.e. it is
assumed that the cross-section Engelfunctions have the form

Of course, any cross-section demand function / can be written in the above form by choosing S= £ and choosing the base function appropriately. However, the idea is that the number of base functions is small compared to the number of commodities, and that the base function are chosen a priori. Let £(b) be the linear function space spanned by the functions b\(p, •),...,b_s(p, •). Since we only consider a fixed price system, we can drop p as a variable.

When looking at a family of base functions b we can for this family and any density
p on IR+, define the S*S matrix B(b,p), by

and the vector a(b,p) eIR5 by

In Grodal and Hildenbrand (1991) the following proposition is proved:

Proposition. The rank of the matrix B(f,p) is not larger than the number S of base functions, and the £*£ matrix B(f,p) is positive semi-definite (on a(f,p)^) for every function fe£(b) if and only if the S*S matrix B(b,p) is positive semi-definite (on a(b,_Pn.

There are two different ways of looking at this Proposition. From a theoretical point of view one may hope to specify families of base functions which imply that B(b,p) is positive semi-definite (or, at least, positive semi-definite on a(b,p))^) for a reasonable large class of expenditure distributions. However, the Proposition also shows that if, for a given data set, one wants to analyze whether the matrix B(f,p) is positive semi-definite, one should not estimate the cross-section demand function, provided one assumes that it is generated by a given family of base functions b. Indeed, one should investigate directly whether the matrix B(b,p) is positive semi-definite for the estimated distribution.

As an example of how the Proposition can be used, look at the base functions
bs(x)=x\ 5=1,..., 5, and the corresponding class of cross-section Engelfunctions £(b).
In this case the matrix B(b,p) will reduce to a "matrix of moments", M(S,p), of the expenditure
and a(b,p) to a vector m(S,p) of moments. The matrix M(S,p)
involves the first 2S-\ moments. Using the proposition above we get that B(f,p) will
be positive semi-definite (on the hyperplan a(J,p)^) if the matrix of moments M(S,p)
is positive semi-definite (on the hyperplan m(S,p)^). Therefore the problem reduces to
the question: For which kinds of income distributions p is the "matrix of moments" positive

First we notice that if S=2 then the matrix M(S,p) is positive semi-definite on
(m(S,p))^ for any density p. For higher S the question can be approached either empirically
theoretically.

Using the U. K. family expenditure survey for the years 1969-83 estimations show that for S less or equal to 4 the estimated matrix turned out to be positive definite. For larger values of S, say 5=7, some eigenvalues turned out to be negative but they were all very small. (These estimations and calculations have been made by H. P. Schmitz, SFB 303, Bonn).

As an example we also directly calculated the matrix M(S,p) for a log normal distribution. this case it turns out that for all S there exists a constant k(S) such that if the coefficient of variation of p is larger than k(S) then M(S,p) is positive definite. For example, we calculated that *(2)=0,36, A(3)=0,50 and *(4)=0,59.

Empirical estimates of the coefficient of variation of income distributions show that
the coefficient of variation is typically around 0,7, i.e. the empirical coefficient of variation
larger than the critical value 0,59, which is obtained for ,S=4.

In conclusion it is seen that there exists reasonable conditions on the distribution of total expenditure, which together with the assumption that cross-section Engelfunctions not bend to quickly (low number of base functions) imply that the delicate matrix mean income effects for a consumption sector is well behaved. The assumption that a low number of base functions can be used is of course closely related to the number commodities i.e. to the definition of a commodity (commodity group), but is in the paper Grodal and Hildenbrand (1991) not directly related to the distribution of agents' characteristics. However, empirical investigations show that Engelfunctions for commodity aggregates are indeed bending rather slowly, see Hildenbrand and Hildenbrand(l9B6).

The above results for the mean income effect matrix A together with the results on the mean Slutsky substitution matrix S indeed show that there exist conditions on the distribution of consumers' characteristics, conditions which partly are supported by empirical evidence, and which give the mean demand function more structure than what can be obtained from assumptions on the individual characteristics.

Diagonal Dominance

In a recent paper, J. M. Grandmont (1991), is given a very beautiful example, which gives even more properties of market demand. As in the papers by E. and H. Dierker and Trockel, Grandmont considers certain transformations of preference relations and distributional assumptions on these transformations.

