Ecanotnic Issl;le,s. Vol. B, PQrt 2, 2O03
A Complete Characterization of Economies
with the Nonsubstitution Property
Takao Fujimoto"*, Carmen Hetrerob,
Ravindra R Ranade",
Jos€ A Silva" and Antonio Villardl
ABSTRACT
A complete clnracterization of econnmies utith tlrc nnnsubstitution propertg is
presented for linear models. In this characterization, a degree of proper ioint Pro'
duction as uell as the existence of durable capital goods is alloued for'
1. INTRoDUcrroN
In Schefold(1978a), it is shown that a constant returns to scale economy with
joint production can have th.e properties of one with single production, if the
eilicient set of processes forms a squate matrix of net products and it has its
nonnegative inverse. (See a-Iso Schefold 1978b)' Then Herrero and Villar (1988)
proved this very property is a necessar5r and suflicient condition for an economy to enjoy the nonsubstitution property due to Georgescu-Roegen (1951)
and Samuelson (1951). This proposition was in fact obtained more than two
decades ago by Dasgupta and Sinha (1979), but its pubLication was delayed
untiL 1992 (see Dasgupta, 1992 and Dasgupta and Sinha, 1992). Then'
Bidard (1991) presented a necessarJr and sufficient condition for a real squane
matrix to have a strictly positive inverse. Following this, Erreygers (1996)
rediscovered propositions due to Dasgupta (79921' and Bidard and Erreygers
(1998) gave a necessary and sulficient condition fot a square matdx to have a
nonnegative inverse.
In the above contributions, though a certain class of joint production is
admitted, an important group of joint products, i.e', durable capital goods, are
in general excluded because the nEt output sqttare mntrix composed of elficient
processes nnnnattg d.oes not haue a nonnegatiue inuerse' In connection with
the nonsubstitution theorem, the existence of durable capital goods is dealt
with in Kurz and Sah.adori (1995, Chapters 7 and 9). Their method is to
require that, in the final demand vector, tlle elements coffesponding to old
durable capital goods be zero. Using this method, Fujimoto, Silva and Villar
(2002a) obtained a necessarJr and sufhcient condition under which a given real
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T
Fuimolo, C Herrero, R R Rarutde, J A Silua J and A Villar
square matrix has an inverse whose columns in a particular subset of indices
are nonnegative: ttrat subset of indices for commodities represents perishable
goods as well as brand new durable capita.l goods.
In this article, we combine the result by Dasgupta (1992), the mettrod in
Kurz and Salvadori (1995) explained above, and a mathematical theorem in
Fujimoto, Silva and Villar (2OO2l to obtain a fuI] characterization of linear economic models in which the nonsubstitution theorem holds good. In section 2,
we explain our model and the nonsubstitution theorem with its proof. The
method of proof is based on duaLity in linear programming, and is due to
Chander [1974). Section 3 includes a numerical example to illustrate our
results. In the final section, several remarks are given.
For more recent contributions, see also Kuga {2001}, Hasfr,rra-Buenaga,
Holder and Stuart (2002), and Villar (2002). For inverse positive matrices
reLated to M-matdces, the reader is referred to Berman and Plemmons (1979).
2. Lrunan MoDELS wlrH THE NoNSUBSTrrurroN pRopERTy
Let us consider an ordinary Linear input-output model with joint production
and durable capital goods. First we explain our notation. The symbol
p nmeans the real Euclidean space of dirnension nln>2J, and Rlthe nonnegative orthant oflR lThe two given real m x n nonnegative matrices A and B map
from R iinto R i. One more symbol l'is the row vector in nX 'whose entries are
all unity.
The inequality signs for vector comparison are as follows:
r y elRi;
x>yiffr yeRi-{0};
x>y iff
rD yill
x
_y
e
int(Ri),
Now in our linear economic model, the matrix A is understood as the input
coeffrcient matrix, and the matrix B as the output coellicient matrix. We
denote B - A as M. Ttre number m is that of comrnoditi.es, while n represents
the number of processes available. We assume m < n. This assumption may
be acceptable when we take into consideration ttrose processes each of which
uses one particula-r commoditJr as input and keeps it for one year using a certain alnount of labour. The index set for commodities, C, is defined as C = {1,
2, . . . , n4. The processes are normalized so that the labour input coeflicient
is unity in each process, imFlying that every process needs the sole primary
input i,abour. Thus, the n-row vector ltt= ll, 7, . . . , 1) stands for the labour
input coe{frcient vector. A column vector x eR i shows a corresponding actiwity vector with its j-th entrJr xj meardng the activity level of process j. Then .,4x
stands for the material input vector, andl'- x for the labour input, while &r
shows the material output vector. The vector, .&r-,{x, then, represents the net
output vector. The sy:nbol (,Ax), means the i-th element of (i4x).
