Income distributions and the axiom of revealed preference

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E U R O P E A N U N IV E R S IT Y I N S T IT U T E , F L O R E N C E DEPARTMENT OF ECONOMICS

kf*

W O R K I N G P A P E R N o . 87/308 INCQHE DISTRIBUTIONS

AND THE AXIÓIW OF REVEALED PREFERENCE

by

Birgit GRODAL* and Werner HILDENBRAND*

•Institut flter Gesellschafts- und Wirtschaftswissenschaften University o f Bonn

This is the revised version o f a paper presented at the Workshop on Mathematical Economics organized by the European University Institute in San Miniato, 8-19 Septem ber 1986. Financial support from the Scientific Affairs Division o f NATO and the hospitality oftheC assa di Risparmio di San Miniato are gratefully acknowledged.

BADIA FIESOLANA, SAN DOMENICO (F I)

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permission of the author.

(C) Birgit Grodal and Werner Hildenbrand Printed in Italy in October 1987 European University Institute

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Italy © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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INCOME DISTRIBUTIONS

AND THE AXIOM OF REVEALED PREFERENCE

by

B. Grodal and W. Hildenbrand

1. Introduction

It was observed by H. Scarf (e.g. Kehoe (1985) or Mas-Colell (1985)) that the weak axiom of revealed preference for the market demand function is the most general condition on the consumption sector of a private ownership economy that alone garantees uniqueness of equilibria for a regular economy if no restriction on the production sector of the economy is imposed. Hence, as long as one does not restrict the form of the total production set, the crucial question for uniqueness of equilibria is whether one can expect the axiom to hold for a reasonably general class of consumption sectors.

It is well-known that the axiom for the market demand function does not follow from the hypothesis of individual utility maximization, even if one places strong assumptions on individual preferences. For example, the axiom is not implied if all individuals have Cobb-Douglas utility functions that are not identical or, if all individuals have identical preferences that are not homothetic. In this note we want to show why in an economy with private ownership

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in endowments and production with constant returns to scale the axiom of revealed preference for market demand, typically, does not hold. The main reason for this, as we shall show, is that the income distribution in such a private ownership economy depends too sensitively on the price system.

2. A . Wald’s Axiom and Uniqueness o f Equilibria

Let p i-> F (p) denote a continuous function of Ht!++ into IR l that satisfies Wa/raj identity, i.e., p ■ F(p) = 0 for every p G The function F is interpreted as the mean net demand function of a private ownership economy with a constant returns to scale production sector.

The following property of the function F was f i r s t formulated by Wald (1936) and la t e r applied to th e individual net demand function by Samuelson (1948) who called this property the "axiom of revealed preferences”. Definition . The function F satisfies W ald’s Axiom i f for every p,p' g XR++ ,

p • F (p ') < 0 and p' ■ F (p) < 0 im plies F (p) = F(p') .

Note that the Axiom implies that the function F is homogeneous (i.e. F(Ap)) = F(p), A > 0).

An economy £ is defined by a mean net demand function F (-) and a total production set Y C IR 1, which we assume to be a closed cone.

Definition . A price vector p* is called an equilibrium for the economy £ = ( F , Y ) i f

F (p *) £ Y and sup p* • y = 0 .

v€Y

It follows directly from the Axiom that F (p ') = F (p') if both, p* and p', are equilibria. Furthermore, one easily shows that the set o f equilibrium price vectors is convex. Hence, if the economy £ has at least one regular equilibrium, then the equilibrium is unique up to normalization.

Conversely, if the function F does not satisfy the Axiom, then, as H. Scarf 2 © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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observed1^, there exists a total production set Y such that the economy E = (F, Y ) has multiple equilibria. Indeed, if the Axiom is not satisfied, then there are price vectors p andp' such that p-F(p') < 0, p '-F (p ) < 0 and F (p) F \p'). Define

Clearly, the economy E — {F,Y ) has at least two equilibria p and p'.

Note, however, that in many applications the total production set Y has a particular structure. For example, if every consumption good j is produced in a "production sector”, then

where every production set Yj - the sector j - has no joint production; the only output is the consumption good j . Furthermore, it is often assumed, that the production set Yj of every sector belongs itself to a certain class, e.g., it is given by a CES-production function.

It is not clear, however, to what extend such restrictions of the total pro duction set weaken Scarf’s negative conclusion.

