On the structure of homogeneous
demand functions
Werner Hildenbrand
∗
Department of Economics, University of Bonn
Lennéstraÿe 37, 53113 Bonn, Germany
[email protected]
17. August 2010
Abstract
We characterize the structure of budget share functions derived
from a homogeneous and continuous demand system: either every
continuous function on
(0, ∞)
is limit of linear combinations of
budget share functions (limits are with respect to uniform convergence on bounded intervals) or every budget share function is
limit of linear combinations of functions of the form
cos(b log(·))
or
sin(b log(·)), b ≥ 0,
stant provided it admits a limit for
and the budget share is con-
x→0
or
x → ∞.
Keywords: demand functions; homogeneity
JEL code:
C60; D01; C02
I would like to thank Felix Otto from the Hausdor Center, University of Bonn, for
very helpful suggestions and discussions; I owe to him the reference to L. Schwarz.
∗
1
1
Introduction
The problem to be studied in this paper is the
linear function space
structure of the
L[fk ] := L[fk (p, ·) ∈ C(0, ∞)| p ∈ (0, ∞)l ]
which is generated by all expansion paths
x 7→ fk (p, x)
for a gi-
ven commodity k at all price systems p of a traditional demand
f (p, x) for l commodities, which is dened for every pril
ce system p ∈ (0, ∞) and every income level x ≥ 0. We always
assume that every expansion path fk (p, ·) is continuous on R+ ,
homogeneous of degree zero, i.e., fk (αp, αx) = fk (p, x), α > 0,
and satises 0 ≤ pk fk (p, x) ≤ x (budget restriction). If the defunction
mand function f is derived from preference maximization under
budget constraints or satises the Axiom of Revealed Preferences,
then the above properties of f are satised. Only the assumption
of homogeneity is restrictive from an economic point of view since
it requires that the demand decision depends on the budget set,
yet not on the way how the budget set is dened (no money illusion or framing eect).
Instead of the expansion path
price system
tion
and
p
fk (p, ·)
for commodity
we consider the corresponding
wk (p, ·), which
k , consider the
is dened by
wk (p, x) :=
k
at the
budget share
func-
pk fk (p,x)
. For given
x
p
linear function space
Wk [p] := L[wk (αp, ·) ∈ C(0, ∞)| α > 0]
wk (αp, ·), for all α > 0. It is
convenient to consider the budget share in log-income, z = log(x).
z
Dene vk (p, z) := wk (p, e ), z ∈ R, the budget share in logincome, and Vk [p] the corresponding linear function space spanspanned by the budget share functions
2
ned by the function
fk (p, x)
Homogeneity of demand
Vk [p] is invariant
under the shift-transformations, i.e., for every v ∈ Vk [p] and every τ ∈ R, the function z 7→ v(z + τ ) also belongs to Vk [p].
in
(p, x)
vk (αp, ·) α > 0.
implies that the linear space
This invariance property is crucial for all results to follow. Indeed, closed and shift-invariant linear subspaces of Cc (R) - the
linear space of complex-valued continuous functions on
R,
endo-
wed with the topology of uniform convergence on compact sets -
have been characterized by Schwartz (1949): every closed and shiftinvariant subspace which is dierent from the complex linear space
Cc (R) admits a topological
n rx
form x 7→ x e , where n is
basis consisting of functions of the
an integer and r a complex number
(Theorème 5, p. 879). The results of L. Schwartz (in particular,
7◦
on p.907) applied to our situation - the complexication of the
real linear space
Vk [p]
- leads to the following basic
Theorem: If lim wk (p, x) exists for x → 0 or x → ∞, then
x
the linear space Wk [p] is either very small or very large, more precisely, either all functions in Wk [p] are constant, hence
dim Wk [p] = 1, or Wk [p] is dense in C(0, ∞) with respect to
the topology of uniform convergence on bounded intervals, hence
dim Wk [p] = ∞.
