Discussion Paper No. 529

THE NON-SUBSTITUTION THEOREM:
MULTIPLE PRIMARY FACTORSAND THE COST FUNCTION APPROACH

Kiyoshi Kuga

March 2001

The Institute of Social and Economic Research
Osaka University
6-1 Mihogaoka, Ibaraki, Osaka 567-0047, Japan

The Non-Substitution Theorem : Multiple
Primary Factors and the Cost Function Approach
By Kiyoshi Kuga
JEL Number: D57 (Input-Output Analysis)
Keyword : Input-Output Analysis, Non-Substitution, Price-Cost Relation,
General Equilibrium

ABSTRACTThe non-substitution theorem asserts that the choice of technique is independent ofpatterns of ﬁnal demand when eﬃciency prevails as to the use of a single primary factor,
say labor, while the asserted constancy of the input-output table no longer holds whenmore than one kind of primary factor is involved. No deﬁnite answer has yet been given
as to whether the commodity price vector is determined independently of ﬁnal demand
patterns when there are multiple primary factors of production. This paper shows thatthe “unit cost = price” relation uniquely determines the commodity price relatively toa given factor price vector. The proof of such a commdity price is provided by the useof Tarski’s ﬁxed point theorem without recourse to topology.

March 2001

Author’s Name: Kiyoshi KUGA
Postal Address: Kiyoshi Kuga,
Institute of Social and Economic Research, Osaka University
6 - 1, Mihogaoka, Ibaraki, Osaka 567-0047, Japan
Email Address: kuga@iser.osaka-u.ac.jp

The Non-Substitution Theorem : Multiple
Primary Factors and the Cost Function Approach
By Kiyoshi Kuga ∗1

Introduction

The end and aim of the Non-Substitution Theorem lies in the justiﬁcation of constancy of an
input-output table, and it asserts(i) that the choice of technique is independent of patterns of ﬁnal demand when eﬃciency prevails
as to the use of a single primary factor, say labor 1) , and(ii) that the commodity price vector is uniquely determined by the production structure, independently of ﬁnal demand patterns 2) .
These two ﬁndings have a solid foundation in the static context, but the dynamic non-substitution
theorems allow various degrees of freedom (Mirrlees (1969), Burmeister and Kuga (1970), Stiglitz(1970), Samuelson (1991) ) in that the input-output table including the capital input coeﬃcients is
stable in the long-run along the wage-proﬁt frontier (Morishima(1964, Chapter 3)).
It is well known that the asserted constancy of the input-output table no longer holds when morethan one kind of primary factor is involved (Koopmans (1951, p.153)). Yet, no question has been
posed so far as to whether the commodity price vector is determined independently of ﬁnal demand
patterns when there are multiple primary factors of production. The purpose of this paper is to
examine the question by exploiting the cost function framework.
The outline of this paper is as follows. Section 2 describes the framework of the model and proves
the existence of a price that equals the unit cost. Section 3 deals with its uniqueness relative to agiven factor price. Section 4 develops the discussion.

∗ Thanks are due to Professor Jun Iritani who inspired the present work by correspondence, and to Dr. Henrietta
Cole Calbonel for improving a draft of this work.
1 ) See Arrow (1951), Georgescu-Roegen (1951), Koopmans (1951b), Samuelson (1951).
2 ) See Morishima (1964, Chapter 3), Mirrlees (1969), Burmeister and Kuga (1970). Stiglitz (1970, p.544) writes
to the eﬀect that the non-substitution theorem is equivalent to the statement that the set of equations p = c(p) has
at most one solution p∗ .

1

2

The Model
defEach industry indexed by j ∈ N = {1, 2, . . ., n} has a no-joint, concave, constant returns toscale technology. The input consists of intermediate outputs from industries in N and primarydeffactors indexed by k ∈ M = {1, 2, . . ., m}. We begin with the description of the unit cost functions.
def
The domain of each cost function is P × W, where P = Rn+ consists of prices of n commoditiesdefp = (p1 , . . ., pn ), and W = Rm
+ \ {0} consists of factor prices w = (w1 , . . . , wm ) of m primaryfactors. Our ﬁrst concern is to examine the preliminary question of whether or not to the setof unit cost functions cj : P × W → Rn+ of industry j ∈ N, there exists a certain price vectorp = (p1 , p2 , . . . , pn ) ∈ P given w = (w1 , w2 , . . ., wm ) ∈ W satisfying
pj = cj (p, w), j ∈ N.

