Title

An infinite-dimensional extension of the von
Neumann model

Author(s)Murakami, Hiromi

Citation

大阪大学経済学. 2023, 73(2・3), p. 111-122

Version Type VoR
URL

https://doi.org/10.18910/93335

rights
Note

Osaka University Knowledge Archive : OUKA
https://ir.library.osaka-u.ac.jp/
Osaka University

Vol. 73 Nos. 2・3

OSAKA ECONOMIC PAPERS

December 2023

An infinite-dimensional extension of the von
Neumann model＊
Hiromi Murakami †Abstract
The equilibrium existence theorem of Kemeny et al. (1956) treats one of the most generalextensions of the von Neumann model. Their method, however, crucially depends on thetheorems of finite-dimensional dual-linear systems, and it has great difficulty in generalization
for infinite-dimensional situations. In this paper, the equilibrium existence theorem is extended tocases where the index sets of commodities and activities are not finite but compact metric spaces
having a certain kind of countable dense subset. This extension enables us to treat models with
continuously many kinds of activities and countably infinite commodities, which is essentiallyequivalent to considering continuous production functions homogeneous of degree

anddifferentiated commodities including intermediate goods for capital accumulation.
JEL Classification: C62, C70, D53, E22
Keywords: von Neumann Model, Economic Growth, Capital Accumulation, Commodity
Differentiation, Input-Output Analysis, Matrix Game, von Neumann Revolution

1 Introduction
The model of von Neumann (1937) brought remarkable progress to economic theories such as
input-output analysis, fixed-point arguments ensuring the existence of general economic equilibrium,
the minimax saddle-point problem for a zero-sum matrix game, theory of capital accumulation,and fundamental concepts and settings for balanced growth theory. In this model, every action in
the economy is treated as a process called “input and output” with an elementary positive time
length. As Morishima studied and strongly emphasized in his books and papers (see, for example,
Morishima1964; Morishima1969), the von Neumann model gives us a general perspective indescribing capital goods under multi-sectoral economic growth and capital accumulation settings. By
treating capital goods at different stages of wear and tear as different goods, an economic situation＊†

This paper is based on research derived from a joint paper by Professors Ken Urai and Satoru Kageyama, for which I
would like to express my special gratitude.
Lecturer, Faculty of Fine Arts, Kyoto City University of Arts, E-mail: h_murakami@kcua.ac.jp

－ 112 －OSAKA ECONOMIC PAPERS

Vol. 73 Nos. 2・3

can be observed where the length of time under which a capital good stays in the same condition
depends on the intensity of its use. Morishima called this the von Neumann revolution.
From a purely mathematical view, extensions of von Neumann’s results are abundant in theliterature, such as fixed-point theorems, minimax arguments, and saddle-point problems. Above all,the models based on the settings of Kemeny et al. (1956) are the most general kind of extension of the
von Neumann model. However, the equilibrium existence theorem of Kemeny et al. (1956) cruciallydepends on the finite-dimensional matrix concepts and tools like the central solutions and theorems on
finite-dimensional dual-linear systems (see, e.g., Tucker1955).1
In this paper, the equilibrium existence theorem of Kemeny et al. (1956) is extended to the case with
continuously many kinds of activities and countably infinite commodities under a boundary condition
on a set of activity or commodity indices.2 The theorem enables us to treat models with infinitely
many kinds of activities and commodities, which is essentially equivalent to considering differentiatedcommodities, including intermediate goods for capital accumulation or continuous production
functions homogeneous of degree .
2 Extension of von Neumann Model
Denote bydenote by, andthe set of non-negative integers and by
the -dimensional Euclidean space. On, such that for eachthe set of real numbers. For each,, consider three types of order relations,

,

and

in

,

, we have, and. Moreover, byand, we denoteand, respectively.
Let

be a compact metric space andbe a countable compact metric space.3 Set
is the index set of goods. Functionis the index set of production processes, andrepresents input and. Variablevalued onandrepresents output. Both are assumed to be non-negative realrepresents an intensity of production, a Borel probability measure on ,represents a level of prices, which is also taken as a Borel probability measure on .

Two variables,and, are used to look for a solution

satisfying the following system of inequalities.

(1)
(2)
1

2

3

Nikaido (1968) proved the existence theorem of Kemeny et al. (1956) under the settings of dual-linear systems infinite-dimensional spaces without using the central solution or any game theoretic term. The linear programming in
infinite-dimensional spaces was discussed by Anderson and Nash (1987) in detail.
There is an abundant literature on the extension of the von Neuaman model with coordinate-free input-output
structures. See, for example, Fujimoto (1986). For models with non coordinate-free matrices, see Urai and Kageyama(2012).
A compact metric space has a countable dense subset, soand

are Polish spaces (separable metric spaces).