Consequently any restriction of the functional form of a budget
share function
wk (p, ·)
that leads to a linear space
se closure is dierent from
Wk [p]
who-
C(0, ∞) implies constant budget share
functions. The simplest example of such a restriction is that the linear space
Wk [p] is nite dimensional1 . Indeed, in this case Wk [p]
1 The
dimension of Wk [p] and the rank of a demand system as dened by Lewbel (1990)
is quite dierent. Lewbel consideres for a given p the dimension of the linear space spanned
3
C(0, ∞). In the case of a
Wk [p], one can characte-
is closed and is certainly dierent from
nite dimensional budget share space
Wk [p] without using the general results of Schwartz (1949).
Indeed, dim Vk [p] < ∞ and invariance of Vk [p] under the shift
translations implies that all functions in Vk [p] are smooth and all
derivatives of function in Vk [p] are also in Vk [p]. But then it follows that vk (p, ·) is a solution of a n-th order homogeneous linear
rize
dierential equation with constant coecients. Since the solutions
of such dierential equations are completely classied one obtains
the claim of the Theorem (for details see section 2).
If
dim Wk [p] = m < ∞,
Wk [p],
commodity k
then there exists a nite basis of
and hence, the partial demand function
fk (p, x)
for
can be represented in the form
(G)
fk (p, x) = α1,k (p)s1 (x) + ... + αm,k (p)sm (x)
which holds for
all p = αp, α > 0
and
all x ∈ (0, ∞).
This is, of
cause, Gorman's well-known assumption in his paper Some Engel curves of 1981 with the crucial dierence that Gorman only
assumes that the representation (G) holds for every commodity
k locally,
i.e., in a neighborhood of a given point
tion, Gorman assumes that
f (p, x)
(p, x).
In addi-
is continuously dierentiable
(p, x) and that the Slutsky-Substitution matrix of the system
f (p, x) is symmetric. Then he obtains a remarkable local result
on the functional form of the expansion path fk (p, ·) in the neighborhood of x. The connection of our approach to Gorman's paper
in
is discussed in section 3.
by wk (p, ·) for all commodities k.
4
2
Finite dimensional budget share space
Wk [p]
In this section we prove the following
Theorem 1:Let w(p, x) be homogeneous of degree zero in (p, x)
and assume that the function w(p, ·) is continuous and bounded
on (0, ∞). If the linear space
W[p] := L[w(λp, ·) ∈ C(0, ∞)|λ > 0]
is nite dimensional, say dim W[p] = m , then w(p, ·) is a linear combination of at most m functions of the form I(x) = 1,
cos(bj log x), sin(bj log x), where the coecients bj are positive
and 1 ≤ j ≤ m2 . In particular, if the limit of w(p, x) for x → 0 or
x → ∞ exists, then it follows that w(p, ·) is constant.
For a budget share function
system
p
wk (p, ·)
of commodity
at the price
derived from a standard demand function one can ass-
ume without loss of economic content that
x→0
k
lim wk (p, x)
x
exists for
x → ∞.
Proof: Let v(p, z) := w(p, ez ), z ∈ R and V [p] := L[v(λp, ·) ∈
C(R)|λ > 0] By assumption, the linear subspace V [p] of C(R)
is nite dimensional and, since w(p, x) is homogeneous, the linear space V [p] is invariant under the shift-transformation x 7→
v(x + τ ) , τ ∈ R. Then it follows that all functions v ∈ V[p] are
(n)
smooth (innitely often dierentiable) and all derivatives v
of
v also belong to V[p]. (This result must be known in the literaor
ture yet we could not nd a reference. Since it follows easily from
known results we give a proof in Appendix 1.) Consequently,
v
is
≤ m) homogeneous linear dierential
coecients. Indeed, the m + 1 functions
a solution of a n-th order (n
equation with constant
v, v 0 , . . . , v (m)
in
V[p]
are linearly dependent and hence there is
5
an integer
n≤m
and coecients
a1 , . . . , a n
in
R
such that
v (n) + a1 v (n−1) + · · · + an v = 0
(1)
which proves our claim.
The solutions of the dierential equation (1) are completely
classied. The form of the solutions depends on the roots of the
characteristic polynomial
λn + a1 λn−1 + . . . + an−1 λ + an .
