A Note on the Decomposition (at a Point) of
Aggregate Excess Demand on the Grassmannian
Piero Gottardi
Universita’di Venezia and Brown University
Andreu Mas-Colell
Universitat Pompeu Fabra
Revised April 1999
Abstract
This paper analyzes the properties of aggregate excess demand functions
for economies with an arbitrary …nite set of N commodities where agents
face trading restrictions of a general, abstract form: their budget set is
de…ned by K dimensional planes in RN : It is shown that, if there are at
least K agents in the economy, the only general property satis…ed by the
value of aggregate excess demand and its derivative, at any arbitrary point,
is Walras Law. The result is established by considering an economy where
agents’preferences are of a ’generalized Leontief’type.
JEL Classi…cation Nos.:D52, C62
Keywords: Market Demand, Disaggregation, Missing Markets
Most of the work for this paper was carried out while the …rst author was visiting Universitat
Pompeu Fabra, in January 1997 and again in January 1998. He wishes to thank that institution
for the warm hospitality and the support.
1. Introduction
In this note we examine the properties of aggregate demand functions when agents
face trading restrictions of the ”missing markets”type. Formally, the agents’budget set will be de…ned by K dimensional planes in <N : This abstract formulation
is quite general. For example, when …nancial markets are incomplete, the budget
equations each agent faces can always be reduced to the condition that his net
trades in commodities have to lie on (or below) a K plane. The speci…cation
of these planes is a rich parameter space, describing both the restrictions arising
from the value of prices and from the form of the trading constraints (thus in the
case of incomplete markets the parameters will include both prices and the level
of asset returns).
When markets are complete and agents face no restriction in their trades,
Sonnenschein (1973), Mantel (1974) and Debreu (1974) showed that if there are
at least N agents in the economy, the aggregate excess demand has no properties,
on a compact set of prices, besides continuity, homogeneity and Walras Law1 .
The validity of a similar result for the case in which the agents’choice problem is
subject to trading restrictions of the general form mentioned above was o¤ered as
a problem in Mas-Colell (1986), and progress has not been obtained until recently.
Some …rst steps towards a solution were made by Bottazzi and Hens (1996)
and Gottardi and Hens (1999), who showed that, in the presence of incomplete
markets, when agents’ demand is written as a function only of commodity and
asset prices, aggregate demand again has no structure2 . Chiappori and Ekeland
1
See Shafer and Sonnenschein (1982) for a survey of the further developments of this literature.
2
In particular, Bottazzi and Hens (1996) look at the properties of aggregate excess demand,
2
(1999) have then shown that any analytic function satisfying the above mentioned
restrictions describing the trading constraints can be decomposed, in the neighborhood of an arbitrary point, as the aggregate excess demand of an economy
with K agents.
In the work reported in this note we o¤er a result which is less general than
Chiappori and Ekeland. Yet our method of proof is rather di¤erent, and is simpler (for a simpler result). Also, the preferences we rely on to rationalize the
given function as an excess demand function are quite distinct and may be of
independent interest.
Speci…cally, what we do here is to solve the linearization-at-a-point problem.
That is, in the context of the general Grassmannian problem, we show that, if there
are at least K agents in the economy, the value of aggregate excess demand and of
its derivative, at a prespeci…ed point, can be arbitrary (except for the restrictions
imposed by Walras Law). We do this by considering a collection of agents whose
preferences are of the ”generalized Leontief”type, as they are de…ned, e¤ectively,
on an a¢ ne subspace of <N of dimension N
K: The characterization of the
agents’preferences and demand in such case, and more generally the analysis of
how the properties of a collection of agents of this type give rise to aggregate
demand, are, we believe, of independent interest.
over a compact set of prices, for the case of real assets, while Gottardi and Hens (1999) examine
the properties of its linear approximation at a point, in the presence of incomplete markets with
nominal assets.
3
2. The Economy
We consider a pure exchange economy with N commodities and H consumers.
