COMPETITIVE EQUILIBRIUM AND FIXED-POINT THEOREM II
WALRAS'
EXISTENCE
BROUWER'S
FIXED-POINT
By
THE PURPOSE of this note
theorems-Walras'
Point Theorem
Walras'
Walrasian
THEOREM
Existence
HIROFUMI
THEOREM*
UZAWA
is to show
Theorem
AND
the
on
equivalence
the
one hand
of two
and
fundamental
Brouwer's
Fixed-
on the other.
theoremD
system
is concerned
of general
with
the
equilibrium
existence
of an equilibrium
and has been a problem
in the
of some
im-
portance in formal economic analysis since his work [12] appeared in 1874-7.
It was, however, not until Wald's contributions,
[10] and [11], that the existence
problem was rigorously treated.
Recent contributions,
in particular
those of
Arrow and Debreu [2], McKenzie [6], Nikaido [7], and Gale [4], have shown
that Walras'
theorem
Point Theorem.
bears
It would
importance
to see that
Brouwer's
indicate
the reason
in the Walrasian
century
1.
a necessary
The latter theorem, 2)first
a fundamental
interest
blem
is essentially
system
consequence
proved
in mathematics.
theorem
that
by Brouwer
general
treatment
Fixed-
[3] in 1911, also
It may be hence
is in fact implied
the
of Brouwer's
by Walras'
of
theorem.
of the existence
had to wait for the development
some
pro-
of the twentieth
mathematics.
According to Gale [4] and Nikaido^ [7], Walras' theorem may be formulated
as follows.
Let there be n commodities,
x=(x1.
......
labeled I. ......
, n, p=(p1,.....
, xn) be a price vector and a commodity
*
This work was supported by the Office of Naval
Stanford University.
I am indebted to Frank H.
the writing of this note.
1) The reader is referred to Schumpeter [8], Part
of Walras' theory of general equilibrium.
2) See, for example, Alexandroff and Hopf [1], pp.
p. 117.
一59一
bundle,
, pn) and
respectively.
Research under Contract NR-047-004 at
Hahn for discussions and for suggesting
IV, especially Chapter 7, for an appraisal
376-8 and p. 480, and Lefschetz
[5],
季 刊
Price vectors
are assumed
are
n-vectors.
arbitrary
all commodity
理
論
経
to be nonzero
Let P and
済
学
vol.
and nonnegative;
XIII,
commodity
X be the sets of all price
No.1
bundles
vectors
and of
bundles:
P=
{p=(p1,
......
, pn)
X=
{x=(x1,
......
,
xn)}
:pi•¬0,
i=l,
......
,
n,
but
p#0}
.
The excess demand function x(p)=[x1(p),......
, xn(p)] is a mapping from
P into X.
A
price
vector P^
is
called
an
equilibrium
if
x1(P-) •¬0,
with
equality
unless P-1=0,
WALRAS'
the
EXISTENCE
following
(A)
(B)
(C)
(i=1,
......
THEOREM.
(i=1,
,
Let
......
,
n)
n),
an
excess
demand
fuction
x(p)
satisfy
conditions:
x(p)
is
x(p)
is
Walras'
a
continuous
homogeneous
law
mapping
of
order
from p
0;
into
i.e.,
X.
x(tp)=x(p),
for
all
t>0
and
p ƒÃP.
holds:
n•¬i=1pixi(p)=0,
for
all
p ƒÃ P.
Then there exists at least an equilibrium price vector p- for x(p).
2. Brouwer's theorem, on the other hand, is concerned with a continuous
mapping on the simplex.
The fundamental (n-1)-simplex II
is the set of all nonnegative n-vectors
whose component sums are one:
II = {ƒÎ=(ƒÎ1,.
.....
, ƒÎn); ƒÎ•¬0, n•¬i=1 ƒÎi=1}
BROUWER'S
FIXED-POINTTHEOREM. Let to(7r) be a continuous mapping from
17 into itself.
Then there is at least a fixed-point >r in 17:
n=(P(~')-
3.
EQUIVALENCE THEOREM.
