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The Review of Economic Studies Ltd.
A Model of General Economic Equilibrium
Author(s): J. V. Neumann
Source: The Review of Economic Studies, Vol. 13, No. 1 (1945 - 1946), pp. 1-9
Published by: Oxford University Press
Stable URL: http://www.jstor.org/stable/2296111 .
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A Model of General Economic Ecuilibrium1
The of this
subject paper is the solution of a typical ec-onomic
equation
The system has the following system.
properties:
are "
(i) Goods produced from
not only natural factors of production," but in
the first from other. These of be
place each processes production i.e.
may circular,
good G1 is produced with the aid of and with the aid of
(2) There be more good G2, G2 of G1. than
may technically
possible processes production goods
and for this reason " of " is of no avail. The is rather
counting equations problem
which be used "
establish and which
to will not
processes actually (being unprofitable").
order
In to be
able to discuss we
(i), (2) quite shall idealise
freely other
elements
of
the situation (see I and Most of these idealisations are but
paragraphs 2). irrelevant,
here.
this not be discussed
question will
in
which
The our
way are leads of
questions put to
necessity a system of
(3)-(8') in paragraph the of a solution of which is inequalities
be proved 3 possibility The not evident, i.e. it
cannot by mathematical is
any qualitative
argument. proof possible
by means of a of Brouwer's Fix-Point i.e. the use only
generalisation Theorem, by of very
fundamental facts. This theorem " "
topological generalised
paragraph 7) is also in itself. fix-point (the lemma of
interesting
The connection with topology may be at
it is in very surprising first, but the author
thinks that natural of this kind. The
problems immediate reason for this
of " "
is the occurrence a certain minimum-maximum problem, familiar from the
of In our the
calculus variations. present question, minimum-maximum
in problem
formulated
been It is
has paragraph related to another
5. closely problem occur-
ring in the theory of footnote i in
games (see paragraph
A direct of the function 6).
interpretation would
role to be similar to that of 0 (X, Y) be highly desirable. Its
appears in
it can be thermodynamic potentials phenomenological
thermodynamics; surmised that the similarity will persist in its full
of our
phenomenological restrictive
generality
feature our (independently idealisations).
Another of theory, so far without interpretation, is the remarkable
duality of the variables
(symmetry) monetary (prices interest factor f) and the
technical variables of yj,
(intensities production xi, coefficient of expansion of the
economy a). This is out in
brought very clearly paragraph in
of 3 (3)-(8') as well as
the minimum-maximum formulation paragraph 5 (7**)-(8**).
Lastly, attention is drawn to the results of ii from which
that the normal paragraph follows,
other mechanism
among things, price about-if our
tions are valid-the most efficient brings assump-
technically intensities of production. This seems
not since we have eliminated
unreasonable all monetary
complications.
was
The read for the first time in the
present paper winter of I932 at the mathe-
of Princeton The
matical seminar reason
University. for its was an
from K. to whom the publication invita-
tion Mr. author
Menger, wishes to his
express thanks.
i. Consider the following problem: there are n goods . . . ,
G1, which can
be produced by m processes F1, . . Pm. Which Gn
and what of the ., processes will be used (as
will obtain The
"-profitable ?
") prices goods problem is evidently
1 This paper was first published in German, under the title Uber ein Okonomisches Gleichungssystem
und eine des Brouwerschen in the
Verallgemeinerung K. Fixpunktsatzes volume entitled Ergebuisse eines Mathe-
matischen Seminars, edited by Menger (Vienna, I938). It was translated into English by G.
A commentary note on this article, by D. G. Champernowne, is printed below. Morgenstern.
I
THE REVIEW OF STUDIES
2 ECONOMIC
one has been
of its the other
either can be after
since answered
non-trivial parts only
is in
i.e. its solution We observe
answered, implicit. particular:
(a) Since it is possible that m> n it cannot be solved through the usual
of
counting equations.
further we
to assume:
In order avoid complications
there are constant returns
That scale);
(b) (to be
the natural of can
That factors labour, expanded
including
(c) production,
in unlimited
quantities. that we wish to is this: are
The essential grasp goods produced
phenomenon we to determine which
other and want
each
from (see processes
equation
(7) (i)
below) be with which the total of
be what the relative will
will used; (ii) velocity quantity
will be.
increases what will obtain; what the rate of interest
goods ; (iii) prices (iv) furthermore:
to isolate this we assume
In order completely
phenomenon the of
of goods takes place only through processes pro-
(d) Consumption and
duction which include necessities of life consumed workers employees.
by of life will be
in of necessities
we that all income excess
words assume
In other
reinvested. the above
kind of models
to what theoretical
It is obvious correspond.
assumptions
2. In each process Pi (i= i, . . ., m) quantities aij (expressed in some units)
are used up, and quantities are produced, of the respective goods Gj (j = I, . . ., n).