(2)
The following result is well-known (e.g. Hirsch and Smale(1979),
chapter 6.6):
Every solution of (1) is a linear combination of the following
n functions:
(α)
the function
z k ezλ , where λ runs through the distinct real
roots of the characteristic polynomial (2), and k is a nonnegative
0 ≤ k < multiplicity of λ; together with
(β) the functions z k eaz cos bz and z k eaz sin bz , where a + bi
runs through the complex roots of (2) having b > 0 and k is a
nonnegative integer in the range 0 ≤ k < multiplicity of a + bi.
integer in the range
υ is a bounded solution of (1). Therefore it can be
expressed as a linear combination of the bounded functions in (α)
and (β), since every linear combination with coecients dierent
from zero of unbounded functions in (α) and (β) is unbounded.
The function
For a proof see Appendix 2.
The only bounded function in
λ = 0 is a root of
form cos bz, sin bz
(α)
is
I(z) = 1,
which requires
that
(2). The bounded functions in
the
provided
6
bi
with
b > 0
(β)
are of
is a root of (2).
Since with
with
3
b>0
bi
also
−bi
is a root, the number of dierent roots
bi
n
must be less than or equal to .
2
Gorman's Some Engel Curves
Terence Gorman posed the following problem in his paper Some
Engel Curves Gorman (1981): if the demand function
a
rational consumer
(G)
f (p, x)
can be expressed in the form
f (p, x) = α1 (p)s1 (x) + · · · + αm (p)sm (x)
what does this imply on the form of the expansion path
fk (p, x)
of
of commodity
k
x 7→
at the price system p?
(G)? This
specication (G) of a
Why is one interested in demand functions of the form
is a natural question since the structural
standard demand function is not the implication of a plausible
or justied behavioral postulate. In pure theory one would never
make such an assumption. Gorman gives two reasons. First, the
representation (G) is
useful for estimating demand systems from
survey data (See Christensen et al (1975), Muellbauer (1975), Deaton and Muellbauer (1980), Jorgenson et al (1982), Blundell, Pashardes and Weber (1993), Banks et al (1997), etc.. For a recent
reference, see the survey paper by Blundell and Stoker(2005)).
Second, in the theory of income aggregation of a heterogeneous
population of consumers, a solution to the problem of 'exact income aggregation' requires demand functions that satisfy
(G)
(Lau
(1982), Heineke and Shefrin (1988) or Hildenbrand (2008)).
If the representation
(G)
holds for a given price system
p,
all
λp, λ > 0, and all income levels x > 0, then the linear space
Wk [p], dened in section 2, has nite dimension and hence by
Theorem 1 the expansion path x 7→ fk (p, x) is linear if lim wk (p, x)
x
7
exists for
2
ons,
x → ∞
or
x → 0.Thus,under
the above assumpti-
Gorman's question has a clear, yet disappointing, answer,
since constant budged share function are too special; they are
not supported by empirical evidence. Gorman's analysis, howe-
(G) holds locally in a
neighborhood of a certain price-income pair (p, x). In addition,
Gorman assumes that the demand function f is continuously differentiable and that the Slutsky-substitution matrix of f is symver, only
assumes
that the representation
metric. Then Gorman proves a remarkable local result:
the matrix
A(p) = (α1 (p), . . . , αm (p)) in the representation (G), where αj (p)
has rank three or less, and
(G) have special forms. In particular, if the
is a vector in the commodity space,
the functions sj
in
A(p) is three, then for every commodity k the expansion
fk (p, ·) coincides locally (in a neighborhood of (p, x)) with
rank of
path
a linear combination of at most three functions of the following
x(log x)nj , nj ∈ N, j = 1, 2, 3, or xmj ,
x sin(bj log x) or x cos(bj log x), all bj ∈ R+ .
form: either
else
all
mj ∈ N,
or
For alternative proofs of Gorman's result using Lie group arguments see Rusell (1983, 1996, 1998) and Jerison (1993).