December 2023

An infinite-dimensional extension of the von Neumann model－ 113 －

(3)(4)Equation (1) implies that at any time period, no more goods can be used than were produced during
the preceding time period. Equation (2) shows that there exist no residual profits, i.e., no process canyield a return in the given time period greater than that which would have been obtained by the goinginterest rate. Equation (3) says that the prices of overproduced goods are zero, and (4) shows that theintensity of deficit processes must be zero.
andIn general, without further assumptions about
these conditions. In von Neumann (1937),

and, there is no solution satisfying
are assumed to be finite matrices, andthe following additional assumption is used.
(5)

Intuitively, this means that every production process must consume or produce a positive amount of
every good. As pointed out in Kemeny et al. (1956), this assumption is highly restrictive.4 We use thefollowing weaker and economically plausible conditions, which are based on the idea of Kemeny etal. (1956).
(6)(i) every row of

has at least one positive entry,(ii) every column ofhas at least one positive entry.

The condition of equation (6) means that the total value of all goods produced in the economy must be
positive. Assumption (i) implies that every process uses some inputs, and (ii) says that every good canbe produced in at least one process. More precisely, sinceandare metric spaces, we must write

conditions (i) and (ii) as follows.
(i )
(ii )

,

,
,

,

,
.

Note that here we use conditions (i) and (ii) with the meanings of (i ) and (ii ).
Ifsatisfies equations (1)–(6) with assumptions (i) and (ii), this will provide a solution to

the economic model that holds in every time period. This is the equilibrium for the von Neumann Model.
Following the same approach in Kemeny et al. (1956; Lemma 1), we can show that
under our settings. Suppose that we have a solution

to (1)–(6). From (6) we see that, and thus by (3) and (4) we have(7)4See, e.g., Takayama (1985).

－ 114 －OSAKA ECONOMIC PAPERS

From the last equation,

Vol. 73 Nos. 2・3, and it follows that.

, i.e., the case where the interest rate isHence, we need to look only for the solutions in whichequal to the growth rate.

, we see that (3) and (4) become identical as the consequences of the conditions inSince

equations (1) and (2). Thus, our equations can be written simply as

(8)
(9)
(10)

. Divide

For technical reasons, we give the next operation tobyand define(note that). Then, we obtain the function, which is a convex combination ofand. Let us defineas, and our system of equations becomes

(11)
(12)
(13)

3 Infinite-Dimensional Zero-Sum Game
It is well known that we can interpret the equilibrium of the von Neumann Model as the saddlepoint solution of a zero-sum matrix game. In this section, we extend the game theoretic concepts andmethod to our infinite-dimensional settings.
Lemma 1: For continuous functionon, the extended zero-sum matrix game defined by, is a continuous function of .5

is strictly determined, and the value of game,
Proof: Letbe the real-valued continuous function onprobability measure on the Borel -algebrabe the whole space of the measure5(resp.

(resp.

) of). Here,. We assume that
(resp. ), and let(resp.

(resp. ) is a
(resp.

)) becomes a compact

Note that this result also implies the existence of the solution of such a zero-sum matrix game defined byforeach . See also Miyazawa (1958) as a reference to the continuity ofwith respect to for a general version of
the zero-sum minimax theorem.

December 2023

An infinite-dimensional extension of the von Neumann model－ 115 －

metric space under the topology of weak convergence.6 Then,(14)is assured by the weak compactness

For (14), the existence ofand continuity of

of

(15)

. Inequality

holds, since

is a Borel set for each. To show the contrary, let us. We see thatdefine. Then,follows from the fact that7

for all. In the same way, for (15), the existence ofis assured by the weak compactness ofand the continuity of. Here,holds,

is a Borel set for eachsince

. To show the contrary, let. Then,andfollow
for allfrom the fact that.

Furthermore, we define
(16)
(17)

The continuity ofandfollows from Parthasarathy (1967; Lemma 1.1, p.57) and Berge’s
andmaximum theorem (see, for example, Ichiishi 1983). Hence, we have the maximum value ofthe minimum value of. From (14) and (15),for all, and we have.
Next, we must show that

. Let us define

andas follows:elaborate correspondence



Then

has a graph with non-empty, closed, and convex values (see Parthasarathy 1967; Theorem 5.7,

p. 35). From the fixed-point theorem of Glicksberg (Glicksberg 1952), the correspondence
6(18)

Consider the case of . Whenis any continuous function on , since

is compact,

is bounded, so that

has a
iff

. See, for example, Parthasarathy (1967; Theorem 6.4, p. 45).
7

See, for example, Fubini’s Theorem in Halmos (1974; Theorem 36.B, p. 147). Note thatwith the value .

is a constant mapping－ 116 －OSAKA ECONOMIC PAPERS.8 Hence, we have

fixed point

Vol. 73 Nos. 2・3

as follows:(19)Thus we establish, or minimax=maximin.