3. W ald’s Axiom and Income Distributions

We now consider a class of economies which have a particular structure; there are initial endowments which are used as factors of production but which are not consumed (i.e., they do not appear in any individual consumption set). We call these commodities "pure factors of production”. Many general equilib rium models in the literature are of this type. For example, Cassel (1918), Wald (1936), McKenzie (1954) or most of the "applied general equilibrium models” see e.g. Scarf and Shoven (1984).

Assume there are m > 2 such "pure factors of production”; e e IRm. The price vector of these commodities is denoted by q e IR++- There are n other

’) Already A. Wald (1936) made a similar remark ”fur die Gultigkeit unseres Theorems (ist) die Bedingung 6 (the Axiom) unentbehrlich”, p. 646. Wald/ however, gave only an example, where the Axiom was not satisfied and then showed that this economy has multiple (actually a continuum of) equilibria.

Y = {y 6 IR l | p ■ y < 0 and p' ■ y < 0} . © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research

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commodities; their price vector is denoted by p G IR"+ . Then the mean net demand function F has the following form:

/•F i (P> ?) \

R(p> 9) = Fn{p,q)_{- e i} _-

e

m n+m

\ —e / where p • i r(p, q) = q ■ e.

If this mean net demand function JP satisfies the Axiom, then the price vector q enters into the demand system F (p, q) only through its effect on the mean income q -e that is derived from the "pure factors of production”. Indeed the Axiom implies that for every (p,q) the function F (p, •) is constant on the set

#?,« = {9 € LR++ | q ■ e = q ■ e} .

In fact q,q' G implies that (p, q) ■ F (p ,q ') < 0 and (p, q') ■ F (p ,q ) < 0, and hence, by the Axiom, we obtain F (p ,q ) = F (p ,q '), i.e., F (p,q) = F(p,

q')-Consequently, there exists an "aggregated” demand function C7(p,mean income) of ffi” + x iR + into IR n such that

F(p, q) = C(p, q ■ e) for all (p, 9) .

Lemma . I I F is differentiable, then F (p, ■) is constant on i f and only i f rank dqF (p ,q ) = 1 for every q 6

ffq,e-Proof.

9qF (p, q) = (d , Fi(p,q)) ,=1....„

‘ k=l....m

(a) The function F (p, •) is constant on if and only if for every u G ZRm with u • e = 0 one has dqF (p, q)(u) = 0.

Since p • F (p, q) = q ■ e for every q G we obtain p T dqF (p, q) = e for all q, i.e., e is a linear combination of the rows of dqF (p, q). Hence

4 © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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(b) rank dqF (p, q) > 1.

Since the rank of dqF (p ,q ) is equal to one, if and only if, all rows of dqF are collinear with e, it follows that dqF (p,q)(u) = 0 for every u € III™ with u ■ e = 0. Hence, by (a), F (p, ■) is constant on H q<c. Conversely, if F (p, •) is constant on HPtt, then by (a) dqF (p, q)(u) = 0 for every u • e = 0 and hence rank dqF (p ,q ) < 1. Consequently, by (b), we obtain rank 8qF (p ,q ) = 1.

□ The argument of this section can be summarized by the following

Proposition . II the differentiable net demand function

*■(?.«) = (f(p;9))

satisfies W ald’s A xiom then rank dqF (p ,q ) = 1.

Remark . It is well-known (e.g. Kihlstrom et al. (1976) or Mitjuschin and Polterovich (1978)) that Wald’s Axiom for the demand function F (p, q) implies that the Jacobian matrix dPtqF (p ,q ) is negative semi-definite on the hyperplane

F (p, q)-1- = {v £ £Rn+m | v ■ F (p, q) = 0}. It is not difficult to show that the

Jacobian matrix dPtqF (p ,q ) is not negative semi-definite on F (p ,q )J- if rank dqF (p ,q ) > 1. This is an alternative proof for the above proposition.

In a standard private ownership economy the condition rank dqF (p, q) = 1 is typically not satisfied. Indeed, define

F(P,1) =

E(/“

•exv.

(p, ? • e“) - ff

where / “(p, 6) denotes the individual demand function of agent o at the price system p and income b and e“ the initial endowments of agent a with pure factors of production. Then

9qF ( P , q ) = ( X > / “( P , 9 - e ° K ) Vagit / •=!, —■»_k=l © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research

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The rank of this matrix is typically greater than one. The two obvious ex ceptional cases, where either the vectors e*,o € A, of initial endowments or the vectors 5 s / “(p ,ç ■ ea),a £ A, are colli near1) can hardly be considererd as reasonable and relevant classes of consumption sectors.