(1)

We may write (1) as p = c(p, w) and call p that satisﬁes (1) the unit cost equating pricerelative to w. To ensure the existence of such a price p ∈ P to (1), we assume(A.1) Given w ∈ W, there is a certain p̄ = (p̄1 , . . . , p̄n ) ∈ P, depending upon w, satisfyingp̄j ≥ cj (p̄, w) for all j ∈ N. 3)(A.2) Given w ∈ W, cj is non-decreasing, i.e., cj (p, w) ≤ cj (p , w) for p ≤ p (p, p ∈ P) and j ∈ N.
We are now ready to state
Theorem 1. Suppose an economy satisﬁes (A.1) and (A.2). Then for each w ∈ W, there is acertain p ∈ P satisfying pj = cj (p, w), j ∈ N.
Proof of Theorem 1.
A subset P ⊂ S × S is a partial order in S if the following three conditions are met: (i) (x, x) ∈P for x ∈ S, (ii) [(x, y) ∈ P ∧ (y, x) ∈ P ] ⇒ x = y, (iii) [(x, y) ∈ P ∧ (y, z) ∈ P ] ⇒ (x, z) ∈ P . A
function f : P → P is called non-decreasing if u ≤ v ⇒ f(u) ≤ f(v). Exploiting these concepts, we
use the well-known Tarski’s ﬁxed point theorem: 4)
Tarski’s Fixed Point Theorem ( Tarski (1955) ): Let (P, ≤) be a conditionally complete partiallyordered set and f a non-decreasing function from P into P . If P has two elements a and b such thata ≤ f(a) ≤ f(b) ≤ b, then there exists an element c ∈ P : c = f(c).
A partially ordered set in which every non-empty subset that is bounded above has a least upperbound is conditionally complete (Alexander (1965, p.178)). P = Rn+ is conditionally complete withrespect to the usual partial order ≤. We show that the required conditions are met in our case. Our(A.2) is the non-decreasingness in Tarski’s Fixed Point Theorem. We use (A.1) for the existence ofan upper bound p̄ satisfying c(p̄, w) ≤ p̄. The lower bound in our case may be taken to be zero.
Then by (A.2), it follows that 0 ≤ c(0, w) ≤ c(p̄, w). In sum, we have 0 ≤ c(0, w) ≤ c(p̄, w) ≤ p̄.

3 ) This assumption may correspond to Assumption 3 in Morishima (1964, p.59).
4 ) For an exposition of the theorem, the reader is referred to Abian (1965, p.182).

2

Remark 1. Fujimoto (1986) opened the way to applying Tarski’s theorem to the solvability of an
inﬁnite dimensional non-linear Leontief system. In his case as well as in our Theorem 1, only theproperty of the partial order is involved, and no topology such as continuity is used. See Abian (1965,
p.182) and Birkhoﬀ (1967, p.115) for the explanation of Tarski’s ﬁxed point theorem.

3

Uniqueness of the Unit Cost Equating Price Relative to a
Factor Price
To proceed further we assume for each j ∈ N(A.3) cj is homogeneous of degree one in (p, w) ∈ P × W, i.e., cj (µp, µw) = µ · cj (p, w) for µ ∈ R++ ,(A.4) cj is concave and continuously diﬀerentiable in the interior of P × W,
(A.5) The primary factors are indispensable, i.e., cj (p, w) > 0 for p ∈ P and w in the interior of W
for j ∈ N, and furthermore w · Aw (p, w) > 0 (component-wise strictly positive) for each (p, w)
in the interior of P × W.
Aw (p, w) above is the factor input coeﬃcient matrix that appears in the ﬁnal part of the following
exposition of notation:p = (p1 , p2 , . . . , pn ), q = (q1 , q2, . . . , qn ), w = (w1 , w2 , . . . , wm)jdef ∂c (p, w)

aij (p, w) =∂pi

, i, j ∈ N

a11 (p, w) a12 (p, w) . . .

a22 (p, w) . . .
a
21 (p, w)
def
A(p, w) = 
..
..
..