Each consumer h = 1; ::; H is characterized by preferences described by the function U h (:); de…ned on <N and assumed increasing and quasi-concave, and by
endowments ! h 2 <N :
Let K < N and G N;K denote the Grassmannian manifold of K planes in <N ;
de…ned by fL
RN : L is a K dimensional linear subspace}; also G+N;K
fL 2
G N;K : L \ <N
+ = f0gg:
The choice problem faced by an arbitrary agent h; for L 2 G+N;K ; is the following
max U h (x)
s:t: x
!h 2 L
(P h )
<N
+
Let xh (L) denote the solution set of the above problem, describing the agent’s
demand at L; similarly the agent’s excess demand set is z h (L) = xh (L)
Note that if the utility function is strictly increasing, then z h (L)
!h.
L:
Remark 1 ( ). The speci…cation of the budget constraints in (P h ) allows us to
capture various kinds of market structures. When K = N
1, we obtain the
case of complete markets, and L is simply the budget hyperplane de…ned by the
price vector; with free disposal the budget set is then the half space lying below
this hyperplane. On the other hand, when K < N
1 the budget constraints
correspond to the case in which markets are incomplete (and N
K
1 is the
number of ’missing markets’)3 , or more generally in which agents face a set of
3
See Mas-Colell (1986), Balasko and Cass (1989) for a more complete argument illustrating
how the budget equations with incomplete markets can be reduced to the more abstract form
considered here.
4
linear restrictions on the level of their net trades. In this case the set L re‡ects
both the level of prices and the speci…cation of asset returns (more generally of
the trading constraints); thus changes in L may correspond to changes both in the
level of prices and in asset returns:
Adding individual agents’ excess demands we obtain the expression of the
aggregate excess demand
z(L) =
X
z h (L)
h
which also satis…es the condition: z(L)
L
N;K
<N
+ for all L 2 G+ : Of course,
with strictly increasing utility functions we have z(L)
L for all L 2 G+N;K , which
we view as the expression of Walras Law in our set-up.
3. The Problem and the Result
We want to examine the problem of whether the optimizing behavior of the agents,
under the above general speci…cation of their budget equations, imposes any restriction on the form of an aggregate di¤erentiable excess demand function, in
addition to Walras Law. We will, however, address this issue in a more restricted
way, by limiting our attention to the values of aggregate excess demand and of its
derivative, at an arbitrary point L 2 G+N;K :
Assume that, in a neighborhood of L; an arbitrary di¤erentiable function,
satisfying Walras Law, is given to us: To provide a more precise formal de…nition of
the problem, it is convenient to introduce and make reference to a local coordinate
system on G+N;K :
For each L 2 G N;K we can always …nd a N
N permutation matrix P
I
IK
and a N K matrix of the form [ K ], such that Sp[P
] = L, where
A
A
5
Sp[:] denotes the linear space generated by the columns of a matrix, IK is the
K dimensional identity matrix and A is a matrix of dimension (N
K)
K:
Hence L is identi…ed by a pair (P ; A); and a …nite parameterization of a small
neighborhood N (L)
G+N;K of L is induced by the elements of the (N
matrix A in a neighborhood of A; V(A)
fA 2 <(N
K) K
: Sp[P
K) K
IK
]2
A
N (L)g:
Using this parameterization, an aggregate excess demand function, in the
neighborhood N (L) of L; can also be written as a function of A; z h (A); for
A 2 V(A): In addition, if the agents’utility function is strictly increasing, Walras
IK
Law assures us that z h (A) = [P
] h (A); for some h : V(A) ! <K : Hence
A
we can naturally take derivatives of z h (:) with respect to A; whenever h (:) is
di¤erentiable. Let DA z h denote the derivative of z h (A) with respect to A.
