- Email: [email protected]
The positiveness and the uniqueness of a solution
Economics Letters North-Holland
THE POSITIVENESS Takao
137
17 (1985) 137-139
AND THE UNIQUENESS
OF A SOLUTION
FUJIMOTO
C’nioersity of Kagawa, Kagawa 760, Japan
Carmen
HERRERO
University of Alicante, Alicante, Spain Received
3 July 1984
This paper presents a sufficient condition equations and inequality systems.
which assures
the positiveness
and uniqueness
of a solution
for various
economic
1. Introduction In this article we present a theorem on the positiveness and uniqueness of a solution for various economic equations and inequality systems. Our theorem is general enough to include several results obtained so far as special cases. Some examples of applications of our result are given in the final section.
2. Notation and the problem Let X be an ordered topological vector space with its positive cone X+ [see Schaefer (1970, p. 222)]. The cone X+ is assumed to have its non-empty interior int X+. For two vectors x, y E X, x 2 y meansx-yEX+; x>ymeansx2yandx#y;x>>ymeansx-yEintX+.LetbdX+denotethe boundary of Xt with the zero vector excluded, i.e., bd Xf = X+ - int X+ - (0). For a given proposition P(x) on x E X+ , we define S = {x E X+ 1P(x) true}. We assume that the proposition P is false for the zero vector, i.e., 0 GGS. Our problem is this: find out a sufficient condition to assure the positiveness (S c int X’) and the uniqueness of an element in S.
3. Theorem First, it is important
to note the following
properties
of our system:
Property P.I. If x is in bd X+, then for any scalar k > 1, kx = x + (k - 1)x = x + u, where u=(k-l)xEbdX+. Property P.2. If x and y are int X+ and x # y, then either (i) kx = y for some positive scalar k, or (ii) there exist a scalar k > 1 and a vector u E bd X+ such that either kx =y + u or x + u = ky. 0165-1765/85/$3.30
0 1985, Elsevier Science Publishers
B.V. (North-Holland)
T. Fujimoto,
138
C. Herrero / Positiveness and uniqueness
of solutron
Property P.l is satisfied because (k - 1)x is in bd X+ whenever x itself is in bd X+ provided k > 1. Property P.2 can be shown to hold as follows: Since x # y, we have either x $ y or x $ y. Suppose the former inequality. Put k = inf{ s ) sx 2 y }. As x is in int X+, it IS radial, and so the set {s 1sx 2 y } is not empty. In an ordered topological vector space, the positive cone is closed [see Schaefer (1970, p. 222)]. Thus, we can rewrite k = min{ s 1sx 2 y }, and clearly k > 1 because x $ y. Therefore, either kx=yorkx=y+v,wherevEbd X’. Now we can prove: Theorem, If we can find a proposition Q(x) on x E X+ such that (i) Q( kx) is true for x E S and any scalar k > 1, and (ii) Q(x + u) is false for any x E S and any v E bd X+, then S c int X+ and S is either a singleton or S = { kx* 1k > 0} for some x* E S. If x E S in in Proof we have a contradiction Suppose there are kx = y + u for some k because q(kx) is true, Sometimes can show:
bd X+, then by Property P.l, kx = x + u for some because Q(kx) is true, while Q(x + u) must be two vectors x, y E S. By Property P.2, either > 1 and u E bd X+, supposing x & y. In the latter while Q(x + u) is false. This completes the proof.
it is helpful
to have another
type of proposition
k > 1 and v E bd X+. Then false. Thus, S c int X+. kx = y for some k > 0, or case we have a contradiction Q.E.D.
to show positiveness
separately.
We
If we can find a proposition Q+( x, y) f or x, y E X+ such that (i) Q+(x, kx) is true for any Corollary. then x E S and any scalar k > 1, and (ii) Q+( x,x+u)isfalseforanyx~Xandanyu~bdX+, Stint X+. Suppose x is in bd X+. By Property P.l, kx = x + u for some k > 1 and u E bd X+. This Proof Q.E.D. leads to a contradiction because Q+(x, kx) is true, while Q+(x, x + u) is false.
4. Applications 4.1. Equation systems in R” Consider the equation x -Ax positive vector in R”. We put P(x)
: x is a solution
= d, where x E R”, A an n by n non-negative
to the equation,
By our theorem, we know that a solution semi-positive, we can put Q as
matrix,
and d a given
Q(x): x-Axed. is positive
and unique.