Point
Theorem
PROOF:
3)
It
are
is
Walras'
Existence
Theorem
and
Brouwer's
Fixed-
equivalent.
well
established3)
that
Brouwer's
See Nikaido^ [7], Gale [4], and Uzawa [9], Appendix.
一60一
theorem
implies
Walras'
Sept.,
1962
theorem
of
implies
H. Uzawa:
the
form
Brouwer's
Let _??_(ƒÎ)
excess
Walras'
Existence
above.
We
Theorem
shall
and Brouwer's
therefore
Fixed-Point
prove
that
Theorem
Walras'
theorem
theorem.
be
any
demand
continuous
function
mapping
x(p)=[xl(p),
(I)
from
......
II
into
, xn(p)]
itself.
We
construct
an
by
xt(P)=_??_i(p/ƒÉ(P)) -piƒÊ(P),
(i=1,
......
, n,
p ƒÃP),
where
ă
(P)=•¬n i=1
a-1
It
of
may
order
It
noted
that _??_i(p/ă(p))
and
piƒÊ(p)
are
both
positively
thus
defined
homogeneous-.
0.
is
tions
be
evident
(A),
that
(B),
librium p-.
the
and
Then,
(C).
by
excess
demand
Hence,
(1),
we
function
applying
Walras'
theorem,
satisfies
there
equality
unless
an
equi-
have
(2) _??_i p-/ƒÉ(p-)•¬p-iƒÊ(p-)
with
is
condi-
(i=1,
......,
n),
P-i=O.
Letting
the
relation
(2)
may
be
written
(3)_??_(ƒÎ-)•¬ƒÀƒÎ-i
with
equality
Summing
=1;
unless ƒÎ-i=0.
(3)
over
i=1,
......
,
n,
and
noticing
that ƒÎ, _??_(ƒÎ-)ƒÃII,
hence,
we
haveƒÀ
(4) _??_(ƒÎ-)•¬ƒÎi
with
The
equality
relation
unless ƒÎi=0.
(4),
again
together
with ƒÎ, _??_(ƒÎ-) ƒÃ
II,
implies
that
Stanford University
一61一
季刊
理
論
経
済
学Vol.
XIII,
No.
1
REFERENCES
1]
[2]
Alexandroff, P. and H. Hopf, Topologie,
[
I, Berlin: Springer, 1935.
Arrow, K. J. and G. Debreu, "Existence of an Equilibriuma for a Competitive Economy,"
Econometrica, Vol. 22 (1945), pp. 256-290.
3] Brouwer, L. E. J., "Uber Abbildung
[
von Mannigfaltigkeiten,"
Mathematische Annalen, Vol.
(1911-12), pp. 97-115.
71
4] Gale, D., "The Law
[
of Supply and Demand," Mathematica Scandinavica, Vol. 3 (1955), pp.
155-69.
5] [ Lefschetz, S., Introduction to Topology, Princeton: Princeton University Press (1949).
(6 ] McKenzie, L. W., "On Equilibrium in Graham's Model of World Trade and Other Competitive Systems," Econometrica, Vol. 22 (1954), pp. 146-61.
[7]
Nikaido, H., "On the Classical Multilateral Exchange Problem," Metroeconomica, Vol. 8
(1956), pp. 135-45.
(81
Schumpeter, J. A., History of Economic Analysis, New York: Oxford University Press
(1945).
j 9 ] Uzawa, H., "Walras' Tatonnement in the Theory of Exchange," Review of Economic Studies,
Vol. 27 (1960), pp. 182-94.
[10] Wald, A., "Uber die eindeutige positive Losbarkeit der neuen Produktionsgleichungen,"
Ergebnissen eines mathematischen Kolloquiums, Vol. 6 (1933-4), pp. 12-20.
[11] Wald, A., "Uber die Produktionsgleichungen der dkonomischen Wertlehre," Ergebnisse eines
mathematischen Kolloquiums, Vol. 7 (1934-5), pp. 1-6.
12] Walras, L., Elements of Pure Economics, translated
[
by W. Jaff6, Homewood: Richard D.
Irwin (1954).
-62-