The problem we intend to analyze is then formally stated as follows:
De…nition 3.1. Let (A; P ) be arbitrarily given and F : V(A) ! <N be a di¤erIK
entiable function satisfying Walras Law: F (A) 2 Sp[P
] for all A 2 V(A):
A
Then we say that the function F (:) can be rationalized, at (A; P ); by H agents if
there exists an economy, as described above, with H agents whose excess demand
is di¤erentiable, at A; and satis…es:
P h
= F (A)
Ph z (A)
h
D
z
(
A)
= DA F (A)
A
h
Our main result is the following:
6
Theorem 3.2. For an arbitrary pair (A; P ); any di¤erentiable function F :
V(A) ! <N satisfying Walras Law can be rationalized, at (A; P ); by K agents
if F (A) 6= 0, or by K + 1 agents if F (A) = 0.
Hence the rational behavior of agents does not impose any restriction on the
value and the derivative of aggregate excess demand, at a point, even with the
present very general speci…cation of the budget set, and the large set of parameters
with respect to which demand is de…ned (which includes, as we saw, not only
prices but also the description of trading constraints, or asset returns in the case
of incomplete markets).
Note also that the minimal number of required consumers is not related to the
dimension of the parameter space but to the dimension of the individual budget
sets. Thus, for K = 1 and K = N
(N
1 the parameter space has the same dimension
1) but, for K = 1; one consumer su¢ ces to rationalize demand, while with
K=N
1 we need N
1 consumers (if F (A) 6= 0):.
4. A Class of Preference Relations
We describe in this section a family UK of monotone, quasi-concave utility functions in <N : This family can be viewed as a generalization of the class of Leontief utility functions. The decomposition result will be established by …nding
economies whose agents’utility functions all lie in UK .
Let E be an arbitrary a¢ ne linear subspace of <N of dimension (N
K) and
let q be an arbitrary N -dimensional vector: The pair (E; q) completely determines
7
a utility function in UK ; U (:; E; q), as follows:
8
max 9q x0 ;
if there is x0 2 E such that x0 6 x :
>
>
> 8
< x0 6 x =
>
<
: x0 2 E ;
U (x; E; q) =
>
>
>
>
:
1;
otherwise
It is easily veri…ed that the utility function U (:; E; q) is monotonic and quasiconcave. We can also see that it is continuous when E is such that for all x 2 <N
there exists a vector x0 2 E such that x0 6 x; evidently, this property does not
hold for all E; 4 and in fact not all the utility functions in the class UK we are
considering are continuous5 . In the special case in which K = N
1 the a¢ ne
subspace E has dimension 1 and the class UK reduces to the class of Leontief
utility functions.
When agents’preferences are described by utility functions lying in the class
UK ; the solution of the agents’choice problem has a simple form, as we show next.
We will focus our attention on the values of demand in a neighborhood of L:
Notice …rst that under the required conditions on the dimensionality of E and
L; the intersection of these two subspaces, L \ E; is a singleton as long as E is
complementary to L, or dim L
E = N (a condition which, given L; is satis…ed
by almost all a¢ ne subspace E of dimension N
to all L su¢ ciently close to L. Let '(L; E)
K). This property extends then
L \ E:
We prove in the following Lemma that, for any (N
K)-dimensional a¢ ne
subspace E which is complementary to L, we can …nd a vector q such that the
4
See also Figure 1, where the shapes of some possible indi¤erence curves in this class are
illustrated for the case N = 2; K = 1:
5
At the cost of a more elaborate description of the value of the agents’ utility functions
outside the space E; we could have also ensured that all preferences in the class we consider are
continuous. All the results we present in this and the following section extend to such case.
8
value of the excess demand of an agent with utility function U (:; E; q), and endowments ! = 0 is given, in a su¢ ciently small neighborhood of L; by the (unique)
intersection point of L and E; '(L; E)6 :
Lemma 4.1. Given L; and E complementary to L; there exists a vector q 2 <N
and a neighborhood N (L; E)
N (L) of L; such that, for all L 2 N (L; E)
'(L; E) 2 arg max U (x; E; q)
fx2L <N+ g
Moreover, the demand set arg max U (x; E; q) is a singleton.
fx2L <N+ g
Proof. Under the assumed speci…cation of the agent’s utility function, for all
x2L
0
0
0
<N
+ either there exists x 2 E; x 6 x; such that U (x ; E; q) = U (x; E; q);
or U (x; E; q) =
1. Hence when the utility function is in the class UK ; to …nd
the solution of the agent’s maximization problem (P h ), we can always limit our
attention to the vectors of excess demand which are not only attainable, i.e. lie
in the budget set L
<N
+ ; but also belong to the subspace E.