When A is indecomposable
and d is
Q(x):x-Ax>d. A generalization of this result is possible into non-linear systems, i.e., Ax is now not necessarily linear but merely isotone and subhomogeneous. That is, Ax 2 Ay if x 2 y, and A(kx) I kAx for k > 1. Such an equation system was used to obtain the LeChatelier-Samuelson principle and the non-substitution theorem [see Fujimoto (1980a,b)].
T. Fujimoto, C. Herrero / Positiveness and uniqueness of solution
4.2. Systems
139
in Banach lattices
Since our result is for an arbitrary ordered topological vector space, we can further generalize the preceding results into function spaces so long as they have a positive cone with its non-empty interior. Here we mention two examples: C(X), the set of continuous functions on a compact Hausdorf space X, and L”(X), the set of essentially bounded measurable functions on a measure space X. 3.3. General equilibrium
model
Let us take up a general E(x)
IO,
x’,!?(x)=0
where E(x)=(E,(x), vector of dimension P(x):
and
E,(x),..., n. We put
x is a solution
Q+(x,r):
equilibrium
E(x)=E(y)>
model
x20, E,,(x))
(1) is a vector of excess demand
functions
with x being a price
to (l), Q(x):
E(x)
= 0.
The property (i) of Q + in our corollary is nothing but the homogeneity of degree zero of excess demand functions. The property (ii) of Q+ is realized if we assume a kind of indecomposability, i.e., E(x) f E(x + u) for any u E bd R’J - (0). Then, a solution is positive. By Walras law, E(x) = 0 for an equilibrium price vector x. Now we can check the properties (i) and (ii) for the proposition Q. This is what is shown in Iritani (1981). 4.4. Complex analysis By a certain mapping, it is possible to transform the unit circle {z 1z E C, IzI 5 l} on the complex plane into the first quadrant {z 1z E C, Re z 2 0, Im z 2 O}. Thus we can translate the uniqueness problem for an equation on the disk into that on the first quadrant, where we may be able to use our result.
References Fujimoto, T., 1980a, Global strong LeChatelier-Samuelson principle, Econometrica 48, 1667-1674. Fujimoto, T., 1980b, Non-substitution theorems and the systems of nonlinear equations, Journal 410-415. Iritani, J., 1981, On uniqueness of general equilibrium, Review of Economic Studies 48, 167-171. Schaefer, H., 1970, Topological vector spaces (Springer, New York).
of Economic
Theory
23,
THE POSITIVENESS Takao
137
17 (1985) 137-139
AND THE UNIQUENESS
OF A SOLUTION
FUJIMOTO
C’nioersity of Kagawa, Kagawa 760, Japan
Carmen
HERRERO
University of Alicante, Alicante, Spain Received
3 July 1984
This paper presents a sufficient condition equations and inequality systems.
which assures
the positiveness
and uniqueness
of a solution
for various
economic
1. Introduction In this article we present a theorem on the positiveness and uniqueness of a solution for various economic equations and inequality systems. Our theorem is general enough to include several results obtained so far as special cases. Some examples of applications of our result are given in the final section.
2. Notation and the problem Let X be an ordered topological vector space with its positive cone X+ [see Schaefer (1970, p. 222)]. The cone X+ is assumed to have its non-empty interior int X+. For two vectors x, y E X, x 2 y meansx-yEX+; x>ymeansx2yandx#y;x>>ymeansx-yEintX+.LetbdX+denotethe boundary of Xt with the zero vector excluded, i.e., bd Xf = X+ - int X+ - (0). For a given proposition P(x) on x E X+ , we define S = {x E X+ 1P(x) true}. We assume that the proposition P is false for the zero vector, i.e., 0 GGS. Our problem is this: find out a sufficient condition to assure the positiveness (S c int X’) and the uniqueness of an element in S.
