.N~
174
LOCAL CONCAVIFIABILITY OF PREFERENCES
. AND DETERMINACY DF EQUILIBRIUM
MARIO RUI PASCOA
E
SERGIO RIBEIRO DA COSTA WERLANG
MAIO DE 1991
l
LOCAL CDNCAVIFIABILITY DF PREFERENCES
AND DETERKlNACY DF EQUlLIBRIlDI
By
"Mario Rui Pascoa
University of Pennsylvania
and
Sergio Ribeiro Da Costa Werlang
Fundacao Getulio Vargas, Rio De Janeiro
Keywords:
Concavifiability of Preferences; Rectifiability of Demand; Local
Uniqueness of Equilibrium Prices
We are thankfuI to Beth Allen, Bernard Cornet, James Dow, Andreu MasColell, Hugo Sonnenschein and Bi11 Zame for helpful conversations and to Paulo
Monteiro for pointing out some errors in an earlier version of this papcr.
are also indebted to Eduarda Figueiredo for the translation of A1exandroff's
(1939) paper.
\Je
i
ABSTRACT
In this paper we consi.der strictly convex monotone continuous complete
preorderings on
function.
nf+
that are local1y representable by a concave utility
By Alexandroff's (1939) theorem. this function is twice
dífferentiable almost
everywhe~e.
We show that if the bordered hessian determinant of a concave utility
representation vanishes on a nul1 set. then demand is countably rectifiable,
that is, except for a null set of bundles, it is a countable union of
manifolds.
C1
This property of consumer demand is enough to guarantee that the
equilibrium prices of apure exchange economy will be locally unique. for
almost every endowment.
We give an example of an economy satisfying these
conditions but not the Katzner (1968) - Debreu (1970. lq72)
conditions.
~~00th~e~~
I
J
1
1.
Introduction
In the past twenty years economic theorists have devoted a considerab1e
effort to the issue of local uniqueness of equilibrium prices.
paper Debreu (1970) showed that apure exchange economy with
In a seminal
m consumers
whose demand functions are smooth is such that, for almost every endowment,
equilibrium prices are locally unique.
demand functions goes to
+
IX)
If, in addition, the norm of the
.as the boundary of the price simplex is
approached, then the set of equilibrium prices associated with each endowment
is compact, and therefore, local uniqueness implies finiteness.
Smoothness of demand is a very strong condition.
Katzner (1968)
considered strict1y quasiconcave twice-continuously differentiable utility
functions, forwhich the marginal utilities are always strictly positive and
the indifference hypersurfaces do not intersect the boundary of the positive
iff the bordered hessian of the utility function does not vanish at
x.
Under
Katzner's assumptions, th1s condition 1s satisfied on an open dense subset of
consumption bundles and on a corresponding open dense subset of ful1 measure
of prices and incomes.
Katzner's condition isequivalent to saying that the
indifference hypersurface has nonvanishing Gaussian curvature at
x.
A more
general result by Debreu (1972) established that for preferences of class
demand is smooth at
curvature at
x
x
iff the indifference hypersurface has nonzero Gaussian
(see Debreu (1976) also, where it is shown that smoothness of
demand does not require the existence of a
2
C
utility representation with no
criticaI point).
Differentiability almost everywhere is therefore the most that can be
expected under very general assumptions.
However this is an interesting
2
property on1y if one can use it to infer the nature of demand from observed
reactions to price changes.
Therefore we need to guarantee that most quantity
variat10ns do not occur on a negligible set of pr1ces.
This argument
motivated Rader (1973) to introduce the condition that demand shou1d map nu11
sets of prices and 1ncomes into nu11 sets of quantities - condition (N).
estab1ished that differentiabi1ity a1most everywhere together
(N) guarantee that, for a1most
~1ith
He
condition
alI endowments, equi1ibrium is 10ca11y unique
(a1though the set of endowments having infinite equilibria might be dense).
Rader (1973) showed a1so that demand is different1able a1most everywhere
in prices if the utility function is concave and demand is continuous and
10ca11y Lipschitzian in 1ncome.
He also estabIished that if demand 1s
uniformly Lipschitzlan income and the incrementaI ratio of pr1ces to quantity
1s bounded from below, then condition (N) is satisf1ed.
In a later paper,
uti11ty function is twice d1fferentiab1e and condition (N) holds if uti11ty 1s
ana1ytlc (derivatives of a11 orders exist and are 10cally subject to a common
bound).
K1einberg (1980) studied the issue of generic finiteness of equi1ibria
using the weaker concept of approximate differentiabi1ity of demando
cons1dered strictly quasi-concave utl1ity functions of class
Vu
15 approximate1y differentiable.
c1
He
for which
In additlon, he imposed some contlnulty
and boundedness condltions on the approximate derivative of
Vu
to guarantee
that the demand function 1s approximately differentiable on the comp1ement of
an at most countable set of prices and incomes.
Kleinberg showed that for
th1s c1sss of demand functions equi1ibria is locally un1que for almost every
endowment.
_____ J
3
Tha purpose of this paper is to examine what can be the contribution of
concavifiabílity of preferences to the issue of local uniqueness of
equilibrium prices.
An assumption on local concavifiability is not too strong
and is already implied by Katzner's (1968) conditions.
(1985) showed (Proposition 2.6.4), if a strictly convex
In fact, as Mas-Colell
C2
preference has
indifference hypersurfaces with nonvanishing Gaussian curvature and can be
representable by a
2
C
compact convex set
K there is a
utility function with nocritica1 point, then for any
point where the restriction to
2
C
utility representation with no criticaI
K is differentiability strict1y concave
(i.e., has a negative definite hessian matrix on
K).
Ye consider monotone continuous complete preorderings that are locally
representable by a concave utility function
u,
which might not be
differentiable throughout the interior of the effective domain.
1:1 wt:ll
kllOll/ll
lhat the funct:ion
u
However, it
is áifferent1able almost everywhere on the
interior of its effective domain and that the partia 1 functions admit
everywhere one-sided derivatives.
Much is also known about the second
derivative of a concave function.
Alexandroff (1939) showed that a concave
function admits almost everywhere a 2nd order expansion.
Moreover, he
introduced extended partial derivatives which lie within the one-sided
derivatives of the partial functions and showed that these extensions are
almost everywhere differentiable, with remainder converging uniformly on the
choice of the extensions and with derivatives given by the matrix of the
quadratic form in the 2nd order expansion.