We characterize next the properties of the convex set E \ (L
<N
+ ) = fe 2
E : 9l 2 L; l > eg: We will show that this set is a displaced pointed cone, whose
vertex is the (unique) intersection point e 2 E \ L.
Let e be an arbitrary vector in E \ (L
<N
+ ) and l 2 L be such that l > e:
Since e 2 E \ L; and E is an a¢ ne linear subspace, for any
e + (e
e) is an element of E; moreover, e + (l
2 <+ the vector
e) 2 L and the inequality
6
h
N
Evidently, it is always possible to …nd a vector ! h 2 <N
++ such that '(L; E) + ! 2 <++ :
It is then easy to verify that '(L; E) de…nes also the excess demand for an agent with utility
function U (:; E; q); endowment ! h ; and consumption space de…ned by the non-negative orthant
<N
+ : All our results extend then immediately to the case in which the agents’consumption plans
are restricted to be non-negative.
9
e + (l
L
e) > e + (e
<N
+ : Hence e + (e
e) implies that the vector e + (e
e) 2 E \ (L
e) also belongs to
<N
+ ) is a
<N
+ ); 8 2 <+ : Thus E \ (L
displaced cone with vertex e:
To conclude that E \ (L
<N
+ ) is a displaced pointed cone, with vertex e; it
<N
+ ) and (e
remains to show that if e + v 2 E \ (L
v = 0: Suppose that e + v 2 E; e
that l > e + v and l0 > e
we get l + l0
2e
v 2 E and there exists a pair l; l0 2 L such
v: Rewrite the second inequality as v > e
0: The three vectors l;
subspace, this yields (l + l0
e
v: Hence e + v = l; e
l0 : Hence
l0 ; e lie in L and, since L is a linear
2e) 2 L: In addition, since L was chosen to be in
0
G+N;K (i.e. L \ <N
+ = f0g); this implies then that l + l
l0
<N
+ ), then
v) 2
= E \ (L
2e = 0: But l
e
v;
v = l0 ; and so it must be that l; l0 2 E: However
E \ L has a single solution, e; so we get l = l0 = e and thus v = 0.
Since E \ (L
<N
+ ) is then a displaced pointed cone, with vertex e; we can
…nd a vector q 2 <N such that q e < q e, 8 for all e 2 E \ (L
<N
+ ); e 6= e:
Hence for the utility function U (:; E; q) in UK ; de…ned by this value of q and the
given subspace E; the maximum utility over L
<N
+ is uniquely achieved at the
point e 2 E \ L:
For all L in a su¢ ciently small neighborhood of L; E \ (L
<N
+ ) remains a
pointed cone and the maximum of U (:; E; q) over L <N
+ is also uniquely achieved
at E \ L.