Now we confirm that the minimax becomes a continuous function ofand. Since, we can combine them as(20)Assume that,and, then we have to show thattends to . Let us consider the following inequation.

(21), tends to

The first term of the right-hand side of (21),under the topology of weak convergence. On the other hand, the second term of the right-hand side
, also tends toof (21),converges to

and

of

, we callConsider two different players
for their pure strategy

and,for allthe saddle-point solution of the zero-sum matrix game.
, and interpretas the profit ofrepresents the profit-, and it is strictly determined. In addition, the value ismatrix, is

as the saddle-point’s profit (resp.

of’s loss) under the mixed strategy, whichand.

as an extended matrix game, then equations (11) and (12) assert that

become the optimal strategies for this game and8

as well as the loss

. Then, we could summarize the above result as follows:

Value of game of the extended zero-sum matrix game, in which

If we interpret▪

becomes a continuous function of .

With the property of the fixed point

represents

uniformly. Hence, we see that

confirm that minimax valueof, sinceandmust be chosen such that the value of the game isSinceis a subset of the vector space generated by whole measures on under the topology of weak convergence,
it depends on the duality with the vector space
generated by a real-valued, bounded, and continuous functionon .

December 2023

An infinite-dimensional extension of the von Neumann model－ 117 －

zero.9 The condition of equation (13) remains as an economic condition.
The above economic assumptions (i) and (ii) can also be reformulated in the zero-sum gameandtheoretic notation. We consider

to be extended matrix games, where the maximizing player

controls the -side and the minimizing player controls theandof each of these games. Since-side. Letandbe the values

are non-negative, we can easily confirm that the conditions(i ) and (ii ) are equivalent to
(i )

,

(ii )

.

4 Boundary Condition
To consider the existence theorem, we introduce the next two assumptions, “Boundary Condition”
and “Minimum Positive Amount Condition for Input-Output Coefficients.”
Boundary Condition
There exist a countable dense subset,

and points

,

,

of
of

with an open neighborhoodof , finite points,, such that each(resp.

(resp.

of,and, we have(B-2) For each

and, we havefor all

One can easily confirm that ifof,) is associated), for), satisfying the following conditions on the restriction(B-1) For each,(resp.
ofon:for all

or
or

or
and

(which are restrictions of

and

on

) satisfy

also satisfies exactly the same requirements in this condition.

the boundary condition, thenMinimum Positive Amount Condition for Input-Output Coefficients
Let
any

and

be the values of functions

and

,

and

or

, respectively, atholds, where. For

is fixed.

Under these assumptions, we show the existence of an economic solution of equations (11), (12)and (13) under (i) and (ii) using the game-theoretic framework. Before moving on to the proof of ourmain theorem, we prove the following lemma.
Lemma 2: Let,, and. Pair9

satisfy the boundary condition and,possess solutions

is called a central solution for matrix gameA game with the value of. Then, the dual systems

is called a fair game.

such that.

－ 118 －OSAKA ECONOMIC PAPERS

Vol. 73 Nos. 2・3

Proof : [First Step] Under the boundary condition, we consider the next subsets of

or :

(22)
(23)
(24)

Note that. Let

be the restriction of, andbe the restriction of

for

.

on

on

,

(25)

be the restriction ofonas the restriction ofon. We also define

We apply Theorem 5 of Tucker (1956) (see also, Tucker 1955; Theorem 6 and Nikaido 1968:of orderCorollary 3, p. 39) to the skew-symmetric matrix

defined by
(26)where the upper left 0 is the zero matrix of order10. We haveand the lower right 0 is the zero matrix of ordersatisfying

,

and
,

,

,

. If we let, the above relations yield the following results:

,

,

,

,

, and

.
[Second Step] We observe that the solutionofis constructed from the solutionofboundary condition, for eachand for eachfor the dual system of the lemma

for the dual system with respect to,orfor all. From

and

have
when

, sincefor all, and we have

and

when

on

. Hence, we have
. From.

,for, we seefor. Now we show that there exist

and(Case 3)

.
and

and

.

in the sense thatand.

For a finite set

such that.