.

.
.
an1 (p, w) an2 (p, w) . . .
jdef ∂c (p, w)

ãkj (p, w) =∂wkã11 (p, w)


a1n (p, w)

a2n (p, w)

..


.
ann (p, w), j ∈ N, k ∈ M...

ã1n (p, w)

ã22 (p, w) . . .
Aw (p, w) = 
..
..
..
.

.
.
ãm1 (p, w) ãm2 (p, w) . . .

ãmn (p, w)ã12 (p, w)

ã21 (p, w)
def 



ã2n (p, w) 
.
..


.

From the former part of (A.5), we may take that each component of the equality between unit
cost and price is strictly positive. Let us now suppose, given w in the interior of W, that there
existed two diﬀerent equilibrium price vectors, p, q ∈ P satisfyingpj=

cj (p, w) > 0, j ∈ N

qj=

cj (q, w) > 0, j ∈ N.
3

Then we have for w in the interior of Wp =c(p, w) > 0 by (A.5)

p =p · A(p, w) + w · Aw (p, w) > 0 by (A.3) and (A.4)
w · Aw (p, w) > 0 by (A.5).

p(I − A(p, w)) =Thus the matrix I − A(p, w), with its oﬀ-diagonals being non-positive, has a non-zero nonnegative
solution p for a positive vector w · Aw (p, w). Thus we have 5)(I − A(p, w))−1≥0.

On the other hand, we also haveq

= c(q, w) = c(q − p + p, w)↓

by the concavity and diﬀerentiability (A.4)≤ c(p, w) + (q − p) · A(p, w)↓
q −p

p = c(p, w)≤ (q − p) · A(p, w)↓(q − p) · (I − A(p, w))

≤ 0
↓

q −p

(I − A(p, w))−1 ≥ 0≤ 0.

We have thus obtained q ≤ p. By a symmetric argument, we also obtain p ≤ q. We may therefore
summarize these results in the following uniqueness theorem
Theorem 2. Suppose an economy satisﬁes (A.1) - (A.5). Then for each w in the interior of W,
there exists a unique positive cost equating price p ∈ P satisfying p = c(p, w).
Remark 2. Our Theorem 2 can be regarded as an extension of Theorem 1 of Stiglitz (1970, p.544).
He assumes that each cost function ci does not contain pi as its arguments, and that there is only
one kind of primary factor, labor. Our Theorem 2 is more general in that ci does contain pi as its
arguments, in addition to the structure in which more than one kind of primary factor is involved.
Remark 3. Fujimoto(1980) deals with the non-substitution theorem targeting an extension of of
Stiglitz’s (1970) as well as of the uniqueness theorem of Nikaido (1968, p.170). However, his costfunction approach does not cover the multiple primary factor case. See also Fujimoto (1987).
Remark 4. Samuelson (1991) deals with the case in which m primary factors are involved. He
leaves open the values of an own rate of interest 1+r, and ratios of factor prices w2 /w1 , . . . , wm /w1 ,5 ) See Nikaido (1968), Chapter 2.

4and asserts determinacy of the input coeﬃcient matrix. This must be qualiﬁed by the fact that theother equilibrium conditions (tastes, etc.) then impinge on the cost and price relations. Our Theorem
2 shows that the unit cost equating price is uniquely determined relative to (w1 , . . ., wm ). But as will
be discussed in the ﬁnal part of our Section 4, the factor price vector depends upon the then existent
combination of the primary factors.
Remark 5. Kurz and Salvadori (1994) discusses the case in which wages are zero and prices are
not uniquely determined. They conclude, however, that the direct and indirect use of labor leads
ﬁnally to the uniqueness result. Their result may be compared with the non-substitution theorem ofMorishima (1964).