We remark that one cannot obtain Gorman's result with the
method of section 2. Suitably adapted to the local setting, this method allows to prove that locally around
z = log x,
the function
vk (p, ·) coincides with a solution of a n-th order homogeneous linear dierential equation with constant coecients. Hence, vk (p, ·)
is in a neighborhood of z a linear combination of at most m functions of the form (α) and (β) of section 2. This conclusion is much
weaker than Gorman's result, however, it requires much weaker
2 The negative implication of a global interpretation of (G) has been already mentioned
in Heinecke and Shefrin (1987)
8
assumptions. The symmetry of the Slutzky substitution matrix is
not needed.
Why should a demand function of a rational consumer allow a
(G) at some (p, x), hence, by homogeneity, also at (γp, γx), γ > 0, but not at other price-income
situations? What is special about (p, x)? In micro-economic theo-
local representation of the form
ry an individual demand function is globally dened, i.e., for every
price system and every income-level. Therefore, one is tempted to
consider the case where for every price-income pair
sentation of the form
depend on
(p, x)).
(G)
(p, x) a repre-
holds locally (the representation might
Then, however, one can show that the abo-
vk (p, ·) implies that the budget share
function wk (p, ·)is constant on R+ , provided the limit of wk (p, x)
exists for x → ∞ or x → 0.
ve local characterization of
Consequently, to escape the undesirable case where Gorman's conclusion collapses to constant budget share functions, one has to
assume for the demand function
price-income pairs
allow
by
(p̃, x̃)
f
in question that there exist
where the demand function
a local representation of the form
Ef .
But if one assumes that
rity of a globally dened
f
Ef 6= ∅
(G);
f does not
denote this domain
- in order to avoid linea-
- , then one has to face an unpleasant
consequence. The explanatory variables of household's demand,
i.e., the demand function
f
the prevailing price system
household level, since one
(or preference relation), income
x and
p, are interdependent on the individual
must require (p, x) ∈
/ Ef 6= ∅!
This interdependence can create diculties for estimation and for
comparative statics analysis. For example, if one considers a population of households with a common demand function, yet heterogeneous income, then a widely spread income distribution imposes
a strong restriction on the common demand function.
9
Appendix 1: Proposition:
Let V be a nite dimensional linear subspace of C(R) which is
translation invariant (v ∈ V and τ ∈ R implies that z 7→ v(z + τ )
belongs to V). Then every function v in V is smooth (derivatives
v (n) of any order n exists) and all derivatives v (n) of v belong to
V.
Proof: The convolution K ∗ v
function
K
of a function
v∈V
with a kernel
is dened by
Z+∞
Z+∞
(K ∗ v)(z) :=
v(z − τ )K(τ )dτ =
v(τ )K(z − τ )dτ.
−∞
−∞
Consequently, if the kernel
K∗v
K
is smooth, then the convolution
is smooth. By denition, the convolution
linear combinations of transformations of
K∗v
belongs to
V
since
V
v.
is limit of
Hence, with
v
also
is closed being a nite dimensional
linear space. Consider now a Dirac sequence
nel functions
K∗v
(Kn )
of smooth ker-
Kn .
It is well known (e.g. Lang (1968), chap XI, Ÿ1 Th. 1) that the
sequence
Kn ∗ v
of convolutions converges uniformly on compact
subsets to the function
sequence in
in
C
(k)
(R),
C(R).
v.
(Kn ∗ v)n
above, (Kn ∗ v)
Consequently,
As we showed
is a Cauchy
is contained
the linear space of k-times continuously dierentiable
functions endowed with the topology of uniform converges on com-
j ≤ k . Since V∩C k (R) is nite
dimensional, it follows (Theorem of Tychono ) that (Kn ∗ v)n is
(k)
also a Cauchy sequence in C
(R). Which proves that v is smooth
and all derivatives of v belong to V.
Remark: If V is a translation - invariant subspace of C(R), then
smooth functions in V are dense in V , where V denotes the clospact sets of all derivatives of order
10
ure of
V
with respect to the topology of uniform convergence on
compact sets.
Appendix 2:
Lemma: (unbounded basic functions do not balance)
Every linear combination s(z) of unbounded basic functions in
(α) and (β) with non-zero coecients is unbounded.