(Case 2)10

on

, we can consider the next three cases.

(Case 1)

We use. Hence,. For this, we extend the above result toand for each

. In addition, fromsuch that,. Thus, we define

by using. From the boundary condition, for each, since, we. Consequently, we obtain[Third Step] Then, we will also defineall. From the, we denote by

the number of elements in.

December 2023

An infinite-dimensional extension of the von Neumann model(Case 1) From equation

and assumption (B-2) in the11boundary condition, we have
, we haveas minfor all
and

and

as

. In addition, we consider an index set,oncorresponds to

when

and

.

,

, and the boundary condition, we havefor all. Thus we define

and

when

, and

represent a constant mapping with the value 1 on

for all(Case 2) From

and

, of

such that. Let. Then, we define

when. Since. Let us define. Then, we introducewhen－ 119 －

onsuch that.

(Case 3) Sinceand, we can confirm whether

is equal to zero or positive for eachfind the firstascorresponding to. Then, wesuch that, and we rename the. In addition, we continue to confirm whether

is equal to zero or positive forcorresponding to someallfind that, we rename the

as

From

, we have

. Let

when

,

when

,

,

. At every time we. We call the set, and we introduce

corresponds toand is not assigned any
mapping with the value 1 on

. Then, we define

For all three cases, we have

and

as

,

. Let

.

on

such that

, and

when

represent a constantfor all.

.

[Fourth Step] In a similar way to the third step, we also defineonby using. From, we can consider the following three cases.
(Case 1 ).

(Case 2 )and.

(Case 3 )We useandand.

in the sense that

and.

(Case 1 ) From equation

and assumption (B-2) in thefor allboundary condition, we have
and

we have. Let us defineand. In addition, we consider an index set,onsuch that

and

, and

as min. Then, we introduce
corresponds torepresent a constant mapping with the value 1 on

. Then, we define.

(Case 2 ) From,
for all

Read assumption (B-2) with

and

,

when

for all

11

as
, of

when

. Let

. Since. Thus we define.

, and the boundary condition, we haveonsuch that

when

and

－ 120 －OSAKA ECONOMIC PAPERS

when

Vol. 73 Nos. 2・3.

(Case 3 ) Sinceand, we can confirm whether

is equal to zero or positive for each

. Then, we find the firstsuch that

corresponding to, and we rename thecorresponding to some
, we rename the

as

,

,

,

as

, and we introduce

whencorresponds to. Let

and is not assigned any
we defineon. From,such that

,

, and

when
when

represent a constant mapping with the value 1 on

. Then,.

for alland

For all three cases, we have.

[Fifth Step] By repeating the same discussion in the fourth step, we can extend the above
tofor. Then we havecan get step by stepfor

we haveand,.

. At every time we find that

. We call the set. Let

we haveasis equal to zero or positive for all

In addition, we continue to confirm whether

and
for

,

,

for

. In the same way, we
for

, and

for

. Hence, finally,▪

.

5 Existence Theorem
Now we can prove the equilibrium existence theorem with a method similar to Thompson (1956),
which treats the von Neumann model with finite matrices. As stated above, we can assume that theboundary condition holds on
Theorem 1: If

is the largest value that makes

optimal strategies for
Proof: Letfor all .

such thata fair game, then there exists a pair.

be the largest value that makes. By assumption,. Hence, from the continuity, there is at least one positive

and

of

such that.
with

Assume that there is no solution to. Then, if

for all , i.e., all such solutions haveis a pair of optimal strategies for, and sinceandall optimal strategies for the gameand, we have, we also have

. Let

. We define

and

and

run over

as

(27)
(28)
We now confirm thatsuch that. For an optimal solution. Then, if we assume that there is no, sinceis countable, we can findsuch that, whereisDecember 2023

An infinite-dimensional extension of the von Neumann model

some open set, we have, and this is a contradiction. Thus, there exists

for some open set. Thisbelongs to. Since there exists someand,since every row of A has at least one positive entry. Ifsuch that. We can also say

must be positive for all

such that, we haveandfor all

By considering the fact thatfor all－ 121 －

for any open set

,
from
.

, we must have

. Again, this is a contradiction.

Hence, now we consider. Take any

such that. We define

as

(29)
Let us define

as follows:where

is an optimal solution assured by Lemma 8.2 foronor. For anymust have either

for all,.

must be strictly positive. From the minimum positive amount condition forHence,

input-output coefficients,

has a minimum value, andThis implies that even if we take a larger , the optimal solution
the game. From the lemma, wewith.
is still the solution of. This contradicts the maximality of .

▪

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