This result was our main
motivation to write this paper.
If, in addition to local concavifiability we assume that the bordered
hessian matrix of the local utility representation is nonsingular almost
rr
J
4
everywhere, then, we can claim that, from the point of v1ew of geometric
measure theory, the range of the demand function behaves like a
c~ manifold.
In fact, except for a null set of bundles, demand 1s a countable union of
submanifolds.
C1
In the terminology of geometric measure theory, the demand
range is countably rectifiable.
We show that this property of demand 1s
enough to guarantee that, in apure exchange economy, equ11ibrium prices are
local1y unique, for a1most every endowment.
We give an examp1e of a
2 x 2
pure exchange transferab1e ut11ity
economy satisfying our assumption, but not the Katzner-Debreu smoothness
conditions, and for which the set of equilibrium prices is finite, for almost
every endowment.
It is interesting to notice that the property of rectifiabi11ty of
demand that we have estab1ished is weaker than Rader's (1973) conditions for
generic local
unl\{uefl~sli>.
In fact, under rectifiahillty thê
cüüditiü~
_'L. __
,","CLt..
demand map null sets into nu11 sets is sat1sfied but demand is not
differentiable in almost every price and income.
The comparison of our
results with Kleinberg's (1980) is more subt1e, s1nce he requires the demand
function to be on1y approx1mately different1able except on a countab1e set and
his assumptions on preferences do not involve local concavifiabi1ity.
However, K1einberg (1980) assumes differentiability of the uti1ity function,
which is an assumption that we have d1spensed with.
2.
RECTIFIABILITY OF DEKAND
2.1
Preliminar1ties
lRn and a preference re1ation R is a complete
-+
x -y if x R y and y R x and 1et x P y if
The consumpt1on set 1s
preordering on
r -n
K+.
Let
5
x R Y but not
y R x.
Ye say that a preference relat10n
x + (l-À) y P y,
À
implies
x P y.
when
x R y
A function
if and only if
x IRn (ii) strictly convex if x R y
+
O < À ~ I and (iii) monotone if x» y
nf+
continuous if it graph is closed on
implies
for any
:nf+ ...
u:
u(x)
~
IR 1s a utility representation for
n-l
d
mapp1ng
int â
The demand set 1s
A preference relation
compact convex set
for
R.
R the demand function for
~ into
m 1
- X
(where
p d(p,y) - Y and
dimensional simplex), satisfying
d(p,y) R z.
R
u(y).
Given a strictly convex preference relation
is a function
R 1s (1)
n 1
d(int â - X
p z
â
S
n-l
R
1s the
y->
I~).
R is locally (strictly) concavifiable if for any
K~ ~
there is a (strictly) concave representation
By Alexandroff's (1939) theorem, this function
~
~
is twice-
differentiable almost everywhere and the hessian matrix, where 1t exists, is a
negative semi-definite matrix.
this theorem.
2.1.2.
Alexandroff's theorem
For technica1 reasons we want to think of alI concave functions as
defined throughout
domain.
nf
and taking the va1ue
u
and by
Denote by
int(dom u)
A proper concave function
subset
outside of the effect1ve
The extended real-valued function obtained thisway 1s ca11ed a
proper concave function.
function
-~
D of
int(doro u)
of measure zero; moreover,
dom u
the effective domain of the
the respective interior.
u
on
Df
is differentiab1e on adense
and the complement of
u
D on
int(dom u)
is a set
is actually continuously differentiable on
D
(see Rockafellar (1970) 25.5)
r
6
A1exandroff (1939) established that a proper concave function
15 a1so twice-differentiable almost everywhere on
domain
int(dom u).
u
Df
on
Since the
D of the first derivative may have an empty interior, we should be
more precise and recall the exact statement of A1exandroff's theorem.
Let
u(xi;x_ ) : IR ~ IR u
i
by setting alI variables but
u~ (x ;x. )
i
1
(-c:o)
xi
be the proper concave function obtained
equa1 to
and the 1eft derivative
u:(Xi;X_ i )
the effective domain of this function and
ui
The right deriva tive
are well defined throughout
of the
u
u:(xi;X_ i )
S u~
(Xi;X_ ) (see
i
A1exandroff (1939) defined an extended partia1
Rockefe11ar (1970) 23).
derivative
•
x_i
E -Il-1.
JK
as any function satisfying the inequa1ity
u:<Xi;X_ i ) S ui(x) S u~(xi;X_i)'
where
coincides with the partia1 derivative
x - (xi;X_ i ).
8u (x)
8x
Note that
when it exists.
i
Iheorem (Alexandroff):
a1most everywhere on
for any direction
where
L(S)/S
y E
the extended partia1 derivative
int(dom u)
ui
i5 differentiab1e
and at any point X of differentiabi1ity,
Df, we have
converges to zero uniform1y on the directions
independent1y of the choice of the extension
H(x) - [Vul(x) ... Vun(x)~
ui'
y
and a1so
Moreover. the matrix
is uniquely determined, lndependent of the choice of
the extens10ns and 1s a symmetric negative semidefinite matrlx.
Actua11y, the matrix
H(x) - [hik(x)]
exists on1y at points where
u
once differentiab1e (see Alexandroff (1939) pp. 5 and 6) but not necessari1y
is
1
7
at any such point.
Furthermore, for any direction given by a normalized
21 y' H(x) Y coincides with the dlrectlonal
second-derivative given b y 1im ( u ( x + sy) - u(x) - ~
L u (x) sYi)/5 2
By a
1
vector
y, the quadratic for
s~O
l-i
result due to Iessen (1929), in the case of a concave function, a d1rectional
n
L
second-derivative is a usual second-der1vat1ve, that 1s, u(x+sy)-u(x)2 n
n
2
i-i
U1 (x)sYi-;hiK(x) YiYk - €S
where €~O as s~O, uniformly on alI
i-1 K-1
.
d1rections y. Then, any concave function admits at almost everypo1nt x a
r r
second-order Taylor expans10n and the matrix of the quadratic form is the
matrix
H(x).
IR2
Busemann and Fel1er (1935) had shown that a concave funct10n in
admits directional second-der1vat1ves a1most everywhere 1n its effective
doma1n.