4

Discussion
The static non-substitution theorem ceases to be true when more than one scarce primary factor

is required (Koopmans (1951, p.153)). Our theorem still asserts that a unique choice of activity
prevails relatively to a given factor price w. When the number of the primary factors exceeds unity,
the factor price w will be determined endogenously by the general equilibrium system. Even thenthere is an advantage in knowing that an input-output matrix is determined by the factor pricevector w alone, and that the commodity price vector is a unique function of the factor price.
Morishima (1964, Chapter 3) developed a dynamic non-substitution theorem in which there is atrade-oﬀ between the real wage rate and the uniform rate of proﬁt. His theorem asserts that the
input-output and capital input coeﬃcients table of a dynamic Leontief system is uniquely determinedwhen either the real wage rate or the uniform rate of proﬁt is given. Two comments are in order.
There is the question of whether or not our formulation is formally equivalent to the logicaldevelopment of Morishima’s dynamic non-substitution theorem. The answer is no. With the linear
homogeneity of cost functions ((A.3)), we could have followed his line of argument by restricting
all our variables (p, w) to the unit simplex, and could have obtained a uniqueness result almost
along his line of argument (p.63). But such a uniqueness result holds only relatively to the simplex
normalization, whereas our theorem asserts the uniqueness result relative to each absolute level ofw ∈ W.
The important question that remains is the economic interpretation. Morishima develops a worldin which an equal rate of proﬁt prevails in a dynamic economy, while our economy can be either
static or dynamic. When the factor price combination in our economy is in a steady state, then ourtheorem can be embedded into his world. When a factor price vector yields diﬀerent rates of return
commoditywise, 6) our theorem can be given a temporary equilibrium interpretation.
Mirrlees (1969)’s Theorem 1 (p.69) asserts that an eﬃciency price is unique up to scale in thetimeless context of no-joint, constant-returns-to-scale economy. Mirrlees’ Theorems 2-4 are attempts
to embed dynamic economies into the timeless world of his Theorem 1. His method of examination
can be extended to the case of many primary factors, but it turns out that the factor prices are6 ) See also Burmeister and Kuga (1970) for the non-substitution theorem for the case in which rates of returns aresectorwise diﬀerent.

5allowed then to vary only along the eﬃciency price contour. By contrast, in our approach, the unit
cost equating price is a function of factor prices that run freely over the whole interior of W.
Mirrlees (1969, p.76) poses a question asking under what circumstances prices in one period aredetermined independently of consumer demand. In a sense, our Theorem 2 is a possible answer tohis problem posing.
One word on the general equilibrium theory. Our cost function approach reduces commodity
prices to a function of factor prices, while there are markets for commodities and rented factors.
How do we interpret these markets which reach equilibrium only through changes in factor prices?
In such a case, the received income consists of factor incomes, and it pays for expenditures. The
demand supply relation of commodities is balanced by output adjustment through the input-output
mechanism, and only the rented factor market remains to be balanced through the variation of factor
prices. In this way, the Walras law for the excess demand functions for factors comes to hold. We
may express this line of reasoning mathematically as
w · [Aw (p(w), w) · (I − A(p(w), w))−1 · f(p(w), w) − L]
= p(w) · f(p(w), w) − w · L= 0
where p(w) denotes the unit cost equating price relative to w, f(p(w), w) the ﬁnal demand vector,
and L is a supply vector of primary factors. In this way, we obtain a link between a large scalemodel and a small scale one, such as Uzawa (1961)’s two sector growth model. Data of the capitalintensity in various times and places may provide a key to evaluating the workings of large scale aswell as miniature models. The interested reader may refer to Kuga (1967) for such values.
When there are only two kinds of primary factors, say capital and labor, the input-output table
including the static capital and labor coeﬃcients remains stable as long as the wage rental ratio does.
In the temporary equilibrium, the short-run wage-rental ratio is determined by the then existentcapital labor ratio as in the economy of Uzawa (1961). When the capital labor ratio (and therefore
the wage rental ratio) remains constant in the steady state, our input-output commodity table keeps
a stable value as in the case of Morishima (1964) ’s dynamic non-substitution theorem.

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