5. Proof of the Main Result
We prove in this section the result stated in Theorem 3.2. More precisely we
will show that, for an arbitrary choice of (A; P ); any di¤erentiable function F :
10
V(A) ! <N satisfying Walras Law can be rationalized, at the point (A; P ); by
an economy with K agents (K + 1 if F (A) = 0); all with zero endowments and
utility functions belonging to the class UK :
We begin by changing the coordinates of the space <N so that in the proof,
without loss of generality, we can restrict our attention to the case A = 0; P = IN :
The excess demand function z h (A) of agent h de…ned, with respect to these
new coordinates in the neighborhood V(0) of A = 0; implicitly de…nes, as we
IK
saw, the function h (A) : z h (A) = [
] h (A): Similarly, since F (A) 2
A
IK
Sp[
] 8A 2 V(0); there exists a function f : V(0) ! RK such that
A
IK
F (A) = [
]f (A): Hence to prove the result it su¢ ces to show that we
A
P
P
can …nd K agents such that the function h h (A) satis…es h h (0) = f (0); and
P
DA ( h h (0)) = DA f (0):
Let us associate to each agent h; h = 1; ::; H; a (N
K) dimensional a¢ ne
linear subspace of <N , E h ; also described by the pair (V h ; v h ); where V h is a
matrix, of dimension N
(N
K); and v h a N dimensional vector, such that
for any e 2 E h there exists a vector
2 <N K : e = V h + v h : The intersection of
IK
]; which with some abuse of notation
E h ; identi…ed by (V h ; v h ); with Sp[
A
we will still denote by '(A; (V h ; v h )); is then obtained as follows:
8
9
< e 2 <N : 9 2 <N K ; 2 <K such that =
IK
(5.1)
'(A; (V h ; v h )) =
: e = V h + vh =
;
A
. I
IK
]; or equivalently rank V h .. K =
0
0
h h
N; the set '(0; (V ; v )) is a singleton. Hence from Lemma 4.1 it follows that we
As long as E h is complementary to Sp[
can …nd a vector q h 2 <N such that '(A; (V h ; v h )) describes the excess demand of
11
an agent with endowment ! h = 0 and preferences described by a utility function
U (:; (V h ; v h ); q h ) in UK ; for all A in a su¢ ciently small neighborhood of A = 0:
Therefore, under this speci…cation of preferences, the behavior of agent h is completely determined by the pair (V h ; v h ).
It is immediate to see that any matrix V h of the form
Bh
; where
IN K
K); satis…es the condition
B h is an arbitrary matrix of dimension K (N
. I
rank V h .. K = N: Moreover, we will now show that when V h takes this form,
0
an explicit expression for the unique element of '(A; (V h ; v h )) can be easily deBh
rived. Substituting
for V h in the equation in (5.1); we obtain
IN K
Bh
IN
+ vh =
K
IK
A
(5.2)
.
Premultiplying then by [IK ..B h ] both sides of (5.2), yields
.
[IK ..B h ]v h = (IK + B h A)
(5.3)
Since for A in a su¢ ciently small neighborhood of A = 0 the matrix (IK + B h A)
.
is invertible, solving (5.3) for we get (IK + B h A) 1 [IK ..B h ]v h : We see therefore
Bh
; the expression in (5.1) simpli…es as follows:
that, for V h =
IN K
'(A; (V h ; v h )) = [
IK
A
] IK + B h A
1
v 1;h + B h v 2;h
(5.4)
where the vector v h has been partitioned so that its …rst K components are
denoted by v 1;h and the last (N
K) components by v 2;h : The expression of
'(A; (V h ; v h )) in (5.4) constitutes then the excess demand function of agent h;
Bh
with utility function U (:; (V h ; v h ); q h ); for V h =
.
IN K
12
From the above argument we readily obtain also the expression of the function
h
(A) :
h
(A) = IK + B h A
1
v 1;h + B h v 2;h
(5.5)
so that its value at A = 0 is given by
h
(0) = v 1;h + B h v 2;h
(5.6)
Di¤erentiating then (5.5) with respect to A; we get:
DA h (A) + DA (B h A h (A)) = 0
(5.7)
Hence DA h (A) is implicitly de…ned by (5.7); to …nd DA h (0) we simply have to
develop the expression DA (B h A h (A)); and evaluate it at A = 0:
Let bhi;j be the (i; j)
th element of the (K
(j; k) th element of the ((N
K)) matrix B h , aj;k the
(N
K) K) matrix A; and
h
k (A)
the k th component
of the K dimensional vector describing the value of the function (A) at A:
th element of B h A h (A); a vector of dimension K; is then given by the
P PN K h
h
following expression: [ K
j=1 bi;j aj;k k (A)]. The derivative of this term with
k=1
The i
respect to aj;k ; evaluated at A = 0; is simply bhi;j
respect to all the (N
[bhi;j
h
k (0);
and the derivative with
K)K elements of A is obtained by letting j and k vary:
h
k (0)]j=1;::;N K;k=1;::;K :
By the same argument we see that, as i varies, this
expression also describes the derivatives of the other components of the vector
B h A h (A).