Alexandroff (1939) extended their result to an arbitrary f1nite
dimension (in part 2 of his paper) and established the relation between the
directionAl
second-der{vativp.~
Rnn
~h~ d~riv~tives
of the extended
p~rti~l
derivatives (in part 3 and 4 of his paper).
2.1.3.
Rectifiabi1ity
Recall that a function
Lipschitzian if for any
18
Ln
M-rectifiable if
1s said to be locally
a E A there ex1st an open bal1
llf(x) - f(y) II
constant M such that
Denote by
f: A ~ . n:F~nf
:S M
"x-yll,
the Lebesgue measure in
Ln(E) < ~
and
n
L -
JIf.'
for" any
B (a)
E.
and a
x,y E B (a).
€
We say that a set
almost alI of
E~
JIf
E 1s contained in
the un10n of the images of countably many loca11y L1pschitz1an function5 from
DF
to
JIf.
almost alI of
sets.
A set
E~
n
IR
is said to be countab1y n-rectif1able 1f
n
L _
E is contained in the union of countably many rectifiab1e
As we wil1 see next, from the point of view of geometric measure
r
8
C1
theory, rectifiable sets behave like
manifo1ds.
Geometric measure theory can be described as differentia1 geometry
genera1ized through measure theory to dea1 with maps and surfaces that are not
smooth.
n
Denote by
E I~
ba11 around a
and a
E
Df;
Df
the Lebesgue measure on
L
with radius
the set
A
6.
Let
86 (a)
and by
for any
6 <
A~
nf.
The approximate 1im sup of
V(>o 3 >0
a
if
f
def1ned on a
6
6.
Conslder an extended real va1ued measurab1e function
measurab1e set
I~
A be a measurable subset of
is'said to have density zero at
such that
the c10sed
f
a E
at
nf
is
deflned as
ap 1im sup f(x) - inf B,
x-+a
where
B - (t E De (x E A: f(x) > t}
From the definition, if the set
has dens1ty zero at
a}
A has density zero at a then
ap lim sup f(x) - -~. It is immediate that if 1im sup exists then 1t 1s
x -+ a
equal to the approx1mate 1im sup.
Sim1IarIy,
zero at
ap 11m inf f(x) - sup (t E IR: (x E A: f(x) < t} has density
a}.
Recall that a function
Lipschitzian at
funct10n
at
a
f: A ~
a
E
f
from
A
~
n
IR
1s said to be pointwise
A if Um sup Ilf(x)-f(a)l!lIx-a~ <~.
S1milarly, a
x ... a
nf ... nf
is said to be approximate1y pointwise Lipschitzian
ap 11m sup IIf(x) -f(a) !Ix-ali < ~.
x ... a
The foI1owing lemmas summarize the most important facts about
E
A 1f
approximate1y Lipschizian functions.
9
temIDa 1 (Federer):
if a function
pointwise Lipschltzian on
f: A ~
A then
sets such that the restriction of
nf
~
nr
A is the countable union of measurab1e
f
to each set Is Lipschitzian and moreover
Is approximate1y differentiab1e a.e. on A:
f
map
nf
L:
~
nr
1s approximately
that Is, there existsa linear
with
ap 11m ~f(x) - f(a) - L(x-a)lI/l1x-all - O
x'" a
(for a proof see Federer (1969) 3.1.8).
Remark
f: A ~ IRn ~
Then, the range of a function
nf
whlch is approximate1y
pointwise Lipschitzian is a countably rectifiable set.
a countable cover of
f(A)
In
f~~t.
by bounded sets and each one is a rectiflab1e set,
slnce the Lipschltzian restriction of
f
can be extended to the who1e
preserving the Lipschitzlan constant (see Federer (1969».
g: B C
nf
f: A ~ B ...
~IRn
nr
admits an approximate1y pointwise Lipschitzian restriction
such that
f(A)
has fuII measure in
g(B),
then
a function
f: A ~
nf ...
nr
In fact, by !emma I,
~
A
is nuI1 set, then
A -
co
U C
K-1 K
f(B n C )
K
g(B)
is
f.
that is approximate1y pointwise
Lipschitzian on A maps nu1I sets into nuI! sets, provlded
B
I~
Moreover, if
a1so countably rectifiab1e through Lipschitzian restrictions of
Lemma 2:
on~ ~an tak~
such that
n
~
m.
is Lipschitzian and if
flc
K
is a1so nu1! and
f(B) -
co
u f(B n CK)·
K-I
Now we wlll estab1ish a result on the rectifiabiIity of the inverse of a
differentiab1e function.
r
j
10
Proposition 1:
1et
f and
IR?;
and taking values in
measurable subset
function.
f'(a)
suppose that the restriction of
D admits the restriction of
Let a
nf
g be functions defined on an open subsets of
be a limit point of
D.
lf
g
f
to
f(D)
f
to a
as an inverse
is differentiable at
a
and
is nonsingular, then
b(b+s) -g(b) 11
ap 11m sup
s .. O
glf(D)
Koreover. lf
Ils~
D is a full measure subset of
dom f
then the above inequality
holds as an equality.
ftQ.Qí:
Inverting the incrementaI ratio we have,
ap Um sup Ilg(b+s) - g(b) 11111511 s .. O
~f(a+x)-f(a)1I
- inf {t e IR: (x e(D-a):
Ilxll
I
<
-t}
has density zero at
a}
~fCa+x) -fCa) II
- l/sup {t e IR: {x e (D-a):
< t}
Ilxll
x"O
:S
IIxll
a}
~f(a+x) -fCa) II
~f(a+x)-f(a)~
- l/ap 11m inf
has denslty zero at
l/11m inf
x .. O
IIx~
flD
when the last inequality follows from the following set inclusion:
~f(a+x)-f(a)1I
{t e IR: (x e(D-a):
~xll
< t}
has density zero at
a}::>
r
11
IIf(a+x) -f(a) II
If
< t}has density zero at a}
~xll
(t e IR: (x e(dom f-a):
D has full measure in
dom f
then these two sets are equal and the
above inequaIity holds as an equaIity.
To fin1sh the proof
derivative
at
a
Oh
recal~
that for a mapping h
at a point a, the
ls equa1 to
1im inf
li (Dh(a» -lrl
w1th nonsingu1ar
of the incrementaI ratio of h
(see Federer (1969) p. 209).
FlnaI1y, we present the results that allow us to think of a rectifiab1e
set almost as a countable union of
C1
manifolds.