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Ecorwmic Issues, VoI. 8, Part 2, 2003
In our model, we al.low for proper joint production and durable capital
goods. Thus, a certain set of colum.rrs of B may include more than one positive entry. We classi$ goods into two categories, C, and Cr: the first category'
q, consists of consumption goods and brand new capital goods, which can
appear in a final demand vector as positive entries, and the second, Cr, of old
durable capital goods, which do not appear in any final demand vector. Final
demands a-re to be r.:lderstood as either for consumption or for exportation.
Demand for capital. accumulation is included in the term g ' .A-rc, where g is a
rate of steady balanced grorvth: this groMh rate is assumed to be zero for simplicity in this article. Naturally, we have C = C, nC".
we first solve in this model the following linear programming problem:
min
l".x
s.t.:
&r
x
eRl
> r4x +
d'
I
(1)
I
l
wtrere d" 2 0 is a given final demand vector such that d," >0 for i € Cr, and
d: >o lor i e C2.
We assume:
Assumptlort A1. There exists an x e lR. I such that (&), t (&)t for i e Ct,
and (&*)i >(Axlifori €C2.
Assumption A1 is a standard productivity assumption, applied to production models with fixed capital, ald gua-rantees the existence of an optimal
solution to the problem (l). Then, we assume:
Assunptlon A2. An optimum solution x" to tlle problem (1) has at least m
positive entries.
Assumption 42 says that a-rly effi.cient choice of techniques must involve at
leaat m processes. This is a necessar5r condition for a full nonsubstitution theorem to hold, as i.t ensures that the efficient techniques may be able to span
the whole commodity space.
Let x" have at least m positive entries as is guaranteed by Assumption A2'
Now create two matrices .B' and A" by collecting those processes (i'e.,
columns) which are actually used in x", and defne M" = B" - A' ' Note that M"
is a matrix with m rows and k columns, n> k> m. Further assume:
Assumptlo! A3. If (M"x),> 0 for i e Cl, and (JWx)r = 0 for f e C2, then there
are at Ieast m positive entries in x.
Assumption A3 is a weak form of inverse isotoneness' It requires that eftrcicient production of strictly positive amounts of all brand new commodities
should utilize at least m processes.
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T
F$imato, C Henero, RR R(lnadq J A Silua J and A Villar
1 Note that uector * in Assumption 42 is a solution to (1) uhich determines M" , u.rhereas uedor x in Assumption A3 is an arbitrary uector so lorLg as
it sdtisfies th.e conditions thereinWe are ready to prove:
Lemma l. Giuen Assumptions A1, A2, and A3, problem (1) hr.s an optimal sotution x* uhich has exactlg m positiue eletEnts.
Proof. By ,&2, x" has at least m positive elements. Since there can be at most
m independent colulnns in M", we can remove columns one by one from M.
while keeping an optimal solution nonnegadve, until we find an x*. f
As above, we create two matrices .B* and 4* by collecting those processes
wlrich are used in x*, and define M* = B* - A* . This matrix M* is m x m, that
is, square,
Lemma 2. Giuen Assumptions A1, A2, and A3, tle matrk M* is regular, and th,e
inuerse of (M*) tns nonnegatiue columns ulen tlair index belongs to Cr
Proof. This is a result in Fujimoto, Silva and Villar (2002a), and the proof is
omitted. !
Here is our main result.
Theorem L. Giuen Assumptiotts 41, A2, and 43, tuhen By z Ay+ d. for some
y €Ri and an arbitrary dep'i suchthat i!, > O forie C| andilr= O for ie C",
a solution yt* to Ba y* =4*y* + d, satisf.es l' y+<l'.y.
Proof. Following Chander(1974), 1et us consider the dual program to {1):
Renark
rnaxp.d"
I
s.t.:pB<pA+/"f
P€RT
l2l
]
An optimal solution to l2], p* > 0, can be obtained by solwing the equation
p*B* = D* A* + l.' .It should be noted that y' €Rl because of Lemma 2 and
the property supposed for d. Then, postmultiply by I both sides of this equation, and we get
From.B
I
=
A*y*
p+Bxyx=p*l*y*+/'.y*
(3)
it follows that
p*E* y* =P*A*l
(4)
+ d,
tlrough premultiplying by p*. From eqs.
!'
.y* =p+.
+p*.d
(3) and (4), we have
d
(5)
On the other hand, from By > Ay + d, premultiplying both sides by p* yields
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E(pnDmic
P*.BY>P*AJr+P*'d
while from the constraint p*B 3 p* A +
l.'