3. Theorem First, it is important
to note the following
properties
of our system:
Property P.I. If x is in bd X+, then for any scalar k > 1, kx = x + (k - 1)x = x + u, where u=(k-l)xEbdX+. Property P.2. If x and y are int X+ and x # y, then either (i) kx = y for some positive scalar k, or (ii) there exist a scalar k > 1 and a vector u E bd X+ such that either kx =y + u or x + u = ky. 0165-1765/85/$3.30
0 1985, Elsevier Science Publishers
B.V. (North-Holland)
T. Fujimoto,
138
C. Herrero / Positiveness and uniqueness
of solutron
Property P.l is satisfied because (k - 1)x is in bd X+ whenever x itself is in bd X+ provided k > 1. Property P.2 can be shown to hold as follows: Since x # y, we have either x $ y or x $ y. Suppose the former inequality. Put k = inf{ s ) sx 2 y }. As x is in int X+, it IS radial, and so the set {s 1sx 2 y } is not empty. In an ordered topological vector space, the positive cone is closed [see Schaefer (1970, p. 222)]. Thus, we can rewrite k = min{ s 1sx 2 y }, and clearly k > 1 because x $ y. Therefore, either kx=yorkx=y+v,wherevEbd X’. Now we can prove: Theorem, If we can find a proposition Q(x) on x E X+ such that (i) Q( kx) is true for x E S and any scalar k > 1, and (ii) Q(x + u) is false for any x E S and any v E bd X+, then S c int X+ and S is either a singleton or S = { kx* 1k > 0} for some x* E S. If x E S in in Proof we have a contradiction Suppose there are kx = y + u for some k because q(kx) is true, Sometimes can show:
bd X+, then by Property P.l, kx = x + u for some because Q(kx) is true, while Q(x + u) must be two vectors x, y E S. By Property P.2, either > 1 and u E bd X+, supposing x & y. In the latter while Q(x + u) is false. This completes the proof.
it is helpful
to have another
type of proposition
k > 1 and v E bd X+. Then false. Thus, S c int X+. kx = y for some k > 0, or case we have a contradiction Q.E.D.
to show positiveness
separately.
We
If we can find a proposition Q+( x, y) f or x, y E X+ such that (i) Q+(x, kx) is true for any Corollary. then x E S and any scalar k > 1, and (ii) Q+( x,x+u)isfalseforanyx~Xandanyu~bdX+, Stint X+. Suppose x is in bd X+. By Property P.l, kx = x + u for some k > 1 and u E bd X+. This Proof Q.E.D. leads to a contradiction because Q+(x, kx) is true, while Q+(x, x + u) is false.
4. Applications 4.1. Equation systems in R” Consider the equation x -Ax positive vector in R”. We put P(x)
: x is a solution
= d, where x E R”, A an n by n non-negative
to the equation,
By our theorem, we know that a solution semi-positive, we can put Q as
matrix,
and d a given
Q(x): x-Axed. is positive
and unique.
When A is indecomposable
and d is
Q(x):x-Ax>d. A generalization of this result is possible into non-linear systems, i.e., Ax is now not necessarily linear but merely isotone and subhomogeneous. That is, Ax 2 Ay if x 2 y, and A(kx) I kAx for k > 1. Such an equation system was used to obtain the LeChatelier-Samuelson principle and the non-substitution theorem [see Fujimoto (1980a,b)].
T. Fujimoto, C. Herrero / Positiveness and uniqueness of solution
4.2. Systems
139
in Banach lattices
Since our result is for an arbitrary ordered topological vector space, we can further generalize the preceding results into function spaces so long as they have a positive cone with its non-empty interior. Here we mention two examples: C(X), the set of continuous functions on a compact Hausdorf space X, and L”(X), the set of essentially bounded measurable functions on a measure space X. 3.3. General equilibrium
model
Let us take up a general E(x)
IO,
x’,!?(x)=0
where E(x)=(E,(x), vector of dimension P(x):
and
E,(x),..., n. We put
x is a solution
Q+(x,r):
equilibrium
E(x)=E(y)>
model
x20, E,,(x))
(1) is a vector of excess demand
functions
with x being a price
to (l), Q(x):
E(x)
= 0.
The property (i) of Q + in our corollary is nothing but the homogeneity of degree zero of excess demand functions. The property (ii) of Q+ is realized if we assume a kind of indecomposability, i.e., E(x) f E(x + u) for any u E bd R’J - (0). Then, a solution is positive. By Walras law, E(x) = 0 for an equilibrium price vector x. Now we can check the properties (i) and (ii) for the proposition Q. This is what is shown in Iritani (1981). 4.4. Complex analysis By a certain mapping, it is possible to transform the unit circle {z 1z E C, IzI 5 l} on the complex plane into the first quadrant {z 1z E C, Re z 2 0, Im z 2 O}. Thus we can translate the uniqueness problem for an equation on the disk into that on the first quadrant, where we may be able to use our result.
References Fujimoto, T., 1980a, Global strong LeChatelier-Samuelson principle, Econometrica 48, 1667-1674. Fujimoto, T., 1980b, Non-substitution theorems and the systems of nonlinear equations, Journal 410-415. Iritani, J., 1981, On uniqueness of general equilibrium, Review of Economic Studies 48, 167-171. Schaefer, H., 1970, Topological vector spaces (Springer, New York).
of Economic
Theory
23,