The Lipschitzian functions
In the definition of a rectifiable set can bereplaced by
1
C
functions due
to the following lemma.
Lemma 3:
(Federer):
f: A ~
a function
Df
~
nF
that ls
approximately polntw1se Lipschitzian is such that for
1
C
map
than
1-(
n
g: 1K ....
-m
lK
identical to
f
a.e.
In
A
(o> O there exists a
on a subset of A of measure greater
(for a proof see Federer (1969) 3.1.16).
Using this lemma, Federer established the fo110wing characterization of
rectifiability.
Lemma 4:
in the definition of a rectifiable set
Llpschitzian functions to be
C1
o
E one can take the
diffeomorphisms on compact domains with
disjoint images whose union coincides with
(see Federer (1969) 3.2.18 and 3.2.29
E a1most everywhere.
and Morgan (1988) 3.11)
r
12
2.2.
Ibe result
2.2.1.
Tbeorem 1:
if a continuous, monotone, strictly convex preference
~
re1ation R on
is locally concavifiab1e and for each representat10n ~
the bordered hess1an matrix 1s singular only in a set of measure zero, then
the- demand function
every va1ue.
d
That is,
n 1
e d(int A - x IR++).
is approx1mately pointwise Lipschitzian for almost
ap 1im sup d(PÓ Y) <
(p,y) ~ (pO,y )
~
for a1most every
xo • d(p O,yO)
In other words, by 1emma 1 and the remark in 2.1.3 demand 1s countab1y
rectifiab1e and, by 1emma 4 in 2.1.3, except for a nu11 set of bundles, it 1s
the union of countably many pairwise disjoint compact
2.2.2
1
C
submanifo1ds.
Proof of the Tbeorem
Consider a countab1e cover of
K ~ ~,R
cube
~
by c10sed cubes.
is represcntab1e by a concave uti1ity function
criticaI point, by monotonicity of
R.
nonempty interior, since otherwise
u
with no
For simpIicity, we assume that
a proper concave function with effective doma in
K.
We assume that
u
K has
poin~ise
Lipschitzian for a1most every
For the same reason, we w111 consider on1y the optima1 interior
so1utions in
K,
since the boundary of
K is a n-nu11 set.
Let N be the nu11 subset of int K when u is not twicedifferentiable.
x e int K\N,
Let
given by
6(x)
is
K would be irre1evant for the purpose of
proving that demand is approximate1y
va1ue.
For each c10sed
be the bordered hess1an determ1nant of
u
at
;
_ .i
13
6(x) -
u
n
(x)
u
nn
(x)
u (x)
n
o
... u (x)
n
Let
CP
be the nuIl subset of
determinant
int K\N where the bordered hessian
6 of u van18hes-.
Let
RP
be the complement of
CP
on
int
K\N.
Recall Debreu's (1972) decomposition of the demand function and adopt it
to the case where
dom Du
g: x ... V u(x)/IIVu(x) 11
of
g(x).
is just a fuIl measure subset of
and
g(x)
a
The restriction
the inverse of the function
1s the composition of
p
a: x
be the vector of the first
of the demand function
d
(g(u), u(x»
~nd
n-1
to
f: x ... (p,y) • (g(x),g(x)-x).
~
int K.
p: (p,v)
d
-1
Let
components
(dom Du)
This function
4
is
f
when
(p,w(v})
e int â n-1 and m(v) - min p.z.
u(z)
g,
Let
and
ã
f
v
~
be any extensions of
g, a
and
f,
respectively
obtained using Alexamdroff's extended partial derivatives.
The Jacobian determinant of ã
n
gn(x)/llvu(x) II ,
gn(x) _
u
n
where
(x)/IIDu(x) 11) .
n
g (x)
S'"
J f(x) - J (fJoã) -
pA
~S
O on RP.
J ã(x) - -6(x)
g(x)
(that is,
The function
fJ
has the
which 18 the inverse of the function
- 6(x)u (x) /IIVu(x) II
n
1s
component of
Now Vv(s) - IIVu(x) li,
max u (z).
pz
m,
x E int K\N
n th
1s the
Then J(a)(x)
same Jacobian determinant as
given by
at
n+2
and
for
J f(x)
x - d(p,s).
pA
v
Therefore
O if and on1y 1f x
E
RP.
r
1.
14
Now. by Proposition 1 in 2.1.3 above. for
x
oE
o O
O
let (p ,y ) - f(x )
RP.
ap lim sup d(p.y) < (1). where
d Id -1 (dom Du). Since dom
(p,y)"'(pO.yO)
has fuH measure in int K we also have ap lim sup d(g'y) < (1), for
(P.y) -+ (pO.y )
O
O O
O
x
and
E
RP.
We have proven that the demand function is
(p .y ) - f(x )
a-
and we have
RP
approximately pointwise Lipschitzian on the inverse image of
a full measure subset of
int K.
and RP
is
To complete the proof. notice that the
countable union of the null complements of each set
cube
Ou
RP
in the respective
K is a null subset of
Q.E.D.
~
~
Notice that the result still holds for a strict1y concave function
which is differentiable almost everywhere and such that the derivative
admits an extension
f
to the whole domain of u
differentiable almost everywhere.
(iH)
(ii)
f'
so that
(i)
exists whenever
f
VU
u
Vu
is itself
exists and
f'
in the case when
u
admits
almost-everywhere a second-order expansion and
the matrix in this quadratic form coincides with
fI.
almost verywhere (as it
1s the case with a concave function).
3.
Local Unigueness
Consider a exchange economy with
dj : int Iln-1
X
1R++'"
endowment of the
nf+
jth
m consumers and
be the demand function and
consumer.
n
goods.
w E ~
j
Let
be the
We keep the demand functions fixed and
parameterize an economy by an endowment vector w - (w1 •...• wm). Given w E
n-l is an equilibrium price vector of the economy
an element p E int Il
nfl1l
++'
m
m
w if
I dj(p.p-wj ) - I wj ' Let E(w) be the set of equilibrium price
j-l
j-l
vectors of the economy w. We say that equilibrium prices are locally unique
r
15
if alI e1ements in
Ibeorem 2:
E(w)
are iso1ated points of this set.