We get so
DA h (0) =
[ h1 (0)B h ; ::;
13
h
h
k (0)B ; ::;
h
h
K (0)B ]
(5.8)
The matrix DA h (0) is of dimension K
rearranged so that the k
the k
th component of
(N
K)K: In (5.8) its terms have been
th block of this matrix,
h
h
h
k (0)B ,
is the derivative of
(A) with respect to all the elements of A:7
On the basis of the expressions of h (0) and DA h (0) derived in (5.6), (5.8), we
P
P
can then compute h h (0) and DA ( h h (0)): To complete the proof we simply
have to show that we can always …nd a collection of K (K + 1 when f (0) = 0)
pairs (B h ; v h ) 2 <K (N K) <N which satisfy the following conditions:
8 P 1;h
< h (v + B h v 2;h ) = f (0)
P
:
h
(v 1;h + B h v 2;h )1 B h ; ::; (v 1;h + B h v 2;h )k B h ; ::; (v 1;h + B h v 2;h )K B h = DA f (0)
(5.9)
where we used (5.6) to substitute for
h
k (0);
and (v 1;h + B h v 2;h )k denotes the k
th
element of the vector (v 1;h + B h v 2;h ); k = 1; ::; K:
Consider …rst the case in which f (0) 6= 0; without loss of generality suppose
fk (0) = 0 for k = 1; ::; k
1 and fk (0) 6= 0 for k = k; ::K: For h = k + 1; ::; K;
set the matrix B h equal to
block, of dimension K
(N
1
fh (0)
[DA f (0)]h ; where [DA f (0)]h denotes the h
th
K); of the matrix [DA f (0)], and the vector v h
such that v 1;h = fh (0)ih ; v 2;h = 0 ; where ih denotes the K dimensional vector (0; ::0; 1; 0; ::0) with 1 in the h
th place: Let then B k =
1
fk (0)
[DA f (0)]k ;
The special structure of DA h (0) in (5.8) depends on the particular speci…cation of preferences we adopted, and on the change in coordinates we made. With utility functions in UK ; as
we saw, the substitution e¤ect is null and demand is simply determined by the level of consumption agents can a¤ord, given their income and the form of the trading constraints described by
A, in the subspace E h describing the agent’s direction of preferences. Hence a change in A will
only in‡uence demand by its e¤ect on the value of the intersection point '(E h ; A) and, for any
choice of one of the K coordinates of , its derivative with respect to A will be an arbitrary
matrix. On the other hand the symmetry of the point, in the new coordinate space, at which
the derivative is evaluated (A = 0) implies that the derivative will be the same for each of the
K generating coordinates.
7
14
v 1;k = fk (0)ik
kP1
ih ; v 2;k = 0
h=1
1
fk (0)
!
and, for h = 1; ::; k
1; B h =
[DA f (0)]h
[DA f (0)]k ; and v 1;h = ih ; v 2;h = 0 : It is immediate to see that when these
values are substituted for (B h ; v h )h=1;::;K in system (5.9), all the equations are
satis…ed.
Finally, when f (0) = 0, to solve system (5.9) (K + 1) agents are needed. In
K
P
such case we can set B K+1 = 0; v 1;(K+1) =
ih ; v 2;K+1 = 0 and, for all
other agents h = 1; ::; K; B h =
h=1
[DA f (0)]h ; and v 1;h = ih ; v 2;h = 0 , to see that
all conditions in (5.9) hold.
This completes the proof of Theorem 3.2.
15
Figure 1
16
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17
[9] Sonnenschein, H. (1973): Do Walras Identity and Continuity Characterize
Excess Demand Functions?, Journal of Economic Theory 6, 345-354.
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