(6)
we have
P*B*Y<P*AY+ !''Y
From inequalities (6) and (7), it follows
(71
that p*' il < !' ' y
(8)
Therefore, [5] and [8] show that our Theorem is valid. l-
3. A
NUMERICAL EXAMPLE
Consider the case in which there are two perishable goods which serve both
for consumption and for production, and one durable capital good (or a
machine), indexed as 1, 2, and 3 respectively. A machine lasts for two years,
and is discarded without any scrap value. One year old machine is indexed as
4. There are frve basic processes described as follows.
,I
o'rl (0
0.2 1 l0
l.A=
o.rl
It
101
110
011
000
0)
[0
0.2?0
0.1 02
0 00 lJ'" = u'''''''u'
100
1 0.8 -2
10.9 1-2
-l 011
1-1 0 0
1
,"o
0.2)
o.r
o.r
I
1
0)
The two categories of commodities are Cr = {f , 2, 3} and q = {4}' Note that commodity 3 is brand new, while commodity 4 is one year old. And its depreciation is not quantitative, i.e., there is no simple way to corrvert a unit of one
year old machine to a certain amount ofa brand new one. Moreover, in every
process, proper joint production is involved, and yet it is not di.flicult to see
that our assumptions are satisfied.
The fifth process turns out to be ineffrcient, and the fust four processes
form the optimal basis lW, and its iaverse is
t)' r0.370... 0.370... 0.370... 0.630... I
0.370... 0.370... {.370...
rM')'=l 1 0'9 t -'l _ | 0.370...
(r
o.s
I
0 I
l-r
\ 1 -1 0
rl
0/
]
0.0r2... 0.346... 0.67e... 0.321...
|
10.3s8... 0.025... 0.6er... 0.309... J
I
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T
F\limoto, C Herrero,
R R Ranade, J A Silua
J and. A Wla.
Since the inverse of lW includes a negative entry, this example cannot be covered by the results in Herrero and Vilar (1988) and Dasgupta (1992). The
equilibrium pdce vector, n* = (1.111..., 1.111..., 2.f 77...,0.889...), which confirms that the fifth process is inefficient, making losses at these prices.
The ftst four processes are modified from an example in Dasgupta (1992),
which is due to Sinha: that is
(t t
I
1l
t r-rl
[-1 1
lJ
This matrix is inverse-positive.
4. R-ouems
Our way of making assumptions step by step as we solve a particular prograrnrning problem makes it possible to a]low for the existence of those
processes which need no material input, which trun up as ineftrcient. This is
illustrated by the fifth process in the numerica_l example.
If Assumption A3 is not satisfied with an index set smaller than C, (and an
index set larger thal Cr), the columns in the index set C, of the inverse (M*)-l
contain at least one negative entry. This is also illustrated in the lourth column of (M*)-r .
Although it has been required that di= O for ie C", in fact, they can be positive to some extent. With respect to the equation fuf z = d, in order for x to
remain nonnegative, d, for i e C, should be bounded from above provided d,
for i e C, are given, because t}le columns of the inverse (Jt4*)-t in the index set
C, contain negative entries.
Our characterization is 'complete'in t]"e sense that linear models with joint
production and durable capital goods should satisf. Assumptions A1, 42, and
A3 when those models are required to have the nonsubstitution propert5r and
every final demand vector, where entdes for old machines are zero, is produced with no slack. The condition of nonnegative invertibility in Herrero and
Villar (1988) is'necessar5r and sufficcient'for the nonsubstitution ttreorem to
hold, only when models can produce any strictly positive fina1 demand vectors, including aged capital goods.
Having presented a complete characterization of the nonsubstitution theorem, it is then possible to extend the model to one in which synergetic external economies or diseconomies operate as is done in Fujimoto, Silva and Villar
(2002b).
Finally, since some kinds of proper joint production and qualitative depreciation a-re allowed for, we ca-nnot, in establishing the nonsubstitulion theo-
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Ecotwmic Isst/Es, Vol.
I,
Part 2, 2OO3
rem, use ttre methods of proof based on cost functions, which are empLoyed in
Morishima (1964), Stigfitz (1970), Johansen (19721 (as amended by Dasgupta
(1974)), Fujimoto (1980), Fujimoto (1987), alrd Kuga (2001)'
Accepted for publication: 10
Apil
2OO3
Enpnore
1. 'Department of Economics, University of Kagawa; bDePartment of Economic
Ana1ysis, University of Alicante, arrd IME; "Department of Economic Analysis,
University of Alicante; dDepaJtment of Economic Analysis, Universi\r of Alicante, IVrE,
a.Ird European Universit5r Institute. "corresponding author: e-mail: [email protected]. Thanks are due for a gra-nt from Mirdsterio de Ciencia y Tecnologia, under
Project BEC2OOI-0535. This paper was started when Takao Fujimoto was at the
Department of Fundamental Economic Analysis, University of Alicante, during the academic year 2OO1, under a sabbatical year professorship given by the Spanish Mirdstry
of Education. He is grateful for the hospitality alld warm errviron@ent there. Thanks
are also due to Professor Dipankaj Dasgupta, who supplied us with his old papers.
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