If the demand functions
dI"'. ,d
of a11 consumers are
m
approximate1y pointwise Lipschitzian for a1most every demand bund1e, then the
equi11brium prices are 1oca11y unique, for a1most every
froof:
Let
F: U ~
nrm
U - lnt
Â
n 1
- x lR++ x
vector.
lR~m-1) and define the function
as in Debreu (1970) by
m
F1(e) - d 1 (p'Y1) +
where
endo~~ent
r
m
di (p,p-w) -
1-2
r
W
1-2
1
u, peF1(e) - Y1' A1so, P is an equi1ibrium price vector
-1
(P'Y1'w2 ,··· .wm) E F (w). We want to show that for a.e. w. every e
e
For
F- 1 (w)
18
a
is local1y iso1ated.
Let us start by c1aim1ng that the range of
m-n-countab1y rectifiab1e set (through Lipsch1tz1an restrictions of
Let
~
K~1
E
Mj
be the full measure subset of
M
of disjoints images of
jk
Denote by
Define
N
C1
its nu11 comp1ement in
j
N1
-
diffeomorphic restrictions of
~.
d1(p'Yl) E N1 }
(w E IR~ : w - F(e), e - (P'yl'w2 •... ,wm)
and
in
F
F)
which 1s the countable union
- (w E ~: w - F(e). c - (P'y1.w2 •...• wm)
and
Nj
~
1ff
d .
j
16
\le want to show that
Nj
and
-
IR~Mj
jê N
1 j
is a nu11 set.
N1 -
Now
~K~\ 1
where
M1 - (w E ~: w E F(e). e - (P.y1.w2' ...• wm)
and
d1 (p'Y1) E MIl
Mj - (w E ~: w E F(e). e - (P'Y1. w2'· ..• wm)
and
\le need to establish that
dj (p •P owj)
Mj (j
E
Mj l • j ,.. 1
0- 1 •...• m)
1s offull measure in
llfID
+ .
Now
wE M
1
iff d1 (p'Y1) E M1 for any (P,Y 1 .w2 •... ,W ) E F -1 (w). that 1s,
m
-1
-1
(P'Y1) E d1 (M 1 ) n 11(F (w» - B1 where 1j is the jth projection mopping
(app1ied in this case to
E
-1
(Mj )' n 'Yj (F
dj
m
j~l
n
Similar1y.
lR).
-1
Let
(w» - Bj.
n
B1- (x E IR : x - (P.pow i ), P
and
Bj1 - (x
M-
Then
i ,.. 1.
j
i!i A
i
iff
E
nf:
where
x -
E
j1(Bj ), w
i
E
n\l
( P 'Y1 ) ,P E j 1 (B)
>
j ' Y1 E IR+.
A{ - Mj +
L
i"'j
di(Bi> -
j
L
A
i,.. 1 i
and
ne,
j
A
i
for
By the Brunn-Minkowski theorem (see Federer (1969). 3.2.4», for any
nonempty sub sets
di(Bji) -
L
A
Aj )l/n,
and
B
of
n
lR
(not necessar11y of finite measure), we
where. on the right-hand-side, at 1east the first term
1~
1,..1
+~. So
1s
mon - countab1y rect1fiab1e through
i
Mj 1s of fu11 measure in ~ and therefore the range of F
F.
To complete the proof of the theorem. notice that the set
regarded as the un10n of countab1y many disjoint
(by taking appropr1ate set differences).
F(U)
can be
mon-rectifiab1e sets
Now. by lemma 5 in
~
2.1.3 (from
Federer (1969», one can take the countably many Lipschitzian functions in the
definition of each rectifiab1e set
~
to be
1
C
diffeomorphisms
DiK
on
11
compact domains with disjoint images, whose union coincides with the
rectifiable set alrnost everywhere.
which is a
Let
N - F(U) \ KQ1 iê
range
l
m-n-null set.
F(u) \ N is such that
Now, any endowment vector on
w e F(u) \ N and any e1ement
E(w)
e e F-1(w)
have neighborhoods
respectively, that are homeornorphic under the restriction to
DiK .
w e F(u) \ N,
Moreover, for each
D
this map
iK
determined, because the reetifiab1e sets are disjoint and the
disjoint images.
Then, for any
O
has a neighborhood
neighborhood
between
4.
O
P
E(w)
p
and
F
-1
w e F(U) \ N,
O
e
ow
and
Oe
O,
e
C1
of some
is uniquely
D
iK
any price vector
where there are no other e1ements of
ls induced by
has on1y
1
C -inverse function theorem, any element
In fact, by toe
isolated points.
map
Dlk'
p
maps have
in
E(w)
E(w) (this
through the one-to-one correspondence
(w».
AN EXAMPLE
Here, we give an exarnp1e of apure exchange economy where utility
functions are concave, but not differentiab1e, and we still have finiteness of
equilibria, for almost every endowment.
In this examp1e the set of endowments
generating infinite equilibria is dense.
Consider an economy with two eonsumers with the sarne cone ave uti1ity
which i5 constructed so that an indifference curve is
function on
differentiable on1y at irrationa1 points.
For each
~
e
Q,
let
f
n
Notice that
now
f
be the characteristlc function of the set
f
n
is nondecreasing and has a jump at
is nondeereasing, at any
q
n
q .
n
e Q it has a jump of
18
magnitude
1/2
n
and it 1s differentiable except at the rationals, with zero
derivative.
Now integrate
f
to obtain a convex increasing function.
Since
only a countable set of discontinuities it 1s Riemann 1ntegrable.
f
Let
has
g be
such that.
g(x) - xIO f(t)dt
The derivative
any irrat10nal
x.
g'
exist& at every continuity point of
Furthermore, at any rationa1
and g'(q ) - f(q +) - f(q ).
+ n
n
n
Notice that
g(ax + (1-a)y) ~ ~x f(t)dt +
since
f
i8 nondecreasing and for
g
~
f,
we have
that is, at
g'(q) - f(q-)
n
n
i8 strictly convex:
(1-a)y
lo
f(t)dt,
O < a < 1 we have
~x f(t)dt < aJ~ f(t)dt
Now 1et h: IR++
~
IR++ be defined by h(x) - g(1/x).
The function
h
1s decreas1ng, convex, differentiable only at irrationa1 points; it wi11 be
used as an indifference" curve.
Let the utility function be
2
u: lR++
~
IR
The preferences in this examp1e satisfy the assumptions of theorems 1 and
2 but not the smoothness conditions of Katzner (1968) or Debreu (1972)-(1976).