can bij in the
The be symbolised following way:
process n
n
Pi: Saij G - bij Gj .................................... (i)
j=i j=-I
It is to be noted: wear tear of
are to be inserted on both sides of and
goods
(e) Capital (i); of wear as
are to be described different stages
capital goods by introducing
a each of
different Pi for these.
using separate
goods, time of duration
Each to be of unit duration. Processes longer
(f) process of if
into unit duration
to down
be broken processes introducing necessary
single
intermediate as additional
products goods. can be
the case where
can describe only
(g) (I) special good Gj produced
with certain viz. its
jointly others, permanent products.
joint with
In the actual economy, these processes Pi, i = i, . . ., m, will be used
certain intensities xi, i = i, . . ., m. That means that for the total production the
write
be We
of must multiplied by xi. symbolically:
quantities equations
m (I)
E = xi Pi.............................................. (2)
- i=i
xi o means that process Pi is not used. without
We are interested in those states where the whole economy expands
change of structure, i.e. where the ratios of the intensities x: .. . .: m remain
. themselves In such a case are
unchanged, although xl, . . xm may change. is the they
factor a unit of time. This factor of
by a common per
multiplied the whole coefficient
of economy.
expansion numerical unknowns of our are: the intensities x1, . . ., xm
3. The problem (i) of the whole
of the P1, . . ., ; the economy a;
processes Pm of expansion
(ii) coefficient the interest factor
(iii) the prices y, . . ., yn of goods G, . . ., Gn; (iv)
z in unit of time. Obviously:
iE z the rate of interest
(= '+ ±, being per
%
> ............ ...
xi o,................ ,j o, (4)
A MODEL OF GENERAL ECONOMIC
EQUILIBRIUM 3
and since a solution with xl - . . . = xm =o, or yi = .. .-yn= o would be
meaningless:
m n
xi > ................ 2 > ........
i=I o, (5) j=i yj o,....... (6)
The economic are now:
equations
a m < m
Zaij xi bii xi, ........ .............................(7)
and if in (7)< applies, =o .......................... .........
yj (7)
ftZaij yj >= ij yj, . ....................... (8)
j=i j=I
and if in (8) > applies,, xi = o................................. (8')
The of is: it is to consume more of a in the
meaning (7), (7') impossible good
total than is is Gj
If, less i.e. if there
process (2) of being produced. however, consumed,
excess a free =
is production becomes and its o.
Gj, good price
The of Gj yj
is: in no can made on
meaning (8') equilibrium profit be any
else (8), process
or the rate of interest would
Pi (or prices rise-it is clear how this abstraction
is to be If there is a loss, i.e. if is then will
understood). = however, Pi unprofitable, Pi
not be used and its intensity xi o.
The quantities aij, are to be taken as given, whereas the xi, a, ft are
unknown. There bij m n 2 but since in the yj, of
the ratios are, then, + + unknowns, case xi, yj
only xl: . . . Xm, : . .: are essential, are reduced to m n.
there are Y yn they +
Against this, m n conditions (7') and As
are not + (7) + (8) + (8'). these,
however, but rather the fact that the
equations, complicated
number of conditions is to the number of inequalities,
that the equal unknowns does not constitute a
guarantee system can be solved.
The dual of of the variables a and of
symmetry equations (3), (5), (7), (7') xi,
the " "
concept unused on the one and of
process hand, equations (4), (6), (8), )8')
of the variables and of the " "
remarkable. yj, Pf concept free good on the other hand seems
Our is
task to solve shall
4. (3)-(8'). We to show:
proceed
Solutions there be several
of exist, solutions with
(3)-(8') always although may
different xl: . . .: or with different . . : The first is since we
have Xm yx: yn. possible
not even excluded the case
where
several the
Pi describe same or where
several combine process
to form The
another. second is since
Pi possible some
goods may
enter Gj
into each Pi in a fixed ratio with some others. But even
from these process only apart
trivial possibilities there may exist-for less obvious reasons-several
solutions . Y . ft
x, : . .: Xm, : . . : Against this it is of importance that a,
should ym. f
have the same value for all solutions; i.e. a, are determined.
uniquely
shall
We even find
that a and be
can characterised in a manner
(see paragraphs 10 and P. directly simple
To ii).
our considerations we
shall
simplify assume that always:
aij + bij > o ............................................ (9)
are > the
(aij, clearly Since be small this
bij always o). aij, may arbitrarily
restriction is not bij
very far-reaching, it must be in order to assure
of as otherwise although imposed
uniqueness a, f W break into disconnected
Consider a might up parts.
now solution a, of If we had in
hypothetical xi, yj, ft (3)-(8'). (7)
then =
we
should o
<, have of in contradiction to
always always (because (7'))
yj (6).
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