As a matter of fact, economic agents differ in preferences and initial endow ments, and therefore, there is no justifiable reason why the above collinearity conditions should hold or why the covariance of thé functions

a i-* fli>/“(?, 9 •«“) and ej should be zero (which, of course, would imply rank dqF = 1).

As a consequence we have to conclude that W ald’s Axiom in a private ownership economy of the type that we have considered in this section has no acceptable justification.

Let us therefore leave the framework of the traditional micro-economic model and define the mean demand function of a lhrge consumption sector by

F (p ,q ) =

J

f(p,b)p(q,b)db

where p(q, •) denote the density of an income distribution and fh{p, •) the statis tical Engelcurve for commodity h at the price vector p, i.e., /( p , 6) is the mean demand vector (at price p) of all economic agents with income b.

Note, if one insists to derive this representation of F in the traditional way from a joint distribution of preferences and initial endowments, then one has to assume that preferences and endowments are independently distributed. The income densities p(q, ■) then depend on all prices q i , . . . , q m

-4

We obtain

dqF (p ,q ) = ( ^ j fi( p ,b ) d ,k p(q,b)dbj

*= 1 ,...,n

fc=l

As before, rank dqF (p, q) = 1 if either the vectors /( p , 6), b £ LR, or the vectors dqp(q,b),b £ IR , are collineari. The first case, i.e., linear statistical Engel- curves, is not well supported by empirical evidence. There seems to be agree ment on this claim in the empirical literature. The second case implies a strong Ì which implies, by Walras identity, that &bfa(p,q-e) is independent of a £ A. i The covariance of the functions /j(p, •) and 8,^ (q, ■) can not be zero, since

this would imply that OqF (p ,q ) = (0). 6 © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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assumption on the dependence of the income distribution on the price vector q. If the dependence of the income density p(q, ■) on the price vector q is described only by an index S(q) G IR (e.g., mean income) i.e.,

(*) p(<l,b) = r){S(q),b)

where q(s,6) is a one-parameter family of income densities and S : 1R!J*+ “ * is a differentiable function then the second case prevails, since

9,p(q,b) = d.T)(S(q),b) ■ d ,S (q ) .

It should be clear that we do not claim that this one-parameter form of income densities can be theoretically justified in the traditional private owner ship economy; the income density p(q, •) depends, in general, on the parameters Empirical analysis of income distributions, however, reveal a sur prisingly strong structural stability over time. This is shown in Figure 1, where every income density is normalized (mean income equal one).

Of course, it is not dear whether in this time series there is suffident variation of the rdative prices that could allow the condusion that the actual income densities depend only on a few parameters. To go from here to condition (*), i.e., one parameter, is surely a big step.

There are perhaps less ad hoc ways to obtain rank dt F (p, q) = 1, which is, as we have shown, a necessary condition for Wald’s Axiom to be satisfied.

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L o g no r m o 1 — F 1 t 8 Figure 1 7 0 © The Author(s). European University Institute. Digitised version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research Repository.

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References

C&ssel, Gustav [1918] Theoretische Sozialòkonomie, Leipzig: Winter.

Kehoe, Th. [1985] Multiplicity o f Equilibria and Comparative Statics, Quarterly Journal of Economics 100 (1), pp. 119-148.

Kihlstrom, R., Mas-Colell, A. and H. Sonnenschein [1976] The Demand Theory o f the

Weak Axiom o f Revealed Preference, Econometrica 44 (5), pp. 971-978.

Mas-Colell, A. [1985] The Theory o f General Economic Equilibrium; A Differential

Approach, Cambridge, University Press: Cambridge.

McKenzie, L. [1954] On Equilibrium in Graham’s Model of World Trade and Other

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Mitjuschin, L.G. and W.M. Polterovich [1978] Criteria for Monotonicity of Demand

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Samuelson, P.A. [1948] Consumption Theory in Terms o f Revealed Preference, Eco nomica N.S. 15 (60), pp. 243-253.

Scarf, H. and J. Shoven [1984] Applied general equilibria analysis, Cambridge Univer sity Press: Cambridge.

Wald, A. [1936] Uber die Produktionsgleichungen der okonomischen Wertlehre, Ergeb- nisse eines mathematischen Kolloquiums, Karl Menger (ed.), 7 (1934-35), pp. 1-6. © The Author(s). European University Institute. version produced by the EUI Library in 2020. Available Open Access on Cadmus, European University Institute Research

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:Working Paper out of print

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