Let the space of consumers' characteristics be the set of pairs (f, x), where / is a demand and jc an expenditure level. The basis of Grandmont's example is that the consumption sector (x can be decomposed in a measure on a set A of types, where a type corresponds to a demand function f_a and an expenditure level x_a , and for each type a, a conditional distribution over the space of all transformations of the demand function/,. A transformation corresponds to a change of units of measurement for the commodities.

Grandmont investigates conditions on the conditional distributions over the transformations, imply nice properties for the market demand function. Among others he shows that if the integral of the absolute value of the derivatives of the conditional densities are sufficiently small, then the market demand for commodity h is a decreasing of its own price, and moreover the Jacobian matrix has approximately a dominant diagonal for a large set of price systems.

One major difference between the previous results and the result by Grandmont in its present form is that the distributional assumptions cannot be tested directly. However, descripton of the consumption sector used by Grandmont implies that for each tyPe (fa> xa) tnere is. f°r eacn price system pelßf_+, a specific distribution of the demands the budget hyperplan determined by p and x_a. For each expenditure level and the actual price system p one has also information on the empirical distribution of demands this budget hyperplan. If one assumes that all individuals with a given total expenditure x have demand functions, which are transformations of each other, then the consequences of distributional assumptions on the transformations can be compared to the empirical distribution on the budget hyperplan.

It will be a subject of further research to link assumptions on the distribution of transformations of a demand function and assumptions on the distributions of demand choices, and to find nice conditions on these last distributions, which will imply the beautiful conclusions in Grandmont's paper.

Distributions and Profit Functions

It is not only of importance to demand theory itself to investigate the properties of the aggregated demand function. If one e.g. looks at models with imperfect competition, is necessary in order to prove existence of equilibria (in pure strategies) that the profit functions are quasi concave. Again, only in trivial examples one obtains this property the theory is based on properties of the individual characteristics. In the recent papers E. Dierker (1991), and Caplin and Nalebuff (1991) it is shown that with distributional on the consumption sector it is possible to obtain quasi concave profit functions for the firms and thereby existence of equilibria.

Conclusion

As shown above distributional assumptions might throw more light on some of the areas where it has not been possible to obtain positive results from only individual assumptions. might also be the case that distributional assumptions can replace some of the assumptions on individuals, of which we are not totally comfortable.

References

Caplin, A. and B. Nalebuff. 1991. Aggregation Imperfect Competition: On the Existence of Equilibrium. Econometrica 59:25-59.

Chiappori, P. A. 1985. Distributions of Income
the "Law of Demand". Econometrica

Dierker, E. 1991. Competition for Customers.
In Equilibrium Theory and Applications,

edt. W. Barnett, B. Cornet, C. D'Aspremont,
Gabszewicz and A. Mas-Colell,
Cambridge, p. 383-402.

Dierker, E., H. Dierker and W. Trockel. 1980. Continuous mean demand functions derived nonconvex preferences. Journal of Mathematical Economics 7:27-33.

Grandmont, J. M. 1991. Transformations of
the Commodity Space. Behavioral Hete-

rogenisity, and the Aggregation Problem.
CEPREMAP.

Grodal, B. and W. Hildenbrand. 1989. The Weak Axiom of Revealed Preference in a Productive Economy. Review of Economic Studies 56:635-639.

Grodal, B. and W. Hildenbrand. 1991. Cross- Section Engelcurves, Expenditure Distributions the "Law of Demand". To appear in Aggregation, Consumption and Trade: Essays in Honor ofH. S. Houthakker, L. Phlips and L. D. Taylor, Amsterdam, 1992.

Hildenbrand, W. 1983. On the "Law of demand".
51:997-1019.

Hildenbrand, W. 1989. The Weak Axiom of
Revealed Preference for Market Demand

is Strong. Econometrica 57:979-985.

Hildenbrand, K. and W. Hildenbrand. 1986. On the Mean Income Effect: A Data Analysis the U. K. Family Expenditure Survey. Contributions to Mathematical Economics, in Honor of Gerard Debreu, edt. Hildenbrand, W. and A. Mas-Colell, p. 247-268.

Hildenbrand, W. 1991. Two Essays on Market
Demand. To appear.

Hårdie, W., W. Hildenbrand and M. Jerison.
1991. Empirical Evidence on the Law of
Demand. Econometrica 59:1525-1549.

Mas-Colell, A. 1985. The theory of general
economic equilibrium. Cambridge.

Trockel, W. 1984. Market Demand. Berlin.