In fact the bordered hessian determinant of
u,
at
(x ,x ) E
1 2
(~Q)
x
~
15
'"
19
o
o
1
- h" (x )
1
o
1
and
h"(x ) - -- g"(l/x ) 1
2
1
xl
Now
AIexandroff's extended partial derivative
g"
is the derivative of the
and therefore
g' - f
+
2
g"
vanishes
1 .
3
o, for any
xl
and the bordered hessian determinant is nonzero on the full measure
identically on its domain
x, E
~Q
set
(~Q)
x
~Q.
Then
h"(x ) - l
f( /x l ) >
]R.
Let us examine the demand functions and the equilibria of this econorny.
implying
except at prices associated with points where
h
is not
differentiabIe and these prices are elements oí the subdifferentials
and generate demand
x
2
-
Y-Plqn
P2
lle have
determined completely the form of the demand function.
It is easy to see that when the endowment vector
w
1
For
(w ,w )
1 2
1s irrational, then the equilibrium prlce ratio 1s equal to
w
1
is such that
-h' (w / ).
l 2
rational, the set of equ11ibrium price ratios is the interval
[-h~(wl/2)' -h:(wl / 2 )]·
That is, aImost every endowment generates finite equiIibrium prices but
the set of endowments giving rise to infinite equilibrium prices 1s dense.
r
· ..!
20
References
A1exandroff, A.D. (1939): nExistence of the Second Differentia1 of a Convex
Function A1most Everywhere in its Domain and Some Related Properties of
Convex Surfaces" (Russian), Leningrad State University, Annah 37,
Mathematical Series, 6, pp. 3-35.
Debreu, G. (1970):
"Economies with a Finite Set of Equilibria", Econometrica
38 .
. Debreu, G. (1972):
"Smooth Preferences", Econometrica 40.
Debreu, G. (1976):
"Smooth Preferences:
Federer, H. (1969):
A Corrigendum", Econometrica 44.
"Geometric Measure Theory", Springer-Verlag
Katzner, D. (1968): nA Note on the Differentiabi1ity of Consumer Demand
Functions" , Econometrica 36, No. 2, pp 415-418.
K1einberg, N. (1980): "Continuous Economies with a Finite Set of Equi1ibria",
Journal of Mathematica1 Economics.
Mas-Colell, A. (1985): "The Theory of General Economic Equi1ibrium",
Cambridge University Press.
Morgan, F. (1988): "Geometric Measure Theory:
Press
A Beginner's Guide" , Academic
Rader, T. (1973):
"Nice Demand Functions" , Econometrica 4.
Rader, T. (1979):
Economics.
"Nice Demand Functions 11", Journal of Mathematica1
Rockafel1ar, T. (1970):
Sard, A. (1958):
No, 2.
Convex Ana1ysis" , Princeton University Press
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jOlJlloU.\c,.:AO GETULIO VARGAS
.uw.Ol IÇA MARtO HK.~WQUE SlMONSr:.
r
ENSAIOS ECONÔMICOS DA EPGE
(a partir do nº 100)
100. JUROS, PREÇOS E DíVIDA P~3LICA VO~U~lE I: ASPECTOS TEÓRICOS
- Marco Antonio C. Martins e Clovis de Faro - 19a7 (esgotaoo)
101. JUROS, ?REÇOS E D11..'IDM PÚBLICA VOLIJr-:E 11: A ECO~Cr':IA 8r:U\SILGRA - ::'~~7VSS
- Ar.-:onic Sa1azar P. arandão, ClÓJis ~ r::>ro e ~8rco,ll.• C. j'''ertir6 - 1767 (e=~1.l-'o)
102. MACROECJNOHIA KALECKIA~A - Rubens Penha Cysne - 1987
103. O PR~MIO JJ DÓL~R ~O ~ERCA~O PARALELO. O 5UBFATURA~ENTO DE
EXPORTAÇ~ES E O SUPERFATURA~~NTO DE IMPCRTAÇdES - Fernando de HQ1anda
Barbosa - Rubens Panh3 Cyene e Marcos Costa rlola~d3 - 19B7 (esgotado)
104. BRAZILIAN EXPERIENCE WITH EXT~RNAL DEBT AND PROSPECTS FOR GRC~THFernando de Hol.anda 8a:-bosa 2nd íJ~an'Je1 S2!lCn8Z de. La Cal - 1987 (esgCJ:sc:»
105. KEYNES ~!A SEDIÇÃO DA t:5COLHri PÚ8LICA- A,-,tonio ~·j'.da Silveira-1987(es;:::'a:o)
106. O TEORU1,n. ::JE FROaENIUS-PER~[ji\ - Carlos Ivan Simoflsen Leal - 1537
107. POPULAÇ~O BRASILEIRA - Jess~ Montello-19B7 (escotedo)
108. MACR.O~Cm~O:'lIA - CAPíTULO VI: "DnlM!DA POR MOEDA:: A CURIJA U,jlT
- Mario H9nrique Simonsen e Rubens Penha Cysne-1987 (esgotado)
109. MACROECOi'·jOf"1IA - CAPíTULO VII: "DGiP.NDA P~REGA.DA E·A CURVL\ :S~:
- Mario Henrique Sií.lons8n e .RL!bens Penha Cysne - 1987 - (esgotacC))
110. MACROECONOMIA - MODELOS DE EQUILíaRIO AGREGATIVO A CURTO ~~~ZD
- Mario Henrique Simonsen e Rubens Penha Cysne-19B7 (esgotado)
ll1.
-
•• -
lHe.
_
.... , ' _
.... -
,.. ...
OHYr..~l.HI·~
-,-, I I" I
r"'\
"
~
""l"" ,.... f i ' -
rUUI'IJUM~~UI\l':';
n
r-
ur
r:--"
I
I T
T"'" 1\1
..JULUI..L.UI"4
r r"''' I r r- n
1..J'-.Ji~L...L-1
T C'
l..J
n
C"
'-'I
r,,"~
C" C
Ufl'IL---..J
c: '
.