Proof: We order the unbounded basic functions of s(z) lexicoλ's and a's, respectively,
z . Let (µ, m) denote the
graphically, rst by the magnitude of the
and then by the degree of the power
k
highest ranked index pair. There might be several highest ranked
basic functions; a basic function
veral basic functions in
(β)
z m eµz
in
(α)
and/or possibly se-
which dier in their b's. Note that
(µ, m) 6= (0, 0).
The function s(z) is unbounded if there exists
s(zn )
m µz
with |zn e n | → +∞ and inf | m µzn | > 0. To
z e
µ
n
a sequence exists we choose the case
a sequence
(zn )
show that such
µ ≥ 0(otherwise
consider
s(−z)). Every basic function which is not top ranked divided by
z m eµz converges to zero for z → +∞. Indeed, z k−m eλ−µ)z converges to zero for z → +∞ since either λ < µ or λ = µ and
k − m < 0.
m µz
A top ranked basic function divided by |z e | is either the constant 1(z) = 1 or of the form cos bj z or sin bj z . The claim of the
lemma now follows since for any non-zero linear combination of
these functions there exists a sequence
inf (α +
n
X
zn → ∞
such that
(βj1 cos bj zn + βj2 sin bj zn ) > 0.
j
11
References
- Banks, J., Blundell, R. and A. Lewbel, Quadratic Engel Curves and Consumer Demand., The Review of Economics and Statistics 79 (1997): 527-539.
- Blundell, R. and T. Stoker, Models of aggregate economic relationship that
account for heterogeneity.", Handbook of Econometrics 6 (2004).
- Christensen, L., D. Jorgenson and L. Lau, Transcendental Logarithmic Utility Functions., American Economic Review 65 (1975): 367-383.
- Deaton, A. and J. Muellbauer, An Almost Ideal Demand System,American
Economic Review, 70 (1980): 312-326
- Gorman, W., Some Engel Curves., in A. Deaton (ed.), Essays in Honour
of Sir Richard Stone, Cambridge: Cambridge University Press, 1981.
- Heineke, J. and H. Shefrin, On some global properties of Gorman class
demand systems., Economic Letters 25 (1987): 155-160.
- Hildenbrand, W., Aggregation: Theory., The New Palgrave, 2nd ed., 2008.
- Hirsch, M. and S. Smale, Dierential Equations, Dynamical Systems and
Linear Algebra., Academic Press, N.Y., San Francisco, London, 1979.
- Jerison, M., Russel on Gorman's Engel Curves: A Correction., Economics
Letters 23: 171 - 175, 1993.
- Jorgenson, D., Lau, L. and T. Stoker, The Transendental Logarithmic Model
of Aggregate Cosumer Behavior., Advances in Econometrics, R.L. Basmann
and G.F. Rhodes, Jr. (eds.), Greenwich: JAI Press.
- Lang, S., Analysis I., Addison-Wesley, 1968.
- Lau, L., A note on the fundalmental theorem of exact aggregation` Econonomic Letters 9 (1982): 119-126.
- Lewbel, A., Characterizing some Gorman systems that satisfy consistent
aggregation., Econometrica 55 (1987a): 1451-1459.
- Lewbel, A., Full Rank Demand Systems, International Economics Review
31 (1990): 289-300.
- Lewbel, A., The Rank of Demand Systems: Theory and Nonparametric Estimation., Econometrica 59 (1991): 711-730.
12
- Muellbauer, J. Aggregation, Income Distribution and Consumer Demand.,
Review of Economc Studies 42 (1975): 525-543.
- Russel, T., On a Theorem of Gorman., Economic Letters 11 (1983): 223 224.
- Russel, T. and F. Farris, The Geometric Structure of Some Systems of Demand Functions., Journal of Mathematical Economics 22 (1993): 309-325.
- Russel, T. and F. Farris, Integrability, Gorman Systems and the Lie Bracket
Structure of the Real Line., Journal of Mathemarical Economics 29 (1998):
183-209.
- Schwartz, L., Théorie Générale des Fonctions Moyenne-Périodiques., Annals of Mathematics 48(1947): 857-929.
13