_=:-g!8
Ribeiro da Costa Werlang e TOf:1my Chir.-Chiu ian - 1987 (esgotado)
112. PREÇOS líqUIDOS (PREÇOS DE VALOR ADICIONADO) E SEUS DETERr1!NANTES;
DE PRODUTOS SELECIONADOS, NO PERíODO 1980/1º Semestre/1986 - Raul Ekermdn - 1987
113. EMPRrSTIMOS BANCARIOS E SALDO-MrDIO: O CASO DE PRESTAÇÔES
- Clovis de Faro - 1988 (esgotaQo)
114. A DINAMICA DA INFLAÇAo - Mario Henrique Simonsen - 1988 (esgotado)
115. UNCERTAINTY AVERSION AND THE OPTIMAL CHOISE DF PORTFOLIO James - Dow e S~rgio Ribeiro da Costa Werlang-198B (esgotaoG)
116. O CICLO ECONÔMICO - Mario Henrique Simonsen - 1988 (esgotado)
117. FOREIGN CAPITAL AND ECONOMIS GROWTH - THE BRAZILIAN CASE ST~JY­
Mario Henrique Simonsen - 1988
118. cor,mO~J KNOWLEDGE - Sérgio Ribeiro da Costa Wer1ang - 1989(e:ljJtJ.:iJ)
119. OS FUNDAMENTOS DA ANALISE ~ACROECONÔMICA-Prcf.Mario Henrique
Simonsen e Prof. Rubens Penha Cysne - 198B (esgotado)
120. CAPíTULO XII - EXPECTATIVASS RACIONAIS - Mario Henrique
Simonsen - 1988 (esgotado)
121. A OFERTA AGREGADA E O MERCADO DE TRABALHO - Prof. Mario Henrique
Sim o n s e n e P r o f. R u b e n s P e nOh a Cy s n e - 1 9 8 8 (e s 9 o t a do)
122. I1J~RCIA INFLA.CIONARIA E HJFLAÇAO HlE?CIAL- - Prof. Mario He'l:-:.;Üf!
Simonen - 1988 (esgot~do)
123. 1'-10DELOS DO rlOf~Eí'l: ECO;iCi':IA E ,ll.o:'lIrnSTRAçÃO - Antonio r:',Jl-ia rLI
Silveira - E1ZjB
124. U~Jll[rnrJ\JllICI;JG DF EXPORiS OVERHJVOIC::!:i~C DF n1PQRTS, M:D TW:
DlJLLf\R PFt=YIUií O r·) THr= flLP.l,K iJ);J.Ri<ET - ?rof. Fe'!:'n~mdo dc flOl.lí'i~:J [i :::!):;;l,
Prof ° f~ut",ói'3 ?e>rikJ Cy:::Jne E: ;';ôrcos Costa Ho12nd3 - 1988 (esgot:-ldo)
I
1 7. ~. O R(J NCl r~ A[, J CO OO CH OQli [ H[ 1 ~ F: OD[l XO - F E' r n a ri d O d (;I Hn 1 ~ ri Ó a B~ ;" b [) S;)
A11 t o ri j o SLI 1 u Z LI r Pe s S D tJ B r a n ~ u o c C1 (l v i s d e F a r O - 1 9 O8 (~ s 9 11 t iJ d Q )
1 2 (,. PL Ar~ ú CRUZ fi DCJ: COI; ~, [ p ç ~ O [
Penh~
-127.
Cysnp -
D E RRO D [
P OL 11 1 CA r I 5 CAL -
Ru b (' n s
1988
lAXfJ, DE JUr:,[t~ rLU1Ur,InE VrRSUS C(lRREÇ1~u ~Otnl.4F;JA O'AS PRESTJiÇUES:
UI-1A Cor';PAR{,ç.~G r~o CASO DO 5JIC [ JrJFLAÇT\O COf,SlhNT[- Clovis úe
furo - 19bG
128. CAPiTULO JJ - t'.[1r~[TAp.y COr.RECTIO!~
- .Rub~n~ P~nh2 Cysnc - 1988
129. CAPiTULO ]~l
Pen~~
Cy~n~
- I~COKE
Ar~~ 'REAL IN1ERESl ACCOUrnli~:;
ANDDEMANO POLICIES
130. CAP11ULO IV - 8RAZ1LIAN ECONOKY IN
CRlSIS - Ruber~ PEnha Cysne - 1908
1 31. TP. E
fj
IN
BRAZIL -
Rubcn~
- 1988
lHE
EIGHTIES AND
F: ;; Z1 L Ir~J! AGR1 CUL TUR~. L P OL I CY E XP ERI EIJ CE:
fUTURC DIR[CTI01;S -
Anionlo Sc:lõzar
lHE
OEBT
RAT I Or~ AL E A1\ D
Pessoa Brandão -
1988
132. r·~ OF\ ". i 6r:: J A 1 In ERl~ A, o 1 v I D,'\ PÚB L I C A E J UF. o 5 RE AIS - t'. a r i õ Si 1 v :l
Ba5to~ !~t.-rqu~s e SÉrgio Ribeiro dê Costa lJerlõng - 1985
133.
CAPlTULO 1>: - TEORIA 00 CRESL:It-',ErJTQ ECOr--;Ôr·~ICD Sirr,ons2n - 1988
2
Mario Henric.;u2
13l; • COI~GELr\l·~EtlTO cm·i p.BOrW SALARIAL GERANDO EXCESSO DE DEr"ÍAr~D;' - JocCiU:lr.l Vieirc. Ferreira Levy e SÉrgio Ribeiro de C.osta IL1erlang - 29:E
135.
AS ORIGENS E CONSEqUtNCIAS DA INrLAç~O NA AMERICA LAT!~A Fernando de Holance Barbosa - 1988
136.
A CO~TA-CORRENTE DO GOVERNO -
1970-1988 -
~
Mario Henrique
137. A REVIEU ON lHE THEORY DF COMKO~ KNOWLEQGE
- S~rgiD Ribeiro da Costa Werlang - 1989
138. ~ACROECG~O~lA
- ferncndc de Holanda 82rbosa -
1989 (esgotadu)
139. TEORIP DO EALAr~çO DE PAGAI-~ENTOS: Ur--'A ABORDAGEr., SIMPLIFICADA - João Luiz Tenreiro Barroso - 1989
l~O.
CO~TABILIDADE
1~1.
"CF:EDIT
COM JUROS REAIS -
PJ\TIOt~lt~G
Af,D THE PERI·:At:[tn
Tommy Tan. Daniel Vicent.
S~rgio
RUBE~S
It~CO!':E:
PENHA CYSNE - 1989
HYPOTHESIS" - Vicente
~·.adriç:éll,
Ribeiro da Costa Wer1ang - 1989
llj2. liA AMAZONIA BRASILEIRA".- Ney Coe· de Oliveira - 1989
143.
D~S~GIO
DAS LFTs E A PROBABILIDADE IMPLICITA DE
MORA1C~IA
Maria Silvia Bastos Karques e S~rgio Ribeiro da Costa ~erlanf - lSSS
1~4. THE
Loe
OEBT PROBLEM: A GAME-THEORETICAL ANALYSIS
Mario Henrique Simonsen e Sérgio Ribeiro da Costa Werlang - 1989
n
1~5. ANALISE CONVEXA NO R - Mario Henrique Simonsen - 1989
.
1~6.
A CONTROVtRSIA MONETARISTA NO· HEMISrERIO NORTE
Fernando de Holanda Barbosa - 1989
147. FISCAL REFORM AND STAB1LIZATION: THE BRAZILI~N EXPERIENCE - Fcrn~ndo de Hol~nda
Barbosa, Antônio ~alazar Pessoa Brand~o e Clovis de F~ro - 1989
148. RETOR::OS Er-1 EDCC1\ÇÃO
Ca::los Ivan sir!cI1s(?n
~,10
BRASIL: 1976-1986
Leal e Sérgio Ribeiro da Costa \'Jerlang - 1929
149.
PREFERE~CES, COMMON K~OWLEDGE,
ANO SPECULATIVE TRADE - James Dow,
vicente Madrigal, S~rgio Ribeiro da Costa Wer1ang - 1990
150.
FDUCAC~O S DIS7RIBUIÇ~O DE RENDA
C él r 1 o.s I v a n S ir~ o 11 :3 '2 n - L e a 1 e S é r 9 i o R i b e i r o d a Co s t a - 1990
151. OBSERVAÇ(:JE3 À l'lARGE~~ DO
Coe de Oliveira - 1990
T~ABALHO
"A M1AZONIA BRASILEIRA"
Ney
152. PLAKO COLLOR: UM GOLPE DE MESTRE CONTRA A INFLAÇÃO?
- Fernando de Holanda Barbosa - 1990
153. O EFEITO DA T,~XA DE JUROS E DA It\CF:RTEZ'" SOBRE A CURVA DE PHILLl?.'3
DA ECONOMIA ERASIL2IRA - Ricardo de Oliveira Cavalcanti - 1990
154. PLANO COLLOR: CO~7RA FACTUALIDADE E SUGESTOES SOBRE Â CONDUÇ~O
DA POLíTICA MONeTÁRIA-FISCAL - Rubens Penha CyGne - 1990
155. DEPOSITOS DO TESOURO: ~O BANCO CENTRAL OU NOS BANCOS COMERCI~IS?
Rubens Penha Cysne - 1990
156. SISTEMA FINA~CEIRO DE HABITAÇAo: A QUESTÃO DO DESEQUILíBRIO
FCVS - Clovis de Faro - 1990
157. COi-1PLE;'lE~JTO DO FAScíCULO N2151 DOS "ENSl\IOS ECONOr-nCOS"
ZONIA BRASILEIRA) - Ney Coe de Oliveira - 1990
DO
(A Mll\-
158. POLíTICA MCNETÂRIA ÓTIMA NO COMBATE À INFLAÇÃO
- Fernando de Holanda Barbosa - 1990
159. TEORIA DOS JOGOS - CONCEITOS BÂSICOS - Mario Henrique Simonscn - 1990
]60. O MERCADO A8ERTO BRASILEIRO: ANÂLISE DOS PROCEDIMENTOS OPERACIO·
~AIS - Fernando de Holanda Barbosa - 1990
161. A
RELAÇ~O
ARBITRAGEM ENTRE A ORTN CAMBIAL E A ORTN MONETÂRIA -
- Luiz Guilherme Schyl!'ura de Oliveira
- 1990
162. SUBADDITIVE PROBABILITIES AND PORTFOLIO INERTIA - Mario
Simonsen e S~rgio Ribeiro d0 Costa Werlang - 1990
163.
Ilenri~ue
~lJ1CROECO;:0:nA COl'l r-14 - Carlos Ivan Simonsen Leal e S~rgio Ribeiro d,:l
C03t.:1
Werlang - lSi90
164.
A RF:-EXl.._:íI~~!"TIO;\ O~-' SOLOíi I S GPOí'iTIl r'lODSL HITH l'... PPL lCA'~'IO'l~
C A P I ':';., L r1 O V E :'; r~ l JT S - i:1c?~] n t r o S () él V e d r.'l R i v <" no - 1 990
'1'"
.. "
PURLIC Cf!Orc:r:: :;;::l1ITH)'J: 'v,"\:-~rr\TIO:~S ml TPE Tflf.l'lE OF' SCTI;~:'n!T"
- Antonio Mario 2~ Silvelre - 1990
lGG. TIrE nCI~LIC CHOIeS pr;;PSrr::CTIVE ]\:~D lZf\lIGIIfT1'S If\lS'1.'I'T'{J'T'IO·\ll'LlS'l' !',J:'!'('
- l\ntonio :1.Jr Ll ua Silv(;irêt '- J 0()Ó
lG5.
'fi;!;
WA~FARS
1G7. 'rIm INDLTi;I:;H::A1'IO:l OI-' Sl':NIOE -
Antonio I/li1r ic1 lla silvciri1 -1. ~)')(J
•
168. JAPANESE DIRECT INVESTMENT IN BRAZIL - Neantro Saavedra Riva
no - 1990
169. A CARTEIRA DE AÇOES DA CORRETORA: UMA ANÁLISE ECONOMICA-Luiz
Guilherme Schymura de Oliveira - 1991
170. PLANO COLLOR: OS PRIMEIROS NOVE MESES - Clovis de Faro -1991
171. PERCALÇOS DA
INDEXAÇ~O
EX-ANTE - Clovis de Faro - 1991
172. NOVE PONTOS SOBRE O PLANO COLLOR 11 - Rubens Penha Cysne-19991
173. A DINÂMICA DA
HIPERINFLAÇ~O
- Fernando de H. Barbosa, Waldyr
Muniz Oliva e E1via Mureb Sa11um - 1991
174. LOCAL
CONCAVIFIABILITY OF PREFERENCES AND DETERMINACY
OF
EQUILIBRIUM - Mario Rui Pascoa e Sérgio Ribeiro da Costa Wer
lang - Maio de 1991
000056535
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